What's wrong with string theory?
Thomas Larsson
an amateur crackpot. Now, technically I am an amateur, since I
perished scientifically long ago, but at least I have more recent
CMP papers than most of my fellow crackpots. Moreover, I believe that
my arguments are quite significant and a wider audience should know
about them, even if not everybody needs to agree.
As is well known, string theory disagrees sharply with present-day
experiments, e.g. on the number of spacetime dimensions and the
existence of supersymmetric partners. This is usually not a problem
for string theory, or at least not for string theorists, since one
can always speculate that these features will show up in future
experiments at higher energy. In contrast, my criticism attacks string
theory on its own turf: intrinsic mathematical beauty.
It turns out that the underlying symmetry, the superconformal algebra,
is a quite ordinary and boring algebraic structure. True, it is one
of the few Lie superalgebras that admits a central extension (it is
not the only one, but the list is quite short). However, there is
in my opinion no reason to restrict oneself to *central* extensions.
The diffeomorphism algebra diff(N) (i.e. algebra of vector fields)
in any number of dimensions admits two independent (in general)
extensions with values in the module of closed dual one-forms, i.e.
closed (N-1)-forms. In N=1 dimensions, a closed dual one-form is a
closed zero-form is a constant function, so the extensions are
central in this case, and both coincide with the Virasoro algebra.
There are no obstructions to superization, so the superdiff algebra
diff(N|M) also has two non-central Virasoro-like extensions. Of
course, one obtains extensions of every subalgebra of diff(N|M) by
restriction, but in some boring cases, the generically non-central
extension reduces to a central one, and in some even more boring
cases it vanishes altogether. It turns out that the superconformal
algebra is such a boring case, although it is not even more boring.
To be more concrete, let me present the extension of diff(N), in
a Fourier basis for simplicity. Start with the Virasoro algebra Vir
which we all know and love:
[L_m, L_n] = (n-m)L_m+n - c/12 (m^3-m) \delta_m+n,
where \delta_m is the Kronecker delta. When c=0, L_m = -i exp(imx) d/dx,
m in Z. The element c is central, meaning that it commutes with all of
Vir. By Schur's lemma, it can therefore be considered as a c-number.
Now rewrite Vir as
[L_m, L_n] = (n-m)L_m+n - c m^2 n S_m+n,
[L_m, S_n] = (n+m)S_m+n,
[S_m, S_n] = 0,
m S_m = 0.
It is easy to see that the two formulations of Vir are equivalent
(I have absorbed the linear cocycle into a redefinition of L_0).
The second formulation immediately generalizes to N dimensions.
The generators are L_j(m) = -i exp(i m_k x^k) d/dx^j and
S^i(m), where x = (x^k), k = 1, 2, ... N, is a point in N-dimensional
space and m = (m_k) labels the Fourier modes. The Einstein convention
is used (repeated indices, one up and one down, are implicitly summed
over). The defining relations are
[L_i(m), L_j(n)] = n_i L_j(m+n) - m_j L_i(m+n)
+ (c_1 m_j n_i + c_2 m_i n_j) m_k S^k(m+n),
[L_i(m), S^j(n)] = n_i S^j(m+n) + \delta^j_i m_k S^k(m+n),
[S^i(m), S^j(n)] = 0,
m_i S^i(m) = 0.
This is an extension of diff(N) by the abelian ideal with basis S^i(m).
Geometrically, we can think of L_i(m) as a vector field and S^i(m) as
a dual one-form; the last condition expresses closedness. The cocycle
proportional to c_1 was first written down by Rao and Moody, whereas
I figured out the c_2 term myself. Here are the references:
S E Rao and R V Moody: Vertex representations for N-toroidal Lie algebras
and a generalization of the Virasoro algebra,
Comm. Math. Phys. 159, 239--264 (1994)
T A Larsson: Central and non-central extensions of multi-graded Lie
algebras, J. Phys. A 25, 1177--1184 (1992)
Rao and Moody also constructed the first interesting representations.
After some thinking (quite a lot of thinking, actually), I understood
what their modules meant geometrically and I was able to write down
essentially the most general Fock modules. This is written up in
T A Larsson: Extended diffeomorphism algebras and trajectories in jet
space, Comm. Math. Phys. 214, 469--491 (2000)
http://www.arxiv.org/abs/math-ph/9810003
The anti-string argument has appeared on the Los Alamos preprint server:
T A Larsson: Fock representations of non-centrally extended
super-diffeomorphism algebras, http://www.arxiv.org/abs/physics/9710022.
I was never able to publish this paper, and now it needs a complete
overhaul in view of later progress on the bosonic algebra. The main
conclusions relevant for string theory remain true, however.
To conclude and maybe save myself some flames, let me emphasize the
difference between fact and fiction. That the centrally extended
superconformal algebra is a subalgebra of the non-centrally extended
superdiff algebra is a mathematical theorem and not really up to
discussion. That this theorem is bad news for string theory is my
personal opinion.
Thomas
zir...@my-deja.com
Thomas Larsson <Thomas....@hdd.se> wrote:
> I have hesitated to make this post, fearing that I will appear as
> an amateur crackpot. Now, technically I am an amateur [...]
Don't forget that Newton, Einstein, etc. made some of their greatest
discoveries as amateurs and that no less than Mendel, Fermat, Darwin,
etc. were amateurs their whole lives. Soon, we may have a major
scientific revolution (which could effect fundamental physics) due to
the work of another amateur scientist named Stephen Wolfram, but this
will depend on the validity of his forthcoming magnum opus- a book
entitled "A New Kind of Science". Although Wolfram can be quite
dismissive of mathematical physics and even natural selection he does
*not* appear to be a crank. (Previously, I emailed him about various
examples of emergent phenomena which, hopefully, he will try to explain
in his book). If you're interested see:
http://www.wolfram.com/news/forbes.html
>
> As is well known, string theory disagrees sharply with present-day
> experiments, [...]
This is too strong a statement (but, anyways, it is not the specific
issue you are arguing).
> To conclude and maybe save myself some flames, let me emphasize the
> difference between fact and fiction. That the centrally extended
> superconformal algebra is a subalgebra of the non-centrally extended
> superdiff algebra is a mathematical theorem and not really up to
> discussion. That this theorem is bad news for string theory is my
> personal opinion.
>
Consider what strings might be made of. Superstring and M-theories can
be described by supersymmetric quantum matrix models (QMM). The Lie
superalgebras of QMM can exist as generalized Witt algebras [1]. Your
super diffeomorphism algebra diff(N+1|M) is also a generalized Witt
algebra W(N+1|M).
[1] http://arxiv.org/abs/hep-th/9907130
Unfortunately, I have not had the time to read your paper or this paper
but if you can figure out if/how the two are related I would be curious
to know.
-------------------------------------------------------
Sent via Deja.com
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Thomas Larsson
> In article <E180A6F82759D211BFDA00E01890523D14C4@HDDNT01>,
> Thomas Larsson <Thomas....@hdd.se> wrote:
> >
> > As is well known, string theory disagrees sharply with present-day
> > experiments, [...]
>
> This is too strong a statement (but, anyways, it is not the specific
> issue you are arguing).
>
appearence of susy partners. As far as I know, these features have not
been seen in present-day experiments. Of course, this does not rule out
the possibility that they will appear at higher energy, but this remains
so far an unconfirmed hypothesis.
Sometimes, disagreement with experiments drives science forward. When
Dirac wrote down his equation in the late 20s, anti-matter had not yet
been seen. Out of pure cowardice, he waited two years to make this
obvious consequence public. He was of course extraordinarily lucky to
have his prediction confirmed soon thereafter.
So disagreement with experiments can sometimes be a virtue, but usually
it is not.
> > To conclude and maybe save myself some flames, let me emphasize the
> > difference between fact and fiction. That the centrally extended
> > superconformal algebra is a subalgebra of the non-centrally extended
> > superdiff algebra is a mathematical theorem and not really up to
> > discussion. That this theorem is bad news for string theory is my
> > personal opinion.
> >
> Consider what strings might be made of. Superstring and M-theories can
> be described by supersymmetric quantum matrix models (QMM). The Lie
> superalgebras of QMM can exist as generalized Witt algebras [1]. Your
> super diffeomorphism algebra diff(N+1|M) is also a generalized Witt
> algebra W(N+1|M).
>
> [1] http://arxiv.org/abs/hep-th/9907130
>
Ouch! 150 pages!! When I skimmed the paper, I made two observations.
1. I didn't see any extensions anywhere. My main point was that the
diff algebra (super or not) have two non-central extensions in any number
of dimensions, and in one bosonic dimension both extensions reduce to
the Virasoro algebra.
Extensions are really crucial, because otherwise you will be hit by
the standard objection that diff symmetry makes all correlations functions
trivial. Like in CFT with c=0.
2. The author Lee is a student of S G Rajeev, who is well known for his
work with Mickelsson on the Mickelsson-Faddeev algebra. For those who
don't know, the MF algebra is an abelian extension of the current algebra
in three dimensions by the gauge connection module. In a Fourier basis,
it looks like this
[J^a(m), J^b(n)] = f^abc J^c(m+n) + d^abc e^ijk m_i n_j A^c_k(m+n)
[J^a(m), A^b_j(n)] = f^abc A^c_j(m+n) + \delta^ab m_j \delta(m+n)
[A^a_i(m), A^b_j(n)] = 0
Here m = (m_j) labels Fourier modes. Space indices i,j,k = 1,2,3.
a,b,c are gauge indices. f^abc are the structure constants.
d^abc are the totally symmetric constants related to the cubic Casimir;
d^abc = 0 for su(2). e^ijk is the epsilon tensor in three dimensions.
The inhomogeneous term in the second line makes A^a_i(m) into a connection.
Mickelsson, together with Rajeev and others, spent a lot of time in the
late 80s to work out the rep theory of the MF algebra, by embedding it
in a really big version of gl(infinity). It was a rather
fatal blow when Pickrell (CMP, late 80s) proved that this algebra, and
perhaps the MF algebra as well, didn't have any nice representations
(unitary reps on a separable Hilbert space?). Mickelsson nevertheless
claims to have some reps in a generalized sense, but despite considerable
effort I never understood him. Since he is here in Stockholm, I have
asked him, but he is just too abstract for me.
zir...@my-deja.com
Thomas Larsson <Thomas....@hdd.se> wrote:
> Sometimes, disagreement with experiments drives science forward. [...]
>
> So disagreement with experiments can sometimes be a virtue, but
> usually it is not.
Indeed. For instance, from 1973-78, experiments indicated at least 3
separate times that the Standard Model was incorrect, but the
experiments were later shown to have been in error.
> Ouch! 150 pages!!
This is Lee's thesis but I found his review article on the same subject
which is half as long:
http://arxiv.org/abs/hep-th/9906060
> When I skimmed the paper, I made two observations.
> 1. I didn't see any extensions anywhere.
Also doing a quick skim, I see that there *are* extensions which are
discussed in Section 3 of the thesis. See, e.g., pages 53, 60, 63 and 71
(referring to his page numbers rather than the ones that your software
might format). Note that the central extension of the Witt algebra is
the Virasoro algebra. The Virasoro algebra describes the conformal
symmetry of string theory (ST) and can elucidate, e.g., the mass
spectrum and S matrix elements of ST.
> 2. The author Lee is a student of S G Rajeev, who is well known for
his work with Mickelsson on the Mickelsson-Faddeev algebra.
It doesn't look as if MV algebras are (directly) involved in Lee's
thesis or review paper. The main point I would like to make is this:
If we are going to consider your observation(s) as a motivation for
possibly rejecting ST then someone should try to figure out if your
arguments are relevant regarding the most fundamental formulation of ST
that we can find- which, AFAIK, is Matrix Theory (and this is why I
referred you to Lee's paper).
Thomas Larsson
<zir...@my-deja.com> wrote in message news:93nkrv$skf$1...@nnrp1.deja.com...
> > When I skimmed the paper, I made two observations.
> > 1. I didn't see any extensions anywhere.
>
> Also doing a quick skim, I see that there *are* extensions which are
> discussed in Section 3 of the thesis. See, e.g., pages 53, 60, 63 and 71
> (referring to his page numbers rather than the ones that your software
> might format).
Lee summarizes what his algebras are on page 71. Let me first quote him and
then add my comments.
"Now, recall that outer derivations of the algebra of functions on a
manifold are just vector fields. We have seen that the Cuntz algebra is
a noncommutative generalization of the algebra of functions on a circle.
Thus our centrix Lie algebra should be thought of as the corresponding
noncommutative generalization of the Lie algebra of vector fields on the
circle. Indeed, in the special case \Lambda=1 we we that out algebra
reduces exactly to the Witt algebra, the central extension of which is
the Virasoro algebra; the Witt algebra is just the algebra of vector fields
on a circle."
Clearly Lee is dealing with some algebraic structure that lives in one
dimension, and in particular his extensions are central. The algebras that
I described are the natural generalizations of Virasoro to higher dimensions:
abelian extensions of the diffeomorphism algebra (= generalized Witt algebra
= algebra of vector fields) by the module of closed dual one-forms. Many
people have studied central extensions of various algebras, but the diff
algebra does not permit central extensions except in one dimension.
When one builds Fock representations, three things might happen, in
increasing order of complexity:
1. No extension.
2. Central extension.
3. Abelian but non-central extension.
The first case typically happens for finite-dimensional algebras and the
second for algebras of linear growth, such as Virasoro and affine Kac-Moody.
The third case appears in algebras of polynomial growth higher than linear
(and sometimes in linear cases as well). Another way to formulate my
criticism of string theory is that it only climbs to the middle step on the
complexity ladder above.
The first non-trivial extensions of the diff algebra in more than one
dimension were constructed by myself in
T A Larsson, Multi-dimensional Virasoro algebra,
Phys. Lett. A 231, 94--96 (1989).
Despite the name of this paper, the extension was not quite the higher-
dimensional analogue of Virasoro, because the projection to *closed* dual
one-forms only appeared a few years later. The first string revolution took
place in 1984, so there is no way the higher-dimensional Virasoro algebras
could have been built into string theory - the math just wasn't there.
> Note that the central extension of the Witt algebra is
> the Virasoro algebra. The Virasoro algebra describes the conformal
> symmetry of string theory (ST) and can elucidate, e.g., the mass
> spectrum and S matrix elements of ST.
I am fully aware of what the Virasoro algebra is.
> If we are going to consider your observation(s) as a motivation for
> possibly rejecting ST then someone should try to figure out if your
> arguments are relevant regarding the most fundamental formulation of ST
> that we can find- which, AFAIK, is Matrix Theory (and this is why I
> referred you to Lee's paper).
>
String theory is a huge construction, and I know very little about what
is going on in the upper floors (dualities, AdS/CFT, etc.). I am only
criticizing the shallowness of its algebraic foundation. On the other hand,
if the foundation is removed, the entire edifice might crumble.
zir...@my-deja.com
Thomas Larsson <Thomas....@hdd.se> wrote:
> Lee summarizes what his algebras are on page 71. Let me first quote
> him and then add my comments.
>
> "Now, recall that outer derivations of the algebra of functions on a
> manifold are just vector fields. We have seen that the Cuntz algebra
> is a noncommutative [multi-dimensional] generalization of the
> [abelian] algebra of functions on a circle. Thus our centrix Lie
> algebra should be thought of as the corresponding noncommutative
> generalization of the Lie algebra of vector fields on the circle.
The "centrix algebra" is the one associated with open strings and the
"cyclix algebra" is the one associated with closed strings. Both of
these algebras are related to generalizations of Witt algebras, but it
is not clear how these generalizations are related. (Matrix Theory does
suggest a duality between open and closed strings). Wow, Lee's thesis is
the longest and most complex paper I have seen, and it would take a HUGE
amount of time for me to try to understand it.
> > Note that the central extension of the Witt algebra is
> > the Virasoro algebra. The Virasoro algebra describes the conformal
> > symmetry of string theory (ST) and can elucidate, e.g., the mass
> > spectrum and S matrix elements of ST.
>
> I am fully aware of what the Virasoro algebra is.
I realized this from your first post but I wrote the above anyways in
case someone else reads this thread.
> String theory is a huge construction, and I know very little about
> what is going on in the upper floors (dualities, AdS/CFT, etc.). I
> am only criticizing the shallowness of its algebraic foundation. On
> the other hand, if the foundation is removed, the entire edifice
> might crumble.
Perhaps you might consider e-mailing some expert (like Lee) with your
argument to see what they think. (BTW, in the latest TWF, John Baez
brought up the subject of q-deformation which made me wonder what a
q-deformed Virasoro algebra might be like and whether it could relate to
string theory. But I would guess that someone has already thought about
this).
Charles Francis
>Although Wolfram can be quite
>dismissive of mathematical physics and even natural selection he does
>*not* appear to be a crank.
Ah, so he disputes both logical thought and observational evidence, and
is not a crank!
--
Regards
Charles Francis
cha...@clef.demon.co.uk
John Baez
>When one builds Fock representations, three things might happen, in
>increasing order of complexity:
>1. No extension.
>2. Central extension.
>3. Abelian but non-central extension.
>The first case typically happens for finite-dimensional algebras and the
>second for algebras of linear growth, such as Virasoro and affine Kac-Moody.
>The third case appears in algebras of polynomial growth higher than linear
>(and sometimes in linear cases as well). Another way to formulate my
>criticism of string theory is that it only climbs to the middle step on the
>complexity ladder above.
How is this a criticism? I like my physical theories to be
mathematically rigorous, philosophically sound, simple and
beautiful, to fit the experimental data, and to make testable
predictions. I don't see why using the most complicated possible
extensions of algebras is in itself desirable.
On the other hand, I would be very interested if diffeomorphism
groups of manifolds of dimension > 1 admitted interesting extensions
of any sort, since (as you noted) these could serve as groups of
gauge symmetries for quantum gravity.
Unlike you, I'd like to use the simplest extensions possible!
No extension at all would be best; a central extension would
be second best; but if I were forced to, I'd go to an abelian
noncentral extension.
You mentioned some nonexistence result on central extensions for
diffeomorphism groups of manifolds of dimension > 1... how does
that theorem go, exactly?
I don't think we have much of an argument, but you seem to be making
a virtue out of what may just be a necessity.
Chris Hillman
> In article <E180A6F82759D211BFDA00E01890523D14C4@HDDNT01>,
> Thomas Larsson <Thomas....@hdd.se> wrote:
>
> > I have hesitated to make this post, fearing that I will appear as
> > an amateur crackpot. Now, technically I am an amateur [...]
>
> Don't forget that Newton, Einstein, etc. made some of their greatest
> discoveries as amateurs
Au contraire, Zirkus.
Newton is sufficiently remote from our time that graduate school as we
know it today had not yet developed, but he had a fine formal education by
the standards of the day and most importantly, read the best available
mathematical books when he was a young man. Furthermore, during his time
as a student, he was personally tutored by the leading mathematician in
England. Also, note that Newton's account of the development of the
calculus and of his physics was inaccurate in many respects (to make a
long story short, he claimed to have figured all this stuff out decades
before he actually did so, during the plague years, no doubt in order to
advance his priority vis a vis Leibniz; in fact, he did most of this work
much later in his life, while he was working as a professor).
Furthermore, during his academic career, Newton read books and papers by
other leading physicists and also published his own work. He was no
"amateur": he was a professional scientist (paid full time, if you like,
to do nothing but science, although by special arrangement he mostly
avoided teaching).
[It is true that Newton later abandoned his academic career for public
service, but that doesn't change the fact that he was a professional
physicist for most his life, and was by no means "entirely self-taught".]
Einstein had an excellent undergraduate and graduate education; he was by
no means "entirely self-taught". It is true he worked for a few years in
the Berne Patent office until he obtained his first academic position, but
he had already essentially completed his Ph.D. dissertation before he took
that job. After this fairly brief period, Einstein worked as a
professional physicist until his death. He worked in academia for the
most productive part of his career (including the most prestigious job in
Europe, Prof. of physics at Berlin), attended conferences, read and
published papers in the best journals of the day, heard and gave
colloquium talks, and generally did everything any professor does (except
teach!). As everyone knows he eventually came to the U.S. to work at the
IAS in Princeton. In no sense could Einstein possibly be called an
"amateur" (paid full time, if you like, to do nothing but science,
although by special arrangement, like Newton, he entirely avoided teaching
during most of his career).
> and that no less than Mendel, Fermat, Darwin,
> etc. were amateurs their whole lives.
If "amateur" means "self-educated", OK, but again note that Fermat and
Darwin were active members of the scientific community during their entire
adult lives--- Fermat corresponded via letters with the best "academic"
mathematicians of the day, and Darwin read papers and published landmark
scientific books, including what we would call research monographs. Note
too that while Fermat "held down a day job", as did Cayley (both worked in
the legal profession), Mendel and Darwin were able to function as full
time scientists.
> Soon, we may have a major scientific revolution (which could effect
> fundamental physics) due to the work of another amateur scientist
> named Stephen Wolfram,
FYI, Stephen Wolfram has a Ph.D. (physics, IIRC) from a top flight
department (maybe Oxfam--- can't remember details), and before founding
Wolfram Research, he published papers, gave talks, etc., just like any
other professional physicist; in no sense could he be called an "amateur
scientist".
Chris Hillman
Home Page: http://www.math.washington.edu/~hillman/
Charles DH Williams
Baez) wrote:
> How is this a criticism? I like my physical theories to be
> mathematically rigorous, philosophically sound, simple and
> beautiful, to fit the experimental data, and to make testable
> predictions.
That rules out everything apart from electromagnetism doesn't it?
:-)
Charles
John Baez
zir...@my-deja.com wrote:
>In article <93voai$kn5$1...@news.state.mn.us>,
>Thomas Larsson <Thomas....@hdd.se> wrote:
>> I am fully aware of what the Virasoro algebra is.
>I realized this from your first post but I wrote the above anyways in
>case someone else reads this thread.
Thanks! That's a good habit to get into: backing up a bit and
explaining terms, to help others learn what's going on.
>(BTW, in the latest TWF, John Baez
>brought up the subject of q-deformation which made me wonder what a
>q-deformed Virasoro algebra might be like and whether it could relate to
>string theory. But I would guess that someone has already thought about
>this).
Yes, they have. Mathematicians and physicists have already been hard
at work on the q-deformed Virasoro algebra. I'm no expert on this subject,
but here are some of the many references available online. The last one
looks especially interesting from the viewpoint of physics:
q-alg/9507034
A Quantum Deformation of the Virasoro Algebra and the Macdonald Symmetric
Functions
Authors: Jun'ichi Shiraishi, Harunobu Kubo, Hidetoshi Awata, Satoru Odake
Comments: 15 pages, latex file
Journal-ref: Lett. Math. Phys. 38 (1996) 33
A quantum deformation of the Virasoro algebra is defined. The Kac
determinants at arbitrary levels are conjectured. We construct a
bosonic realization of the quantum deformed Virasoro algebra.
Singular vectors are expressed by the Macdonald symmetric
functions. This is proved by constructing screening currents
acting on the bosonic Fock space.
hep-th/9410232
Generalised q-deformed oscillators and their statistics
Authors: Dao Vong Duc
We consider a version of generalised q-oscillators and some of
their applications. The generalisation includes also "quons" of
infinite statistics and deformed oscillators of parastatistics.
The statistical distributions for different q-oscillators are
derived for their corresponding Fock space representations. The
deformed Virasoro algebra and SU(2) algebra are also treated.
hep-th/0003270
Field Theory on Noncommutative Space-Time and the Deformed Virasoro Algebra
Authors: M. Chaichian, A. Demichev, P. Presnajder
We consider a field theoretical model on the noncommutative
cylinder which leads to a discrete-time evolution. Its Euclidean
version is shown to be equivalent to a model on the complex
q-plane. We reveal a direct link between the model on a
noncommutative cylinder and the deformed Virasoro algebra
constructed earlier on an abstract mathematical background. As it
was shown, the deformed Virasoro generators necessarily carry a
second index (in addition to the usual one), whose meaning,
however, remained unknown. The present field theoretical approach
allows one to ascribe a clear meaning to this second index: its
origin is related to the noncommutativity of the underlying
space-time. The problems with the supersymmetric extension of the
model on a noncommutative super-space are briefly discussed.
John Baez
Though you meant that as a joke, being a pedant I can't resist a
serious reply. This is how the pedant gets back at the humorist.
I regard Newtonian mechanics to have all these qualities, as well
as general relativity and nonrelativistic quantum mechanics. Other
theories tend to be lacking in one or more of these qualities. For
example, quantum field theory of the sort used in the Standard Model
has not yet been made rigorous, and the Standard Model is not so
simple, either. However, I don't DEMAND theories to have ALL of
these qualities - I just LIKE them better when they have MORE of
these qualities.
Thomas Larsson
> >increasing order of complexity:
> >1. No extension.
> >2. Central extension.
> >3. Abelian but non-central extension.
> >The first case typically happens for finite-dimensional algebras and the
> >second for algebras of linear growth, such as Virasoro and affine Kac-Moody.
> >The third case appears in algebras of polynomial growth higher than linear
> >(and sometimes in linear cases as well). Another way to formulate my
> >criticism of string theory is that it only climbs to the middle step on the
> >complexity ladder above.
>
> mathematically rigorous, philosophically sound, simple and
> beautiful, to fit the experimental data, and to make testable
> extensions of algebras is in itself desirable.
It wouldn't be a criticism if string theory had convincing
experimental support. However, if you base your theory on intrinsic
mathematical beauty alone, you will be in trouble if someone shows
that it is in fact ugly. My result removes the distinguished status
from the underlying algebraic structure. At least the superconformal
algebra was the algebraic structure underlying string theory in the
1980s, when I could still understand it.
To avoid confusion, let me emphasize that I am only criticizing string
theory proper, as applied to four- and (less interestingly)
ten-dimensional physics. In contrast, I greatly admire the work of
some string theorists on conformal field theory and its application to
two-dimensional critical phenomena. This is probably the greatest
theoretical advance in the last twenty years, and it is undoubtably
physics.
>
> On the other hand, I would be very interested if diffeomorphism
> groups of manifolds of dimension > 1 admitted interesting extensions
> of any sort, since (as you noted) these could serve as groups of
> gauge symmetries for quantum gravity.
>
> Unlike you, I'd like to use the simplest extensions possible!
> No extension at all would be best; a central extension would
> be second best; but if I were forced to, I'd go to an abelian
> noncentral extension.
Which one of the two extremes is preferable might be a matter of
taste. Being stuck in the middle does not appear to be a good idea,
though.
But there is really no choice. The standard objection is that
diffeomorphism symmetry imply trivial correlation functions. The same
argument applies to the infinite conformal symmetry in two dimensions,
because conf(2) = diff(1) + diff(1) in an obvious notation. CFT with
c=0 is not very interesting.
In physics one is usually interested in lowest-energy
representations. If the algebra is infinite-dimensional, you must
normal order to avoid infinities; normal ordering brings in
extensions. However, you cannot naively duplicate the recipe from one
dimension, because more infinities arise. The central charge in
N-dimensional spacetime turns out to be proportional to the number of
time-independent functions, which is finite when N=1 only. There is
really no way around this. Trust me, I was stuck on this point between
1988 and 1995.
The way to proceed was discovered by R V Moody (of Kac-Moody fame) and
Senapati Eswara Rao, building on analogous work on current algebras
together with T Yokonoma. This work spurred a new field of mathematics
called Toroidal Lie algebras, which now is attracting a growing number
of mathematicians. Check e.g. the recent paper by Steve Berman, Yuly
Billig and J Szmigielski,
http://www.arxiv.org/abs/math.QA/0101094. However, I will phrase the
construction in my own formalism, partly because that is what I
understand and partly because I think that I am ahead of these guys in
certain respects.
The main idea is to fix a one-dimensional curve in spacetime ("the
observer's trajectory"), expand all fields in a multi-dimensional
Taylor series around the points on this curve, and throw away all
Taylor coefficients of order higher than p. We now have a (non-linear)
realization of diff(N) on finitely many functions of a single variable
(the parameter along the trajectory), and this is precisely the
situation where the Fock construction works: add canonical momenta and
normal order. Because the original realization was non-linear, the
resulting extension is non-central.
An interesting point is that the "abelian charges", i.e. the
parameters multiplying the cocycles, diverge with p. A natural
challenge is to find lowest-energy modules (necessarily not Fock) with
a finite p -> infinity limit.
>
> You mentioned some nonexistence result on central extensions for
> diffeomorphism groups of manifolds of dimension > 1... how does
> that theorem go, exactly?
dim H^2(diff(N), C) = \delta_N,1.
Probably vector fields of formal Laurent series. This was proved by
Askar Dzhumadildaev in the early 90s. I can dig up the exact
reference, but not today. A physics proof either appeared in
E Ramos, C H Sah and R E Shrock, Algebras of diffeomorphisms of the N-torus,
J. Math. Phys. {\bf 31}, 1805--1816 (1989).
or in a subsequent preprint by Ramos and Figuieroa-O'Farrill, probably
published in Int J Mod Phys A. The proof is not difficult: If L_j(m)
denote the diff(N) generators (see my original post) and a^i and b_j
are two constant vectors such that a^i b_i = 1, then a^j L_j(k b), k
\in Z, satisfies diff(1). Because diff(N) has so many diff(1)
subalgebras, and the extension of each of these can only be Virasoro,
any central extension of diff(N) must be cubic. If you write down the
most general cubic extension and turn the crank, the Jacobi identities
fail unless N=1.
More generally, let Z^1(N) denote the diff(N) module of closed dual
one-forms. Then dim H^2(diff(N), Z^1(N)) = 2 - \delta_N,1.
>
> I don't think we have much of an argument, but you seem to be making
> a virtue out of what may just be a necessity.
Of course, what I'm doing is trying to provoke a reaction.
Kevin A. Scaldeferri
Chris Hillman <hil...@math.washington.edu> wrote:
Caltech, actually. His tenure here is somewhat legendary. (He was 20
when he got his Ph.D.)
--
======================================================================
Kevin Scaldeferri Calif. Institute of Technology
The INTJ's Prayer:
Lord keep me open to others' ideas, WRONG though they may be.
Stephen Speicher
> On 11 Jan 2001 zir...@my-deja.com wrote:
>
> >
> > Don't forget that Newton, Einstein, etc. made some of their
> > greatest discoveries as amateurs
>
> Au contraire, Zirkus.
>
> Einstein had an excellent undergraduate and graduate education;
> he was by no means "entirely self-taught". It is true he
> worked for a few years in the Berne Patent office until he
> obtained his first academic position, but he had already
> essentially completed his Ph.D. dissertation before he took
> that job. After this fairly brief period, Einstein worked as a
> professional physicist until his death. He worked in academia
> for the most productive part of his career (including the most
> prestigious job in Europe, Prof. of physics at Berlin),
> attended conferences, read and published papers in the best
> journals of the day, heard and gave colloquium talks, and
> generally did everything any professor does (except teach!).
> As everyone knows he eventually came to the U.S. to work at the
> IAS in Princeton. In no sense could Einstein possibly be
> called an "amateur" (paid full time, if you like, to do nothing
> but science, although by special arrangement, like Newton, he
> entirely avoided teaching during most of his career).
>
However, it is interesting to note how the early Einstein held
such a differing view from the academic life portrayed above. In
May of 1901 Einstein took a job for two months teaching high
school. In a letter to Professor Jost Winteler, dated 8 July
1901, Einstein speaks of his job and other activities.
"I have been quite exceptionally pleased with my
activities here. It had never occurred to me that I
would enjoy teaching as much as it has proved to be the
case. After having taught 5 or 6 classes in the
morning, I am still quite fresh and work in the
afternoon either in the library on furthering my
education or at home on interesting problems. I cannot
tell you how happy I would feel in such a job. I have
completely given up my ambition to get a position at a
university, since I see that even as it is, I have
enough strength and desire left for scientific
endeavor."
Note that Einstein continued work at the patent office well after
his seminal 1905 paper on relativity. It wasn't until 1908 that
he finally taught an academic course, and that was on Saturday
and Tuesday mornings, while still working for the patent office
in his full capacity. It wasn't until almost 1910 that Einstein
finally resigned from the patent office and began work as an
associate professor at the University of Zurich.
Einstein had done his light-quantum paper, his special relativity
paper, his paper on Brownian motion, the status of radiation
paper, the nature of radiation paper (of which Pauli stated some
forty years later that it "can be considered as one of the
landmarks in the development of theoretical physics), etc., all
of this (and more) without any academic affiliation or
collaboration -- all before his first full academic job. In
fact, it was in 1907, while still working at the patent office,
that Einstein had "The happiest thought of my life" -- the
principle of equivalence.
Indeed, many years later, speaking of education, Einstein would
say:
"The development of general ability for independent
thinking and judgment should always be placed foremost,
not the acquisition of special knowledge. If a person
masters the fundamentals of his subject and has learned
to think and work independently, he will surely find
his way and besides will better be able to adapt
himself to progress and changes than the person whose
training principally consists in the acquiring of
detailed knowledge."
I in no way mean to imply (as Zirkus seems to do above) that
Einstein should be considered as an 'amateur' in his early work,
but most certainly some of his most major accomplishments were
made early, outside of academic affiliation. Even in his later
work, I don't think Einstein ever would consider himself an
'academic' in the sense in which it is applied today.
> > Soon, we may have a major scientific revolution (which could
> > effect fundamental physics) due to the work of another
> > amateur scientist named Stephen Wolfram,
>
> FYI, Stephen Wolfram has a Ph.D. (physics, IIRC) from a top
> flight department (maybe Oxfam--- can't remember details),
Ph.D. in theoretical physics from Caltech, thank you. :)
Stephen
s...@compbio.caltech.edu
You can always tell a pioneer by the arrows in his back.
Printed using 100% recycled electrons.
--------------------------------------------------------
zir...@my-deja.com
<Pine.OSF.4.21.010112...@goedel3.math.washington.edu>,
Chris Hillman <hil...@math.washington.edu> wrote:
> Also, note that Newton's account of the development of the
> calculus and of his physics was inaccurate in many respects (to make a
> long story short, he claimed to have figured all this stuff out
decades
> before he actually did so, during the plague years, [...]
I had not known this. I thought Newton had made important physical
insights (if not mathematical ones) while still an undergraduate.
> Einstein had an excellent undergraduate and graduate education; he was
by
> no means "entirely self-taught". It is true he worked for a few years
in
> the Berne Patent office until he obtained his first academic position,
[...]
Yes, and while there he certainly made important physical insights as an
amateur scientist. By "amateur scientist" I mean someone who does
science but is not specifically employed as a scientific researcher or
teacher. I believe that this is what most people would mean by this
term.
> FYI, Stephen Wolfram has a Ph.D. (physics, IIRC) from a top flight
> department (maybe Oxfam--- can't remember details), and before
founding
> Wolfram Research, he published papers, gave talks, etc., just like any
> other professional physicist; in no sense could he be called an
"amateur
> scientist".
He has a Ph.D. from Caltech, was on the Caltech faculty and worked at
the IAS at Princeton (among other places). But then he became an
entrepeneur and is now the CEO of Wolfram Research, the company that
makes the Mathematica software. I guess you could call him a scientist
but he is not specifically employed as one.
Jacques Distler
one another of having "gone off the deep end." -TB]
In <944vvo$3lo$1...@news.state.mn.us> Thomas Larsson wrote:
>> How is this a criticism? I like my physical theories to be
>> mathematically rigorous, philosophically sound, simple and
>> beautiful, to fit the experimental data, and to make testable
>> predictions. I don't see why using the most complicated possible
>> extensions of algebras is in itself desirable.
>
>It wouldn't be a criticism if string theory had convincing
>experimental support. However, if you base your theory on intrinsic
>mathematical beauty alone, you will be in trouble if someone shows
>that it is in fact ugly. My result removes the distinguished status
>from the underlying algebraic structure. At least the superconformal
>algebra was the algebraic structure underlying string theory in the
>1980s, when I could still understand it.
>
>To avoid confusion, let me emphasize that I am only criticizing string
>theory proper, as applied to four- and (less interestingly)
>ten-dimensional physics. In contrast, I greatly admire the work of
>some string theorists on conformal field theory and its application to
>two-dimensional critical phenomena. This is probably the greatest
>theoretical advance in the last twenty years, and it is undoubtably
>physics.
Now you have gone off the deep end.
The (super)virasoro algebra (which you consider to be too "trivial") is the
underlying algebraic structure of 2d (super)conformal field theory. 2d
superconformal field theory is the basis for string PERTURBATION THEORY.
You profess to LIKE 2d conformal field theory ("the greatest theoretical
advance in the last
twenty years"), DESPITE the "triviality" of its underlying mathematical
framework. But you
DISLIKE string perturbation theory because it is based on 2d conformal field
theory (whose
underlying (super)virasoro algebra is "too trivial".)
Let me say it again: the ONLY relation between string theory and the virasoro
algebra is via the intermediary of 2d conformal field theory. If being based
on virasoro taints a physical theory, then your criticism is directed towards
2d conformal field theory and only indirectly (by inheritance) at string
PERTURBATION theory.
I emphasized "perturbation theory" because, at best, your criticism (were it
not contradicted by your own statement above) would be a criticism of
perturbative string theory.
NONperturbatively, there is nothing fundamental about 2d conformal field
theory (and hence not even a ROLE for the supervirasoro algebra) in string
theory.
A *valid* criticism of string theory -- the lack of any experimental support
-- can be leveled at any approach quantum gravity.
String theory (in contrast to any of its competitors), at least, has the
virtue that it is KNOWN to admit large, weakly-curved macroscopic spacetimes
and is known to have a well-defined
perturbation expansion around such backgrounds (and other, far less trivial
backgrounds).
That's not exactly resounding experimental support (though 50 years of
attempts to construct theories of quantum gravity have failed -- so far -- to
produce an alternative that passes this simplest of tests), and in the
absense of something more persuasive, I can respect a healthy scepticism of
string theory.
But to argue against string theory because it's based on trivial mathematics
is a surprising line of attack to anyone who has attempted to read the papers
appearing on hep-th nowadays.
Perhaps you are made of stronger stuff than I.
Jacques Distler
--
PGP public key: http://golem.ph.utexas.edu/~distler/distler.asc
Charles Francis
<ba...@galaxy.ucr.edu>:
>In article <C.D.H.Williams-...@cw-mac.ex.ac.uk>,
>Charles DH Williams <C.D.H.W...@exeter.ac.uk> wrote:
>>In article <942c1a$orv$1...@news.state.mn.us>, ba...@galaxy.ucr.edu (John
>>Baez) wrote:
>>> How is this a criticism? I like my physical theories to be
>>> mathematically rigorous, philosophically sound, simple and
>>> beautiful, to fit the experimental data, and to make testable
>>> predictions.
>
>>That rules out everything apart from electromagnetism doesn't it?
>I regard Newtonian mechanics to have all these qualities,
The idea of an infinite flat background never was philosophically sound.
Both Liebniz and Gauss picked holes, which cannot be patched without
invoking qm on the one hand and gtr on the other.
>as well
>as general relativity
General relativity encompasses a relativity principle, and yet is still
based on a manifold described in absolute terms. That is not a
philosophically sound position.
>and nonrelativistic quantum mechanics.
Einstein discovered relativity from philosophical considerations about
he meaning of motion. Surely a model which does not preserve the
homogeneity of physical law cannot be regarded as philosophically sound
> Other
>theories tend to be lacking in one or more of these qualities. For
>example, quantum field theory of the sort used in the Standard Model
>has not yet been made rigorous, and the Standard Model is not so
>simple, either. However, I don't DEMAND theories to have ALL of
>these qualities - I just LIKE them better when they have MORE of
>these qualities.
I think we should demand all these qualities. Not that we should expect
to find them in existing theories. But it is by niggling away at the
manner in which existing theories fail to satisfy them that we will know
what to research.
Thomas Larsson
> > Thomas Larsson <Thomas....@hdd.se> wrote:
> >
> > > I have hesitated to make this post, fearing that I will appear as
> > > an amateur crackpot. Now, technically I am an amateur [...]
> >
> > discoveries as amateurs
>
> Au contraire, Zirkus.
>
I said that I feared to appear as a crank; I didn't say that I thought that
I was one.
Most cranks do not have two recent publications in CMP, as well as some fifteen
older papers in lesser ranking journals such as JPA, PLA, IJMPA, PRB and NPB.
My wife outnumbers me by a factor 15, though, but she is in medicine ;-)
Most cranks have not completed a postdoc.
Most cranks do not deliver speaches at the Fields Institute on their vacation.
Most cranks do fill their postings with equations.
Most cranks do not refer to themselves as cranks.
Chris Hillman
On Thu, 18 Jan 2001 zir...@my-deja.com wrote:
> In article
> <Pine.OSF.4.21.010112...@goedel3.math.washington.edu>,
> Chris Hillman <hil...@math.washington.edu> wrote:
>
> > Also, note that Newton's account of the development of the
> > calculus and of his physics was inaccurate in many respects (to make a
> > long story short, he claimed to have figured all this stuff out
> > decades
> > before he actually did so, during the plague years, [...]
>
> I had not known this. I thought Newton had made important physical
> insights (if not mathematical ones) while still an undergraduate.
Sure, he did!
(Dunno about physics, but it is known that N -did- have many important
insights concerning pure mathematics as an undergraduate, including key
elements of what would become known as the calculus, and going far beyond
what the best mathematicians of the day were capable of in many respects.)
All I was saying is that if you read a modern biography, you'll find that
Newtonian physics and the calculus did not spring fullblown from Newton's
brow during the plague years, as he later claimed. His scientific ideas
in fact took several decades to fully mature. When the key pieces finally
-did- come together in his mind, it -is- true that he composed the
Principia in an amazingly short amount of time. (Perhaps a bit less
amazingly once you realize that he had probably been thinking about
writing this book for decades.)
> By "amateur scientist" I mean someone who does science but is not
> specifically employed as a scientific researcher or teacher. I believe
> that this is what most people would mean by this term.
You may be right. If one goes by the dictionary you certainly would be---
but most people -don't- look up words in Websters before they employ them
in their speech or writing! In any case, there have been so many wild
claims concerning AE that it is best to spell out what exactly you mean
when making claims concerning the career of AE.
Chris Hillman
On 18 Jan 2001, Kevin A. Scaldeferri wrote:
> In article
> <Pine.OSF.4.21.010112...@goedel3.math.washington.edu>,
> Chris Hillman <hil...@math.washington.edu> wrote:
>
> >FYI, Stephen Wolfram has a Ph.D. (physics, IIRC) from a top flight
> >department (maybe Oxfam--- can't remember details)
^^^^^
Oh dear, I meant -Oxbridge-, of course. Dr. Freud can make of my slip
what he will :-/
> Caltech, actually.
OK, I -really- meant Caltech :-/
> His tenure here is somewhat legendary. (He was 20 when he got his
> Ph.D.)
Right, I remember that too, now that you mention it.
John Baez
Charles Francis <cha...@clef.demon.co.uk> wrote:
>In article <94506h$7kc$1...@news.state.mn.us>, thus spake John Baez
><ba...@galaxy.ucr.edu>:
>In article <C.D.H.Williams-...@cw-mac.ex.ac.uk>,
>>Charles DH Williams <C.D.H.W...@exeter.ac.uk> wrote:
>>>In article <942c1a$orv$1...@news.state.mn.us>, ba...@galaxy.ucr.edu (John
>>>Baez) wrote:
>>>> I like my physical theories to be
>>>> mathematically rigorous, philosophically sound, simple and
>>>> beautiful, to fit the experimental data, and to make testable
>>>> predictions.
>>>That rules out everything apart from electromagnetism doesn't it?
>>I regard Newtonian mechanics to have all these qualities,
>>as well as general relativity and nonrelativistic quantum mechanics.
>The idea of an infinite flat background never was philosophically sound.
You're right. As I said a while back, I've become too busy to
respond to most of your posts. But this is an important issue...
I shouldn't have implied that I believe Newtonian mechanics and
nonrelativistic quantum mechanics are philosophically sound. I
regard their reliance on a fixed background metric as unsound.
My philosophical qualms about general relativity are less strong.
But it's a classical field theory, and no classical field theory
can explain the matter we see around us, so it clearly needs to
be combined with quantum theory somehow.
I regard philosophical soundness as a matter of degree and only
expect the final theory of everything to be truly sound. Of
course, we may or may not reach such a theory.
Ralph E. Frost
[unnecessary quoted text deleted by angry gods]
> I regard philosophical soundness as a matter of degree and only
> expect the final theory of everything to be truly sound. Of
> course, we may or may not reach such a theory.
Like a goal or an ideal? So we set the objectives of scratching out a
theory of a couple things; then a theory of parts and pieces; then a
theory of a large chunk of stuff; then a theory of quite a bit of a certain
tyep of something; then a theory of something; then something else; then a
slightly more unified theory; then a theory of .....
Ralph Hartley
>
> In article <94506h$7kc$1...@news.state.mn.us>, thus spake John Baez
> <ba...@galaxy.ucr.edu>:
>
> The idea of an infinite flat background never was philosophically sound.
> Both Liebniz and Gauss picked holes, which cannot be patched without
> invoking qm on the one hand and gtr on the other.
I don't see how an infinite flat background is unsound at all.
It may be WRONG, but there is no way that could have been known on
"philosophical" grounds.
One could very easily imagine an infinite flat universe in which the
concepts of absolute space and time are perfectly well defined. Such a
universe could be perfectly self-consistent. There is absolutely no
way to decide a priori that we do not inhabit such a universe, you
have to do experiments.
What if the results of, say, the Michelson-Morley experiment were
different? Do you think the experiment never really needed to be done?
Philosophical considerations may be a useful guide, when good
experiments are not available, but considering their past performance,
I wouldn't trust them very far.
Ralph Hartley
Charles Francis
Hartley <har...@aic.nrl.navy.mil>
>Charles Francis wrote:
>>
>> In article <94506h$7kc$1...@news.state.mn.us>, thus spake John Baez
>> <ba...@galaxy.ucr.edu>:
>> >I regard Newtonian mechanics to have all these qualities,
>>
>> The idea of an infinite flat background never was philosophically sound.
>> Both Liebniz and Gauss picked holes, which cannot be patched without
>> invoking qm on the one hand and gtr on the other.
>
>I don't see how an infinite flat background is unsound at all.
>
>It may be WRONG, but there is no way that could have been known on
>"philosophical" grounds.
>
>One could very easily imagine an infinite flat universe in which the
>concepts of absolute space and time are perfectly well defined. Such a
>universe could be perfectly self-consistent. There is absolutely no
>way to decide a priori that we do not inhabit such a universe, you
>have to do experiments.
Your 'philosophical' grounds seem to be limited to the criterion of
mathematical consistency. Even when I was a schoolboy, and we talked of
what happens at the edge of the universe, it was clear that an
infinitely extending universe is not a sensible idea.
But on a practical level the only experiment you have to do is to check
the prediction of a Newtonian universe infinitely extending in time and
space that all matter would have already collapsed to a point under the
force of gravity. Last time I checked that was quite wrong.
>What if the results of, say, the Michelson-Morley experiment were
>different? Do you think the experiment never really needed to be done?
Strictly speaking yes. The result is nothing but tautology. You cannot
set up an apparatus such as Michelson-Morley without, in some way,
calibrating it from the speed of light. So the result is a foregone
conclusion. But don't just take my word for it. Read Hermann Bondi,
Assumption and Myth in Physical Theory.
>Philosophical considerations may be a useful guide, when good
>experiments are not available, but considering their past performance,
>I wouldn't trust them very far.
I would say that the invention of non-Euclidean geometry, motivated by
the philosophical ground that space could not be Euclidean, and
culminating in general relativity was a quite extraordinary success. But
you are right in so far as human beings applying philosophical argument
are prone to error and fantasy. So it is worth doing practical checks
like the Michelson-Morley experiment. But that is not an indictment on
the form of argument, so much as on our intellectual capacity to perform
it correctly.
Ralph Hartley
>
> In article <3A6F0D89...@aic.nrl.navy.mil>, thus spake Ralph
> Hartley <har...@aic.nrl.navy.mil>
> >One could very easily imagine an infinite flat universe in which the
> >concepts of absolute space and time are perfectly well defined. Such a
> >universe could be perfectly self-consistent. There is absolutely no
> >way to decide a priori that we do not inhabit such a universe, you
> >have to do experiments.
>
> Your 'philosophical' grounds seem to be limited to the criterion of
> mathematical consistency.
That's about as far as I'm willing to trust such arguments.
> Even when I was a schoolboy, and we talked of
> what happens at the edge of the universe, it was clear that an
> infinitely extending universe is not a sensible idea.
Even universes with an edge are not really out of the question, but an
infinately extending one? Of course Obers paradox means it can't be
uniform.
> But on a practical level the only experiment you have to do is to check
> the prediction of a Newtonian universe infinitely extending in time and
> space that all matter would have already collapsed to a point under the
> force of gravity. Last time I checked that was quite wrong.
Unless there was some force or proccess that prevented that. In any
case I don't thing that's what happens to a infinite universe with a
finite amout of matter at a finite temperature (everything boils away
instead).
> >What if the results of, say, the Michelson-Morley experiment were
> >different? Do you think the experiment never really needed to be done?
>
> Strictly speaking yes. The result is nothing but tautology. You cannot
> set up an apparatus such as Michelson-Morley without, in some way,
> calibrating it from the speed of light. So the result is a foregone
> conclusion. But don't just take my word for it. Read Hermann Bondi,
> Assumption and Myth in Physical Theory.
But could we have figured that out without knowing how atoms behave
etc? Why didn't the ancient greeks know that then?
> >Philosophical considerations may be a useful guide, when good
> >experiments are not available, but considering their past performance,
> >I wouldn't trust them very far.
>
> I would say that the invention of non-Euclidean geometry, motivated by
> the philosophical ground that space could not be Euclidean, and
> culminating in general relativity was a quite extraordinary success.
And a quite extraordinary distortion of history too. If I am not
mistaken, it was quite a shock to mathematicians when it was
discovered that space NEAD not be Euclidian (after many attempts to
prove that it was).
> But
> you are right in so far as human beings applying philosophical argument
> are prone to error and fantasy. So it is worth doing practical checks
> like the Michelson-Morley experiment. But that is not an indictment on
> the form of argument, so much as on our intellectual capacity to perform
> it correctly.
OK. In the old days they mis-applied the method or used incorrect
principles, but NOW that we know the correct philosophical principles,
and if we are very carefull ...
As far as I can tell, philosophical methods have not, on the whole,
done much better than chance. For formulating a theory that's really
good enough, especially if you are willing to abandon the results when
the universe proves otherwise. A traditional way to do that is to
formulate a new philosphical argument that proves that the old one was
completely inconsistant, and the new experiment could not POSSIBLY
have had any other result.
Ralph Hartley
John Baez
Ralph Hartley <har...@aic.nrl.navy.mil> wrote:
>I don't see how an infinite flat background is unsound at all.
I do.
>It may be WRONG, but there is no way that could have been known on
>"philosophical" grounds.
It couldn't have been *known*... but it could have been *guessed*.
And in fact, it *was* guessed: Leibniz and Mach were unhappy with
Newtonian physics on philosophical grounds. Einstein took their
reasons for unhappiness, thought about them a lot, added a bunch
more good ideas, and eventually came up with general relativity.
For more about this try
http://math.ucr.edu/home/baez/backround.html
>One could very easily imagine an infinite flat universe in which the
>concepts of absolute space and time are perfectly well defined. Such a
>universe could be perfectly self-consistent.
True. It's a consistent theory, but I don't think it's "philosophically
sound" - for that, I want more than mere consistency.
>There is absolutely no
>way to decide a priori that we do not inhabit such a universe, you
>have to do experiments.
I don't think of philosophy as a source of a priori knowledge about
physics. I think it's a bit more like "taste". Everyone has a set
of prejudices concerning what the universe "ought to be like", and
these guide our theorizing whether we like it or not. Sometimes
these prejudices help us, sometimes they hinder us - but it's well-nigh
impossible not to have them! Studying philosophy and physics can
help us become more aware of these prejudices... and not eliminate
them, but *refine* them.
I have a carefully developed set of prejudices about physics, which
come from reading and thinking about physics, math, and philosophy.
I've spent years developing these. If a theory fits these prejudices
(or better yet, comes along and changes them) I am inclined to say it
is "philosophically sound" - which is basically a fancy way of saying
it appeals to my taste. I don't regard this judgement as infallible.
Nonetheless, as a theorist, I am forced to make such judgements when
deciding what to work on.
ba...@rosencrantz.stcloudstate.edu
Charles Francis <cha...@clef.demon.co.uk> wrote:
>In article <3A6F0D89...@aic.nrl.navy.mil>, thus spake Ralph
>Hartley <har...@aic.nrl.navy.mil>
>>I don't see how an infinite flat background is unsound at all.
>>
>>It may be WRONG, but there is no way that could have been known on
>>"philosophical" grounds.
>>
>>One could very easily imagine an infinite flat universe in which the
>>concepts of absolute space and time are perfectly well defined. Such a
>>universe could be perfectly self-consistent. There is absolutely no
>>way to decide a priori that we do not inhabit such a universe, you
>>have to do experiments.
>Your 'philosophical' grounds seem to be limited to the criterion of
>mathematical consistency. Even when I was a schoolboy, and we talked of
>what happens at the edge of the universe, it was clear that an
>infinitely extending universe is not a sensible idea.
As far as I'm concerned, it's a perfectly sensible idea. In fact, I
know of no reason (philosophical or experimental) to suppose that
we don't actually live in precisely such a Universe.
I honestly can't guess what philosophical grounds you have in mind
for supposing that such a thing is impossible. One may decide
that this possibility is unaesthetic or distasteful, but that's
far short of impossible: the Universe never promised to pay
attention to our aesthetics!
You seem to mean something more than this, but I'm at a loss to guess
what.
>>What if the results of, say, the Michelson-Morley experiment were
>>different? Do you think the experiment never really needed to be done?
>Strictly speaking yes. The result is nothing but tautology.
This is just false. It could have happened that we lived in
a Universe with a preferred frame, and the Michelson-Morley
experiment would have gotten a positive result.
>You cannot
>set up an apparatus such as Michelson-Morley without, in some way,
>calibrating it from the speed of light.
I can't imagine what you mean by this. You can imagine experiments
(like one-way speed of light experiments) that depend on things like
the Einstein synchronization convention, but the Michelson-Morley
experiment isn't one of them.
-Ted
Paul Colby
wrote in message news:94qf4q$9re$1...@mortar.ucr.edu...
> In article <3A6F0D89...@aic.nrl.navy.mil>,
> Ralph Hartley <har...@aic.nrl.navy.mil> wrote:
>
> >I don't see how an infinite flat background is unsound at all.
>
> I do.
>
> >It may be WRONG, but there is no way that could have been known on
> >"philosophical" grounds.
>
> It couldn't have been *known*... but it could have been *guessed*.
>
> I don't think of philosophy as a source of a priori
> knowledge about physics. I think it's a bit more like
> "taste". Everyone has a set of prejudices concerning
> what the universe "ought to be like", and these guide
> our theorizing whether we like it or not. Sometimes
> these prejudices help us, sometimes they hinder us -
> but it's well-nigh impossible not to have them!
> Studying philosophy and physics can help us
> become more aware of these prejudices... and not
> eliminate them, but *refine* them.
>
I for one have a very deep seated mistrust of many
"philosophical arguments". If anything should be
learned from the development of quantum mechanics
and the relativities is that the world makes very little
"sense" to us humans. I'll bet a cream filled donut that
it's not philosophy that guides the next big breakthrough
but simple persistence and dumb luck. Being clever
may well count against researchers in quantum gravity
(hey, I have to find some way to rationalize an
advantage for myself after all). The only thing I feel is
certain is that what ever *is* found to work will be
labeled philosophically sound when the dust clears.
Regards
Paul Colby
Michael Weiss
: I don't think of philosophy as a source of a priori knowledge about
: physics. I think it's a bit more like "taste".
Yeah. There's a passage where Feynman, one of the most pragmatic of
physicists, talks about the role of philosophy in physics, and he gives a
pragmatic justification for philosophical prejudices: when you're trying to
come up with brand new ideas to break the logjam, any source is fair game.
Since the final test of theory is experiment, it doesn't really matter
*where* you get your ideas from.
Which reminds me. Asimov, as science fiction writer, was constantly asked,
"Where do you get your ideas?" Real scientists don't seem to be asked this!
As if it was much harder to dream up some patter about the science of the
next century, unconstrained having to make the computations work out *and*
make accurate predictions, than to do it for real.
How good is philosophy as a source for new science? It's hard to say....
our perceptions are naturally colored by a selection effect. The famous
successful cases (Einstein and SR, Heisenberg and QM) get most of the press.
And, to be fair, a few prominent flubs on the other side get an undue amount
of attention (Hegel "proving" that there can be no more than 7 planets; Kant
assuring us that Euclidean geometry was "a priori" true).
It is interesting that even the success stories show cracks when put under a
microscope. When Heisenberg told Einstein how much he admired the way
Einstein supposedly came up with SR --- "If you can't measure it, it doesn't
belong in your theory" --- Einstein replied that that was nonsense. "Oh, I
may have written it, but it's nonsense all the same. It's the *theory* that
tells you what is in principle measurable, not the other way around."
Indeed, a close examination of Einstein's famous paper shows that the
symmetries in Maxwell's equations played a more significant role.
When van der Waerden looked closely at Heisenberg's first paper on QM, he
found a major disconnect between the philosophy espoused in the intro and
the actual execution in the body of the paper. Heisenberg argues that what
we actually measure are the frequencies and intensities of the spectral
lines, but in the body of the paper he makes free use of the phase, in
principle unobservable. Heisenberg was steeped in experimental results and
also in classical mathematical physics (thanks to Born and Sommerfeld);
formal analogies with Fourier series played a more significant role than
operationalism.
As the ever-pragmatic Feynman remarks, if you can't think up a way to
measure X, that means you don't *have* to include X in your shiny new
theory --- it doesn't mean you *can't*.
Charles Francis
<ba...@galaxy.ucr.edu>
>I have a carefully developed set of prejudices about physics, which
>come from reading and thinking about physics, math, and philosophy.
>I've spent years developing these. If a theory fits these prejudices
>(or better yet, comes along and changes them) I am inclined to say it
>is "philosophically sound" - which is basically a fancy way of saying
>it appeals to my taste.
One can, and should, further refine the notion of "philosophically
sound". If we can go from "aesthetically pleasing" to "functional and
efficient" and from there to a practical criteria of "completeness" for
which I gave the definition:
A complete model should have no attributes which cannot be explained in
terms of the fundamental assumptions of the model
Now we may ask what other practical criteria might we find that will
help us to shape physical law.
Traditionally there have been two main themes, that a theory should be
empirically founded, and/or it should be metaphysically viable. The
completeness requirement places conditions on whether a theory is
metaphysically viable. I would cite Bohmian mechanics as an example of a
theory which is not viable, because it asserts a local effect caused by
a non-local property without providing any mechanism for it. That means
that as a model Bohmian mechanics is not complete by the above
definition.
Of course completeness does not tell us whether a model in impossible,
or whether it is merely inadequate. Either way the model is not
philosophically sound, and it is a matter of the researchers judgement
as to whether he will reject the model or try to fix it.
We can also apply empiricist criteria to the notion of philosophically
sound. Ideally the assumptions of a model should be based on the
directly observable properties of the universe. It may be ambitious to
think that we can necessarily construct a complete model such that all
of its assumptions are open to direct verification by observation, but,
given the option, an empirically founded model would be preferable to
one which depends on metaphysics which cannot be observed.
Special relativity is a good example of a theory which is empirically
based. To understand special relativity we merely have to recognise that
the numbers for time and position coming out of the measurement are
defined by the means of measurement, the physical process of timing two
way photon exchange. Once we understand that it this physical process is
which gives us Minkowsky space time, we are ready to understand also
that there is no reason to suppose that space-time should not be curved.
In fact if we assume that the physical process of reflection takes time,
then we are forced to conclude that the metric based on the empirical
process of measurement cannot be flat.
We may also use this criterion to evaluate interpretations of quantum
mechanics. If we take a strict Von Neumann/Dirac approach then quantum
mechanics is a strictly empirically based theory. In this kind of
approach we cannot talk of matter waves or wave-particle duality because
they are not observed. What we observe are states in measurement, and
relations between fore and after states.
The approach becomes really interesting when applied to relativistic
quantum mechanics, because now we put together two empirically based
theories of measurement (sr & qm) and we get the real properties of
electron, photons and the electromagnetic force.
Charles Francis
<ba...@galaxy.ucr.edu>
>
>
>I don't think of philosophy as a source of a priori knowledge about
>physics. I think it's a bit more like "taste". Everyone has a set
>of prejudices concerning what the universe "ought to be like", and
>these guide our theorizing whether we like it or not. Sometimes
>these prejudices help us, sometimes they hinder us - but it's well-nigh
>impossible not to have them! Studying philosophy and physics can
>help us become more aware of these prejudices... and not eliminate
>them, but *refine* them.
>
>I have a carefully developed set of prejudices about physics, which
>come from reading and thinking about physics, math, and philosophy.
>I've spent years developing these. If a theory fits these prejudices
>(or better yet, comes along and changes them) I am inclined to say it
>is "philosophically sound" - which is basically a fancy way of saying
>it appeals to my taste. I don't regard this judgement as infallible.
>Nonetheless, as a theorist, I am forced to make such judgements when
>deciding what to work on.
Talking of refining ideas, surely there is more to aesthetics than an
appeal to the taste of an individual. Harmony, balance, functionality,
efficiency all play a role. If a theory asks (or, worse, overlooks)
unanswered questions, I suspect you would find that ugly. I think it is
proper to require that a "philosophically sound" theory must be
complete, in that it should have no unexplained attributes.
Completeness is clearly a difficult issue. That is to say it is a notion
which is in need of refinement. Einstein probably thought he knew what
he meant when he said quantum mechanics could not be complete, but most
of the discussions I have read seem to ask more questions than they
answer. I am not sure that anyone has ever written down an adequate
explanation of what completeness would mean in a scientific theory. My
expression of the notion is that a complete model should have no
attributes which cannot be explained in terms of the fundamental
assumptions of the model. But do you have any thoughts on the issue.
Like you say, the more we refine these ideas the more we may hope our
judgement on what to work on is accurate.
Charles Francis
Hartley <har...@aic.nrl.navy.mil>
>
>> But on a practical level the only experiment you have to do is to check
>> the prediction of a Newtonian universe infinitely extending in time and
>> space that all matter would have already collapsed to a point under the
>> force of gravity. Last time I checked that was quite wrong.
>
>Unless there was some force or proccess that prevented that.
This is an example of what I would call a philosophical inadequacy. The
theory, Newtonian gravity, is ok as far as it goes but when we look at
all its predictions we find something completely unexplained, something
that forces us to hypothesise a whole new area of science, which fits
with no other evidence whatsoever. It's not a proof that the theory is
wrong, as such, but it shows that the theory is inadequate and
incomplete. The goal of science is to have a theory with no such
unanswered questions. Of course we may never find such a theory, but
who's to say we won't?
>In any
>case I don't thing that's what happens to a infinite universe with a
>finite amout of matter at a finite temperature (everything boils away
>instead).
A universe under Newtonian gravity is subject to collapse. Just as stars
and galaxies form from interstellar gas (a perfectly uniform gas would
be a state of unstable equilibrium).
>> >What if the results of, say, the Michelson-Morley experiment were
>> >different? Do you think the experiment never really needed to be done?
>>
>> Strictly speaking yes. The result is nothing but tautology. You cannot
>> set up an apparatus such as Michelson-Morley without, in some way,
>> calibrating it from the speed of light. So the result is a foregone
>> conclusion. But don't just take my word for it. Read Hermann Bondi,
>> Assumption and Myth in Physical Theory.
>
>But could we have figured that out without knowing how atoms behave
>etc? Why didn't the ancient greeks know that then?
We aren't smart enough.
>> >Philosophical considerations may be a useful guide, when good
>> >experiments are not available, but considering their past performance,
>> >I wouldn't trust them very far.
>>
>> I would say that the invention of non-Euclidean geometry, motivated by
>> the philosophical ground that space could not be Euclidean, and
>> culminating in general relativity was a quite extraordinary success.
>
>And a quite extraordinary distortion of history too. If I am not
>mistaken, it was quite a shock to mathematicians when it was
>discovered that space NEAD not be Euclidian (after many attempts to
>prove that it was).
It is an illustration of just how smart we would need to be.
Mathematicians worked on Euclid's fifth postulate for two thousand years
(ok with a break in the dark ages) without suspecting that it was wrong.
But the culmination of this two thousand years of study was that one of
the greatest mathematicians in history, Gauss, first began to suspect it
was wrong, and actually convinced himself it could not be right. And
that without experimental evidence.
>> But
>> you are right in so far as human beings applying philosophical argument
>> are prone to error and fantasy. So it is worth doing practical checks
>> like the Michelson-Morley experiment. But that is not an indictment on
>> the form of argument, so much as on our intellectual capacity to perform
>> it correctly.
>
>OK. In the old days they mis-applied the method or used incorrect
>principles, but NOW that we know the correct philosophical principles,
>and if we are very carefull ...
>
>As far as I can tell, philosophical methods have not, on the whole,
>done much better than chance. For formulating a theory that's really
>good enough, especially if you are willing to abandon the results when
>the universe proves otherwise. A traditional way to do that is to
>formulate a new philosphical argument that proves that the old one was
>completely inconsistant, and the new experiment could not POSSIBLY
>have had any other result.
Philosophical methods may tell us that a model is not adequate. For
example general relativity contains a number of problems which I would
expect to be resolved by a GUT. For one thing it is based on a
philosophy of measurement in which velocity is recognised as relative,
but in which measurements of position can be made with respect to an
absolute starting point, or origin of co-ordinates. For another it
predicts singularities in which the behaviour of matter cannot be
described, so it is certainly not complete.
Charles Francis
ba...@rosencrantz.stcloudstate.edu
>In article <94psfo$k85$1...@news.state.mn.us>,
>Charles Francis <cha...@clef.demon.co.uk> wrote:
>
> Even when I was a schoolboy, and we talked of
>>what happens at the edge of the universe, it was clear that an
>>infinitely extending universe is not a sensible idea.
>
>As far as I'm concerned, it's a perfectly sensible idea. In fact, I
>know of no reason (philosophical or experimental) to suppose that
>we don't actually live in precisely such a Universe.
>
>I honestly can't guess what philosophical grounds you have in mind
>for supposing that such a thing is impossible. One may decide
>that this possibility is unaesthetic or distasteful, but that's
>far short of impossible: the Universe never promised to pay
>attention to our aesthetics!
>
>You seem to mean something more than this, but I'm at a loss to guess
>what.
Yes I mean more. It is nice, for a change, to find myself on the same
side of a discussion as John Baez. John talks of aesthetics, but you
rightly surmise that I mean something more. So while I I'm hoping I can
keep John on my side of the discussion, I expect him to come up and
clobber me at any time for going too far.
Like John I have recognised that, whether we intend it or not, our ideas
about physics are shaped by philosophy. If we have not put some
considerable effort into refining our philosophy, then it is likely that
our ideas are shaped by prejudice, or that they are based on half baked
notions of what could be.
In this instance I think we can go a bit further than talk of
aesthetics. Aesthetics are often determined by functionality, as in the
design of sailing ships and many bridges. Many a physical theory might
be consistent, but if it raises unanswered or even ananswerable
questions then I would say it is philosophically unsound, on the ground
that it is incomplete.
Of course just because a model is incomplete does it not mean it is
necessarily wrong. It may just need fixing. But until it is fixed I do
not think it can be described as philosophically sound.
> It could have happened that we lived in
>a Universe with a preferred frame, and the Michelson-Morley
>experiment would have gotten a positive result.
Perhaps we could live in a universe which is totally different from the
one we observe, or maybe there really is essentially only one model of a
universe which is complete as well as consistent. We do not even have
one model which is complete, and the models which come closest, quantum
field theories, are plagued with consistency issues.
Until such time as we have one complete consistent model of the universe
it seems hardly worth putting too much effort into ascertaining whether
other such models could exist. But given the difficulty in constructing
one model I certainly think it would be an extravagant claim to say that
others certainly exist, and particularly extravagant to think that they
could exist with properties which are at odds with observation.
>>You cannot
>>set up an apparatus such as Michelson-Morley without, in some way,
>>calibrating it from the speed of light.
>
>I can't imagine what you mean by this. You can imagine experiments
>(like one-way speed of light experiments) that depend on things like
>the Einstein synchronization convention, but the Michelson-Morley
>experiment isn't one of them.
Of course it is! The MM experiment measures two way light speed. To
conclude that the speed of light is constant you have to assume that the
time of reflection is exactly half way between emission and absorption.
On a deeper level there is the question of what was actually being
measured when they calculated the distance travelled by light. I would
say that they were measuring a property of the electromagnetic structure
of matter as described in quantum electrodynamics. The scale such
structures is determined by the speed of light and the fine structure
constant.
Indeed if you think carefully enough about what we are doing in any
measurement, then on the basis of empirical argument only you can say
that we are measuring properties found in the structures of matter, but
you cannot give any empirical argument that we are measuring prior
distances in "empty" space. So there are philosophical and empirical
arguments that we are not living in a universe based on absolute space.
t...@rosencrantz.stcloudstate.edu
>In article <3A6F0D89...@aic.nrl.navy.mil>, thus spake Ralph
>Hartley <har...@aic.nrl.navy.mil>
>>I don't see how an infinite flat background is unsound at all.
>>
>>It may be WRONG, but there is no way that could have been known on
>>"philosophical" grounds.
>>
>>One could very easily imagine an infinite flat universe in which the
>>concepts of absolute space and time are perfectly well defined. Such a
>>universe could be perfectly self-consistent. There is absolutely no
>>way to decide a priori that we do not inhabit such a universe, you
>>have to do experiments.
>Your 'philosophical' grounds seem to be limited to the criterion of
>mathematical consistency. Even when I was a schoolboy, and we talked of
>what happens at the edge of the universe, it was clear that an
>infinitely extending universe is not a sensible idea.
As far as I'm concerned, it's a perfectly sensible idea. In fact, I
know of no reason (philosophical or experimental) to suppose that
we don't actually live in precisely such a Universe.
I honestly can't guess what philosophical grounds you have in mind
for supposing that such a thing is impossible. One may decide
that this possibility is unaesthetic or distasteful, but that's
far short of impossible: the Universe never promised to pay
attention to our aesthetics!
You seem to mean something more than this, but I'm at a loss to guess
what.
>>What if the results of, say, the Michelson-Morley experiment were
>>different? Do you think the experiment never really needed to be done?
>Strictly speaking yes. The result is nothing but tautology.
This is just false. It could have happened that we lived in
a Universe with a preferred frame, and the Michelson-Morley
experiment would have gotten a positive result.
>You cannot
>set up an apparatus such as Michelson-Morley without, in some way,
>calibrating it from the speed of light.
I can't imagine what you mean by this. You can imagine experiments
(like one-way speed of light experiments) that depend on things like
the Einstein synchronization convention, but the Michelson-Morley
experiment isn't one of them.
-Ted
[Moderator's note: yet again, I appear to have screwed up and posted
this article by Ted in such a way that it superficially appeared to be
from me. Sorry, Ted! - jb]
Ralph E. Frost
Charles Francis <cha...@clef.demon.co.uk> wrote in message
news:g1WpnKA5...@clef.demon.co.uk...
> In article <94qf4q$9re$1...@mortar.ucr.edu>, thus spake John Baez
> <ba...@galaxy.ucr.edu>
>
> >I have a carefully developed set of prejudices about physics, which
..
> Special relativity is a good example of a theory which is empirically
> based. To understand special relativity we merely have to recognise that
> the numbers for time and position coming out of the measurement are
> defined by the means of measurement, the physical process of timing two
> way photon exchange. Once we understand that it this physical process is
> which gives us Minkowsky space time, we are ready to understand also
> that there is no reason to suppose that space-time should not be curved.
> In fact if we assume that the physical process of reflection takes time,
> then we are forced to conclude that the metric based on the empirical
> process of measurement cannot be flat.
I was reading something on the gravitational anomalies egroup (now on
Yahoo!) about Michaelson-Morely, or some person named Miller -- the gist
being that MM tests don't actually have a null result but shown some kind of
periodic "swaying back and forth" [my paraphrase].
Would that empirical finding conflict with the Einstein descriptions?
Would it influence the philosophical soundness you and John are discussing?
You _may_ find out more about that group at :
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..snipped from a
The folks that write and investigate this alternative approach are further
up the food chain than I am so please don't associate any prejudices you may
have with my simplistic notions to these other people (automatically). If
their wild-eyed crackpots, I'm sure I will be informaed of that fact
promptly...
C. M. Heard
>One can, and should, further refine the notion of "philosophically
>sound". If we can go from "aesthetically pleasing" to "functional and
>efficient" and from there to a practical criteria of "completeness" for
>which I gave the definition:
>
>A complete model should have no attributes which cannot be explained in
>terms of the fundamental assumptions of the model.
That's fair enough.
>We may also use this criterion to evaluate interpretations of quantum
>mechanics. If we take a strict Von Neumann/Dirac approach then quantum
>mechanics is a strictly empirically based theory. In this kind of
>approach we cannot talk of matter waves or wave-particle duality because
>they are not observed. What we observe are states in measurement, and
>relations between fore and after states.
If the words "the strict Von Neumann/Dirac" refer to the standard
Copenhagen interpretation, with its axiom that a measurement causes
a non-unitary "reduction of the wavepacket", then it is _far_ from
complete! There is no sharp criterion that for deciding when the
unitary Schrodinger equation applies and when the non-unitary reduction
procedure applies. Furthermore, there are no details (i.e., equations)
describing the reduction process.
So, yes, we can use the completeness criterion to evaluate interpretations
of quantum mechanics, and under the strict Von Neumann/Dirac approach
quantum mechanics is an incomplete theory.
Mike
Paul Colby
wrote in message news:9525tk$irg$1...@news.state.mn.us...
> In article <94q3hu$ehvn8$1...@ID-28113.news.dfncis.de>, thus spake
> t...@rosencrantz.stcloudstate.edu
>>In article <94psfo$k85$1...@news.state.mn.us>,
>>Charles Francis <cha...@clef.demon.co.uk> wrote:
>>>You cannot set up an apparatus such as Michelson-Morley
>>>without, in some way, calibrating it from the speed of light.
>>I can't imagine what you mean by this. You can imagine
>>experiments (like one-way speed of light experiments)
>>that depend on things like the Einstein synchronization
>>convention, but the Michelson-Morley experiment isn't
>>one of them.
> Of course it is! The MM experiment measures two way
> light speed. To conclude that the speed of light is
> constant you have to assume that the time of reflection
> is exactly half way between emission and absorption.
I've been taking part in "discussions" of this very topic
on sci.physics.relativity. Sadly, the proponents of
the two-way-speed clock synchronization issue haven't
been able to address the changes that would
appear to be needed in Maxwell's equations to accommodate
such a possibility. I would think that if we assume a
different left-going and right-going light speed then one
must have an anisotropy epsilon and mu which depends
on a vector quantity (the dreaded absolute velocity). What
form would this epsilon and mu take and how would it
depend on velocity?
Regards
Paul Colby
t...@rosencrantz.stcloudstate.edu
Charles Francis <cha...@clef.demon.co.uk> wrote:
>In article <94q3hu$ehvn8$1...@ID-28113.news.dfncis.de>, thus spake
>ba...@rosencrantz.stcloudstate.edu
(Actually, it was me, Ted, who wrote that. The article
got posted under Baez's from line instead of mine by mistake.)
>>As far as I'm concerned, it's a perfectly sensible idea. In fact, I
>>know of no reason (philosophical or experimental) to suppose that
>>we don't actually live in precisely such a Universe.
>>
>But do you know of a complete mathematical model that works like that?
I'm not entirely sure what the question means. If you're asking
whether I know of a complete mathematical model that agrees with all
experimental evidence and has a spatially infinite Universe, the
answer is No. But then I don't know of a complete mathematical model
that agrees with all experimental evidence and has a finite Universe
either. Nor, I claim, do you. Physics isn't over yet; we haven't
reached the theory of everything.
I do claim that, if we stay away from the quantum gravity
regime (near the big-bang and black-hole singularities), theories
based on general relativity work, and they work equally well
with finite and infinite Universes. I know of precisely
no grounds to distinguish between those two options.
Of course, eventually we'll have to supplant general relativity
with another theory that plays well with quantum mechanics and
is free of singularity problems. *Maybe* it'll turn out
to be the case that such a theory will require the Universe
to be finite. But as of now I know of absolutely no evidence
that it will do so.
So I stand by my claim: at the moment, I know of no evidence,
experimental or philosophical, against the hypothesis that the
Universe is spatially infinite. I understand that you have some
philosophical objection to an infinite Universe, and that that
objection is more than mere aesthetics, but as far as I can tell you
still haven't dropped any hints about the nature of that objection.
>>>You cannot
>>>set up an apparatus such as Michelson-Morley without, in some way,
>>>calibrating it from the speed of light.
>>
>>I can't imagine what you mean by this. You can imagine experiments
>>(like one-way speed of light experiments) that depend on things like
>>the Einstein synchronization convention, but the Michelson-Morley
>>experiment isn't one of them.
>
>Of course it is! The MM experiment measures two way light speed. To
>conclude that the speed of light is constant you have to assume that the
>time of reflection is exactly half way between emission and absorption.
To recap: You said that the Michelson-Morley null result was
tautological, so the experiment didn't need to be done. I disagree.
I still haven't seen any justification for your claim. It could have
happened that Michelson and Morley saw fringe shifts when they rotated
the interferometer. They didn't, and the fact that they didn't told
us important and non-tautological things about the Universe.
The observation that the fringes didn't shift in no way depended
on "calibrating" anything to the speed of light.
-Ted
Michael Weiss
: (ok with a break in the dark ages) without suspecting that it was wrong.
: But the culmination of this two thousand years of study was that one of
: the greatest mathematicians in history, Gauss, first began to suspect it
: was wrong, and actually convinced himself it could not be right. And
: that without experimental evidence.
Gauss, as is well-known, concluded that Euclid's fifth postulate was not a
priori true, and that its truth value in our universe was a matter for
experimental test. Supposedly he attempted to test it, by measuring the
angles between the tops of three mountains (though I'm not sure how well
authenticated this story is.)
However, there is a logical distinction between saying that Euclidean
geometry *might* not hold for our universe, and saying it *cannot* hold. As
far as I know, Gauss never gave any indication of believing that Euclidean
geometry *had* to be false for our universe.
Can you back up your historical claim with a citation? (I trust that you
see the distinction.)
Gordon D. Pusch
> I was reading something on the gravitational anomalies egroup (now on
> Yahoo!) about Michaelson-Morely, or some person named Miller -- the gist
> being that MM tests don't actually have a null result but shown some kind of
> periodic "swaying back and forth" [my paraphrase].
Yes, Miller published this claim in _Rev. Mod. Phys._ in the late 1950's,
IIRC; his deduced ``drift velocity'' for the Earth was nearly parallel to
its rotation axis, so he claimed that it had not been previously observed
because the diurnal variation of the fringe-shift nearly vanishes.
Subsequent higher-precision repetitions of the MM experiment of course
failed to replicate Miller's claimed diurnal variation, suggesting that
Miller's observed ``aether drift'' was probably a spurious systematic
effect --- but this does not stop cranks from still citing his paper
as ``proof'' that the aether exists. (A common crank tactic, actually:
cite early experiments with positive or ambiguous results, while ignoring
later experiments definitively _disproving_ their pet thesis...)
> Would that empirical finding conflict with the Einstein descriptions?
Obviously. An unambiguous, reproducible experimental observation of an
``aether drift'' would relegate Einstein's Special Relativity to at best
the same position as Newtonian Gravitation: A useful approximation that
breaks down in some limit. (But don't hold your breathe waiting for this
to happen...)
> Would it influence the philosophical soundness you and John are discussing?
Of course, since experiment disproof invalidates _ANY_ amount of philosophy.
``Philosophical soundness'' is only relevant if you have competing theories
with identical predictions for all known experiments (to current experimental
error), and one needs to choose between them --- e.g., Einstein's Special
Relativity versus Lorentz's Aether Theory. However, the moment even a single
philosophically unaesthetic but reliably reproducible experimental result is
observed that falsifies the more ``philosophically sound'' theory, then that
theory, no matter how elegant or beautiful it is thought to be, must be
relegated to the rubbish-heap of history.
Because ultimately, philosophy is nothing but endless wordy explication of
the things one has merely _ASSUMED_ to be true as axioms --- it is utterly
incapable of self-validation. If an experiment proves that something some
windbag philosopher thinks is ``self-evidently true'' is in fact false, then
the windbag philosopher was simply _WRONG_ about it being ``self evident,''
_period_. Experiment always trumps theory, but it _ANNIHILATES_ an infinite
amount of philosophical hot air.
-- Gordon D. Pusch
perl -e '$_ = "gdpusch\@NO.xnet.SPAM.com\n"; s/NO\.//; s/SPAM\.//; print;'
Stephen Speicher
> "Ralph E. Frost" <ref...@dcwi.com> writes:
>
> > I was reading something on the gravitational anomalies egroup
> > (now on Yahoo!) about Michaelson-Morely, or some person named
> > Miller -- the gist being that MM tests don't actually have a
> > null result but shown some kind of periodic "swaying back and
> > forth" [my paraphrase].
>
> Yes, Miller published this claim in _Rev. Mod. Phys._ in the
> late 1950's,
Actually, it was published in 1933 (D. C. Miller, _Revs. Mod.
Phys._, 5, 1933, 203).
>
> Subsequent higher-precision repetitions of the MM experiment of
> course failed to replicate Miller's claimed diurnal variation,
> suggesting that Miller's observed ``aether drift'' was probably
> a spurious systematic effect --- but this does not stop cranks
> from still citing his paper as ``proof'' that the aether
> exists.
Yes, the effect in the experiment was due to statistical
fluctations and temperature variation, as reported in the 1955
paper by R. S. Shankland et al., _Revs. Mod. Phys._, 27, 1955,
167. Undoubtedly, this was the "1950's" paper you had in mind.
Stephen
s...@compbio.caltech.edu
Welcome to California. Bring your own batteries.
Printed using 100% recycled electrons.
--------------------------------------------------------
Charles Francis
spake Paul Colby <paulc...@earthlink.net>
>> Of course it is! The MM experiment measures two way
>> light speed. To conclude that the speed of light is
>> constant you have to assume that the time of reflection
>> is exactly half way between emission and absorption.
>
>I've been taking part in "discussions" of this very topic
>on sci.physics.relativity. Sadly,
Sadly... That was my conclusion about such "discussions" too.
> the proponents of
>the two-way-speed clock synchronization issue haven't
>been able to address the changes that would
>appear to be needed in Maxwell's equations to accommodate
>such a possibility. I would think that if we assume a
>different left-going and right-going light speed then one
>must have an anisotropy epsilon and mu which depends
>on a vector quantity (the dreaded absolute velocity). What
>form would this epsilon and mu take and how would it
>depend on velocity?
I don't think it is sensible to choose some other definition of a co-
ordinate system. But technically it is possible. In gtr people study
whole families of possible co-ordinate systems, which could contort
Maxwell's equations in all sorts of horrible ways. But at the end of the
day we know that all these co-ordinate systems must contain exactly the
same physics, and they must all be transformable into one in which the
time of reflection is exactly half way between emission and absorption.
The point is that space-time co-ordinates are actually *defined* like
this. Each observer has a clock which is, without loss of generality,
the origin of his co-ordinate system. He then defines a measurement
procedure, which could be
(a) the radar method
(b) something equivalent to the radar method (such as Einstein's
original rigid rod & synchronisation)
(c) something which could be transformed into the radar method
(d) something completely different.
But if he were to use (d) then he would not be defining the same
quantities which we use at all.
Of course, as I have just being saying in response to Ted, there are
circumstances in which we cannot define a space-time co-ordinate by
means of the radar method. Then we get quantum mechanical effects. But
the point is not that the definition by two way light speed is wrong,
but that ultimately there is no other definition. When we have no
possibility of measurement, then the quantity does not exist.
Charles Francis
dstate.edu
>So I stand by my claim: at the moment, I know of no evidence,
>experimental or philosophical, against the hypothesis that the
>Universe is spatially infinite. I understand that you have some
>philosophical objection to an infinite Universe, and that that
>objection is more than mere aesthetics, but as far as I can tell you
>still haven't dropped any hints about the nature of that objection.
To be honest in this thread I have been steering away from my
philosophical objections to the infinite, and looking at other reasons
why we know that existing theories are not a final complete theory. I
have three main objections, which I can summarise briefly as follows.
One objection is to the idea that the universe can be twice as big as
itself (and all similar bizarre properties of the infinite). You may say
it is not inconsistent. "Green ideas sleep furiously" (Leon Chomsky) is
not inconsistent either. But nor does it make sense.
Secondly a differentiable manifold is a concept brimming with structure.
I find it impossible to think that physical structure can exist without
some mechanism. A mechanism which is homomorphic to itself under change
of scale would indeed be a strange thing.
Finally even if the universe were infinite, there could never be any
scientific way of saying so. Empirically we can only test for the
finite, so a spatially infinite universe would always remain speculative
and unscientific. In fact I would say it is you that require evidence to
support the hypothesis of a spatially infinite universe, rather than I.
I do not need to assume that the universe is not infinite. I merely do
not assume that it is infinite. Since this appears to be sufficient to
describe physics, there is a philosophical objection to any further
unsupported hypothesis.
>>>>You cannot
>>>>set up an apparatus such as Michelson-Morley without, in some way,
>>>>calibrating it from the speed of light.
>>>I can't imagine what you mean by this. You can imagine experiments
>>>(like one-way speed of light experiments) that depend on things like
>>>the Einstein synchronization convention, but the Michelson-Morley
>>>experiment isn't one of them.
>>Of course it is! The MM experiment measures two way light speed. To
>>conclude that the speed of light is constant you have to assume that the
>>time of reflection is exactly half way between emission and absorption.
>To recap: You said that the Michelson-Morley null result was
>tautological, so the experiment didn't need to be done. I disagree.
>I still haven't seen any justification for your claim. It could have
>happened that Michelson and Morley saw fringe shifts when they rotated
>the interferometer. They didn't, and the fact that they didn't told
>us important and non-tautological things about the Universe.
>The observation that the fringes didn't shift in no way depended
>on "calibrating" anything to the speed of light.
It tells us that the definition of the space-time co-ordinate of the
point of reflection as ( (t1+t2)/2, (t2-t1)/2 ) is possible. That is
both important and non-tautological, but having made the definition that
this is the co-ordinate, the constancy of the speed of light is
tautology.
Of course there are experiments in which the definition of a space-time
co-ordinate cannot be made. And then we do see interference fringes, or
other such quantum mechanical effects. Nor can we define the classical
concept of speed in such circumstances.
But can you propose some alternative. If we do not define co-ordinates
by the radar method, or by some means equivalent to the radar method,
then I have no idea how co-ordinates could be defined at all. Certainly
you can assume a prior manifold. But how would you measure it? And the
moment you do measure it, how could you know that you are indeed
measuring it, rather than defining a new quantity by the physical
processes involved in the measurement?
Charles Francis
<mic...@spamfree.net>
I thought I had read this in a book which I could reasonably trust. But
now I cannot find the passage, and I fear I may have read it in
Hyperspace, by Michio Kaku. Kaku also tells the story of the expedition
in the mountains. I only skimmed Hyperspace, but having now looked at
Kaku & Thompson, Beyond Einstein, I now know that Kaku is perfectly
happy with fabrication, both scientific and historical, if it sells
books.
E. T. Bell gets a lot of stick for not being 100% historical. But so far
no one has ever given me an instance of a story that E. T. Bell has
certainly got wrong, or one where he claimed that an undocumented story
is history - he does tell unauthenticated stories but it seems to me he
makes it clear when he is doing so. Notably E. T. Bell does not
attribute this realisation to Gauss, though I think Bell did read
Gauss's notes which I think would have been the only source. And he
certainly had a reasonable grasp on non-Euclidean geometry.
It seems unlikely that Bell missed many decent stories, though he did
fail to spot that Gauss and Lobachewsky shared a teacher, Johann Martin
Bartels (any relation, Toby?) It seems likely that Bartels took the
idea of non-Euclidean geometry from Brunswick to Kazan. I am however
forced to withdraw the claim that Gauss believed Euclidean geometry had
to be false for our universe. On the other hand this was before Cantor
had given a semblance of respectability to the infinite and Gauss was
one of those who mistrusted infinity in mathematics.
As for the map making, Gauss was employed as a cartographer as part of
his duty as scientific advisor to the Hanoverian and Danish Governments.
The development of differential geometry for curved surfaces was closely
tied in with the problem of mapping the curved surface of the earth.
Gauss had plenty of survey information without needing to organise a
special expedition, and it would have been a matter of moments to add
the angles of a triangle. It is difficult to suppose that he did not
think to do so. He probably had to calculate the angles anyway as part
of the survey.
J. J. Lodder
> Actually, it was published in 1933 (D. C. Miller, _Revs. Mod.
> Phys._, 5, 1933, 203).
>
> >
> > Subsequent higher-precision repetitions of the MM experiment of
> > course failed to replicate Miller's claimed diurnal variation,
> > suggesting that Miller's observed ``aether drift'' was probably
> > a spurious systematic effect --- but this does not stop cranks
> > from still citing his paper as ``proof'' that the aether
> > exists.
>
> Yes, the effect in the experiment was due to statistical
> fluctations and temperature variation, as reported in the 1955
> paper by R. S. Shankland et al., _Revs. Mod. Phys._, 27, 1955,
> 167. Undoubtedly, this was the "1950's" paper you had in mind.
And don't forget to mention that Miller's claim resulted in the most
memorable quote in all of Physics;
"Schwierig ist der Herrgott, aber boshaft ist er nicht",
Perhaps better known as "Subtle is the Lord..............."
In other words, Einstein did not for a moment take Miller seriously,
even tough it took another 20 years to find out
what precisely had gone wrong with the experiment.
(It was merely a falsification, after all.
Philosophically quite unsound, of course :-)
Best,
Jan
--
"No experiment should be believed until it has been confirmed by a
reliable theory" (Edington)
Jonathan Thornburg
Charles Francis <cha...@clef.demon.co.uk> wrote:
>As for the map making, Gauss was employed as a cartographer as part of
>his duty as scientific advisor to the Hanoverian and Danish Governments.
>The development of differential geometry for curved surfaces was closely
>tied in with the problem of mapping the curved surface of the earth.
>Gauss had plenty of survey information without needing to organise a
>special expedition, and it would have been a matter of moments to add
>the angles of a triangle. It is difficult to suppose that he did not
>think to do so. He probably had to calculate the angles anyway as part
>of the survey.
The current German 10 mark bill features Gauss and his work. It
shows a picture of him, a Gaussian and its equation (!), and on the
other side, a diagram of a surveying network he set up in northern
Germany. The diagram is small enough that it's hard to read the
city names, but the coastline is identifyably that of northwest
Germany.
Alas, there doesn't seem to be anything on the bill about his work
in geometry or number theory.
--
-- Jonathan Thornburg <jth...@thp.univie.ac.at>
http://www.thp.univie.ac.at/~jthorn/home.html
Universitaet Wien (Vienna, Austria) / Institut fuer Theoretische Physik
Q: Only 7 countries have the death penalty for children. Which are they?
A: Congo, Iran, Nigeria, Pakistan[*], Saudi Arabia, United States, Yemen
[*] Pakistan moved to end this in July 2000. -- Amnesty International,
http://www.web.amnesty.org/ai.nsf/index/AMR511392000
Michael Weiss
: As for the map making, Gauss was employed as a cartographer as part of
: his duty as scientific advisor to the Hanoverian and Danish Governments.
: The development of differential geometry for curved surfaces was closely
: tied in with the problem of mapping the curved surface of the earth.
: Gauss had plenty of survey information without needing to organise a
: special expedition, and it would have been a matter of moments to add
: the angles of a triangle. It is difficult to suppose that he did not
: think to do so. He probably had to calculate the angles anyway as part
: of the survey.
I believe that the standard methods of surveying simply *assume* that the
geometry of 3-space is Euclidean, and use that assumption to compute
distances, etc. Of course it is possible that the raw data contains
redundancies which Gauss could have used to test this assumption.
Robert Hill
Jonathan Thornburg wrote in message <95bltv$gek$1...@mach.thp.univie.ac.at>...
> The current German 10 mark bill features Gauss and his work. It
> shows a picture of him, a Gaussian and its equation (!), and on the
> other side, a diagram of a surveying network he set up in northern
> Germany. The diagram is small enough that it's hard to read the
> city names, but the coastline is identifyably that of northwest
> Germany.
Some time ago John Baez drew the attention of readers of this group
to the website http://www.kleinbottle.com .
There you can not only read a lot of learned jokes and buy Klein bottles,
but also see pictures of the Gauss 10 mark bill and a list of some
of the place names. You can even buy one of the bills for 5 dollars.
In another message in the same thread, Michael Weiss writes:
> I believe that the standard methods of surveying simply *assume* that the
> geometry of 3-space is Euclidean, and use that assumption to compute
> distances, etc. Of course it is possible that the raw data contains
> redundancies which Gauss could have used to test this assumption.
I think it's more than possible. For one thing, surveyors may take
Euclidean geometry for granted, but I suspect that even then they didn't
take perfection in their own measurements for granted. They probably
used over-determined data.
This is especially likely for the angles of a triangle.
If you've gone to the trouble of lugging equipment up a mountain
to set up a trig station, you'll want to capitalise on the effort by
measuring the azimuths of all the other stations you can see.
So you'll soon have a network of triangles, in many of which you've
measured all 3 angles. If the angles of a triangle add up to something
a bit different from 180 degrees, you can "distribute the error" among
the 3 angles in an obvious way.
In more elaborate situations - when a chain of triangles closes -
you might want a more sophisticated technique for handling
overdetermined data. I don't know enough biographical details
of Gauss to rule out that it was his surveying experience
that led him to invent least squares.
Charles Francis
<mic...@spamfree.net>
>Charles Francis:
>
>: As for the map making,
Sorry, that was supposed to read "as for the expedition to test the
fifth postulate". The point I was making was that Gauss developed
techniques of non-Euclidean geometry to map the curved 2d surface of the
earth. The story that he organised an expedition to test for non-
Euclidean geometry of 3-space sounds to me perfectly silly, a daft
corruption of what he was really doing with the non-Euclidean geometry
of curved surfaces. If he tested for non-Euclidean 3-space, he had data
without need of a special expedition. Of course it would help if I wrote
what I intended.
>Gauss was employed as a cartographer as part of
>: his duty as scientific advisor to the Hanoverian and Danish Governments.
[...]
--
Regards
Charles Francis
cha...@clef.demon.co.uk
[Moderator's note: Quoted text trimmed. -MM]
Peter Fred
our presently accepted theory of gravity?
That is what Copernicus did when he questioned the
Ptolemaic theory of eocentric planetary motion.
That is what Darwin did when he dared to question
the biblical account of creation.
The first step that Newton made or someone before him
made was that gravity is DUE To MASS. This view is
the holy of holies. It is sacrosanct. Maybe it is
our unquestioned faith that gravity is due to mass is
"What's wrong with gravity?"
What could it be related to? I will not suggest what it
might related to because the moderator will nix this
posting because he will claim I am being overly speculative.
Nevertheless, be that as it may
I remain,
Peter B. Fred
[Moderator's note: in general relativity, gravity is caused by
mass. It's caused by the flow of energy and momentum. - jb]
J. J. Lodder
> nos...@de-ster.demon.nl (J. J. Lodder) writes:
> > And don't forget to mention that Miller's claim resulted in the most
> > memorable quote in all of Physics;
> > "Schwierig ist der Herrgott, aber boshaft ist er nicht",
> > Perhaps better known as "Subtle is the Lord..............."
> >
> > In other words, Einstein did not for a moment take Miller seriously,
> > even tough it took another 20 years to find out
> > what precisely had gone wrong with the experiment.
> > (It was merely a falsification, after all.
> > Philosophically quite unsound, of course :-)
>
> I think few would consider a single, _IRREPRODUCIBLE_ result to constitute
> a ``falsification'' --- which is why I always qualified ``experimental
> result'' with words like ``reliable,'' ``reproducible,'' and
> ``unambiguous.''
Sure, or "in agreement with a reliable theory",
for example.
What Miller (and Kaufmann before him) illustrate
is that 'falsification' is no less problematic in actual practice
than 'verification'.
Miller was generally disbelieved
(excepting some fanatical anti-Einstein propagandists)
Eddington, on the other hand, who verified Einstein's light deflection
prediction in an experimentally very questionable way,
was immediately believed.
(Read Pais' account of Einstein's canonization on basis of Eddington :-)
What it usually boils down to is that acceptance of a verification
or a falsification depends very much on the amount of confidence
we have in the theory in question.
> Miller's claimed result could _NOT_ be independently replicated by
> other researchers, and hence did not qualify as a ``falsification,'' IMO.
> *IF* other experimenters in different parts of the world _had_ observed
> diurnal fringe-shifts comparable to Miller's claims, and *IF* they had
> been able to infer the same ``aether drift'' velocity- vector from them
> to within experimental error, it would have been quite a different story
> --- but that's NOT what happened. A single irreproducible result can't
> be considered a ``falsification,'' IMO.
Sure, it is all very very clear, many many years later,
Jan
Jim Carr
in the same thread, Michael Weiss writes:
}
} I believe that the standard methods of surveying simply *assume* that the
} geometry of 3-space is Euclidean, and use that assumption to compute
} distances, etc.
Only true for small-scale surveying. The effects of the earth's
curvature are easily seen when doing large scale (township or
county sized) surveys at 40 degrees N latitude. It is necessary
to insert "correction lines" to deal with the fact that parallel
lines want to meet. More below at the end for those interested.
} Of course it is possible that the raw data contains
} redundancies which Gauss could have used to test this assumption.
In article <G8576...@leeds.ac.uk>
"Robert Hill" <R.H...@leeds.ac.uk> writes:
>
>I think it's more than possible. For one thing, surveyors may take
>Euclidean geometry for granted, but I suspect that even then they didn't
>take perfection in their own measurements for granted. They probably
>used over-determined data.
I don't know enough of the history of surveying to say what was
standard practice in the time of Gauss, but "closure" of a survey
has been the standard practice for a very long time. When using
both angles and distances, you measure all of the sides and angles.
When using triangulation, you measure more than just one baseline.
Small errors are distributed, large errors are remeasured. However,
effects of spherical geometry are not ignored in large-scale surveys.
>This is especially likely for the angles of a triangle.
>If you've gone to the trouble of lugging equipment up a mountain
>to set up a trig station, you'll want to capitalise on the effort by
>measuring the azimuths of all the other stations you can see.
>So you'll soon have a network of triangles, in many of which you've
>measured all 3 angles. If the angles of a triangle add up to something
>a bit different from 180 degrees, you can "distribute the error" among
>the 3 angles in an obvious way.
Or not, and use spherical trig. One famous survey ran across India
from coast to coast and set the location and elevation of the peaks
in the Himalayas. I don't feel like looking up the numbers, but
compare a chord to a great circle to see what error using just plane
geometry would introduce over that distance.
>In more elaborate situations - when a chain of triangles closes -
>you might want a more sophisticated technique for handling
>overdetermined data. I don't know enough biographical details
>of Gauss to rule out that it was his surveying experience
>that led him to invent least squares.
Interesting question!
----
Now, concerning effects of earth's curvature: Surveys in the "western"
US (the states that started as territories west of the 13 colonies)
generally use a rectangular grid system. Surveyors run an EW baseline
and a NS meridian line as the primary references for (say) a state.
If you know of a "Meridian Road", that is usually on or near the prime
meridian for a state survey.
Square mile sections are laid out in 6x6 blocks by running NS lines
ever 6 miles up from the baseline and then breaking it down by
working EW within a block, starting from one corner. As you go
north, the distance across the top becomes significantly less than
6 miles between the lines, and the last sections get cockeyed.
(Rectangular rather than square.) At that point (IIRC it is every
36 miles) you run a *new* EW baseline and start over.
What this shows is that two lines run at 90 degrees to a common
baseline on the earth's surface are not parallel.
NS roads that run along section lines will not align properly at
such "correction" lines. If you are in an area quite a bit west of
the meridian and at high latitude -- for example, in the northwest
part of lower Michigan around 45 N -- these corrections are large
enough to notice on country roads where no one has bothered to
hide the alignment shift.
You can also see this if you look closely at a topographic map that
shows the section lines for such an area. On a topo map you can also
see that the USGS surveys (based on triangulation) run on a completely
independent system from that used for land surveys.
--
James Carr <j...@scri.fsu.edu> http://www.scri.fsu.edu/~jac/
"The half of knowledge is knowing where to find knowledge" - Anon.
Motto over the entrance to Dodd Hall, former library at FSCW.
Michael Weiss
: }
: } I believe that the standard methods of surveying simply *assume* that
the
: } geometry of 3-space is Euclidean, and use that assumption to compute
: } distances, etc.
Jim Carr replied:
: Only true for small-scale surveying. The effects of the earth's
: curvature are easily seen when doing large scale (township or
: county sized) surveys at 40 degrees N latitude.
Please, Jim! I said the geometry of 3-space! Of course the *Earth's*
surface is curved. But the fact that 3-space is non-Euclidean is something
surveyors can ignore (unless they're using the positions of stars during a
solar eclipse to calibrate their instruments :-)
Ralph Hartley
>
> >I don't see how an infinite flat background is unsound at all.
[Snip]
> For more about this try
>
> http://math.ucr.edu/home/baez/backround.html
>
> >One could very easily imagine an infinite flat universe in which the
> >concepts of absolute space and time are perfectly well defined. Such a
> >universe could be perfectly self-consistent.
>
> True. It's a consistent theory, but I don't think it's "philosophically
> sound" - for that, I want more than mere consistency.
I have to admit that I too would be rather upset by almost any fixed
background metric. That is to say, any background that wasn't flat (we
know NOW that a. It all boils down to a preference for simpler models.
Suppose you have a model for some phenomenon, and the model contains a
dimensionless constant. You wouldn't really be happy with the model if
it didn't explain why the constant had a particular value. There are a
couple of ways a model could do that. One is for the "constant" to
actually be a variable, which the model can predict. But there is
another way. There are some numbers that we consider special,
particularly 0 and 1. If the value of the constant was one of those, we
might be relatively happy.
What we really want is a simple model, and adding a simple number to a
model doesn't make it much more complex, while a number like 1/137.03599
practically screams for an explanation (if anybody thinks that this is
the only possible number on philosophical grounds ...).
Similarly, there are lots of possible metrics, but the flat metric is
distinct from all the others. Unless quantum gravity is a MUCH simpler
theory than I have been lead to believe, it will always be more
complicated than "the metric is flat". In order to be acceptable it will
have to be a much more ACCURATE theory (hopefully by having GR as an
approximation).
The universe in the back of my head through this discussion is Conway's
game of life. That universe by definition IS infinite and flat. A
physicist inhabiting that universe would have more right than we have to
conclude that his laws of physics are a philosophical necessity.
The rules of that universe are known to allow computation, and probably
life (though it is not known if life would arise spontaneously in
life). I would be surprised if the rules of this universe were simpler
than the rules of that one, but it contains phenomena at least as
complex. Maybe more so. Being infinite, the life universe can contain an
asymtotically accurate computational model of our universe, but if
current cosmology is correct, the reverse is not true.
This is actually important because when evaluating the simplicity of a
theory, the correct measure involves comparing it to all the theories
that are NOT true.
John Baez wrote:
> I have a carefully developed set of prejudices about physics, which
> come from reading and thinking about physics, math, and philosophy.
> I've spent years developing these. If a theory fits these prejudices
> (or better yet, comes along and changes them) I am inclined to say it
> is "philosophically sound" - which is basically a fancy way of saying
> it appeals to my taste. I don't regard this judgement as infallible.
> Nonetheless, as a theorist, I am forced to make such judgements when
> deciding what to work on.
As I said:
> > As far as I can tell, philosophical methods have not, on the whole,
> > done much better than chance. For formulating a theory that's really
> > good enough, especially if you are willing to abandon the results when
> >the universe proves otherwise.
Your job IS to formulate a theory, so doing as well as chance IS good
enough. As long as everyone doesn't have the SAME prejudices, we should
be OK.
Good Luck.
Ralph Hartley
Myrddin, Son of Morfryn
<har...@aic.nrl.navy.mil> stretched forth his arms and cast the
incantation:
>I have to admit that I too would be rather upset by almost any fixed
>background metric. That is to say, any background that wasn't flat (we
>know NOW that a. It all boils down to a preference for simpler models.
Why is it different whether the background is flat or not flat? Is the
inner structure of a flat sheet of steel so much different from that of
a pressed panel? Does the curvature tensor of flat space have fewer
elements? It merely has degenerate elements, but that does not make it
simpler per se. Surely the idea is to dispense with background
altogether. A model with no background should be simpler in its internal
relations.
>Similarly, there are lots of possible metrics, but the flat metric is
>distinct from all the others. Unless quantum gravity is a MUCH simpler
>theory than I have been lead to believe, it will always be more
>complicated than "the metric is flat".
I cannot answer for much of what goes by the title of quantum gravity.
But in my view quantum gravity means studying metric relations arising
from particle interactions in the absence of any background. That means
a very simple basic structure, though of course macroscopic behaviours
come from very complex configurations in this structure.
--
Myrddin, Son of Morfryn
Drawing his cloak about him against the cold of the wizard's gaze, Myrddin
retreats and vanishes once more into the Welsh mist.
Cedric Beny
>
> The universe in the back of my head through this discussion is Conway's
> game of life. That universe by definition IS infinite and flat. A
> physicist inhabiting that universe would have more right than we have to
> conclude that his laws of physics are a philosophical necessity.
Are really cellular automata defined as flat universes ? What is
certainly flat is the 2D representation of the result on a grid. But
this has strictly no importance for anything inhabiting that universe.
What defines that universe is just the set of rules, or interactions,
between the elementary components. A kind of geometry defined on this
set of rules may not be "flat". Is it right ?
Cedric Beny
Squark
>I have to admit that I too would be rather upset by almost any fixed
>background metric. That is to say, any background that wasn't flat (we
>know NOW that a. It all boils down to a preference for simpler models.
>
>dimensionless constant. You wouldn't really be happy with the model if
>it didn't explain why the constant had a particular value. There are a
>couple of ways a model could do that. One is for the "constant" to
>actually be a variable, which the model can predict. But there is
>another way. There are some numbers that we consider special,
>particularly 0 and 1. If the value of the constant was one of those, we
>might be relatively happy.
>Similarly, there are lots of possible metrics, but the flat metric is
>distinct from all the others. Unless quantum gravity is a MUCH simpler
>theory than I have been lead to believe, it will always be more
>complicated than "the metric is flat".
Nevertheless, you can't use Occam's razor blindly - or you can chop off too
much! After all, the most simple theories are the trivial ones. And, for
instance, can you see that GR is simpler than SR? More elegant, maybe, but
not
simpler. There are other criterions, after all...
>Being infinite, the life universe can contain an
>asymtotically accurate computational model of our universe, but if
>current cosmology is correct, the reverse is not true.
Hmm, I dunno, does the current cosmology say the universe contains a finite
amount of information?? I don't think so...
Best regards,
squark.
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Squark
>Are really cellular automata defined as flat universes ? What is
>certainly flat is the 2D representation of the result on a grid. But
>this has strictly no importance for anything inhabiting that universe.
>What defines that universe is just the set of rules, or interactions,
>between the elementary components. A kind of geometry defined on this
>set of rules may not be "flat". Is it right ?
Yeah, a good point, but this universe is defintely homogeneous, so would it
not be flat, it would have an invariant curvature - i.e. some intrinsic
parameter it doesn't seem to posses. :-)
[Disclaimer: of course, the above game of physico-mathematical life may or
may
not have meaning :-) ]
Ralph Hartley
It may be possible to have a set of rules defined on a square grid
with a natural geometry that is not flat, but I don't think Conway's
life is one of those. In that universe there is a natural definition
of lines, points, etc., and for any line and a point there is exactly
one parallel line through the point.
The geometry is actually a bit different from ours, but not by being
curved. Unlike our universe, the life universe DOES have preferred
directions, both the rules and their consequences are very direction
dependent.
It isn't clear that a "life" physicist would consider "rotation by an
angle theta" a particularly meaningful concept, compared to "rotation
by n quarter turns".
Ralph Hartley
Ralph Hartley
the same company) wrote:
>
> In spell <3A772E0D...@aic.nrl.navy.mil>, Ralph Hartley
> <har...@aic.nrl.navy.mil> stretched forth his arms and cast the
> incantation:
>
> >I have to admit that I too would be rather upset by almost any fixed
> >background metric. That is to say, any background that wasn't flat (we
> >down to a preference for simpler models.
>
> Why is it different whether the background is flat or not flat? Is the
> inner structure of a flat sheet of steel so much different from that of
> a pressed panel?
Of course it is. Geometry on a flat space obeys Euclid's axioms, on a
curved space it doesn't. Hence the term "Euclidean". There are a lot
more ways to be non Euclidean than there are to be Euclidean.
> Does the curvature tensor of flat space have fewer
> elements? It merely has degenerate elements, but that does not make it
> simpler per se. Surely the idea is to dispense with background
> altogether. A model with no background should be simpler in its internal
> relations.
Do theories in which the foo factor is one not still have a foo
factor? Are theories that never mention the foo factor (or the bar
term) philosophically inferior to those in which the foo factor
interacts dynamically with the bar term?
Of course the curvature tensor of flat space has the same number of
elements, but if space was really flat, the curvature tensor would
never appear in any (reasonable) theory. A lot of equations and
variables that are needed to deal with non zero curvature would just
be left out of the theory. The equations might still HOLD, but being
tautologies in flat space, they wouldn't be part of any theory.
That's one reason why I'm not sure that string theory is wrong just
because it "uses a flat background" (it may still be wrong). If the
"dimensions" of string theory are not directly identified with space
and time (and they can't really be, because there are more than four),
there is no real reason to expect that they have the same sort of
geometry.
Now it may well be true that string theory with a dynamic metric (if
it could be formulated) might give a good theory, or not. If it isn't
too horrible it is something that has to be considered, along with
other "background free" approaches. But flat geometry really is
special, and it shouldn't be dismissed out of hand.
In the sense of being simple, flat geometry has less background than a
theory with dynamical geometry.
This is not to say that it is not desirable to have a "taste" for one
sort of theory or another (If I were a professional physicist I might
prefer dynamical metrics of some kind), as long as you know that is
what you have, and don't try to force the universe to follow your
taste.
Ralph Hartley
Alejandro Rivero
http://xxx.unizar.es/abs/physics/0102051
String Theory: An Evaluation
Authors: Peter Woit (Dept. of Mathematics, Columbia University)
Ralph Hartley
> >Similarly, there are lots of possible metrics, but the flat metric is
> >distinct from all the others. Unless quantum gravity is a MUCH simpler
> >theory than I have been lead to believe, it will always be more
> >complicated than "the metric is flat".
>
> Nevertheless, you can't use Occam's razor blindly - or you can chop off too
> much! After all, the most simple theories are the trivial ones. And, for
> instance, can you see that GR is simpler than SR? More elegant, maybe, but
> not
> simpler. There are other criterions, after all...
Of course! Besides being simple a theory has to be accurate (in
agreeing with experiment). Also, a theory that deals with a larger set
of phenomena, would be forgiven for more complex.
GR beats SR on both counts. It predicts a lot of observations where SR
(flat space time) either gives wrong answers, or more often, no
answers at all. GR also deals with gravity, while SR doesn't.
For both reasons most people (including myself) consider GR a BETTER
theory than SR. That doesn't make it a SIMPLER theory, nor is it in
any way "philosophically necessary". What I am saying is that if we
had no observations of gravity, and no evidence of space-time
curvature (but we do have those), then there would be no reason to
consider GR over SR as a physical theory.
> >Being infinite, the life universe can contain an
> >asymtotically accurate computational model of our universe, but if
> >current cosmology is correct, the reverse is not true.
>
> Hmm, I dunno, does the current cosmology say the universe contains a finite
> amount of information?? I don't think so...
Depends on which current cosmology.
If there is a non zero cosmological constant, the amount of
computation (causally connected to an observer) seems to be finite.
There was a discussion on this recently, see
http://math.ucr.edu/home/baez/end.html . Apparently, the cosmological
constant puts a lower bound on temperature, and thermal fluctuations
will eventually break any computer (but eventually is a long time).
An open universe with no cosmological constant may allow infinite
computation.
Ralph
Ilja Schmelzer
>> I don't see how an infinite flat background is unsound at all.
>
>
>> It may be WRONG, but there is no way that could have been known on
>> "philosophical" grounds.
>
> It couldn't have been *known*... but it could have been *guessed*.
> And in fact, it *was* guessed: Leibniz and Mach were unhappy with
> Newtonian physics on philosophical grounds.
Hm. Mach is one of the main positivists. But positivism is dead, at
least in philosophy of science. And if we start with Popper's
methodology, the classical positivistic arguments are no longer very
sound.
> Einstein took their reasons for unhappiness, thought about them a
> lot, added a bunch more good ideas, and eventually came up with
> general relativity.
And later wrote on Bohr's positivism to Heisenberg: "Perhaps I did use
such a philosophy earlier, and even wrote it, but it is nonsense all
the same."
> For more about this try
> http://math.ucr.edu/home/baez/backround.html
Once "action equals reaction" can be derived from the Lagrange
formalism, it may be considered as a less fundamental, derived
principle.
I would argue that principles like classical realism (EPR principle)
are much more sound and more important. Last not least Newton's
theory was a fine and successful theory without being background-free,
so, incompatibility is nothing very bad for a physical theory.
>> One could very easily imagine an infinite flat universe in which
>> the concepts of absolute space and time are perfectly well
>> defined. Such a universe could be perfectly self-consistent.
> True. It's a consistent theory, but I don't think it's
> "philosophically sound" - for that, I want more than mere
> consistency.
If we start to argue with "philosophically sound" I would start with
classical EPR realism, causality and so on.
>> There is absolutely no way to decide a priori that we do not
>> inhabit such a universe, you have to do experiments.
> I don't think of philosophy as a source of a priori knowledge about
> physics.
I agree.
> I think it's a bit more like "taste".
But not too much. The conceptual problems of classical positivism are
objective problems, not a matter of taste. Thus its rejection is not
a matter of taste but a matter of internal consistency.
Of course, there is a lot related with "taste" - but I don't think
rational argumentation about these issues is as impossible as
argumentation about taste.
> I have a carefully developed set of prejudices about physics, which
> come from reading and thinking about physics, math, and philosophy.
> I've spent years developing these. If a theory fits these prejudices
> (or better yet, comes along and changes them) I am inclined to say it
> is "philosophically sound" - which is basically a fancy way of saying
> it appeals to my taste. I don't regard this judgement as infallible.
> Nonetheless, as a theorist, I am forced to make such judgements when
> deciding what to work on.
Agreement.
Ilja
--
I. Schmelzer, <il...@ilja-schmelzer.net> , http://ilja-schmelzer.net
Chris Hillman
On 20 Feb 2001, Cedric Beny wrote:
> Are really cellular automata defined as flat universes ? What is
> certainly flat is the 2D representation of the result on a grid. But
> this has strictly no importance for anything inhabiting that universe.
> What defines that universe is just the set of rules, or interactions,
> between the elementary components. A kind of geometry defined on this
> set of rules may not be "flat". Is it right ?
For a rare example of a study of cellular automata in which the evolution
map is allowed to change the grid itself (including the topology of
"space"), see David Hillman (no relation), Combinatorial Spacetimes
(General Relativity), Ph.D. thesis, University of Pittsburgh, 1995.
Abstract: "Combinatorial spacetimes are a class of dynamical systems in
which finite pieces of spacetime contain finite amounts of information.
Most of the guiding principles for designing these systems are drawn from
general relativity: the systems are deterministic; spacetime may be
foliated into Cauchy surfaces; the law of evolution is local (there is a
light-cone structure); and the geometry evolves locally (curvature may be
present; big bangs are possible). However, the systems differ from general
relativity in that spacetime is a combinatorial object, constructed by
piecing together copies of finitely many types of allowed neighborhoods in
a prescribed manner. Hence at least initially there is no metric, no
concept of continuity or diffeomorphism. The role of diffeomorphism,
however, is played by something called a 'local equivalence map.' Here I
attempt to begin to lay the mathematical foundations for the study of
these systems. (Examples of such systems already exist in the literature.
The most obvious is reversible cellular automata, which are flat
combinatorial spacetimes. Other related systems are structurally dynamic
cellular automata, L systems and parallel graph grammars.) In the
1+1-dimensional oriented case, sets of spaces may be described
equivalently by matrices of nonnegative integers, directed graphs, or
symmetric tensors; local equivalences between space sets are generated by
simple matrix transformations. These equivalence maps turn out to be
closely related to the flow equivalence maps between subshifts of finite
type studied in symbolic dynamics. Also, the symmetric tensor algebra
generated by equivalence transformations turns out to be isomorphic to the
abstract tensor algebra generated by commutative cocommutative bialgebras.
In higher dimensions I attempt to follow the same basic model, which is to
define the class of n-dimensional space set descriptions and then generate
local equivalences between these descriptions using elementary
transformations. Here I study the case where space is a special type of
colored graph (discovered by Pezzana) which may be interpreted as an
n-dimensional pseudomanifold. Finally, I show how one may study the
behavior of combinatorial spacetimes by searching for constants of motion,
which typically are associated with local flows and often may be
interpreted in terms of particles."
(Note: flow equivalence, one dimensional shifts of finite type,
nonnegative integer matrices, etc., are fundamental in one dimensional
symbolic dynamics.)
The reformulation of gtr in terms of Regge calculus is well known (see
MTW, Gravitation). If time and energy permit, I hope to write an
expository paper called "What is Cartan Geometry?", and if possible to
follow that up by explaining the connection (no pun intended) between
"dislocations" in tilings, which are -multidimensional- symbolic dynamical
phenomena (compare the ideas in D. Hillman's thesis), and the teleparallel
reformulation of gtr.
Chris Hillman
Home Page: http://www.math.washington.edu/~hillman/
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
NOTE TO WOULD-BE CORRESPONDENTS: I have installed a mail filter which
deletes incoming messages not from the "*.edu" or "*.gov" domains, but
also deletes messages from some bad actors whose emails happen to be in
the "*.edu" domain and "passes" messages from a few friends with email
addresses in other domains.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Toby Bartels
>John Baez wrote:
>>It couldn't have been *known*... but it could have been *guessed*.
>>And in fact, it *was* guessed: Leibniz and Mach were unhappy with
>>Newtonian physics on philosophical grounds.
>Hm. Mach is one of the main positivists. But positivism is dead, at
>least in philosophy of science. And if we start with Popper's
>methodology, the classical positivistic arguments are no longer very
>sound.
But do Mach's objections depend on positivism? I don't think so.
Leibniz was around before positivism, but he objected too.
(And I'd rather be a positivist than a Popperian anyway ^_^.)
>Once "action equals reaction" can be derived from the Lagrange
>formalism, it may be considered as a less fundamental, derived
>principle.
But you *can't* derive it from the Lagrange formalism.
What you *can* derive it from is the Lagrange formalism *together*with*
the requirement that any field appearing in the Lagrangian be varied.
SR, of course, fails this requirement.
>I would argue that principles like classical realism (EPR principle)
>are much more sound and more important.
IOW, your tastes differ from John's (and mine).
We like our tastes because they suggest GR from SR.
>Last not least Newton's
>theory was a fine and successful theory without being background-free,
>so, incompatibility is nothing very bad for a physical theory.
I think the useful applications of these philosophical principles
are in justifying *changes* to experimentally sound theories,
not in justifying rejection of a theory with nothing to replace it.
They justify these changes in 2 ways: by suggesting further research
and by suggesting which theory to believe when there is competition.
The experiments speak louder, of course, but if they happen to be silent ...
>If we start to argue with "philosophically sound" I would start with
>classical EPR realism, causality and so on.
And I don't think there's anything wrong with such arguments,
if those happen to be the philosophical principles that appeal to you.
They don't appeal to me, so such arguments won't get anywhere with me,
but that's only relevant if you care about my beliefs personally.
>>I think [philosophy]'s a bit more like "taste".
>But not too much. The conceptual problems of classical positivism are
>objective problems, not a matter of taste. Thus its rejection is not
>a matter of taste but a matter of internal consistency.
I'm not sure what you're talking about here re: positivism.
All the same, I'll bet that, once you explain it,
we'll find that the *taste* for positivism can be preserved
and that the internal contradictions may be found only by
taking things to extremes. (Well, we'll see, if you explain it.)
>Of course, there is a lot related with "taste" - but I don't think
>rational argumentation about these issues is as impossible as
>argumentation about taste.
There *is* a lot of rational argumentation possible about taste.
If people hold fundamentally different tastes, then not much can be done
(just as no logical argument can be done without agreement on premisses);
however, we can argue about what the tastes mean in certain situations
(even if we don't have those tastes! -- try saying something like
<Well, a positivist would think --- about ---.> sometime).
-- Toby
to...@math.ucr.edu
Mark William Hopkins
<har...@aic.nrl.navy.mil> stretched forth his arms and cast the
incantation:
>I have to admit that I too would be rather upset by almost any fixed
>background metric. That is to say, any background that wasn't flat (we
>know NOW that a. It all boils down to a preference for simpler models.
There's nothing wrong with formulating Quantum Field Theory (QFT) over a
fixed background and making THAT the unified theory if a situation like
the following ultimately occurs.
It's possible that you could ultimately have a situation where one is able to
write down a general QFT in a general curved space (including the non globally
hyperbolic ones), but where the requirement of finiteness alone (e.g. the
summability of the S-matrix) actually FORCES underlying metric to satisfy
Einstein's equations with respect to the vacuum expectation of the stress
tensor (for instance).
In other words: it could very well be that QFT, alone(!) will force GR
to come out for free solely by the requirement of self-consistency. No
need to even explain gravity, then, or to have anything like Quantum Gravity.
You'll already have a unified theory solely within QFT.
Another place where self-consistency enters the picture, pertinent to the
above situation:
One might argue that the stress tensor does not have a well-defined
vacuum expectation because the Zero-Point energy density diverges. But
that's not generally true! It can be finite if the particles' masses
satisfy certain mass-constraint formulae:
sum_A ((+/-)(m_A)^n) = 0, n = 0, 2, 4
Sum taken over all field degrees of freedom, A,
+ for bosons, - for fermions.
So here too self-consistency clamps down on the admissible theories,
in this case, admissible particle models. The Standard Model fails the
mass-constraints; but satisfies the required charge-constraint formulae
sum_A q_A
so as to make
:psi-bar D psi: = psi-bar D psi,
where D is the covariant derivative of gauge theory and psi is the entire
fermion spinor (e.g. psi has 90 components in the Standard Model). There
are similar constraints for the rest of the terms in the interaction
Lagrangian and it is generally true (already) for the free fields that
L_free = :L_free:
since <0| L_free |0> = 0 for both the fermion and boson free-field
Lagrangians.
Jim Carr
: } I believe that the standard methods of surveying simply *assume* that the
: } geometry of 3-space is Euclidean, and use that assumption to compute
: } distances, etc.
Jim Carr replied:
: Only true for small-scale surveying. The effects of the earth's
: curvature are easily seen when doing large scale (township or
: county sized) surveys at 40 degrees N latitude.
In article <96b8sl$gns$1...@bob.news.rcn.net>
"Michael Weiss" <mic...@spamfree.net> writes:
>Please, Jim! I said the geometry of 3-space! Of course the *Earth's*
>surface is curved. But the fact that 3-space is non-Euclidean is something
>surveyors can ignore (unless they're using the positions of stars during a
>solar eclipse to calibrate their instruments :-)
Ah, but the 3-space they use is not Euclidean. It consists of a
curved 2-space plus an orthogonal 3rd dimension.
--
James Carr <j...@scri.fsu.edu> http://www.scri.fsu.edu/~jac/
Dopeler Effect: The tendency of stupid ideas to seem smarter when they
come at you rapidly. (anon source via e-chain-letter)
Toby Bartels
>Michael Weiss wrote:
>>Please, Jim! I said the geometry of 3-space! Of course the *Earth's*
>>surface is curved. But the fact that 3-space is non-Euclidean is something
>>surveyors can ignore (unless they're using the positions of stars during a
>>solar eclipse to calibrate their instruments :-)
>Ah, but the 3-space they use is not Euclidean. It consists of a
>curved 2-space plus an orthogonal 3rd dimension.
That doesn't necessarily result in a curved 3space.
In this case, it doesn't; I'm sure surveyors use \R^3.
They just don't use Cartesian coordinates on \R^3.
Their Gammas aren't all 0, but that's an artifact of their coordinate system;
in a different coordinate system, the Gammas could all be 0,
and the space is still flat.
-- Toby
to...@math.ucr.edu
Ilja Schmelzer
>> Ilja, though failing to properly cite himself, was the one who wrote:
>> John Baez, though not properly cited by Ilja, was the one who wrote:
>>> It couldn't have been *known*... but it could have been *guessed*.
>>> And in fact, it *was* guessed: Leibniz and Mach were unhappy with
>>> Newtonian physics on philosophical grounds.
>> Hm. Mach is one of the main positivists. But positivism is dead, at
>> least in philosophy of science. And if we start with Popper's
>> methodology, the classical positivistic arguments are no longer very
>> sound.
> But do Mach's objections depend on positivism? I don't think so.
> Leibniz was around before positivism, but he objected too.
Of course some positivistic ideas have been developed earlier.
Essentially the search for absolute, positive truth based on
observation has been part of the "Aufklärung" (enlightment). To argue
against the claims of absolute truth of the Church they have tried to
derive absolute truth from observation. A quite natural error.
> (And I'd rather be a positivist than a Popperian anyway ^_^.)
If you like to base science on an inconsistent philosophy ....
>> Once "action equals reaction" can be derived from the Lagrange
>> formalism, it may be considered as a less fundamental, derived
>> principle.
> But you *can't* derive it from the Lagrange formalism. What you
> *can* derive it from is the Lagrange formalism *together*with* the
> requirement that any field appearing in the Lagrangian be varied.
> SR, of course, fails this requirement.
If you want to formulate "action equals reaction" as a universal
fundamental principle. Its not clear if this can be done in a
consistent way, the problem that there are always "laws of physics"
which remain fixed has been already mentioned.
My claim was about the non-universal but very special appearences of
the principle "action equals reaction" which appear already in
Newtonian mechanics and are derived from the Lagrange formalism. And
I think such particular symmetries which are already explained by
derivation from another principle are a bad justification in favour
of universality of this particular symmetry principle.
>> I would argue that principles like classical realism (EPR principle)
>> are much more sound and more important.
> IOW, your tastes differ from John's (and mine). We like our tastes
> because they suggest GR from SR.
And I like my tastes, because they have suggested me another quite
interesting theory of gravity, which gives GR in some limit.
See gr-qc/0001101
>> Last not least Newton's theory was a fine and successful theory
>> without being background-free, so, incompatibility is nothing very
>> bad for a physical theory.
> I think the useful applications of these philosophical principles
> are in justifying *changes* to experimentally sound theories,
> not in justifying rejection of a theory with nothing to replace it.
I agree. That's the way I apply the principle of realism too.
> They justify these changes in 2 ways: by suggesting further research
> and by suggesting which theory to believe when there is competition.
> The experiments speak louder, of course, but if they happen to be
> silent ...
Full agreement.
>> If we start to argue with "philosophically sound" I would start with
>> classical EPR realism, causality and so on.
> And I don't think there's anything wrong with such arguments,
> if those happen to be the philosophical principles that appeal to you.
> They don't appeal to me, so such arguments won't get anywhere with me,
> but that's only relevant if you care about my beliefs personally.
Do you think it is impossible to argue rationally about such
principles?
>>> I think [philosophy]'s a bit more like "taste".
>> But not too much. The conceptual problems of classical positivism are
>> objective problems, not a matter of taste. Thus its rejection is not
>> a matter of taste but a matter of internal consistency.
> I'm not sure what you're talking about here re: positivism.
> All the same, I'll bet that, once you explain it,
> we'll find that the *taste* for positivism can be preserved
> and that the internal contradictions may be found only by
> taking things to extremes. (Well, we'll see, if you explain it.)
It would be better if you read Popper (different from many other
philosophers he writes very clear, I'm not sure I can explain all this
as nice as he has done).
Positivism tries to derive theories from observation. This fails
because observations are never pure, they depend on theoretical
assumptions (for example about how the experimental devices work).
Popper says theories are guesses which can be falsified, and science
evolves by survival of the fittest theory. Popper makes no claims
that the best theory is certainly true (= there is no positive
knowledge of truth). There is no criterion of truth which allows to
establish the truth of something, even if it is true.
> There *is* a lot of rational argumentation possible about taste.
Fine, let's try ;-)
Toby Bartels
>Toby Bartels wrote:
>>But do Mach's objections depend on positivism? I don't think so.
>>Leibniz was around before positivism, but he objected too.
>Of course some positivistic ideas have been developed earlier.
>Essentially the search for absolute, positive truth based on
>observation has been part of the "Aufklärung" (enlightment). To argue
>against the claims of absolute truth of the Church they have tried to
>derive absolute truth from observation. A quite natural error.
Since when is the existence of an absoulte truth part of positivism?
And what has that got to do with the objection to Newton anyway?
>>(And I'd rather be a positivist than a Popperian anyway ^_^.)
>If you like to base science on an inconsistent philosophy ....
That remains to be seen.
>>Ilja Schmelzer wrote:
>>>Once "action equals reaction" can be derived from the Lagrange
>>>formalism, it may be considered as a less fundamental, derived
>>>principle.
>>But you *can't* derive it from the Lagrange formalism. What you
>>*can* derive it from is the Lagrange formalism *together*with* the
>>requirement that any field appearing in the Lagrangian be varied.
>>SR, of course, fails this requirement.
>If you want to formulate "action equals reaction" as a universal
>fundamental principle. Its not clear if this can be done in a
>consistent way, the problem that there are always "laws of physics"
>which remain fixed has been already mentioned.
It's not a matter of insisting on an unattainable idea.
But if there are competing theories of physics,
one with more back reaction than the other,
then I'll prefer the one over the other.
>My claim was about the non-universal but very special appearences of
>the principle "action equals reaction" which appear already in
>Newtonian mechanics and are derived from the Lagrange formalism. And
>I think such particular symmetries which are already explained by
>derivation from another principle are a bad justification in favour
>of universality of this particular symmetry principle.
Well, Newton's principle of action and reaction
is already explained by conservation of momentum
(to the extent that his principle is true at all).
I'm not justifying my prejudices by appeal to Newton's 3rd law;
in fact, GR gives a more profound argument to me.
(I don't know that I would have had the foresight
to object to mere SR on the basis of such a principle.)
>>>If we start to argue with "philosophically sound" I would start with
>>>classical EPR realism, causality and so on.
>>And I don't think there's anything wrong with such arguments,
>>if those happen to be the philosophical principles that appeal to you.
>>They don't appeal to me, so such arguments won't get anywhere with me,
>>but that's only relevant if you care about my beliefs personally.
>Do you think it is impossible to argue rationally about such
>principles?
It's impossible if the arguers don't already agree on something else.
>>>But not too much. The conceptual problems of classical positivism are
>>>objective problems, not a matter of taste. Thus its rejection is not
>>>a matter of taste but a matter of internal consistency.
>>I'm not sure what you're talking about here re: positivism.
>>All the same, I'll bet that, once you explain it,
>>we'll find that the *taste* for positivism can be preserved
>>and that the internal contradictions may be found only by
>>taking things to extremes. (Well, we'll see, if you explain it.)
>Positivism tries to derive theories from observation. This fails
>because observations are never pure, they depend on theoretical
>assumptions (for example about how the experimental devices work).
>Popper says theories are guesses which can be falsified, and science
>evolves by survival of the fittest theory. Popper makes no claims
>that the best theory is certainly true (= there is no positive
>knowledge of truth). There is no criterion of truth which allows to
>establish the truth of something, even if it is true.
Wait, since when is the theory's being certainly true part of positivism?
Positivism is about rejection of metaphysical questions as meaningless
and includes the empiricist notion that meaningful statements are testable.
I'm not aware of Popper's criticising positivism as such at all,
although I know that he did criticise the Wien school (Logical Positivism)
for advocating a verifiability test instead of a falsifiability test.
But in fact, as has been discussed here before (search the archive),
falsifiability alone is as useless as verifiability alone, for dual reasons --
and, in any case, the argument is over how to go about realising empiricism,
which is entirely compatible with positivism, so that there is no conflict.
-- Toby
to...@math.ucr.edu
Ilja Schmelzer
>> Of course some positivistic ideas have been developed earlier.
>> Essentially the search for absolute, positive truth based on
>> observation has been part of the "Aufklärung" (enlightment). To argue
>> against the claims of absolute truth of the Church they have tried to
>> derive absolute truth from observation. A quite natural error.
> Since when is the existence of an absoulte truth part of positivism?
Not the existence, but its strong (certain) derivation from
observation.
> And what has that got to do with the objection to Newton anyway?
Newton's theory with absolute space makes all predictions and is quite
fine from point of view of Popper's methodology. There is nothing to
criticize from this point of view.
>>> (And I'd rather be a positivist than a Popperian anyway ^_^.)
>
>> If you like to base science on an inconsistent philosophy ....
>
> That remains to be seen.
Positivism has not reached his goals. That's quite obvious.
>>> But you *can't* derive it from the Lagrange formalism.
>> If you want to formulate "action equals reaction" as a universal
>> fundamental principle. Its not clear if this can be done ...
> It's not a matter of insisting on an unattainable idea.
> But if there are competing theories of physics,
> one with more back reaction than the other,
> then I'll prefer the one over the other.
Me too. But until now we have no quantum gravity based on GR
philosophy, while with a preferred frame most quantization problems
disappear.
>> My claim was about the non-universal but very special appearences of
>> the principle "action equals reaction" which appear already in
>> Newtonian mechanics and are derived from the Lagrange formalism. And
>> I think such particular symmetries which are already explained by
>> derivation from another principle are a bad justification in favour
>> of universality of this particular symmetry principle.
> Well, Newton's principle of action and reaction
> is already explained by conservation of momentum
> (to the extent that his principle is true at all).
> I'm not justifying my prejudices by appeal to Newton's 3rd law;
> in fact, GR gives a more profound argument to me.
The point is not Newton's 3rd law, but the appeal to "action =
reaction" as a general principle, while it is a simple consequence of
the Lagrange formalism used as in Newtons theory as in GR.
> (I don't know that I would have had the foresight
> to object to mere SR on the basis of such a principle.)
Not to SR, but to the Lorentz ether it was a quite obvious (AFAIU even
known) objection.
Essentially one of the main differences between Lorentz ether and SR
is that this obvious objection against the (material) Lorentz ether
who should follow the "action equals reaction" law according to NM
sounds much less serious in SR.
>>>> If we start to argue with "philosophically sound" I would start with
>>>> classical EPR realism, causality and so on.
>
>>> And I don't think there's anything wrong with such arguments,
>>> if those happen to be the philosophical principles that appeal to you.
>>> They don't appeal to me, so such arguments won't get anywhere with me,
>>> but that's only relevant if you care about my beliefs personally.
>
>> Do you think it is impossible to argue rationally about such
>> principles?
>
> It's impossible if the arguers don't already agree on something else.
No problem, we agree on something else - for example how to spell
"something else" ;-).
>> Positivism tries to derive theories from observation. This fails
>> because observations are never pure, they depend on theoretical
>> assumptions (for example about how the experimental devices work).
>> Popper says theories are guesses which can be falsified, and science
>> evolves by survival of the fittest theory. Popper makes no claims
>> that the best theory is certainly true (= there is no positive
>> knowledge of truth). There is no criterion of truth which allows to
>> establish the truth of something, even if it is true.
> Wait, since when is the theory's being certainly true part of
> positivism?
Positivism is derived from "positive knowledge", in the sense of
certain knowledge.
The main argument against absolute NM space is that we cannot be
certain which of the various NM explanations of our world (which
differ in the choice of the absolute rest) is the true one. Once we
do not care for certainty, that's not a point.
> Positivism is about rejection of metaphysical questions as
> meaningless and includes the empiricist notion that meaningful
> statements are testable.
Yep, but Popper's methodology is quite different. Meaningful
statements are all statements of a physical theory, but certainly not
all of them are testable.
> I'm not aware of Popper's criticising positivism as such at all,
> although I know that he did criticise the Wien school (Logical
> Positivism) for advocating a verifiability test instead of a
> falsifiability test. But in fact, as has been discussed here before
> (search the archive), falsifiability alone is as useless as
> verifiability alone, for dual reasons --
Sounds like the triviality that theories about the existence of
something cannot be falsified but only verified?
> and, in any case, the argument is over how to go about realising
> empiricism, which is entirely compatible with positivism, so that
> there is no conflict.
Empiricism AFAIU is priority of observation, Popper is priority of
theory.
But observation is always theory-laden, and therefore cannot be prior
to theory. Theory-independent observation is a myth. So where do you
want to start with empirical observations?
Toby Bartels
How about taking this to e-mail? -TB]
Ilja Schmelzer wrote:
>Toby Bartels wrote:
>>Ilja Schmelzer wrote:
>>>Of course some positivistic ideas have been developed earlier.
>>>Essentially the search for absolute, positive truth based on
>>>observation has been part of the "Aufklärung" (enlightment). To argue
>>>against the claims of absolute truth of the Church they have tried to
>>>derive absolute truth from observation. A quite natural error.
>>Since when is the existence of an absoulte truth part of positivism?
>Not the existence, but its strong (certain) derivation from
>observation.
So, the point is that truth comes from observation.
I would call this "empiricism",
and I would agree that empiricism is part of positivism.
But absoluteness is a completely orthogonal issue.
Surely you don't call the derivation of truth from observation an error?
>>And what has that got to do with the objection to Newton anyway?
>Newton's theory with absolute space makes all predictions and is quite
>fine from point of view of Popper's methodology. There is nothing to
>criticize from this point of view.
Then the positivists had something over Popper!
>Positivism has not reached his goals. That's quite obvious.
Not to me. You're going to have to explain these things.
(I'll admit that Comte's original goals were not achieved,
but the philosophy has outgrown him.)
>>If there are competing theories of physics,
>>one with more back reaction than the other,
>>then I'll prefer the one over the other.
>Me too. But until now we have no quantum gravity based on GR
>philosophy, while with a preferred frame most quantization problems
>disappear.
I don't know a theory of quantum gravity with a preferred frame.
In the meantime, I like the approaches that have most the flavour of GR,
while retaining the principles of QM.
>>>My claim was about the non-universal but very special appearences of
>>>the principle "action equals reaction" which appear already in
>>>Newtonian mechanics and are derived from the Lagrange formalism. And
>>>I think such particular symmetries which are already explained by
>>>derivation from another principle are a bad justification in favour
>>>of universality of this particular symmetry principle.
>>Well, Newton's principle of action and reaction
>>is already explained by conservation of momentum
>>(to the extent that his principle is true at all).
>>I'm not justifying my prejudices by appeal to Newton's 3rd law;
>>in fact, GR gives a more profound argument to me.
>The point is not Newton's 3rd law, but the appeal to "action =
>reaction" as a general principle, while it is a simple consequence of
>the Lagrange formalism used as in Newtons theory as in GR.
You'll have to clarify for me. First you say that
your claim was about the appearance of the principle in Newton,
which appearance is Newton's 3rd law;
then you say that Newton's 3rd law is not the point.
Which is it?
>>(I don't know that I would have had the foresight
>>to object to mere SR on the basis of such a principle.)
>Not to SR, but to the Lorentz ether it was a quite obvious (AFAIU even
>known) objection.
Certainly the objection is much easier to make in that case.
But SR also contains a preferred background structure,
to which I also object on philosophical grounds.
>Essentially one of the main differences between Lorentz ether and SR
>is that this obvious objection against the (material) Lorentz ether
>who should follow the "action equals reaction" law according to NM
>sounds much less serious in SR.
I don't believe that the aether violated Newton's 3rd law
(which is how the action/reaction principle appears in Newton).
>>Wait, since when is the theory's being certainly true part of
>>positivism?
>Positivism is derived from "positive knowledge", in the sense of
>certain knowledge.
None of my references on positivism interpret "postive" in this way.
Indeed, I see that "matter-of-fact" is the preferred synonym.
Since when do matters of fact have to be certain?
>The main argument against absolute NM space is that we cannot be
>certain which of the various NM explanations of our world (which
>differ in the choice of the absolute rest) is the true one. Once we
>do not care for certainty, that's not a point.
Well, that's not how I'd phrase the objection!
I'd say that, if the absolute space makes no difference,
then it ought not to appear in the theory at all.
It's not a matter of merely being uncertain about something;
the thing makes no difference whatsoever!
>>Positivism is about rejection of metaphysical questions as
>>meaningless and includes the empiricist notion that meaningful
>>statements are testable.
>Yep, but Popper's methodology is quite different. Meaningful
>statements are all statements of a physical theory, but certainly not
>all of them are testable.
Well, Popper can be as different as he likes.
I still don't see why positivism is so bankrupt.
>>I'm not aware of Popper's criticising positivism as such at all,
>>although I know that he did criticise the Wien school (Logical
>>Positivism) for advocating a verifiability test instead of a
>>falsifiability test. But in fact, as has been discussed here before
>>(search the archive), falsifiability alone is as useless as
>>verifiability alone, for dual reasons --
>Sounds like the triviality that theories about the existence of
>something cannot be falsified but only verified?
Well, John Baez put it in more dramatic terms:
Statements beginning with universal quantifiers must be falsified,
while statements beginning with existential quantifiers must be verified.
Trivial, yes -- he originally brought it up by looking at negations --
but it shows that both are needed.
>>and, in any case, the argument is over how to go about realising
>>empiricism, which is entirely compatible with positivism, so that
>>there is no conflict.
>Empiricism AFAIU is priority of observation, Popper is priority of
>theory.
OK, so do you object not just to positivism but to empiricism?
>But observation is always theory-laden, and therefore cannot be prior
>to theory. Theory-independent observation is a myth. So where do you
>want to start with empirical observations?
Since when does empiricism require theory independent observation?
Is this extreme strawman supposed to be the objection to Mach --
he was a positivist, therefore he required theory independent observation,
therefore he was a buffoon, therefore we shouldn't trust his opinions?
-- Toby
to...@math.ucr.edu
Urs Schreiber
> >The main argument against absolute NM space is that we cannot be
> >certain which of the various NM explanations of our world (which
> >differ in the choice of the absolute rest) is the true one. Once we
> >do not care for certainty, that's not a point.
>
> Well, that's not how I'd phrase the objection!
> I'd say that, if the absolute space makes no difference,
> then it ought not to appear in the theory at all.
> It's not a matter of merely being uncertain about something;
> the thing makes no difference whatsoever!
A related question: As far as I understand, QFT for massless spin-2 particles
lives in flat Minkowksi-space, but, since the theory includes GR, this flat
space is predicted to be unobservable. Thus it seems to me, that flat Minkowsi
space "makes no difference" here, but is still essential for the theory. No?
Urs Schreiber
Urs Schreiber
> >The main argument against absolute NM space is that we cannot be
> >certain which of the various NM explanations of our world (which
> >differ in the choice of the absolute rest) is the true one. Once we
> >do not care for certainty, that's not a point.
Toby Bartels wrote (in current thread):
> Well, that's not how I'd phrase the objection!
> I'd say that, if the absolute space makes no difference,
> then it ought not to appear in the theory at all.
> It's not a matter of merely being uncertain about something;
> the thing makes no difference whatsoever!
This made me ask (in unpublished posting and by e-mail):
> As far as I understand, QFT of massless spin-2 particles in flat
> Minkowski space produces GR and thus predicts that the flat
> space is not observable. It could be any of the local Lorentz
> frames of GR. So this flat space "makes no difference" but is
> still essential to the theory. No?
And Toby Bartels was so kind to answer (by e-mail):
> First of all, massless spin 2 QFT does not give GR.
> It doesn't give anything; the theory doesn't work.
> Superstring theory tries to get around the problems
> by making a transition from particles to strings;
> the loop quantum gravity that John Baez works on
> tries to get around the problems by expressing GR
> in terms of fundamental elements other than the metric.
> But neither of these has been worked out fully.
>
> However, even if masselss spin 2 QFT was just dandy,
> I would object to it on philosophical grounds.
> I would say <This theory uses a fixed Minkowski background
> which is completely unobservable. That can't be right.
> There must be another way to express this theory,
> more beautiful, more elegant, and with no unobservable background.>,
> and I would encourage or even participate in
> attempts to come up with alternative formulations.
> And, if these alternative formulations turned out to be
> actually different theories with slightly different predictions,
> then I'd put my metaphorical money on the alternatives.
>
> >I had posted this question to s.p.r, but it did not show up.
>
> If you phrased it there as you phrased it to me,
> I see no reason why it shouldn't get in.
To this I would like to reply now and here:
Toby Bartels wrote:
> First of all, massless spin 2 QFT does not give GR.
[snip]
> However, even if masselss spin 2 QFT was just dandy,
> I would object to it on philosophical grounds.
[snip]
Ah, ok, you would object here, too. Thanks for your answer.
By the way, what about "working" theories (i.e. such not defective like
"GR by spin 2") that contain unobservable elements? For example ordinary
QM. It seems that its "absolute time" is not really observable, due to
absence of a time operator and clocks possibly running backwards, etc?
Sure, ordinary QM is not "the truth" either, but its a sound framework.
I am just trying to see how common unobservable notions are in our
current understanding of physics.
Are there perhaps even unobservable elements in classical mechanics?
Probably not, at least I cannot think of any right now.
You know, I am really inclined to follow your view that a theory should
better not contain unobservable features. But, for example, thoughts
like those expressed in a recent thread on s.p.r about "universes that
are cellular automata" made me wonder, that it is pretty likely that any
"concious" (somehow reflecting its environment, anyway) structure
arising in cellular automata would not be able to observe the
fundamental structure of the automata. Such "observers" could perhaps
_infer_ the primitive rules and the cellular design, but probably not
measure them (at least not all of them).
There is this (famous?) example by Feynman (given in his "Lectures on
Gravitation") of the person who is measuring lengths with an iron ruler
in a space of inhomogenous temperature. The space is "really" flat, but
the ruler has different length (with respect to the flat background) at
different points in space, due to thermal expansion. Thus the person,
being insensitive to the heat otherwise, observes a curved space. The
only way to observe the flat background space in this example is to
employ devices unaffacted by heat, such as radar, that are external to
the "heat and metal ruler"-universe.
It seems unlikely, though, that there is always an "external"
interaction (in the above sense) to every aspect of our universe.
Indeed, _our_ "metal ruler" may well be the graviton and the analog of
"heat" in the above example then is "energy" and hence pretty much
everything there is.
The point I see here is, that while both, the "curved observer" and the
"flat" one, come up with physical theories that are different in
formulation, but agree on everything measurable by the curved guy and
are thus equivalent, the "flat" theory will very likely be the simpler
one and more aesthetically pleasing. It may at least be the one that can
be *found* and *understood* much earlier than the probably much more
involved "curved" theory.
We are talking about philosophical prejudices here, that do not affect
physics, but that are important to us (as John Baez has put it on one of
the "What's wrong with [fill in blank]?" threads) in order to decide
where to work and where, for example, to look for quantum gravity. So I
am asking:
Could it not be that we are more likely to find _a_ formulation of _the_
TOE which _does_ contain unobservable elements? Simply because _our_ TOE
is what is the set of primitive rules of, e.g., cellular automata to
their "inhabitants" and it seems to _us_, who are external to these
hypothetical cellular automata, that these inhabitants could _not_
observe the primitive rules, which are still the most concise
(easiest!?) "theory" of a particluar automat?
I am considering "cellular automata" here just to have a specific
example of any "universe having a primitive set of rules" (such as our
should better have if we are ever to understand it properly).
> >I had posted this question to s.p.r, but it did not show
> >up.
> If you phrased it there as you phrased it to me,
> I see no reason why it shouldn't get in.
The phrasing was very similar. My suspicion is that the formatting may
not have been approved. I had been writing the posting on a machine
foreign to me and had tried, but not managed, to turn off automatic HTML
formatting. Both posting that were involuntarily HTMLed this way did not
show up, so...
Urs Schreiber
--
eMail: Urs.Sc...@uni-essen.de
[Moderator's note: if a post of yours to sci.physics.research is
rejected, and your "From" line includes a working email address,
you will receive a rejection notice. - jb]
A.J. Tolland
> And Toby Bartels was so kind to answer (by e-mail):
> > First of all, massless spin 2 QFT does not give GR.
> > It doesn't give anything; the theory doesn't work.
You might as well say that massless spin 1 QFT for gauge group
U(1) doesn't give anything. After all, we have pretty strong indications
that QED alone doesn't even _exist_. Nonetheless, you can extract
trustworthy physical predictions about the behavior of non-renormalizable
couplings by doing effective field theory.
The classic example of this (if I understand my QFT notes) is the
anomalous magnetic moment. If you start with a generic action that looks
like S_qed + non-renormalizable terms and wave your hands while shouting
"Wilson!", you find that it flows down to a theory which has a small but
definitely non-zero value for the coefficient of the non-renormalizable
term psibar [Y^u,Y^v] psi F_uv. (The Y's are meant to be gamma matrices.)
This result is independent of what you start with, so you can reasonably
claim that straight QED + effective field theory give a solid physical
prediction despite QED's lack of microscopic existence.
The situation w/ massless spin 2 particles and GR seems to me to
be about the same: At low energies, the theory gives GR cuz it gives
Einstein-Hilbert action. As Sean Carroll pointed out to me yesterday, the
situation is even less subtle for GR: Curvature terms get surpressed by
inverse powers of energy, and the first order curvature term in the
Lagrangian is unique because it's the only thing you can construct by
contracting the Riemann tensor. So perhaps I should have made the analogy
between the universality of Yang-Mills type theories and the universality
of GR... You get them at low energy no matter what you start with.
There's a 2nd related point to be made: The massless spin 2 thing
isn't really tied to Minkowski space. You can construct these theories in
curved spaces using the vielbein formalism; all you need is a Lorentz
structure on the tangent space and maybe some topological conditions on
your manifold. The graviton should be regarded as the fluctuation of the
metric around any fixed background, not just around Minkowski space.
Minky space is distinguished by the fact that we know how to calculate
things there; empty space is a far easier background to cope with than the
Kerr-NUT-Newman-Penrose-Geroch-Wald-Bagnoud-Bergman-Baez-Taub-Witten-
Ashtekar-Rubbia-Choptuik metric or anything else of similar complexity.
The point is that we're not really introducing some God-given
unobservable Minkowski space background to define the graviton. We're
just trying to study fluctuations away from some average background.
--A.J.
Squark
> The point is that we're not really introducing some God-given
>unobservable Minkowski space background to define the graviton. We're
>just trying to study fluctuations away from some average background.
This claim always seemed doubtful to me and this is why - we always impose
microcasuality on our QFT, or, more generally, use the background metric to
define the canonical commutation relations or an analogue* thereof (as
themselves tend to be even less meaningful than the rest of QFT...). Now,
when the metric itself becomes a dynamical variable, it seems inappropriate to
impose on it the algebra required by whatever we perturb it around. You
may claim that to some first orders, this bears no significance, and you might
even be right (though you'll have to prove me that :-) ), but this must be
in any case recognized, that such a framework can serve a very limited purpose
only.
*We might be able to render them into a more mathematically intelligible
form by some heuristic manipulations, if I am not mistaken, but this is
getting off-topic...
Best regards,
squark.
Ilja Schmelzer
> [Moderator's note: I think we've drifted away from physics proper.
> How about taking this to e-mail? -TB]
I will try to come back to physics.
> So, the point is that truth comes from observation. I would call
> this "empiricism", and I would agree that empiricism is part of
> positivism. But absoluteness is a completely orthogonal issue.
> Surely you don't call the derivation of truth from observation an
> error?
Surely I call the derivation of truth from observation an error.
Observation allows to falsify false theories, that's the only role of
observation in science.
>> Newton's theory with absolute space makes all predictions and is quite
>> fine from point of view of Popper's methodology. There is nothing to
>> criticize from this point of view.
> Then the positivists had something over Popper!
No, the positivists have had only irrelevant criticism of Newton's
theory. As well their criticism of Lorentz ether and hidden variable
theories is irrelevant.
>> Positivism has not reached his goals. That's quite obvious.
>
> Not to me. You're going to have to explain these things.
Take a simple observation: where is a glass of water. Ask simple
questions of type "how do you know its water?" "What means glass" and
so on. Its easy to observe that this game can be played infinitely.
And during this game you will have to admit that your "observation" of
a glass of water is based on a large amount of condensed matter theory,
chemistry, and so on.
Thus, observations are not the fixed point where you can start your
theory building.
> I don't know a theory of quantum gravity with a preferred frame.
Its too easy and rejected by Baez at al for not being beautiful
enough. In some sense known as "regularization of the GR Lagrangian".
To regularize a theory is simple if you have a preferred background,
moreover a preferred frame on this background, and therefore don't
have to obey nor global nor local Lorentz symmetry in the regularized
theory.
If we name such a regularization "theory of quantum gravity" or give
them names is already metaphysics.
> In the meantime, I like the approaches that have most the flavour of GR,
> while retaining the principles of QM.
You may like them, but they are not yet really a theory. And they will
never be, I'm sure. I believe that a preferred background is
necessary for the classical limit.
> You'll have to clarify for me. First you say that your claim was
> about the appearance of the principle in Newton, which appearance is
> Newton's 3rd law; then you say that Newton's 3rd law is not the
> point. Which is it?
The question is if its reasonable to argue for background-free
theories using the "action=reaction" principle. My argument is that
it is unreasonable, because "action=reaction" as it appears in
classical theories is a derived principle, follows from Lagrange
formalism. Thus, its much more reasonable to argue that Lagrange
formalism is fundamental, instead of arguing that its derivation, the
"action=reaction" principle, is fundamental.
>> Essentially one of the main differences between Lorentz ether and SR
>> is that this obvious objection against the (material) Lorentz ether
>> who should follow the "action equals reaction" law according to NM
>> sounds much less serious in SR.
> I don't believe that the aether violated Newton's 3rd law
> (which is how the action/reaction principle appears in Newton).
The Lorentz ether is stationary and incompressible, but interacts with
usual matter via light. The violation is quite obvious. And the
obvious solution of this problem is to make the ether compressible and
instationary, IOW to replace the Minkowski metric which defines the
state of the Lorentz ether by a variable metric.
Instead, you need a genius to make the SR spacetime variable.
> Since when do matters of fact have to be certain?
If they are uncertain they are not matters of fact but guesses.
>> The main argument against absolute NM space is that we cannot be
>> certain which of the various NM explanations of our world (which
>> differ in the choice of the absolute rest) is the true one. Once we
>> do not care for certainty, that's not a point.
> Well, that's not how I'd phrase the objection! I'd say that, if the
> absolute space makes no difference, then it ought not to appear in
> the theory at all.
A completely unreasonable argumentation from point of view of common
sense. For example, in everyday live it makes no difference which
units we use. Nonetheless they appear in our descriptions of
measurements.
>> Yep, but Popper's methodology is quite different. Meaningful
>> statements are all statements of a physical theory, but certainly
>> not all of them are testable.
> Well, Popper can be as different as he likes.
> I still don't see why positivism is so bankrupt.
Because in reality a lot of meaningful statements are not falsifiable
as separate statements.
Try to falsify G_ij = T_ij. You cannot, whatever you observe may be
explained away with dark matter defined as T_ij^dark = G_ij -
T_ij^obs. Does this mean that G_ij = T_ij is meaningless? No.
> Well, John Baez put it in more dramatic terms:
> Statements beginning with universal quantifiers must be falsified,
> while statements beginning with existential quantifiers must be verified.
> Trivial, yes -- he originally brought it up by looking at negations --
> but it shows that both are needed.
Not at all dramatic, and you can find this in Popper's writings too.
>> Empiricism AFAIU is priority of observation, Popper is priority of
>> theory.
> OK, so do you object not just to positivism but to empiricism?
At least to your version of empiricism.
>> But observation is always theory-laden, and therefore cannot be prior
>> to theory. Theory-independent observation is a myth. So where do you
>> want to start with empirical observations?
> Since when does empiricism require theory independent observation?
Else you derive not theories from observation but theories from other
theories. I would not name a methodology which derives theories from
other theories and observation empiricism.
> Is this extreme strawman supposed to be the objection to Mach -- he
> was a positivist, therefore he required theory independent
> observation, therefore he was a buffoon, therefore we shouldn't
> trust his opinions?
I do not object to Mach, I object to his arguments. Inconclusive and
misleading arguments which remain important now, and lead to ignorance
of hidden variable theories, QG theories with preferred backgrounds
and so on.
To reject arguments, I have any right to take them to extremes to show
their absurdity in these extremal situations.
John Baez
A.J. Tolland <a...@hep.uchicago.edu> wrote:
>On Fri, 9 Mar 2001, Urs Schreiber wrote:
>> And Toby Bartels was so kind to answer (by e-mail):
>> > First of all, massless spin 2 QFT does not give GR.
>> > It doesn't give anything; the theory doesn't work.
> You might as well say that massless spin 1 QFT for gauge group
>U(1) doesn't give anything. After all, we have pretty strong indications
>that QED alone doesn't even _exist_. Nonetheless, you can extract
>trustworthy physical predictions about the behavior of non-renormalizable
>couplings by doing effective field theory.
Gravity from spin-2 field theory on Minkowski spacetime is worse
than QED, in my opinion. It's not just that the theory has bad
behavior at high energies. It's that in a quantum field theory
on Minkowski spacetime influences propagate within the Minkowski
lightcones, which are fixed from the start, while in general relativity
influences propagate within lightcones that depend on the state one
is considering.
This problem can't be seen order-by-order in perturbation theory,
but it's perfectly obvious nonetheless, and it screws up our description
of things like the solar system - not just Planck-scale physics.
The lightcone structure in our solar system is not that of Minkowski
spacetime, so no theory which has influences propagating within the
Minkowski lightcones can be right.
This is why people like me whine so much about the importance of
doing background-free physics. In the real world, the causal
structure of spacetime is dynamical - it depends on the state of
the universe! This is a profoundly cool fact, and we should not
neglect it when trying to quantize gravity.
On a far more technical and less important note, quantum gravity
from spin-2 field theory is worse than QED because it goes bad at
the Planck scale (10^{-35} meters) while QED goes bad at a much
smaller length scale - somewhere around 10^{-500} meters, give or
take a few dozen orders of magnitude. This length scale is so
absurdly small that not even us quantum graviters can take it seriously!
Steve Carlip
> Gravity from spin-2 field theory on Minkowski spacetime is worse
> than QED, in my opinion. It's not just that the theory has bad
> behavior at high energies. It's that in a quantum field theory
> on Minkowski spacetime influences propagate within the Minkowski
> lightcones, which are fixed from the start, while in general relativity
> influences propagate within lightcones that depend on the state one
> is considering.
> This problem can't be seen order-by-order in perturbation theory,
> but it's perfectly obvious nonetheless
Is it? I'd love to see a proof (read ``convincing argument'') for this.
First look at the classical case, starting out with a Minkowski metric g_0
and a massless spin 2 field h. It can be proven that the complete action,
once you include the stress-energy tensor for h as a source for h, is a
function of g = g_0 + h only, with no independent appearance of g_0.
So classically, gravitational influences propagate on the light cones
determined by g. (So do electromagnetic influences if you include a
coupling to electromagnetism.)
Now look at the quantum theory. There are several ways to proceed. We
can compute the effective action, for instance, starting with a background
metric g_0 and integrating out quantum fluctuations h. The theory is
nonrenormalizable, so there will be an infinite number of terms in the
effective action, but a general result of the background field method is
that each term will depend only on the full metric; the effective action
will be a power series R + R^2 +..., with no remaining dependence on g_0
or the Minkowski light cone.
Now, it's true that the initial spin two field h has commutators determined
by the Minkowski light cones. But remember, h is not an observable. In
fact, any observable---any quantity that commutes with the constraints
---must be a *nonlocal* function of h. So it's not clear that the algebra
of h implies anything very stong about the algebra of observables. (Note
that the constraints are still there even in the spin 2 field formulation;
they express the gauge invariance that keeps the field massless and pure
spin 2.)
Am I missing something?
Steve Carlip
A.J. Tolland
> On a far more technical and less important note, quantum gravity
> from spin-2 field theory is worse than QED because it goes bad at
> the Planck scale (10^{-35} meters) while QED goes bad at a much
> smaller length scale - somewhere around 10^{-500} meters, give or
> take a few dozen orders of magnitude. This length scale is so
> absurdly small that not even us quantum graviters can take it seriously!
I don't think that QED and spin-2 theory go wrong in exactly the
same way. Perturbative QED blows up cuz it suffers from a Landau pole.
Perturbative spin-2 blows up because it's not renormalizable. I've always
been under the impression that, because of this difference, gravity is
more likely to make sense at the non-perturbative level. Seems to me that
non-renormalizability is an indication of more complicated high energy
behavior, a non-triviality in the fundamental theory which difficult to
understand by studying behavior at finite momentum scales through
perturbation theory. By contrast, I'm told that the Landau pole is
usually taken as a sign that bare QED is garbage and nonsense.
So I don't think your second criticism necessarily applies.
Then again I have a long and distinguished history of error.
--A.J.
John Baez
Steve Carlip <sjca...@ucdavis.edu> wrote:
>John Baez <ba...@galaxy.ucr.edu> wrote:
>> Gravity from spin-2 field theory on Minkowski spacetime is worse
>> than QED, in my opinion. It's not just that the theory has bad
>> behavior at high energies. It's that in a quantum field theory
>> on Minkowski spacetime influences propagate within the Minkowski
>> lightcones, which are fixed from the start, while in general relativity
>> influences propagate within lightcones that depend on the state one
>> is considering.
>> This problem can't be seen order-by-order in perturbation theory,
>> but it's perfectly obvious nonetheless
>Is it? I'd love to see a proof (read ``convincing argument'') for this.
I'm sorry, I was expressing myself very poorly when I said "this
problem can't be seen order-by-order in perturbation theory". What
I meant was practically the opposite: if you work to a given order
in perturbation theory, you will have a theory where influences
propagate in the lightcones determined by the Minkowski metric.
Now you've gotten me worried, but *this* is true, right?
As you say, it's profitable to think about this in the classical
case:
>First look at the classical case, starting out with a Minkowski metric g_0
>and a massless spin 2 field h. It can be proven that the complete action,
>once you include the stress-energy tensor for h as a source for h, is a
>function of g = g_0 + h only, with no independent appearance of g_0.
>So classically, gravitational influences propagate on the light cones
>determined by g. (So do electromagnetic influences if you include a
>coupling to electromagnetism.)
Let's see if I understand this.
If we approach this problem perturbatively, writing
g = g_0 + epsilon g_1 + epsilon^2 g_2 + ...
we can get the "complete action" by an iterative procedure
where we:
first linearize the Einstein action about g_0 and treat
epsilon g_1 as a field propagating on the background g_0 with
equations of motion determined by this linearized action
then work out the stress-energy tensor of epsilon g_1
and get something proportional to epsilon^2
then treat this stress energy tensor as a source for the
metric and get a correction epsilon^2 g_2
then work out the stress-energy tensor of epsilon^2 g_2
and get something proportional to epsilon^3
then treat this stress energy tensor as a source for the
metric and get a correction epsilon^3 g_3
and so on.
Now, if we stop this game after any finite number of terms,
I think we get a field theory where influences propagate
in the lightcones determined by the metric g_0. Is this right?
(It's possible that I'm mixed up, and the field theory for
g_0 + epsilon g_1 + epsilon^2 g_2 + ... epsilon^n g_n
propagates in the lightcones determined by
g_0 + epsilon g_1 + epsilon^2 g_2 + ... epsilon^{n-1} g_{n-1}
But I've never heard about something like this going on
in perturbative *quantum* gravity, so I'd be surprised if
it happened classically.)
Anyway, now suppose we sum the whole power series. You're saying
that we get the ordinary Einstein equations for the metric g, so
influences propagate in the lightcomes determined by g.
Okay. This isn't incompatible with what I think, though I'd like
to understand better how they fit together. I've seen a paper on this,
but it was a long time ago.
>Now look at the quantum theory. There are several ways to proceed. We
>can compute the effective action, for instance, starting with a background
>metric g_0 and integrating out quantum fluctuations h. The theory is
>nonrenormalizable, so there will be an infinite number of terms in the
>effective action, but a general result of the background field method is
>that each term will depend only on the full metric; the effective action
>will be a power series R + R^2 +..., with no remaining dependence on g_0
>or the Minkowski light cone.
Hmm, I want to make sure we agree on the classical situation before
I get into this.
Steve Carlip
> Let's see if I understand this.
> If we approach this problem perturbatively, writing
> g = g_0 + epsilon g_1 + epsilon^2 g_2 + ...
> we can get the "complete action" by an iterative procedure
> where we:
> first linearize the Einstein action about g_0 and treat
> epsilon g_1 as a field propagating on the background g_0 with
> equations of motion determined by this linearized action
> then work out the stress-energy tensor of epsilon g_1
> and get something proportional to epsilon^2
> then treat this stress energy tensor as a source for the
> metric and get a correction epsilon^2 g_2
[...]
Well, this is probably equivalent to the standard approach, but
it's not completely clear; since I don't know what epsilon is,
I'm not sure how to decide in advance what g_1, g_2, etc. are.
Try this:
Start with a background metric g_0 and a massless spin two
field h propagating in the background geometry. Note that
the condition that h is massless and pure spin two requires a
gauge invariance, and that if h is minimally coupled to matter,
the corresponding Noether current is the matter stress-energy
tensor. Note also that gauge invariance determines a unique
free field action I_2 for h, which is equivalent to the linearized
Einstein action for g_0 + h. (I'm labeling this I_2 to remind
myself that it's quadratic in h.)
Now require that h couple to its own stress-energy tensor.
This stress-energy tensor T_2 can be obtained from I_2, by
Noether methods for instance, and is quadratic in h. To
obtain a field equation with such a quadratic source, we need
to add a piece I_3, cubic in h, to the action.
But then T_2 is no longer the full stress-energy tensor of h,
which should be obtained from the corrected action I_2 + I_3.
This gives a new contribution T_3 to the stress-energy tensor
(and also requires a correction to the gauge transformations,
but that's a technical issue).
But now to get a field equation with T_2 + T_3 as a source
for h, we need an extra piece I_4 in the action. This in turn
requires a T_4, etc. When you iterate all the way and sum the
resulting series, you get an action I_infinity that is just the
Einstein-Hilbert action for the metric g_0 + h.
> Now, if we stop this game after any finite number of terms,
> I think we get a field theory where influences propagate
> in the lightcones determined by the metric g_0. Is this right?
In your formulation, it probably is right: at order n, the only
term in the field equations linear in g_n must depend only on
g_0. But I'm not sure that's the right question.
Go back to the approach I just described, and stop at order n.
You then have some action I_2 + I_3 + ... + I_n, which can be
obtained from the Einstein-Hilbert action I_infinity[g_0+h]
by throwing out all terms with more than n powers of h. I could
be persuaded otherwise, but I think the question to ask is this:
given this action, how do small disturbances in h propagate?
In other words, given this action, consider a small change
h -> h + j. What are the linearized equations for j?
I can't write down the answer to this off the top of my head,
but it's certainly not that j propagates along the light cones of
g_0. Instead, the characteristics should converge to the light
cones of g_0 + h as n increases. To see this, note that one way
to get the equations for j is to write down the full Einstein
equations for g_0 + h + j, find the linearized equation for j,
and then expand the result in powers of h, keeping only terms
with fewer than n powers of h. The first step, linearizing in j,
gives a wave equation in the background g_0 + h. The second
step mangles this equation, but I don't see how it could possibly
give back the g_0 light cones.
(Note that at finite order, the result is probably not completely
self-consistent, either: you lose gauge invariance if you don't
go all the way out to infinity.)
Steve Carlip
Steve Carlip
One added note: as I've described this here, you have to sum an
infinite series. But there's actually a clever choice of variables
---basically a Palatini formulation with the connection and the
densitized metric as independent variables---for which the series
terminates after a few terms. See Deser, Class. Quant. Grav. 4,
L99 (1987).
Steve Carlip
John Baez
Steve Carlip <sjca...@ucdavis.edu> wrote:
>Try this:
>
>Start with a background metric g_0 and a massless spin two
>field h propagating in the background geometry. Note that
>the condition that h is massless and pure spin two requires a
>gauge invariance, and that if h is minimally coupled to matter,
>the corresponding Noether current is the matter stress-energy
>tensor.
Okay. On a completely digressive note, in what ways does this
massless spin-2 linear field equation work better when g_0 is a
solution of Einstein's equation? Of course, only in this case
is it a linearization of Einstein's equation! But I'm assuming
you can write down this linear equation for any background metric
g_0 (please correct me if I'm wrong), and I'm wondering what extra
nice properties the equation has when g_0 satisfies Einstein's
equation.
You see, Madhavan Varadarajan has figured out how to express the
ordinary Fock space for photons in terms of loop variables:
Madhavan Varadarajan
Fock representations from U(1) holonomy algebras
http://xxx.lanl.gov/abs/gr-qc/0001050
and I've been talking about this with Ashtekar. It's sort of
exciting, because previously nobody had noticed that the good
old Fock representation of the quantized Maxwell equations gave
rise to a representation of the "U(1) holonomy algebra" on which
loop quantization is based. This lead us to dream of doing a
similar thing for the linearized Einstein equations, thus getting
a firmer connection between the Fock space for gravitons to loop
quantum gravity.
I had naively suggested doing this starting with an arbitrary
Riemannian 3-manifold as "space", but Ashtekar point out that
this is sort of silly unless this space comes from 1) a solution
of Einstein's equations, and 2) a static solution. Restriction 1)
comes from the fact that we can't linearize Einstein's equations
except about a *solution* of Einstein's equations - and this is
the reason for my question here. 2) comes from the fact that
there's no obvious best choice of Fock space unless we have time
translation invariance (which we use to determine the complex
structure on the one-graviton Hilbert space).
Anyway, that was just a huge digression.
>Note also that gauge invariance determines a unique
>free field action I_2 for h, which is equivalent to the linearized
>Einstein action for g_0 + h. (I'm labeling this I_2 to remind
>myself that it's quadratic in h.)
Yup.
>Now require that h couple to its own stress-energy tensor.
>This stress-energy tensor T_2 can be obtained from I_2, by
>Noether methods for instance, and is quadratic in h. To
>obtain a field equation with such a quadratic source, we need
>to add a piece I_3, cubic in h, to the action.
Yup.
>But then T_2 is no longer the full stress-energy tensor of h,
>which should be obtained from the corrected action I_2 + I_3.
>This gives a new contribution T_3 to the stress-energy tensor
>(and also requires a correction to the gauge transformations,
>but that's a technical issue).
>
>But now to get a field equation with T_2 + T_3 as a source
>for h, we need an extra piece I_4 in the action.
Yup. I'm getting sleepy; wake me up when you get to infinity. :-)
>This in turn requires a T_4, etc.
zzzz....
>When you iterate all the way and sum the
>resulting series, you get an action I_infinity that is just the
>Einstein-Hilbert action for the metric g_0 + h.
zzzz... hey! That was mighty quick.
>> Now, if we stop this game after any finite number of terms,
>> I think we get a field theory where influences propagate
>> in the lightcones determined by the metric g_0. Is this right?
>In your formulation, it probably is right: at order n, the only
>term in the field equations linear in g_n must depend only on
>g_0. But I'm not sure that's the right question.
Hmm, I guess not.
>Go back to the approach I just described, and stop at order n.
>You then have some action I_2 + I_3 + ... + I_n, which can be
>obtained from the Einstein-Hilbert action I_infinity[g_0+h]
>by throwing out all terms with more than n powers of h. I could
>be persuaded otherwise, but I think the question to ask is this:
>given this action, how do small disturbances in h propagate?
It's definitely an interesting question, and maybe even "the" question.
>In other words, given this action, consider a small change
>h -> h + j. What are the linearized equations for j?
>
>I can't write down the answer to this off the top of my head,
>but it's certainly not that j propagates along the light cones of
>g_0. Instead, the characteristics should converge to the light
>cones of g_0 + h as n increases. To see this, note that one way
>to get the equations for j is to write down the full Einstein
>equations for g_0 + h + j, find the linearized equation for j,
>and then expand the result in powers of h, keeping only terms
>with fewer than n powers of h. The first step, linearizing in j,
>gives a wave equation in the background g_0 + h. The second
>step mangles this equation, but I don't see how it could possibly
>give back the g_0 light cones.
Okay, great! Thanks very much for disabusing me of a serious
confusion! And now for a *quantum* question. In your previous
article you outlined a rather clever approach to seeing how
causality worked in the quantum case. But let me consider another
approach. Lots of people spent lots of time seeing if quantum
gravity was perturbatively renormalizable as a field theory on
Minkowski spacetime. The calculation blew up in their face at
two loops. But suppose it worked up to n loops. To avoid
contrafactual hypotheses we can either set n = 1, or switch
to a supergravity theory that works up to 2 loops. The resulting
theory would still have influences propagating within the Minkowski
lightcones, right?
Steve Carlip
> Okay. On a completely digressive note, in what ways does this
> massless spin-2 linear field equation work better when g_0 is a
> solution of Einstein's equation?
Um... I don't remember the details. It's a consistency condition.
The reference is Deser, Class. Quant. Grav. 4, L99 (1987).
> And now for a *quantum* question. In your previous
> article you outlined a rather clever approach to seeing how
> causality worked in the quantum case. But let me consider another
> approach. Lots of people spent lots of time seeing if quantum
> gravity was perturbatively renormalizable as a field theory on
> Minkowski spacetime. The calculation blew up in their face at
> two loops. But suppose it worked up to n loops. To avoid
> contrafactual hypotheses we can either set n = 1, or switch
> to a supergravity theory that works up to 2 loops. The resulting
> theory would still have influences propagating within the Minkowski
> lightcones, right?
I don't know. I know it's a trickier question than it seems, though.
Here's why:
The obvious way to try to get an answer would be to look at a correlation
function like <h h>. The result would certainly look like one for a
theory with influences propagating on Minkowski light cones. But
in computing this correlation function, you have to have gauge-fixed
the theory. If you use a standard gauge (harmonic gauge, say) and work
backwards, you find that your gauge-fixed h is really a complicated and
nonlocal function of the curvature. You shouldn't expect correlation
functions of nonlocal variables to tell you much, at least directly, about
the causal structure. And you can't even get away with going off to
infinity and arguing that the S matrix doesn't depend on the choice
of interpolating fields---it's invariant under field redefinitions, but
only if they're local.
More generally, even in a straightforward naive quantum field theory
approach to quantum gravity, you need to look at genuine observables
to determine the propagation speed/causal structure. (Diffeomorphisms
can travel at the speed of thought.) But Torre has shown that true
observables in quantum gravity---quantities that commute with the
constraints---are necessarily nonlocal. So it's a mess, and I don't
think anyone really knows how to even start (though Tsamis and
Woodard have certainly thought about these issues).
Steve Carlip
Robert C. Helling
On 27 Mar 2001 21:00:24 GMT, John Baez <ba...@galaxy.ucr.edu> wrote:
>In article <99gm4b$m40$1...@woodrow.ucdavis.edu>,
>Steve Carlip <sjca...@ucdavis.edu> wrote:
>
>>Try this:
>>
>>Start with a background metric g_0 and a massless spin two
>>field h propagating in the background geometry. Note that
>>the condition that h is massless and pure spin two requires a
>>gauge invariance, and that if h is minimally coupled to matter,
>>the corresponding Noether current is the matter stress-energy
>>tensor.
>
>Okay. On a completely digressive note, in what ways does this
>massless spin-2 linear field equation work better when g_0 is a
>solution of Einstein's equation? Of course, only in this case
>is it a linearization of Einstein's equation! But I'm assuming
>you can write down this linear equation for any background metric
>g_0 (please correct me if I'm wrong), and I'm wondering what extra
>nice properties the equation has when g_0 satisfies Einstein's
>equation.
>
This is well known from the background field method: For any field-
theory, split the field as F = f + phi and then "Taylor expand"
the action around f:
S[f+phi] = S[f] + dS/df[f] phi + 1/2 phi d^2S/df[f] phi + ...
where d is the functional derivative. You can ignore the first term, as
it is just a constant. Note, that, in the second term, dS/df is nothing
but the classical equation of motion (here Einstein's equation). This
is absent if the background is on-shell. The third term provides the
kinetic term for phi (this is all what is needed for a one-loop
calculation) and the suppressed terms are the interactions.
Usually, you don't want terms that are linear in the dynamic field
because they give rise to tadpoles (lines in the Feynman diagram
can just end in the vacuum). You could try to resum them by replacing
the pure propagator by a geometric series as
======= = -------- + ----T----- + --T--T-- + ...
| | |
| | |
but this is nothing but effectively bringing the background f back
on-shell (remember that the 'pure' propagator is built using the
kinetic term that depends non-trivially on f).
Robert
--
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Robert C. Helling Institut fuer Physik
Humboldt-Universitaet zu Berlin
print "Just another Fon +49 30 2093 7964
stupid .sig\n"; http://www.aei-potsdam.mpg.de/~helling