Did Minkowski's geometry really need complex numbers for use in Relativity?
pol...@postoffice.pacbell.net
incorporate complex numbers in order to be employed in:
l. Special Relativity?
2. General Relativity?
This issue was recently discussed on Guardian Unlimited Talk. The claim
was made that Minkowski's original formulation using "t" for the time
dimension would have worked fine in the math for both Special and
General Relativity ... the employment of complex numbers made
computation and formulation much simpler, but nothing more than that.
It was not essential to the theory's development.
Einstein's own popular book about Relativity (Chapters 17 and 26) is
vulnerable to interpretation on this score, it seems. Might just be the
translation into English.
Charles Francis
pol...@postoffice.pacbell.net writes
>I'll refine that: Did Minkowski's space-time geometry really need to
>incorporate complex numbers in order to be employed in:
>
>l. Special Relativity?
>
>2. General Relativity?
>
>This issue was recently discussed on Guardian Unlimited Talk. The claim
>was made that Minkowski's original formulation using "t" for the time
>dimension would have worked fine in the math for both Special and
>General Relativity ... the employment of complex numbers made
>computation and formulation much simpler, but nothing more than that.
>It was not essential to the theory's development.
These days we do not normally use formulations with complex time.
Instead we use a metric. I for one find this much simpler conceptually
than imaginary time. The critical formula we need and on which physical
results depends involves the square of time and space elements (small
amounts dt, dx, dy, dz of time and distance)
ds^2 = -dt^2 + dx^2 + dy^2 + dz^2
You can get this formula by using imaginary time and Pythagoras theorem
in four dimensions, but these days we generally use a matrix
multiplication
(dt,dx,dy,dz)( -1 0 0 0 ) ( dt )
( 0 1 0 0 ) ( dx )
( 0 0 1 0 ) ( dy )
( 0 0 0 1 ) ( dz )
These kind of matrix calculations can be written very neatly in index
notation.
Regards
--
Charles Francis
ueb
> l. Special Relativity?
> 2. General Relativity?
I believe, you underestimate the importance of Minkowski's step with
this question. In fact, time _is_ imaginary length, and conversely.
As well, that has nothing to do with complex numbers depictable with
Gauss' plane of numbers. Einstein uses this insight consequently
in his ``Four lectures on theory of relativity'' (I read the back-
translation ``Grundz\"uge der Relativit\"atstheorie''). Thus one
can grasp the geometric nature of gravitation, and of electromagnetism
too. :-) Why else become the formulae much simpler, when that
is not true ?
Ulrich Bruchholz
www.markt-2000.de/patent
John Baez
<pol...@pacbell.net> wrote:
>I'll refine that: Did Minkowski's space-time geometry really need to
>incorporate complex numbers in order to be employed in:
>
>l. Special Relativity?
>
>2. General Relativity?
No. Proof: most people these days *don't* use complex numbers to
describe the metric in special and general relativity. See the
discussion of this issue in "Gravitation" by Misner, Thorne and
Wheeler.
To me, the more interesting question is whether Minkowski's use of i
to relate time and space was merely a trick or has some deep meaning.
If you ask physicists about the significance of "Wick rotation", you'll
get an outpouring of drastically divergent opinions on this subject.
In his "Brief History of Time", Hawking claims that this imaginary
time stuff is really profound. Many other wise heads hold the opposite
view.
Time will tell.
Danny Ross Lunsford
> I'll refine that: Did Minkowski's space-time geometry really need to
> incorporate complex numbers in order to be employed in:
>
> l. Special Relativity?
>
> 2. General Relativity?
It isn't a question of "need". The light cone is most naturally described
as a locus of isotropic lines in the sense of projective geometry, and
these are willy-nilly associated with complex numbers. To take the case of
the Euclidean plane, where in homogeneous coordinates the distance function
is
R = sqrt( (x1y3 - x3y1)^2 + (x2y3 - x3y2)^2 ) / x3y3
the isotropic lines are the ones along which the distance is zero, so there
are two solutions
(x1x3 - x3y1) = (+-)(x2y3 - x3y2)
or in affine coordinates
x + iy = const
x - iy = const
Through every point of the Euclidean plane pass two isotropic lines.
Similar analysis can be applied to spacetime and there, the light cone is
the locus of isotropic lines.
So what good are they, you say? They allow one to invariantly define the
concept of angle, or better, phases. Complex numbers essentially arise any
time phases are important. In relativity, phases are certainly important
(Penrose-Terrell rotation, Thomas precession, Doppler effect etc.)
In the case of the Euclidean plane, the invariant definition of the angle
of two lines through a point is defined in terms of the cross-ratio they
form with the two isotropic lines through that point. To be precise,
a = (i/2) log XR(u1,u2;I1,I2)
where the arguments represent the four lines (in homogeneous line
coordinates). This is probably the most fundamental way in which "i"
appears in geometry. In the euclidean plane, the isotropic lines are
imaginary so the angle is real. In Minkowski space, the isotropic lines
(the light cone) are real so the angles are imaginary, cos and sin become
cosh and sinh, and we are in the world of boosts instead of rotations.
I intend to write a paper reviewing all the ways "i" shows up in physics,
but in a word, it boils down to phases, and phases mean matter.
--
-drl
Danny Ross Lunsford
> I believe, you underestimate the importance of Minkowski's step with
> this question. In fact, time _is_ imaginary length
Time is NOT imaginary length - it is real time. The topology of Minkowski
space is absolutely not the same as that of Euclidean space.
As mentioned in another post, the "i" in relativity is related to the locus
of isotropic lines in the sense of projective geometry.
One can use "it" and a pseudo-Euclidean metric only if parity is not
considered. But even then, it is a bad idea in my opinion.
--
-drl
AB for MDH
> <pol...@pacbell.net> wrote:
> >I'll refine that: Did Minkowski's space-time geometry really need to
> >incorporate complex numbers in order to be employed in:
> >
> >l. Special Relativity?
> >
> >2. General Relativity?
> No. Proof: most people these days *don't* use complex numbers to
> describe the metric in special and general relativity. See the
> discussion of this issue in "Gravitation" by Misner, Thorne and
> Wheeler.
If you are talking about the ict farewell box, it is too small to be
satisfying.
> To me, the more interesting question is whether Minkowski's use of i
> to relate time and space was merely a trick or has some deep meaning.
> If you ask physicists about the significance of "Wick rotation", you'll
> get an outpouring of drastically divergent opinions on this subject.
> In his "Brief History of Time", Hawking claims that this imaginary
> time stuff is really profound. Many other wise heads hold the opposite
> view.
>
> Time will tell.
All I can tell you is that the seven days of creation is my "preferred canon
of simultaneity."_1 I have recently coached myself that if you mix time
with Hamiltonian operators, Hamilton becomes problematic.
I had you in mind, John, as well when I posted to Steve:
Okay, you got me cracking two books and wondering how much more spending I
need to increase my library.
"Thus, perhaps the most important group in all of physics is seen to emerge
at the very primitive level of topology, i.e., from just an appropriate
definition of "nearby" events.
The fine topology is, however, from the technical point of view, rather
difficult to work with and the arguments in '67 Zeeman are by no means
simple.
In 1976, Hawking, King and McCarthy described another topology on Minkowski
space
which seemed physically even more natural, had precisely the same
homeomorphism group as Zeeman's fine topology and required for the proof
...._2"
Now, looking at the dates I can see why I have never heard of this in high
school. When will topology as a subject be high school appropriate?
Wick's Time Rotation wasn't easy to find. Although Landau(1996) does not
have any AA Robb or JA Winnie as _1 does; I found this coach Wiz :-),
"Although some philosophical questions about quantum mechanics are
inherently easier to address with the path integral formulation, there are
still fundamental difficulties with it. For example, a trajectory and the
integral along a trajectory formulation is well defined in the Minkowski
space of special relativity. In practise, the theory becomes difficult to
work with analytically when nongaussian integrals are encountered or when
the Hamiltonian is complicated, and if the Hamiltonian contains nonlinear
terms in the momentum, the noncommuting aspects of position and momentum can
result in difficulties."
I found this statement helpful but probably redundant to most readers here:
"The action is an integral over this path," in whatever a spacetime lattice
is "while the "path integral" is a sum of integrals over all paths."
So, may the Wiz please the crowd, and fill in a glossary of terms by using
the definition in successive terms. And if it cannot be done may a fireball
be thrown.
Hermitian Hamiltonian
lattice space
compactified complexified Minkowski space
(local) trivialization of bundle
Al
1Peter Hodgson & John Lucas page 108.
2Greg Naber.
Chris Hillman
pol...@postoffice.pacbell.net asked:
> > Did Minkowski's space-time geometry really need to
> > incorporate complex numbers in order to be employed in:
> >
> > l. Special Relativity?
> >
> > 2. General Relativity?
"ueb" replied:
> In fact, time _is_ imaginary length, and conversely.
Several people have already pointed out that this is a very dubious claim,
and that the best short answer to the OP's question is "no". I'd just add
that the best place to learn why the best short answer is "no" is the very
readable and classic undergraduate textbook by Taylor and Wheeler,
Spacetime Physics (the second edition appeared a few years ago).
> As well, that has nothing to do with complex numbers depictable with
> Gauss' plane of numbers.
[Presumably meaning the algebra of complex numbers, which is a
two-dimensional real linear associative algebra and in fact a Clifford
algebra, and even better, a commutative Cayley-Dickson algebra--- which is
a very special thing indeed (there are only three of these, the reals,
complexes, and one described in the next paragraph)!]
Just thought I'd point out that if you consider "two-dimensional Minkowski
spacetime" E^(1,1), then this corresponds not to the complex numbers but
to a closely related two-dimensional commutative Cayley-Dickson algebra,
the Minkowski numbers, just as "two-dimensional flat Newtonian spacetime"
E^(1,0) corresponds to another two-dimensional commutative algebra, the
Gallilei numbers. One way to understand this is to consider the
fundamental second order partial differential operator respecting the
symmetries of E^2, the harmonic operator d^2/dx^2 + d^2/dy^2. Following
Dirac, we can try to factor this into two first-order operators
d^2/dx^2 + d^2/dy^2 = (d/dx + i d/dy ) (d/dx - i d/dy)
where of course this only works if we take i to be a non-real "number"
satisfying i^2 = -1, where the RHS -is- a real number. Similarly, the
fundamental second order partial differential operator respecting the
symmetries of E^(1,1), the wave operator d^2/dt^2 - d^2/dx^2, can be
factored
d^2/dt^2 + d^2/dx^2 = (d/dt + e d/dx ) (d/dt - e d/dx)
where we take e to be a non-real "number" satisfying e^2 = 1, where again
the RHS -is- a real number. The geometry and algebra of these
little-known two-dimensional algebras (Minkowski and Galilei numbers) are
apparently hard to find, but there are two or three old but very readable
books by I. M. Iaglom (also spelt Yaglom) written for bright Soviet high
school students(!), which explain not only the algebra and geometry of
Minkowski and Galilei numbers but also the physical application. (The
Galilei numbers is just the two-dimensional real Grassmann algebra.)
However, IIRC, these books do not discuss the above factorization, which
was suggested to me by the discussion in the classic textbook by Ahlfors,
Complex Analysis. I believe there is a book by Blaine Lawson (?) which
non-mathematicians won't find very easy, with a title including the word
"Calibrations", the first chapter of which gives a concise discussion of
these algebras (including the finite list of Cayley-Dickson algebras, of
which the reals, complexes, quaternions and [non-associative] octonions
are the best known, but there are a few more) and IIRC generalizations to
Clifford algebras. Pertti Lounesto no doubt can give more information
about Clifford algebras (see his recent textbook).
It is very natural to now try to develop the analogues in Minkowski
numbers of the Cauchy-Riemann equations and Cauchy's integral theorem,
Gauss's mean-value theorem, etc., which are formulation in the complex
numbers, and more generally to develop the analogue of the theory of
functions of one complex variable. This program quickly runs into
obstacles which caused severe difficulties to people as smart as Lars
Ahlfors (the Field's medalist). Again, Pertti Lounesto has on past
occasions discussed some recent insights which appear to have clarified
these difficulties, so I'll let him comment on that if he wishes.
Chris Hillman
Home page: http://www.math.washington.edu/~hillman/personal.html
Chris Hillman
On Mon, 28 Jan 2002, Danny Ross Lunsford wrote:
> Time is NOT imaginary length - it is real time.
Needless to say, I agree with that, but...
>.The topology of Minkowski space is absolutely not the same as that of
> Euclidean space.
Uh oh, at the very least this requires explanation. In Lorentzian
four-manifolds, each point has a small neighborhood homeomorphic to R^4,
so Minkowski spacetime, considered as a Lorentzian manifold, -is- locally
euclidean (by the definition, which requires a Lorentzian manifold to be a
topological manifold with special properties--- see the textbook by
Barrett O'Neill, Semi-Riemannian Geometry.). Indeed, E^(1,3) -is-
homeomorphic to E^4 with the usual metric topology, aka "Euclidean
topology" on R^4. It is true that E^(1,3) and E^4 have very different
geometry (as inner product spaces, or as semi-Riemannian manifolds) but
they have the -same- topology. Indeed, E^(1,3) and E^4 are just R^4 (as a
topological manifold) equipped with two different semi-Riemannian
geometries.
I think what Danny meant is that the Minkowski "metric" (better, quadratic
form or inner product) on E^(1,3) is not a "metric" in the sense in which
that word is used in "metric topology". (This terminological conflict is
well-known to be very confusing to students, and has fairly recently been
discussed here in at least two other threads.)
One -can- try to pursue the idea that a suitable pair of hyperbolae (two
branches each), the locus of constant Minkowksi squared norm from some
fixed event, can be taken to define a kind of (noncompact!)
"neighborhood" of said event. Needless to say, this leads to a highly
non-euclidean notion of when two events are "close"; any pair of events
which are "almost null-separated" would be "close", but this certainly
isn't true of the standard topology on E^(1,3) (the Euclidean topology)! I
am not very familiar with this stuff, but I know that several
mathematicians have worked on variations of these ideas. So far this work
seems to be very obscure both in math and in physics, even though IIRC one
of the people who has worked on this stuff is Steve Smale (the Fields'
Medalist).
ueb
> ueb wrote:
>> I believe, you underestimate the importance of Minkowski's step with
>> this question. In fact, time _is_ imaginary length
> Time is NOT imaginary length - it is real time.
I think, time is imaginary length even since it is real time. ;-)
The fourth coordinate is `it' then, for example 1s = -i3E8m .
> The topology of Minkowski
> space is absolutely not the same as that of Euclidean space.
> As mentioned in another post, the "i" in relativity is related to the locus
> of isotropic lines in the sense of projective geometry.
> One can use "it" and a pseudo-Euclidean metric only if parity is not
> considered. But even then, it is a bad idea in my opinion.
May be that this view is not sufficient for projective geometry,
but I mentioned that "bad idea" in the sense of SR/GR with involved
Riemannian geometry, and in the light of Einstein's success
with it.
But it is no problem at all to use the real time `t' as fourth
coordinate, we get in *fully* equivalent description either
negative g_00 (ds^2 < 0) or negative g_11, g_22, g_33 (ds^2 > 0)
then. (The g_0k, k>0, keep always imaginary.)
Ulrich Bruchholz
www.markt-2000.de/patent
John Baez
Danny Ross Lunsford <antima...@yahoo.com> wrote:
>I intend to write a paper reviewing all the ways "i"
>shows up in physics [....]
Great! There's already a nice paper on this subject:
Andrzej Trautman, On complex structures in physics,
Chapter 34 of _On Einstein's Path_, Springer, New York, 1999,
also available as info.fuw.edu.pl/~amt/4schuck.ps
but there is a lot left to say.
David Hillman
Chris Hillman wrote:
> One -can- try to pursue the idea that a suitable pair of hyperbolae (two
> branches each), the locus of constant Minkowksi squared norm from some
> fixed event, can be taken to define a kind of (noncompact!)
> "neighborhood" of said event. Needless to say, this leads to a highly
> non-euclidean notion of when two events are "close"; any pair of events
> which are "almost null-separated" would be "close", but this certainly
> isn't true of the standard topology on E^(1,3) (the Euclidean topology)! I
> am not very familiar with this stuff, but I know that several
> mathematicians have worked on variations of these ideas. So far this work
> seems to be very obscure both in math and in physics, even though IIRC one
> of the people who has worked on this stuff is Steve Smale (the Fields'
> Medalist).
Interesting. I've vaguely pondered this question myself. If you can recall any
more details, let me know.
Is it true that twistor theory follows this program? There if I recall the
points are light rays (which makes sense from this point of view: all points
on ray are distance zero from one another), which (I think, but memory is
foggy here) makes spacetime six-dimensional.
Danny Ross Lunsford
> Andrzej Trautman, On complex structures in physics,
> Chapter 34 of _On Einstein's Path_, Springer, New York, 1999,
> also available as info.fuw.edu.pl/~amt/4schuck.ps
Thanks - there is another interesting paper by Nurowski and Trautman here:
http://arxiv.org/abs/math.DG/0201266
I'd like to learn more about these Robinson congruences. Any pointers? Just
a speculation, but I wonder is these things are differential-geometric
generalizations of the idea of a ruled surface?
--
-drl
John Baez
AB for MDH <mdh...@bright.net> wrote:
>So, may the Wiz please the crowd, and fill in a glossary of terms by using
>the definition in successive terms. And if it cannot be done may a fireball
>be thrown.
>
>Hermitian Hamiltonian
>
>lattice space
>
>compactified complexified Minkowski space
>
>(local) trivialization of bundle
Alas, the Wiz is out on a mission to another universe trying
to convince Oz to return to sci.physics.research. And even
were he around, he would probably react violently, because
you're basically asking him for a mathematics and physics
graduate education. Here's my quick take on your questions:
1) Hamiltonian = energy. Why the fancy word? Learn how once
you know the formula for energy, all else follows. Learn the
Hamiltonian formulation of classical and quantum physics!
2) Hermitian = self-adjoint. Did that help? No? Learn matrix
algebra, and stir in quantum theory to see why it's important.
3) lattice space = a grid, like a checkerboard, but possibly
in higher dimensions. That was easy.
4) Minkowski space = 4d spacetime a la special relativity.
5) compactified = with some extra points at infinity stuck on.
Learn a bit of conformal geometry to see why massless particles
are happier when you compactify Minkowski spacetime.
6) complexified = with the real numbers replaced everywhere by
the complex numbers. Read Penrose and Rindler's "Spinors and
Spacetime" to see why this extra trick makes those massless particles
even happier.
7) bundle = a bunch of copies of one space sitting over another in
a nice way. Huh? Learn differential geometry, and throw in a dash
of gauge field theory to see why modern physics describes all forces
using bundles. I have a book for sale on this stuff! Operators are
standing by....
8) local trivialization = handy way to study a bundle if you're
only interested in a little piece at a time. Good when it's time
start calculating.
Danny Ross Lunsford
> It is very natural to now try to develop the analogues in Minkowski
> numbers of the Cauchy-Riemann equations and Cauchy's integral theorem,
> Gauss's mean-value theorem, etc., which are formulation in the complex
> numbers, and more generally to develop the analogue of the theory of
> functions of one complex variable.
I would guess that the missing link here is Liouville's theorem, a humble
result that supports almost all the mighty edifice of function theory.
Liouville's theorem works because the complex numbers are a division
algebra. Incidentally, this theorem also has roots in projective geometry,
where the completed plane is very naturally a projective complex line. I
can't see an analogue to this in the Minkowski plane. Indeed, the complex
plane is in a sense *one-dimensional*, while the program you outlined seems
to me to be inherently two-dimensional.
It seems to me that Minkowski space would be best understood as a support
for a theory of analytic functions in terms of the next division algebra,
quaternions. But there you are going to run into the problem of lack of
unique division, as did Hamilton.
--
-drl
Robert Low
> Is it true that twistor theory follows this program? There if I recall the
> points are light rays (which makes sense from this point of view: all points
> on ray are distance zero from one another), which (I think, but memory is
> foggy here) makes spacetime six-dimensional.
This is a bit muddled. You can form the space of all light rays
in Minkowski space, which is naturally a 5-dimensional manifold
(it's R^3 x S^2). But there's more structure lying around. It
turns out that this naturally sits as a real submanifold of
(real) codimension one in a certain complex manifold of (complex)
dimension three---in fact, it sits inside complex projective three-space
(the space of complex lines in C^4). This complex manifold is
twistor space, and has real dimension six. Space-time remains
four dimensional.
---
Rob. http://www.mis.coventry.ac.uk/~mtx014/
Paul Reilly
>
> > One -can- try to pursue the idea that a suitable pair of hyperbolae ...
> > can be taken to define a kind of (noncompact!) "neighborhood" of [an] event. > > Needless to say, this leads to a highly
> > non-euclidean notion of when two events are "close"; any pair of events
> > which are "almost null-separated" would be "close", but this certainly
> > isn't true of the standard topology on E^(1,3) (the Euclidean topology)! I
> > So far this work
> > seems to be very obscure both in math and in physics,
Yes. I am very interested in it but I don't know anyone else locally
who is.
David Hillman <d...@cablespeed.com> responded in message
news:<3C588B5D...@cablespeed.com>:
> Interesting. I've vaguely pondered this question myself. ...
I ponder this all the time. If I had substantial results I'd
certainly post -
I might post some airy speculations later though, as this is a
newsgroup and
not a journal. I think in terms of "matching data is close" so that
two
surfaces joined by a null congruence are indeed close. If people
don't mind some spam and a few (many?) crackpot points I'll try to
post at length.
> Is it true that twistor theory follows this program? There if I recall the
> points are light rays (which makes sense from this point of view: all points
> on ray are distance zero from one another), which (I think, but memory is
> foggy here) makes spacetime six-dimensional.
'Classic' twistor space is a complex projective 3-space, so
six-dimensional if you count one complex dimension as two real ones.
It is the space of null geodesics. I (as a simple minded physicist)
think of light coming through a window: the point at which light comes
through gives us 2 dimensions, the angle at which it comes through
gives 2 more, the time at which it comes gives 1 more, and the phase
when it comes gives one more - that's 6 dimensions and is enough to
construct a holographic movie. Our usual 4-D manifold is the
holographic movie.
Andrew Hodges has added an extra (complex) dimension for reasons I
don't undersand yet. I think Roger Penrose is using a 4-complex
dimensional space now also.
My simple picture for understanding twistor space has been described
as simplistic but not really wrong by people who really know what they
are talking about, so take it with a grain of salt.
Points in our usual space map to spheres (S^2 hollow spheres, not
balls) in twistor space. In my view, points are a misleading construct
so I think of small patches of our usual space corresponding to fuzzy
spheres in twistor space. If you ignore quantum mechanics you can
think of an S2 family of null geodesics passing through a point of
spacetime - that's the sphere in twistor space. Again, this is a
simplified picture and I don't claim to be an expert. Andrew Hodges
has a nice introduction to twistors page at (consulting Google):
http://www.wadham.ox.ac.uk/~ahodges/twistors.html
There are many links to good books there.
Paul Reilly
Chris Hillman
On Thu, 31 Jan 2002, David Hillman wrote:
> Chris Hillman wrote:
> > One -can- try to pursue the idea that a suitable pair of hyperbolae (two
> > branches each), the locus of constant Minkowksi squared norm from some
> > fixed event, can be taken to define a kind of (noncompact!)
> > "neighborhood" of said event.
[snip]
> Is it true that twistor theory follows this program?
Hmm.... I probably ought to know this, but I don't. Maybe someone else
knows?
There is a little book called Twistor Theory which IIRC is by Baston and
Tod, in the London Mathematical Society student text series published by
Cambridge University Press, which you can look for-- should be in most
university libraries on either side of the pond. I don't recall anything
in there about the kind of "neighborhood" I mentioned above, but I haven't
looked at it in a long time.
Chris Hillman
Home page: http://www.math.washington.edu/~hillman/personal.html
[Moderator's note: twistor theory does not make use of the alternate
topology on Minkowski spacetime described above. - jb]
Danny Ross Lunsford
> Hmm.... I probably ought to know this, but I don't. Maybe someone else
> knows?
(Note: This is my understanding of the matter. It may be wrong.)
Twistors are a sort of generalization of the Plucker line coordinates for
Minkowski space. Recall that in Euclidean three space, the totality of
lines is a six dimensional projective manifold. Penrose's idea was to find
primitive objects to represent the generators of the light cone as the
basic geometric elements of spacetime, as opposed to events. An event is
then the locus of all these elements incident with that event. Instead of a
line being a locus of points, a point "is" a bundle of lines.
Twistor space turns out to be a 3-dimensional complex projective manifold
(4 homogenous complex coordinates). So, the relationship of Plucker's line
coordinates to Penrose's twistors is like the relation of two-spinors as
representations of the rotation group to Dirac spinors as representations
of the Lorentz group.
Here is an amusing fact.
When I was in school my favorite thing was to haunt the stacks of the old
journals. We had the Royal Society Proceedings going far back into the
1800s, and similarly for all the classic science journals. (We had an
original copy of a certain 1905 volume of Annalen der Physik, sitting out
on the shelf...) I decided to look up Plucker's original work on line
coordinates - as it happened, he published a paper called "On A New Theory
of Space" (as I recall it) in the Philosophical Transactions of the Royal
Society of London, 1865. I soon found the paper and slogged through the
obtuse style of the day. Satisifed, I began to look through the other works
in that volume. One caught my eye right off..
"On a Dynamical Theory of the Electromagnetic Field" - James Clerk Maxwell
This was the first appearence of Maxwell's equations!
A very strange coincidence.
--
-drl
Chris Hillman
On Fri, 1 Feb 2002, Danny Ross Lunsford wrote:
> Chris Hillman wrote:
> > It is very natural to now try to develop the analogues in Minkowski
> > numbers of the Cauchy-Riemann equations and Cauchy's integral theorem,
> > Gauss's mean-value theorem, etc., which are formulation in the complex
> > numbers, and more generally to develop the analogue of the theory of
> > functions of one complex variable.
> I would guess that the missing link here is Liouville's theorem, a humble
> result that supports almost all the mighty edifice of function theory.
Not to give too much away, but it is easy to figure out analogues of the
C-R equations; as Danny knows (or guessed), the tricky part is obtaining
-useful- analogues of Cauchy's theorem (e.g. ideally one naively wants to
obtain an analogue of residue theory). If I am not mistaken, in fact the
"missing piece" (which causes naive attempts to create a coherent theory
to fall apart) is currently held to be the "bifurcation" of various
natural alternative formulations of the very notion of "holomorphic
function", when you pass from complex to Minkowksi variables, etc. But
Pertti can probably say much more about this than I can.
> Liouville's theorem works because the complex numbers are a division
> algebra. Incidentally, this theorem also has roots in projective
> geometry, where the completed plane is very naturally a projective
> complex line. I can't see an analogue to this in the Minkowski plane.
> Indeed, the complex plane is in a sense *one-dimensional*, while the
> program you outlined seems to me to be inherently two-dimensional.
In all three cases, you have algebraico-geometric structure which are
two-dimensional algebras, considered as real associative linear algebras,
and indeed are two-dimensional Cayley-Dickson algebras (a much rarer
aves). But it is certainly very natural to attempt to treat them as
one-dimensional "Minkowksi" or "Galilei" spaces and see how far you can
get. Again, I don't know much about this (I once knew more, but have long
since forgotten almost everything), but for a long time, many tried and
(as far as I could tell, circa 1992, when I skimmed the book by Alhfors on
generalized Moebious transformations and various papers, all dating back
to the previous decade or more) all attempts failed to obtain elegant or
even very interesting theories. However, IIRC, for purposes of the
differential geometry part of Lorentzian manifold type stuff, using a C-R
notion of "holomorphic function", it is straightforward to define
meaningful analogues of holomorphic complex manifolds, e.g.
"two-dimensional Lorentzian manifolds" do become "one dimensional
holomorphic Minkowksi manifolds". But yes, if you follow this route, lots
of stuff which works out for holomorphic complex manifolds won't work out
for "holomorphic Minkowksi manifolds" (at least not if the latter are
defined in this way); in particular, integral theorems and related "local
to global phenomena".
As for the projective plane RP^2 and its relation to Minkowksi and Galilei
numbers, au contaire, "projective metrics" goes right back to Cayley and
Klein, and were extensively developed by a Russian school in the first
half of the twentieth century. Basically, Cayley and Klein introduced
what we'd call a (possibly indefinite or even indeterminate) inner product
on the two real dimensional plane using a conic they called "the
Absolute", and eventually wind up "completing" the geometry of the
Cayley-Dickson algebras of complex, Minkowski, or Galilei numbers to
obtain the usual projective plane, RP^2. You are correct in suspecting
that this is a bit tricky. This stuff is now (AFAIK) very obscure, at
least in the English language literature, although in the late nineteenth
and early twentieth century it was apparently regarded as having a central
importance, until in the first half of the last century, attention turned
away from projective geometry toward the intensive development (by
algebraists) of the theory of rings, modules, and Galois theory a la E.
Artin, etc., and (by geometers) of increasingly abstract algebraic
geometry (on the one hand) and differential geometry (on the other).
The only sources I know for these geometries are translations of the
(elementary!) books (which don't discuss "projective metrics" as such, but
IIRC do at least hint at the "completions") and a few (nonelementary!)
research papers by Iaglom and his students. To read the papers I found
(no, I don't remember the citation or even the name of the authors), IIRC,
you unfortunately need a fair degree of background in classical papers on
projective geometry, background which is hard for contemporary students to
acquire, unless perhaps if they are widely read students of algebraic
geometry (which I certainly was not). For what it's worth, you can find a
brief attempt by myself to explain the basic idea of "the Absolute" as one
of the examples in "What is a Concept?", an expository eprint you can find
by following the links from my home page (see url below). This
explanation was reverse-engineered from various sources--- none of which
quite made sense to me!--- but AFAIK it does describe in terms
comprehensible to contemporary graduate students the definition originally
due to Cayley (which was generalized and further developed by Klein and
his students, and as I said, much later still further developed by the
Russian school).
Going back to "holomorphic functions", as I said, I understand that recent
developments (1990s?), of which I was unaware circa 1992, have apparently
led to something of a breakthrough in this area. IIRC, the first clue is
that someone finally explained clearly how various alternative
formulations of "holomorphic" in the theory of functions of one complex
variable (e.g., C-R equations) give -distinct- notions in the theory of
functions of one Minkowski variable, etc., and the second clue is that one
of the more subtle alternatives, IIRC, allegedly -does- give a workable
analog of the integral theorems in the classical theory of holomorphic
functions of one complex variable. But unless I've forgotten something,
I've never seen the details of how this works, and I have no idea where
they may be found. I think it's still fairly obscure, but apparently
highly touted by those who know about it.
Again, Pertti can probably say much more than I can about all of this.
Also, perhaps this discussion should move to sci.math.research, since we
seem to be talking about pure math rather than physics!
> It seems to me that Minkowski space would be best understood as a support
> for a theory of analytic functions in terms of the next division algebra,
> quaternions.
Well, certainly "many tried and all failed" to produce an elegant or even
very interesting theory of "holomorphic" functions of one quaternionic
variable, at least until fairly recently. Even such a simple idea as
Moebius transformations of one Clifford variable seemed to tax the
ingenuity of someone as smart as Ahlfors (whose monograph I found almost
unreadable, in contrast to the lucidity of his famous textbook on complex
analysis).
> But there you are going to run into the problem of lack of unique
> division, as did Hamilton.
Indeed, anyone who's read much about "geometric algebra" knows that in
dealing with a more general idea, functions of one Clifford algebra
variable, the algebraic structure alone requires the introduction of quite
a bit of rather exotic appearing operations, essentially because many
useful notions in the theory of complex numbers split up into various
distinct notions in more general Clifford algebras. Indeed, so much
exotic superstructure appears that most who have sipped the waters
(including me) have been rather quickly dissauded from attempting to
master it all.
Pertti Lounesto
> Pertti can probably say much more about this than I can.
All differentiable functions satisfy the Stokes' theorem, and its
special cases (Gauss, Green). Cauchy's theorem (and Cauchy's
integral formula) are much more restrictive; functions satisfying
them are determined by a small portion, for which reason they
are called monogenic (= generalization of complex analytic).
Cauchy-Riemann equations can be generalized in several ways
to higher dimensions, but such generalizations are interesting
only if the functions satisfy Cauchy's theorem. Basically, one
is then driven to so-called Clifford analysis.
> Again, Pertti can probably say much more than I can about all of this.
Clifford analysis of means study of functions f: R^n -> Cl_n
that are monogenic, that is, satisfy Df = 0. Their study was
initiated in integral form by Moisil 1931 and Fueter 1935, who
studied the special of quaternion analysis. Iftimie 1965 was
the first one to study monogenic functions. Habetha 1983
showed that if Cauchy's integral formula is satisfied, then
non-zero vectors must have inverses; this implied that the
algebra involved must be (almost) the Clifford algebra. Clifford
algebra exists for dimensions n, its dimension being 2^n.
As for division algebras, there are only four real division
algebras satisfying |ab| = |a||b| and having 1, namely R,C,H,O,
where the octonion algebra O is 8-dimensional non-associative
algebra. Non-associativity makes any function theory awkward.
ueb
I get the unauthoritative (probably wrong) impression that
mathematicians tend to complicate things which unbiased persons
see quite simply. ;-) The mentioned view as physical reality seems
me to be the simplest possible. In the reply to Danny Ross Lunsford
I mentioned already the equivalent possibility to take the real
time as fourth coordinate, which changes solely signs in the
formulae. But there keep imaginary components, so no difference
whether I put i in the coordinate or to metrics.
[ueb]
>> As well, that has nothing to do with complex numbers depictable with
>> Gauss' plane of numbers.
I must add that it refers to each several coordinate, and each tensor
component.
Chris Hillman <hil...@math.washington.edu> wrote:
> It is very natural to now try to develop the analogues in Minkowski
> numbers of the Cauchy-Riemann equations and Cauchy's integral theorem,
> Gauss's mean-value theorem, etc., which are formulation in the complex
> numbers, and more generally to develop the analogue of the theory of
> functions of one complex variable. This program quickly runs into
> obstacles which caused severe difficulties to people as smart as Lars
> Ahlfors (the Field's medalist). Again, Pertti Lounesto has on past
> occasions discussed some recent insights which appear to have clarified
> these difficulties, so I'll let him comment on that if he wishes.
I know such stuff regarding images (transform origin to complex image,
calculate with image, and back transform to origin - a popular method
in engineering), but I believe that might be not very relevant to
SR/GR. The mentioned analogues refer to two coordinates commonly,
when I correctly understood that. But I spoke always about several
coordinates.
Ulrich Bruchholz
www.markt-2000.de/patent