How active is research on other quantum gravity theories than loops or strings
EvT
and Peter Woit), both string theory and loop quantum
gravity have not made much progress recently.
How active are other approaches like noncommutative
geometry, euclidean quantum gravity, discrete
approaches (Lorentzian, Regge calculus, ...), twistor
theory, topos theory, supergravity, Ads/CFT, emerging
properties (Robert Laughlin)...?
______________________________________________________
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I.Vecchi
> According to some physicists (for instance John Baez
> and Peter Woit), both string theory and loop quantum
> gravity have not made much progress recently.
>
> How active are other approaches like noncommutative
> geometry, euclidean quantum gravity, discrete
> approaches (Lorentzian, Regge calculus, ...), twistor
> theory, topos theory, supergravity, Ads/CFT, emerging
> properties (Robert Laughlin)...?
>
Very interesting question, to which I would like to append my own.
Is there (or better, on the basis of the current knowledge can one make
an educated guess about) a common issue underlying the diffuculties of
the different theories?
And here is another volley.
In 1984 Wald wrote ([1]) "... consider a state of matter where , with
probability 1/2, all the matter is located in a certain region O1 of
spacetime and , with probability 1/2, the matter is located in a region
O2 disjoint from O1 ... Suppose now that we make a measurement of the
location of the matter. We then find the matter to be entirely in O1 or
in O2 ... after we have resolved the quantum state of the matter by
this measurement , then the gravitational field must change in a
discontinuous , acausal manner ... These difficulties apparently can be
avoided only by treating the space time metric in a probabilistic
fashion , i.e. by quantising the gravitational field".
My further questions are :
Is Wald's formulation of the problem still considered appropriate?
If yes, which of the currently fashionable theories of gravity provide
a concrete answer to the issue above?
Are there explicit (possibly simplified) models available?
Is the dearth of relevant experimental results due only to the fact
that we don't know how to put planets or stars into the quantum state
described by Wald and that with smaller objects the effects are too
small to measure?
Cheers,
IV
[1] R.M. Wald "General Relativity", 14.1
J. Horta
> My further questions are :
> Is Wald's formulation of the problem still considered appropriate?
> If yes, which of the currently fashionable theories of gravity provide
> a concrete answer to the issue above?
> Are there explicit (possibly simplified) models available?
> Is the dearth of relevant experimental results due only to the fact
> that we don't know how to put planets or stars into the quantum state
> described by Wald and that with smaller objects the effects are too
> small to measure?
>
> Cheers,
>
> IV
>
> [1] R.M. Wald "General Relativity", 14.1
I don't see how Wald's thought experiment is any different if formulated
in terms of say charge and electric field. In effect he's only
pointing out the effects of gravity propagate at finite speeds. Now
it is unclear to me that one may conclude that gravity must therefore be
quantized like any other field. Last I checked quantum theory is not a
consequence of relativity.
Igor Khavkine
> EvT ha scritto:
>
> > According to some physicists (for instance John Baez
> > and Peter Woit), both string theory and loop quantum
> > gravity have not made much progress recently.
> >
> > How active are other approaches like noncommutative
> > geometry, euclidean quantum gravity, discrete
> > approaches (Lorentzian, Regge calculus, ...), twistor
> > theory, topos theory, supergravity, Ads/CFT, emerging
> > properties (Robert Laughlin)...?
> >
>
> Very interesting question, to which I would like to append my own.
>
> Is there (or better, on the basis of the current knowledge can one make
> an educated guess about) a common issue underlying the diffuculties of
> the different theories?
There are two fundamentally different kinds of approaches, continuous
ones and discrete ones. I'd say that for any kind of discrete theory,
the biggest problem is showing that the classical limit of smooth
space-time can be recovered. While for continuous models, the biggest
problem is most likely the parametrization of the dynamical degrees of
freedom in a diffeomorphism invariant way (or equivalently, the
solution of constraints imposed by diffeomorphism invariance). Now,
these are technical problems. In general, there are always problems
with interpretation and recovering known physics in the limits of
various combinations of G, c, and hbar being small.
> And here is another volley.
> In 1984 Wald wrote ([1]) "... consider a state of matter where , with
> probability 1/2, all the matter is located in a certain region O1 of
> spacetime and , with probability 1/2, the matter is located in a region
> O2 disjoint from O1 ... Suppose now that we make a measurement of the
> location of the matter. We then find the matter to be entirely in O1 or
> in O2 ... after we have resolved the quantum state of the matter by
> this measurement , then the gravitational field must change in a
> discontinuous , acausal manner ... These difficulties apparently can be
> avoided only by treating the space time metric in a probabilistic
> fashion , i.e. by quantising the gravitational field".
>
>
> My further questions are :
> Is Wald's formulation of the problem still considered appropriate?
I see nothing wrong with Wald's thought experiment, extrapolating from
what we actually know.
> If yes, which of the currently fashionable theories of gravity provide
> a concrete answer to the issue above?
None as far as I know.
> Are there explicit (possibly simplified) models available?
> Is the dearth of relevant experimental results due only to the fact
> that we don't know how to put planets or stars into the quantum state
> described by Wald and that with smaller objects the effects are too
> small to measure?
Yes. The basic problem is that gravity is so many orders of magnitude
weaker than all the other forces. Direct experiments in a regime in
which neither gravity nor quantum mechanics can be neglected are rather
unlikely in the foreseeable future. The best we can hope for is some
sort of indirect evidence from astronomical observations or very large
particle colliders (like the LHC) where, hypothetically, miniature
black holes can be created. Either is yet to be seen (or conclusively
recognized).
> [1] R.M. Wald "General Relativity", 14.1
Igor
Igor Khavkine
> According to some physicists (for instance John Baez
> and Peter Woit), both string theory and loop quantum
> gravity have not made much progress recently.
>
> How active are other approaches like noncommutative
> geometry,
There are definitely a few people working on this. From what I
understand, not many are trying to construct a quantum theory of
gravity from some basic principles assuming noncommutative geometry
(although there are some, John Madore being an example). Instead,
people seek to express some sector or limit of an underlying theory
(like strings or loops) in terms of the language of noncommutative
geometry. This is how noncommutative field theory makes an appearance
in string theory or even in condensed matter physics.
> euclidean quantum gravity,
Don't know a whole lot about this, except that it's Stephen Hawking's
favorite way of looking at the problem.
> discrete approaches (Lorentzian, Regge calculus, ...),
The recent work on Lorentzian dynamical triangulations by Loll,
Ambjorn, and Jurkiewicz seems to be showing some promise like emergence
of a smooth large scale limit. I don't know that very many people
outside their group are working on it.
> twistor theory, topos theory, supergravity,
Sorry, don't know much about those.
> Ads/CFT,
Judging merely from the number of citations that Maldacena's paper has
gathered since its appearance, it is definitely an active area of
research. Unfortunately, I don't know enough about it to say whether
it's leading anywhere or not.
> emerging properties (Robert Laughlin)...?
These ideas are mostly favored by physicists with a condensed matter
background. There are definitely examples where features like Lorentz
invariance make an appearance as a low energy emergent phenomenon.
However, I have not seen any proposals of this sort of a significantly
greater sophistication. You can add a few more names to the list of
supporters of such ideas. These include Volovik, who's tried to say
something about the cosmological constant problem, and Xiao-Gang Wen,
who is trying to construct the standard model as an emergent system.
One approach that you haven't mentioned is that of causal sets. Again,
there is a small group of people working on such models centered around
Rafael Sorkin. From what I've seen, the main attraction of these models
is there generality (all you need is discreteness with a built-in
notion of causality) and potential for richness (as a theory). However,
definite results that can be thought of as progress toward quantum
gravity are in short supply.
Hope this helps.
Igor
Eugene Stefanovich
> > Is there (or better, on the basis of the current knowledge can one make
> > an educated guess about) a common issue underlying the diffuculties of
> > the different theories?
In my opinion, the fundamental problem facing modern theoretical physics is
the deep contradiction between quantum mechanics and Einsteinian
relativity (both special and general relativity). In Einsteinian relativity
space and time are interchangeable. In quantum mechanics,
position is an observable that depends on the state of the system, and time
is a numerical parameter. A big difference.
A consistent theory cannot keep these two conflicting views at the same
time.
Something's got to give. I think that eventually the QM approach to
time and position will prevail.
Eugene.
J. Horta
> I.Vecchi wrote:
> In my opinion, the fundamental problem facing modern theoretical physics is
> the deep contradiction between quantum mechanics and Einsteinian
> relativity (both special and general relativity). In Einsteinian relativity
> space and time are interchangeable. In quantum mechanics,
> position is an observable that depends on the state of the system, and time
> is a numerical parameter. A big difference.
>
> Eugene.
Is this strickly true in Quantum Field Theory? As far as I can tell
quantum fields are defined as operator valued distributions parameterized
by space and time. Space and time appear on an equal footing in QFT and
are not operators as say partical position x in (first quantized) QM.
I don't see the conflict.
Eugene Stefanovich
There is a conflict. Just try to follow this logic:
1. We are trying to describe observables (like position) in
multiparticle systems. This is true for both QM (=fixed number
of pareticles) and QFT (=variable number of particles).
2. Then we need to have a position operator R_i for each particle i in
the system. Then, the state of an n-particle system can be described
as a position-space wave function, i.e., a complex function
on the common spectrum of eigenvalues of n commuting operators R_i:
phi(r_1, r_2, ..., r_n)
3. Question: how these position operators are defined in QFT?
One thing I understand is that these position operators have nothing
to do with the quantum fields psi(x,t).
In QFT, quantum fields psi(x,t) are operator functions defined
on the 4D
Minkowski space-time. These operators act on the Fock space
(= Hilbert space for the variable number of particles). However,
quantum fields has no direct relationship to the wave function
phi(r_1, r_2, ..., r_n) of
the considered multi-particle system. The parameter x has no
relationship to the particles' positions. For example, it is wrong to
interpret x as eigenvalues of the position operator.
The Minkowski space-time on which the quantum fields are defined is an
abstract concept.
The only role of quantum
fields psi(x,t) is to provide "building blocks" for the interaction
operator in the Hamiltonian. It is explained well in Weinberg's vol. 1
how relativistically invariant interaction operator V in the Hamiltonian
H = H_0 + V can be built as x-integral of certain polynomials of quantum
fields. In this construction, the variable x plays a role of integration
variable, so it never shows up in any final answer.
The correct way to define particle position observables in each
n-particle sector of the Fock space is to use the non-interacting
representation of the Poincare group there. This representation
has the form
U_g^0 = U_g^1 (x) U_g^2 (x) ... (x) U_g^n
where U_g^i are irreducible representations of the Poincare group
related to each particle, and (x) is the tensor product sign
(add (anti)symmetrization whenever needed).
For each U_g^i one can find 10 generators P_i, J_i, K_i, H_i.
The Newton-Wigner position operator R_i for the particle i is then
constructed as a function of these generators. For example, for
a massless particle i
R_i = -1/2(K_i H_i^{-1} + H_i^{-1} K_i)
The Hermitian operator R_i directly corresponds to the observable
"position of the particle i in the n-particle system". There is no
operator of time "T_i" that can make a 4-th component of the
"position-time 4-vector" together with R_i.
Of course, the interpretation presented here is different from what you
can read in textbooks. However, this is the only way I can make sense of
QFT for myself.
Eugene.
Ilja Schmelzer
> In 1984 Wald wrote ([1]) "... consider a state of matter where , with
> probability 1/2, all the matter is located in a certain region O1 of
> spacetime and , with probability 1/2, the matter is located in a region
> O2 disjoint from O1 ... Suppose now that we make a measurement of the
> location of the matter. We then find the matter to be entirely in O1 or
> in O2 ... after we have resolved the quantum state of the matter by
> this measurement , then the gravitational field must change in a
> discontinuous , acausal manner ... These difficulties apparently can be
> avoided only by treating the space time metric in a probabilistic
> fashion , i.e. by quantising the gravitational field".
>
>
> My further questions are :
> Is Wald's formulation of the problem still considered appropriate?
I think it is appropriate.
> If yes, which of the currently fashionable theories of gravity provide
> a concrete answer to the issue above?
In gr-qc/0001101 I consider superpositions of gravitational fields in a
more specific thought experiment. My conclusion is that quantum gravity
needs a common background.
Ilja
Ilja Schmelzer
> > If yes, which of the currently fashionable theories of gravity provide
> > a concrete answer to the issue above?
>
> None as far as I know.
What do you think about gr-qc/0205035?
Ilja
J. Horta
Thanks for the reply. First I don't see the burning need for R_i operators
in the field theory case. Also, it is less clear that such operators have
an unambiguous definition given particle creation and annihilation. At low
energies where it makes sense to talk about a fixed number of particles
let me write the QM expression on the left and the corresponding QFT one
on the right. The QM 1 particle wave function is
<x|psi> = <0|phi(x)|psi>
where phi(x) is the annihilation operator and |psi> is a single particle
state vector. Wouldn't a location operator for this single particle be
something like
<x|x|psi> = <0|x phi(x)|psi> ?
For two particles we would have
<x_1,x_2|psi_1,psi_2> = <0| x_1 psi(x_1) x_2 psi(x_2) |psi_1,psi_2>
<snip>
Let me respond to the rest when I get longer to think.
Thanks
Paul C.
> Eugene.
Charles Francis
<vant...@yahoo.com> writes
>According to some physicists (for instance John Baez
>and Peter Woit), both string theory and loop quantum
>gravity have not made much progress recently.
>
>How active are other approaches like noncommutative
>geometry, euclidean quantum gravity, discrete
>approaches (Lorentzian, Regge calculus, ...), twistor
>theory, topos theory, supergravity, Ads/CFT, emerging
>properties (Robert Laughlin)...?
>
I have a paper currently in the hands of referees, so I suppose that is
as active as I can make it, using quantum logic and a relational
interpretation. As far as I know, no one has seriously developed this
approach since Von Neumann, though Rovelli has worked on it a bit.
Twistors gave way to spin networks which lead on to LQG as I recall. I
think some of this may be useful, ultimately; mostly spin network bit. I
don't think topos theory got very far. Chris Isham was working on it
with Jeremy Butterfield, but I don't think he is now.
Regards
--
Charles Francis
Igor Khavkine
> Thanks for the reply. First I don't see the burning need for R_i operators
> in the field theory case. Also, it is less clear that such operators have
> an unambiguous definition given particle creation and annihilation. At low
> energies where it makes sense to talk about a fixed number of particles
> let me write the QM expression on the left and the corresponding QFT one
> on the right. The QM 1 particle wave function is
>
> <x|psi> = <0|phi(x)|psi>
>
> where phi(x) is the annihilation operator and |psi> is a single particle
> state vector. Wouldn't a location operator for this single particle be
> something like
>
> <x|x|psi> = <0|x phi(x)|psi> ?
Right, although the notation could be a bit clearer:
<x|X|psi> = x<x|psi> = x <0| phi(x) |psi>.
Here X is the single particle position operator, with |x> and x being an
eigenvector an its correspondint egenvalue of X. Now, if you actually
want to calculate the expectation value of X you must do the following:
<psi|X|psi> = int <psi|x><x|X|psi> dx
= int x <psi|phi*(x)|0><0|phi(x)|psi> dx
= <psi| (int phi*(x)|0><0|phi(x) dx) |psi>
If you only care about |psi> being in the 1-particle subspace, then we
can replace |0><0| by a resolution of identity sum_n |n><n| over the
Fock space, since all the terms other than the one above will vanish
(why?). Then the expression becomes even simpler:
= <psi| (int phi*(x) x phi(x) dx) |psi>.
In other words, the position operator can be identified with
X = int phi*(x) x phi(x) dx.
> For two particles we would have
>
> <x_1,x_2|psi_1,psi_2> = <0| x_1 psi(x_1) x_2 psi(x_2) |psi_1,psi_2>
I think you meant
<x_1,x_2| X_1 X_2 |psi_1,psi_2> = ...
in the above.
Igor
J. Horta
Thanks for the affirmation and cleaning up the notation. Basically
what this suggests to me is the the role space-time plays in QFT is
very much the same as in classical field theory, as parameters.
Or am I missing some subtlety?
Paul C.
Eugene Stefanovich
>>Wouldn't a location operator for this single particle be
>>something like
>>
>> <x|x|psi> = <0|x phi(x)|psi> ?
>
>
> Right, although the notation could be a bit clearer:
>
> <x|X|psi> = x<x|psi> = x <0| phi(x) |psi>.
>
> Here X is the single particle position operator, with |x> and x being an
> eigenvector an its correspondint egenvalue of X. Now, if you actually
> want to calculate the expectation value of X you must do the following:
>
> <psi|X|psi> = int <psi|x><x|X|psi> dx
> = int x <psi|phi*(x)|0><0|phi(x)|psi> dx
> = <psi| (int phi*(x)|0><0|phi(x) dx) |psi>
>
> If you only care about |psi> being in the 1-particle subspace, then we
> can replace |0><0| by a resolution of identity sum_n |n><n| over the
> Fock space, since all the terms other than the one above will vanish
> (why?). Then the expression becomes even simpler:
>
> = <psi| (int phi*(x) x phi(x) dx) |psi>.
>
> In other words, the position operator can be identified with
> X = int phi*(x) x phi(x) dx.
This does not qualify as a position operator even in the 1-electron
subspace. You wrote
|x> = phi*(x)|0>
This formula implies that phi*(x) is a creation operator
for the electron with definite position x. This is not true.
The Dirac field phi*(x) cannot be interpreted in this way.
Eugene.
Igor Khavkine
> Igor Khavkine wrote:
> > In other words, the position operator can be identified with
> > X = int phi*(x) x phi(x) dx.
>
> This does not qualify as a position operator even in the 1-electron
> subspace. You wrote
>
> |x> = phi*(x)|0>
>
> This formula implies that phi*(x) is a creation operator
> for the electron with definite position x. This is not true.
> The Dirac field phi*(x) cannot be interpreted in this way.
It is indeed interpreted this way. I've already explained this in the
past. But you don't have to believe me in particular. These people say
the same thing as well:
Dirac, _Principles of QM_, Sections 59-65
Peskin & Schroeder, _Intro to QFT_, Section 2.3,
especially around Equation 2.41
Mahan, _Many-particle Physics_, Chapter 1
Abrikosov et al., _QFT Methods in Statistical Physics_, Chapter 2,
Section 6
There are many others, these are the ones I could check off hand. Be
careful when you make unsubstantiated statements, especially if they
are wrong. It goes against your credibility.
Igor
mark...@yahoo.com
> How active are other approaches like noncommutative
> geometry, euclidean quantum gravity, discrete
> approaches (Lorentzian, Regge calculus, ...), twistor
> theory, topos theory, supergravity, Ads/CFT, emerging
> properties (Robert Laughlin)...?
AdS/CFT is an approach to supergravity and string theory is widely
regarded as the quantization of supergravity; all 3 are bundled
together under the same header, along with Super Yang Mills. They're
not separate approaches.
Loop quantum gravity -- more properly named the "loop approach to GAUGE
theory" (not gravity theory) -- is orthogonal to all these. All 4
combinations can be conceived {no loop, yes loop} x {no
string/supergravity/AdS/CFT, yes string/supergravity/AdS/CFT}. If
there's a gauge theory anywhere, it can be loopized.
Noncommutative geometry is orthogonal to all of these. So, for the
most part, all 8 combinations can be conceived.
Twistors emerge naturally from string theory -- a point made by Witten
in a recent journal article this year.
Eugene Stefanovich
> Eugene Stefanovich wrote:
>
>>Igor Khavkine wrote:
>
>
>>>In other words, the position operator can be identified with
>>>X = int phi*(x) x phi(x) dx.
>>
>>This does not qualify as a position operator even in the 1-electron
>>subspace. You wrote
>>
>>|x> = phi*(x)|0>
>>
>>This formula implies that phi*(x) is a creation operator
>>for the electron with definite position x. This is not true.
>>The Dirac field phi*(x) cannot be interpreted in this way.
>
>
> It is indeed interpreted this way. I've already explained this in the
> past. But you don't have to believe me in particular. These people say
> the same thing as well:
>
> Dirac, _Principles of QM_, Sections 59-65
>
> Peskin & Schroeder, _Intro to QFT_, Section 2.3,
> especially around Equation 2.41
>
> Mahan, _Many-particle Physics_, Chapter 1
>
> Abrikosov et al., _QFT Methods in Statistical Physics_, Chapter 2,
> Section 6
>
> There are many others, these are the ones I could check off hand.
My statement was about relativistic electron-positron quantum field
psi(x).
Mahan's and Abrikosov's books are about non-relativistic QFT in
condensed matter physics, so they are not relevant for this discussion.
I couldn't find Peskin & Schroeder in my library.
There is no discussion of electron-positron field in sections 59-65
of the Dirac's books. I think, section 78 is more relevant. I don't know
if this section was present in the first (1930) edition of the book,
but certainly it appeared before Wigner's theory of irreducible
representations (1939). Dirac's presentation of the electron-positron
field is rather cumbersome. For one thing, his momentum-space
creation-annihilation operators of electrons and positrons have
4 components, while there are only two spin degrees of freedom for
these particles. Dirac realizes that and introduces projection
operators to eliminate extra components. This is not my ideal of
clear physical interpretation.
If you want a crisp clear introduction to the construction of
relativistic quantum fields (both motivation and derivation), I can
recommend chapter 5 in Weinberg's "The quantum theory of fields",
vol. 1. The Dirac's field psi(x,t) is written in eq. (5.5.34) there.
It is a linear combination of electron creation operators and
positron annihilation operators (note that these operators depend
on true quantum numbers: momentum p and 2-component spin sigma).
The field psi(x,t) itself is a 4-component quantity, so its
interpretation as creation operator of the electron at point x
doesn't look plausible to me. Such interpretation would imply that
sum_{sigma} u_l (p, sigma) exp(ipx -i omega_p t)
is the momentum-space wavefunction for a state localized at point x.
This is not correct.
Besides simply the number of components, there is another reason
why psi(x,t) cannot be interpreted as an operator analog of
the position-space wave function. As you know, in QM wave functions
are defined on the sets of eigenvalues of mutually commuting operators.
Boost transformations of the Dirac's field mix x and t arguments
psi(x,t) --> L psi(x', t') (1)
where L is a 4x4 matrix, and x', t' are related to x,t by Lorentz
formulas. In QM this implies
existence of the operator of time T (whose eigenvalue is t) that
commutes with the operator of position X, but does not commute with
the generator of boost.
In QM, there is no operator of time. t is just a numerical parameter,
and correct boost transformation of the position-space wavefunction
must be
psi(x,t) --> L' psi(x'', t) (2)
Of course, L' and x'' in (2) are different from L and x' in (1).
For free relativistic particles, position-space wave functions
with correct number of components and correct transformation
laws (1) can be easily defined. I would recommend you to read chapter 7
of my book, but I doubt that you'll follow my advice.
Eugene.
I.Vecchi
..
> Twistors emerge naturally from string theory -- a point made by Witten
> in a recent journal article this year.
Maybe you are referring to [1], maybe not.
Anyways, such "natural emergence" reminds me of the marvellous
explanatory power of Aristotelian science.
I wonder whether there is ANYTHING that does not arise naturally from
string theory.
IV
[1] http://www.arxiv.org/abs/hep-th/0403199
-------------------------
"In that Empire the Art of Cartography attained such Perfection that
the map of one Province covered one City and the map of the Empire
covered one Province. With time, those immense Maps were not considered
satisfactory anymore and the College of Cartographers raised a Map of
the Empire which was as extended as the Empire itself and coincided
exactly with it."
"En aquel Imperio, el Arte de la Cartografía logró tal Perfección
que el mapa de una sola Provincia ocupaba toda una Ciudad, y el mapa
del Iimperio toda una Provincia. Con el tiempo, esos Mapas Desmesurados
no satisfacieron y los Colegios de Cartógrafos levantaron un Mapa del
Imperio, que tenía el tamaño del Imperio y coincidía puntualmente
con él."
Jorge Luis Borges. 'Del rigor de la ciencia,' in El Hacedor.
[1] http://www.arxiv.org/abs/hep-th/0403199
Igor Khavkine
> Igor Khavkine wrote:
> >>You wrote
> >>
> >>|x> = phi*(x)|0>
> >>
> >>This formula implies that phi*(x) is a creation operator
> >>for the electron with definite position x. This is not true.
> >>The Dirac field phi*(x) cannot be interpreted in this way.
> >
> > It is indeed interpreted this way. I've already explained this in the
> > past. But you don't have to believe me in particular. These people say
> > the same thing as well:
> >
> > Dirac, _Principles of QM_, Sections 59-65
> >
> > Peskin & Schroeder, _Intro to QFT_, Section 2.3,
> > especially around Equation 2.41
> >
> > Mahan, _Many-particle Physics_, Chapter 1
> >
> > Abrikosov et al., _QFT Methods in Statistical Physics_, Chapter 2,
> > Section 6
> >
> > There are many others, these are the ones I could check off hand.
>
> My statement was about relativistic electron-positron quantum field
> psi(x).
While my statement was about the field operators in any field theory.
> Mahan's and Abrikosov's books are about non-relativistic QFT in
> condensed matter physics, so they are not relevant for this discussion.
> I couldn't find Peskin & Schroeder in my library.
QFT has a range of application much wider than just QED. And in all
instances, its application is the same. This includes relativistic,
non-relativistic and condensed matter applications.
> There is no discussion of electron-positron field in sections 59-65
> of the Dirac's books. [...]
That is not why I pointed to those sections. They contain the essense
of second quantization, the translation between wave functions and
field operators. The field operators serve precisely the role of
creation/annihilation operators.
> Besides simply the number of components, there is another reason
> why psi(x,t) cannot be interpreted as an operator analog of
> the position-space wave function.
>From what you've said, I can only conclude that your definition of wave
function is not the same as other people's. So if you want to
contradict what I wrote at the top of this post, you'd have to specify
what you are actually contradicting.
> As you know, in QM wave functions
> are defined on the sets of eigenvalues of mutually commuting operators.
> Boost transformations of the Dirac's field mix x and t arguments
>
> psi(x,t) --> L psi(x', t') (1)
>
> where L is a 4x4 matrix, and x', t' are related to x,t by Lorentz
> formulas. In QM this implies
> existence of the operator of time T (whose eigenvalue is t) that
> commutes with the operator of position X, but does not commute with
> the generator of boost.
>
> In QM, there is no operator of time. t is just a numerical parameter,
> and correct boost transformation of the position-space wavefunction
> must be
>
> psi(x,t) --> L' psi(x'', t) (2)
>
> Of course, L' and x'' in (2) are different from L and x' in (1).
Of course they are not the same. L and L' differ by a judicious
application of time translation, which is accomplished by the
Hamiltonian. There is no contradiction here.
Igor
Juan R.
> >>The Dirac field phi*(x) cannot be interpreted in this way.
you say is completely right. There is NOT position quantum operator in
RQFT. The imposibility for defining x in a relativisitc framework was
already proven by Landau and is the reason that RQM was abandoned and
substituted by RQFT where x is a parameter, NOT an observable.
Of course, this does RQFT incompatible with NRQM where x is an
observable and can be measured. This incompatibility was already noted
by Dirac. In fact, Dirac CLEARLY states that RQFT IS wrong due its
incompatibility with NRQM.
Dirac. Mathematical Foundations of Quantum Theory (Academic Press, Inc,
1978)
That RQFT is incompatible with NRQM is very well known for anyone who
studied RQFT, RQM, and QM with a few of detail.
I agree with you in that we need a NEW formulation of relativistic
quantum mechanics (already stated by Dirac in his above work, and even
partially predicted by Landau if one interprets his 'reject' of RQFT),
however i do not agree with many details of your specific theory.
In some time i will begin a new topic on this here.
*********************************************
Regarding research in quantum gravity all hyphotesis proposed UNTIL
TODAY are in a 'dead end'. Absolutely no significative advance has been
done.
None of alternatives to LQG and ST have provided significant advances.
NC geometry does not work, since there is not link between the
noncommutativity and quantum character, in fact, mathematicians call
'quantum' that physicists call 'classic'.
Penrose theory does not advance, other theories do not advance. Only
recently triangulation has achieved new advances in the study of
microscopic structure of spacetime. Curiously, models predict
REDUCCTION of dimensionality to 2, which contradices the increase
'predicted' by ST.
However, it is really difficult believe that triangulation models or
similar will be a quantum theory of gravitation (however, could sure
provide some useful thecniques).
Juan R.
Center for CANONICAL |SCIENCE)
Juan R.
> markw...@yahoo.com ha scritto:
>
> ..
>
> > Twistors emerge naturally from string theory -- a point made by Witten
> > in a recent journal article this year.
>
> Maybe you are referring to [1], maybe not.
> Anyways, such "natural emergence" reminds me of the marvellous
> explanatory power of Aristotelian science.
> I wonder whether there is ANYTHING that does not arise naturally from
> string theory.
>
> IV
>
> [1] http://www.arxiv.org/abs/hep-th/0403199
> -------------------------
Well, the term 'arises naturally' has, in string theory community, a
different connotation that in the rest of scientific commuities. In the
rest of science, 'arises naturally' means that can be derived from
first principles on any underlying theory. In the string world, means
other thing. In the own words of string theorists Seiberg:
"string theorists are arrogant enough that whatever comes up in their
research, they will call it string theory."
page 6 of
http://www.canonicalscience.com/stringcriticism.pdf
In
http://www.math.columbia.edu/~woit/blog/archives/000161.html
this quote is incorrectly attributed to string theorist Maldacena. But
Peter Woit corrects this below.
It is clear that Twistor theory has been introduced in recent versions
of string theory, and thus, it now 'arises naturally'...
Hontas F. Farmer III
> According to some physicists (for instance John Baez
> and Peter Woit), both string theory and loop quantum
> gravity have not made much progress recently.
>
> How active are other approaches like noncommutative
> geometry, euclidean quantum gravity, discrete
> approaches (Lorentzian, Regge calculus, ...), twistor
> theory, topos theory, supergravity, Ads/CFT, emerging
> properties (Robert Laughlin)...?
>
On a scale of 1-10. Where 1 is almost no activity, and ten
would have everyone researching the subject. I would say
alternative theories of gravity are at 3 but rising rapidly.
I say this based on:
The number of proposals for new theories
of gravity seen on this group.
The number of proposed alternative theories that have been published
in recognized journals.
The growing dissatisfaction with Strings and Loop inability to
deliver results after all this time.
In short pursue an alternative theory of quantum gravity as a future
career. The future belongs to something new.
John Baez
EvT <vant...@yahoo.com> wrote:
>According to some physicists (for instance John Baez
>and Peter Woit), both string theory and loop quantum
>gravity have not made much progress recently.
>
>How active are other approaches like noncommutative
>geometry, euclidean quantum gravity, discrete
>approaches (Lorentzian, Regge calculus, ...), twistor
>theory, topos theory, supergravity, Ads/CFT, emerging
>properties (Robert Laughlin)...?
Ultimately what matters most is not whether an approach
is "active", but whether it's getting somewhere. A big
bandwagon can make a lot of noise just by spinning its wheels
in the mud.
As far as I'm concerned, the one approach that's making
the most progress now is Causal Dynamical Triangulations,
which is a variant of the Regge calculus.
Not many people are working on this yet, in part because
it requires computer simulations, and most researchers
in quantum gravity still prefer pencil-and-paper work.
But, the results so far are impressive. They've numerically
simulated quantum gravity, and found something surprising:
their spacetimes act 4-dimensional at large scales but
2-dimensional at small scales!
The three main people working on Causal Dynamical Triangulations
are Ambjorn, Jurkiewicz and Loll. Here's a nice simple review
article:
http://arxiv.org/abs/hep-th/0509010
and here's a more technical one:
http://arxiv.org/abs/hep-th/0505154
Sophisticated work on perturbative quantum gravity by Lauscher,
Reuter and others adds evidence for this idea that quantum gravity
makes spacetime effectively 2-dimensional at short distance scales.
For a review with lots of references, try:
http://arxiv.org/abs/hep-th/0508202
So, technically speaking, the old problem of the nonrenormalizability
of quantum gravity may be solved by an ultraviolet fixed point of
surprising kind!
Of course I'm optimistic that this 2d small-scale behavior
is ultimately due to a spin foam model: imagine a bunch of
"soap bubbles" (2d surfaces) forming a "spacetime foam" that
mimics a 4d continuum at length scales much larger than the
Planck length. But, this is just speculation at this point.
I hope there will be some discussion about this idea when I
talk about it at Loops '05 next week, where Loll will also be
speaking:
thomas_l...@hotmail.com
> the most progress now is Causal Dynamical Triangulations,
> which is a variant of the Regge calculus.
> Not many people are working on this yet, in part because
> it requires computer simulations, and most researchers
> in quantum gravity still prefer pencil-and-paper work.
> But, the results so far are impressive. They've numerically
> simulated quantum gravity, and found something surprising:
> their spacetimes act 4-dimensional at large scales but
> 2-dimensional at small scales!
I tend to think about CDT as an experimental rather than
theoretical approach. In the absense of real experiments in quantum
gravity, we have to settle for numerical ones. This is a situation
quite familiar in statistical physics, where real experiments are
possible, but computer experiments are often of higher quality.
Already there seems to be at least one qualitative question which
CDT has answered, more or less conclusively: whether causality
holds strictly or not. There are two ways to quantize the metric:
summing over metrics which are compatible with spacetime being a
cylinder, and summing over all metrics irrespective of topology.
AJL makes the distinction between Lorentzian vs Euclidean
quantization; one could also talk about canonical vs path-integral
quantization. The two agree in flat space, but it seems that they
differ in QG, basically because the path-integral includes
trajectories that move backwards in time. If they differ, then both
cannot be right (both could be wrong, of course). CDT seems to say
that canonical quantization is the right choice.
Some time ago I looked at another of AJL's papers,
http://www.arxiv.org/abs/hep-lat/9909129 , in which they dealt
with the 2D Ising model coupled to 2D gravity. In Euclidean
quantization, the critical exponents change (to those of the spherical
model, I think), but in Lorentzian quantization they stay at the
Onsager values that they have in flat space. This is highly desirable,
IMO. The Ising model is used for various phenomena in condensed matter
physics, i.e. for systems completely governed by electromagnetism. If
there is one thing we know for sure about QG, it is that it is
unimportant compared to EM, the whole idea that we can ignore gravity
in condensed matter is based on that. If QG would modify critical
exponents, then it would not be possible to ignore gravity, which
would invalidate most of 20th century physics.
I'm less optimistic about CDT's prospects at producing quantitative
results in our lifetime. The reason for this pessimism is that
lattice gauge theory has only started to produce high-quality data
over the last five years or so, and I am unsure whether LGT
calculations are really used by experimentalists, more than as a
test of the feasibility of the method itself. There is no reason
to expect that simulations in the much less understood QG will
proceed faster.
I.Vecchi
>
> In short pursue an alternative theory of quantum gravity as a future
> career. The future belongs to something new.
I agree, but the relevant question is where to look.
>From my rather touristy viewpoint I find it perplexing that the
measurement problem is not widely recognised as crucial for any theory
of quantum gravity (it isn't, is it?). I read the Wald quote in [3] as
telling us that space-time metrics are really big and nasty cats. If we
do not know how to handle superposed cats (and observers) I don't see
how we can talk meaningfully about quantum GR.
On the other side, among the people whose chosen job is to tackle the
measurement problem, David Mermin, hearing that "photons are clicks in
photon counters" (widely attributed to Anton Zeilinger), asks "[are]
electrons ... clicks on electron counters? Are fullerene clicks on
fullerene counters? Is Anton a click in an Anton counter?" ([2]).
My answer is an enthusiastic "Yes, of course!", but I doubt it will go
down well with the GR tribes.
Cheers,
IV
[1] Conceptual and epistemic issues seem to be widely ignored by the GR
mainstream, although some exceptionally smart AND bold people (Wald ,
Hawking, Rovelli among others) appear well-aware of them. A guy who's
been consistently looking for a solution starting from conceptual
fundamentals is Isham. While I dislike his foundational choices, he
seems at least fully aware of their importance and his surveys provide,
in my opinion, a very valuable thread through the current conceptual
fog.
[2] http://xxx.lanl.gov/abs/quant-ph/0505187
[3] http://groups.google.com/group/sci.physics.research/msg/83ff473481ceb6ce
----------------------------------------
"To a kid with a hammer, everything looks like a nail"
Tim Josling
> measurement problem, David Mermin, hearing that "photons are clicks
> in photon counters" (widely attributed to Anton Zeilinger), asks
> "[are] electrons ... clicks on electron counters? Are fullerene clicks
> on fullerene counters? Is Anton a click in an Anton counter?" ([2]).
> My answer is an enthusiastic "Yes, of course!", but I doubt it will go
> down well with the GR tribes. Cheers, IV
So, what is an Anton counter? Is it a click in an "Anton counter
counter". So, an Anton counter counter would be? Perhaps a click on an
Anton counter counter counter. And so on. There seems to be an element
of circularity in all this.
Tim Josling
I.Vecchi
> So, what is an Anton counter?
>Is it a click in an "Anton counter counter". So, an Anton counter counter
>would be? Perhaps a click on an Anton counter counter counter. And so on.
>There seems to be an element of circularity in all this.
A click is what you hear or see. The buck stops at the observer.
However, you rightly point out the underlying semantic problem.
Physical "objects", from photons onward, are construed by semantic
frameworks.
I added the Mermin quote to stress that even in standard quantum theory
the measurement issue is far from settled. I am not sure that we need a
full-fledged semantic quantum theory in order to tackle quantum
gravity, but the problem of what kind of measurement
outcomes/perceptions correspond to the fancy mathematical objects that
GR theories have been churning out is pretty much there.
Cheers,
IV