Problem of time in quantum field theory?

[TomServo]

TL;DR Summary

Does the "problem of time in quantum mechanics" go for Lorentz-invariant quantum mechanical theories like QED?

Summary: Does the "problem of time in quantum mechanics" go for Lorentz-invariant quantum mechanical theories like QED?

Everything I read about "the problem of time in quantum mechanics," i.e. absolute time in QM clashing with relativity's relative time coordinate and relativity of simultaneity, invokes non-relativistic QM to explain what the problem is. However, while I'm not that good at QFT I know that it is a Lorentz invariant theory, correct? (except for non-relativistic versions used in condensed matter systems) Thus QFT doesn't, or shouldn't, have an absolute time but rather an infinite number of Lorentz frames which are equivalent to each other as far as the laws of physics are concerned, but which have different time coordinates.

So what exactly is the problem? I'm assuming simultaneity has something to do with it but I can't quite put my finger on it. I'm not that up on QFT, does it in fact require an absolute time? Please help me.

 

[vanhees71]

There is no problem of time in quantum mechanics, neither in non-relativistic nor in relativistic theory. Relativistic QT is most conveniently formulated in terms of quantum field theory, QFT, since at relativistic collision energies usually particles get created and destroyed.

One heuristic way to formulate QFTs is to use "canonical quantization" starting from classical field theories based on an analysis of the unitary representations of (the covering group of) the proper orthochronous Lorentz group. To guarantee the correct causality structure, i.e., the linked-cluster principle, you find that you can use massive $p^2>0$ and massless $p^2=0$ representations and then build local fields. Together with the demand that energy should be bounded from below you get the spin-statistics theorem (half-integer-spin fields have to be quantized as fermions integer-spin fields as bosons), and microcausality follows, i.e., operators for local observables (energy, momentum, angular-momentum density, several charge- and current-densities) commute at spacelike separations of its arguments. This leads also to a Hamiltonian which implies the validity of the linked-cluster principle.

For details see

https://www.physicsforums.com/kbibtex%3Afilter%3Aauthor=Weinberg: https://www.physicsforums.com/kbibtex%3Afilter%3Atitle=The%20Quantum%20Theory%20of%20Fields , volume https://www.physicsforums.com/kbibtex%3Afilter%3Avolume=1, https://www.physicsforums.com/kbibtex%3Afilter%3Apublisher=Cambridge%20University%20Press, https://www.physicsforums.com/kbibtex%3Afilter%3Ayear=1995
https://www.physicsforums.com/kbibtex%3Afilter%3Aauthor=Duncan: https://www.physicsforums.com/kbibtex%3Afilter%3Atitle=The%20conceptual%20framework%20of%20quantum%20field%20theory , https://www.physicsforums.com/kbibtex%3Afilter%3Apublisher=Oxford%20University%20Press, https://www.physicsforums.com/kbibtex%3Afilter%3Ayear=2012

 

[A. Neumaier]

Everything I read about "the problem of time in quantum mechanics,"

You should let us know where you have read this, to provide the context necessary to understand your query.

 

[TomServo]

Well okay, but why do people say that there is a problem of time in QM if there isn't?

 

[TomServo]

You should let us know where you have read this, to provide the context necessary to understand your query.

Well today on arXiv was posted this pre-print:

https://arxiv.org/abs/1905.09860
And that's what got me thinking, although it's a notion I've seen here and there before (often in the popular press, which of course doesn't explain it). The first ten references seem to be relevant to defining the "problem."

 

[A. Neumaier]

Well today on arXiv was posted this pre-print:

This is about quantum gravity, not about quantum mechanics or quantum field theory! Quantum gravity is an area of physics where unsolved problems abound.

The ''problem of time'' is that in general relativity, time is not well-defined, as it depends on the curvilinear coordinate system used to coordinatize spacetime. If one considers spacetime itself as emergent (which is the case in some approaches to quantum gravity), there is the problem of how the notion of time emerges.

 

[A. Neumaier]

QFT, does it in fact require an absolute time?

Relativistic QFT has no absolute time; it is Lorentz invariant.

 

[dextercioby]

Well today on arXiv was posted this pre-print:

https://arxiv.org/abs/1905.09860
And that's what got me thinking, although it's a notion I've seen here and there before (often in the popular press, which of course doesn't explain it). The first ten references seem to be relevant to defining the "problem."

Thank you for the reference. This is a very short note with strong bibliography, which is actually the asset of the paper.

 

[arivero]

Is it the problem of not having an observable for time? This is turn is because the energy is a positive operator, on the other hand momentum has positive and negative values and then we have still a good operator for position.

I am not sure how the problem resolves itself in relativistic quantum mechanics. It could be said that the energy becomes negative too, or it could be argued that we also lose the position operator.

 

[vanhees71]

I think it's again in the sense of thinking not "quantum theoretically" enough. There's this example often envoked using a Mach-Zehnder interferometer tuned such that with two beam splitters in place one exit has no signal due to destructive interference and the conclusion that then there'd be no "which-way information" on the photon. Taking out the beam splitter then leads to 50:50 chance for the photon to end up in either detector and then it's concluded that you have which-way information after the detection of the photon in one of the detectors, i.e., you apparently draw conclusions about the "past of the photon".

This is of course thinking in a classical-wave or classical-particle picture about the photon and then in terms of the idea of "complementarity" or "wave-particle dualism" of old quantum mechanics. Particularly for photons that's utterly misleading. It's not even possible to define a proper position observable for the photon. So the "which-way information" doesn't make sense to begin with.

In quantum theory, in this case particularly QED, the state of a photon is given by a statistical operator, which includes the information about the probabilities for detection in one or the other of the detectors in the Mach-Zehnder interferometer setups. You can also interpret it as measuring the energy density of the electromagnetic field, being prepared in a single-photon state (in general not a pure but a mixed state of course). The corresponding detection probabilities are all you can know about the photon in this setup, and you can calculate the time evolution of these probabilities using the standard dynamics of QED. Of course, it doesn't refer to the time evolution of some observable since the observable may not be determined due to the preparation of the state of the photon at the beginning of the experiment.

 

[Auto-Didact]

The problem of time in QG is the problem that time in QM is itself a background dependent notion just like in Newtonian mechanics (i.e. you can evolve a wavefunction using the TDSE) in stark contrast to the notion of time in GR.

This conceptual inconsistency between these two different notions of time makes the direct quantization of GR mathematically intractable, if not impossible.

 

[Lynch101]

Summary: Does the "problem of time in quantum mechanics" go for Lorentz-invariant quantum mechanical theories like QED?

Summary: Does the "problem of time in quantum mechanics" go for Lorentz-invariant quantum mechanical theories like QED?

Everything I read about "the problem of time in quantum mechanics," i.e. absolute time in QM clashing with relativity's relative time coordinate and relativity of simultaneity, invokes non-relativistic QM to explain what the problem is.

There's a very comprehensive book on the subject called The Problem of Time: Quantum Mechanics versus General Relativity by Dr.Edward Anderson. He's also got a number of preprints on the subject (which appear to form the basis of the book). There's one titled:
Problem of Time and Background Independence: the Individual Facets

In the book he suggests that "the Problem of Time is, in greater generality, a consequence of the mismatch between Background Dependent and Background Independent Paradigms of Physics".

He goes on to say that it "has nine facets--closely following [the review by] Isham and Kuchaˇr--resulting from nine corresponding aspects of Background Independence".

However, while I'm not that good at QFT I know that it is a Lorentz invariant theory, correct? (except for non-relativistic versions used in condensed matter systems) Thus QFT doesn't, or shouldn't, have an absolute time but rather an infinite number of Lorentz frames which are equivalent to each other as far as the laws of physics are concerned, but which have different time coordinates.

So what exactly is the problem? I'm assuming simultaneity has something to do with it but I can't quite put my finger on it. I'm not that up on QFT, does it in fact require an absolute time? Please help me.

In their paper, The problem with ‘The Problem of Time’, Bryan and Medved say that "the Minkowski metric of quantum field theory is generally regarded as a mathematical construct and not a real physical object." This would seem to imply that the absolute notion of time is retained in QFT.

 

[Demystifier]

Summary: Does the "problem of time in quantum mechanics" go for Lorentz-invariant quantum mechanical theories like QED?

So what exactly is the problem?

There are several problems of time in quantum theory.

1. Time in non-relativistic QM: Time can be observed, but unlike other quantum observables it is not described by a hermitian operator. Why is time different?

2. Time in relativistic QM: The Klein-Gordon equation does not define a probability density in space conserved in time. A common interpretation of that is that there is no relativistic position operator, but then where does the position operator (in 1. above) come from in the non-relativistic limit?

3. Time in relativistic QFT: In the Schrodinger picture the state $|\psi(t)\rangle$ does not satisfy a Lorentz-covariant equation, so how exactly is relativistic QFT Lorentz covariant in that picture?

4. Time in canonical quantum gravity: The time-reparametrization invariance of classical gravity implies the Hamiltonian constraint $H=0$, which in the quantum case implies $H|\psi\rangle=0$. Combining it with the Schrodinger equation $H|\psi\rangle=i\partial_t|\psi\rangle$ implies that the state $|\psi\rangle$ does not depend on time. So where does the change of Nature with time come from?

All those problems have some proposed solutions, but neither of those solutions seems completely satisfying. People are still working on it.

 

[vanhees71]

Generally, time is not an observable in physics but a parameter that labels the causal order of events. You can infer about time only by measurements involving some observables (e.g., in classical mechanics the uniform rectilinear motion of a particle, by taking its position as a function of counting cycles of a pendulum or whatever more accurate clock you prefer).

In QT (at least for cases 1-3; I can't say anything about 4), because I've no clue about this branch of physics) the reason simply is that if time would be treated as an observable in the usual sense it should be the canonical conjugate to the Hamiltonian, which then wouldn't be bounded from below and we'd have no stable world we could live in and discuss about physics ;-)).

Ad 2. For massive particles you can construct position operators in standard QFT; you can't for massless particles with spin $\geq 1$ (i.e., in the real world for photons), but that's not a bug but a feature since it precisely describes how photons behave. Relativistic QM simply doesn't exist nor make it physical sense beyond the usual approximation in situations, where you are close to non-relativistic physics anyway.

Ad 3. That's not surprising since you break manifest Poincare covariance by just going to the Hamiltonian description. You don't, of course, really use Poincare covariance for observable quantities like S-matrix elements etc. Of course, it's often easier to keep everything manifestly covariant, e.g., by using the path-integral formalism and stick to the Lagrangian formulation (however only after carefully checking whether it's really applicable by first using the Hamltonian path-integral formalism!).

 

[Demystifier]

For massive particles you can construct position operators in standard QFT; you can't for massless particles with spin $\geq 1$ (i.e., in the real world for photons)

So can you write down the explicit construction of the position operator for the massless spin-0 particle? And what exactly goes wrong when you attempt to generalize it to spin-1?

 

[vanhees71]

See @A. Neumaier 's FAQ,

https://arnold-neumaier.at/physfaq/topics/position.html

and the references therein.

In which sense photons are "localized" though there's no position operator, see any good textbook on Quantum Optics. In lack of a position observable, not the photons themselves can be localized but there are only detection probabilities of a given detector which, as a massive object, can of course be localized. A good discussion can be found in

J. C. Garrison, R. Y. Chiao, Quantum Optics, Oxford Univ. Press (2008)

 

[Demystifier]

See @A. Neumaier 's FAQ,

https://arnold-neumaier.at/physfaq/topics/position.html

and the references therein.

Unfortunately, I don't quite understand the arguments in those references. What goes wrong if I simply propose that, for any mass and spin, the position operator in the momentum basis is
$$q^k=i\frac{\partial}{\partial p^k}$$
where $p^k$ are components of the 3-momentum?

 

[Demystifier]

I get it now. All the problems start from the Lorentz invariant measure
$$\frac{d^3p}{p_0}$$
The position operator defined as being proportional to the momentum derivative in the post above then creates an additional term due to the momentum derivative acting on $1/p_0$, making the position operator non-hermitian (that's nicely explained in the Schweber's QFT book). For that reason one must redefine the position operator by adding a compensating term that renders the position operator hermitian. It is this compensating term that depends on spin and mass and eventually makes it non-existent for zero mass and spin larger than $1/2$.

But there is a way out of this problem. Instead of using the Lorentz invariant measure above, one can use the canonical measure $d^3p$ as in non-relativistic QM. Sure, the position operator will not be Lorentz covariant, but as Newton and Wigner have shown, the position operator is not Lorentz covariant even when the Lorentz invariant measure above is used.

So what is lost by using the canonical measure $d^3p$? Newton and Wigner propose 4 axioms that a "reasonable" position operator should satisfy, called axioms (a), (b), (c) and (d). Their theorem (completed by others) says that there is no position operator satisfying all 4 axioms for zero mass and spin larger than $1/2$. The axioms (a), (b) and (c) are axioms which are valid also in non-relativistic QM, so they are not violated by taking the canonical measure $d^3p$. What is lost is the axiom (d), which is a certain regularity condition inspired by the requirement of Lorentz covariance. But since the resulting (Newton Wigner) position operator violates Lorentz covariance anyway, it does not seem that much more is lost when (d) is violated too. Hence it seems reasonable to give up the axiom (d) and construct the position operator basically as in non-relativistic QM. In this way the position operator becomes simple and does not depend on mass and spin. And the absence of Lorentz covariance makes perfect sense from the operational point of view, because the position operator is supposed to describe a situation in which the position of the particle is measured by an apparatus which defines a preferred Lorentz frame in which the apparatus is at rest.

To conclude, it is not correct to say that position operator does not exist for zero mass and spin larger than $1/2$. The correct statement is that position operator satisfying Newton-Wigner axioms does not exist for zero mass and spin larger than $1/2$. What I propose here is to give up one of those axioms (axiom (d)), in which case the position operator exists, does not depend on mass and spin, and is basically the same as in non-relativistic QM.

 

[Lynch101]

Relevant to the OP

Problem of localization in a quantum field theory. Schroedinger’s equation evolves wave- functions in a non-local way, so there seems to be a problem with superluminal propagation.
...
Quantum mechanics has one thing, time, which is absolute. But [special and] general relativity tells us that space and time are both dynamical so there is a big contradiction there.

https://www.perimeterinstitute.ca/research/conferences/convergence/roundtable-discussion-questions/what-are-lessons-quantum
square brackets are my own addition.

Current versions of quantum field theory do a fine job explaining how individual particles or small systems of particles behave, but they fail to take into account what is needed to have a sensible theory of the cosmos as a whole.

Lee Smolin Time Reborn p.142

Relativity gives nonsensical answers when you try to scale it down to quantum size, eventually descending to infinite values in its description of gravity. Likewise, quantum mechanics runs into serious trouble when you blow it up to cosmic dimensions. Quantum fields carry a certain amount of energy, even in seemingly empty space, and the amount of energy gets bigger as the fields get bigger. According to Einstein, energy and mass are equivalent (that’s the message of E=mc2), so piling up energy is exactly like piling up mass. Go big enough, and the amount of energy in the quantum fields becomes so great that it creates a black hole that causes the universe to fold in on itself. Oops.

Guardian article

The root of all the evil was clearly special relativity. All these paradoxes resulted from well-known effects such as length contraction, time dilation, or E=mc2, all basic predictions of special relativity. And all denied the possibility of establishing a well-defined border, common to all observers, capable of containing new quantum gravitational effects. Quantum gravity seemed to lack a dam—its effects wanted to spill out all over the place; and the underlying reason was none other than special relativity.

João Magueijo Faster than the Speed of Light p.250

despite the successes of quantum field theory, many physicists, beginning with Einstein, have wanted to go beyond it to a deeper theory that gives a complete description of each individual experiment--which, as we have seen, no quantum theory does. Their searches have consistently found an irreconcilable conflict between quantum physics and special relativity.

Lee Smolin Time Reborn p.142

 

[A. Neumaier]

Newton and Wigner propose 4 axioms that a "reasonable" position operator should satisfy, called axioms (a), (b), (c) and (d). Their theorem (completed by others) says that there is no position operator satisfying all 4 axioms for zero mass and spin larger than $1/2$. The axioms (a), (b) and (c) are axioms which are valid also in non-relativistic QM, so they are not violated by taking the canonical measure $d^3p$. What is lost is the axiom (d), which is a certain regularity condition inspired by the requirement of Lorentz covariance. But since the resulting (Newton Wigner) position operator violates Lorentz covariance anyway, it does not seem that much more is lost when (d) is violated too. Hence it seems reasonable to give up the axiom (d) and construct the position operator basically as in non-relativistic QM.

Newton and Wignenr's paper is here. Do you use their labeling of the axioms, from p.401? Or what is the axiom (d) that you are prepared to give up?

What I propose here is to give up one of those axioms (axiom (d)), in which case the position operator exists, does not depend on mass and spin, and is basically the same as in non-relativistic QM.

Which operator do you propose? Is it uniquely defined by (a)-(c)?
If yes, is it one of the Pryce operators? If no, why should Nature pick that particular one?

 

[Jimster41]

I get it now. All the problems start from the Lorentz invariant measure
$$\frac{d^3p}{p_0}$$
The position operator defined as being proportional to the momentum derivative in the post above then creates an additional term due to the momentum derivative acting on $1/p_0$, making the position operator non-hermitian (that's nicely explained in the Schweber's QFT book). For that reason one must redefine the position operator by adding a compensating term that renders the position operator hermitian. It is this compensating term that depends on spin and mass and eventually makes it non-existent for zero mass and spin larger than $1/2$.

But there is a way out of this problem. Instead of using the Lorentz invariant measure above, one can use the canonical measure $d^3p$ as in non-relativistic QM. Sure, the position operator will not be Lorentz covariant, but as Newton and Wigner have shown, the position operator is not Lorentz covariant even when the Lorentz invariant measure above is used.

So what is lost by using the canonical measure $d^3p$? Newton and Wigner propose 4 axioms that a "reasonable" position operator should satisfy, called axioms (a), (b), (c) and (d). Their theorem (completed by others) says that there is no position operator satisfying all 4 axioms for zero mass and spin larger than $1/2$. The axioms (a), (b) and (c) are axioms which are valid also in non-relativistic QM, so they are not violated by taking the canonical measure $d^3p$. What is lost is the axiom (d), which is a certain regularity condition inspired by the requirement of Lorentz covariance. But since the resulting (Newton Wigner) position operator violates Lorentz covariance anyway, it does not seem that much more is lost when (d) is violated too. Hence it seems reasonable to give up the axiom (d) and construct the position operator basically as in non-relativistic QM. In this way the position operator becomes simple and does not depend on mass and spin. And the absence of Lorentz covariance makes perfect sense from the operational point of view, because the position operator is supposed to describe a situation in which the position of the particle is measured by an apparatus which defines a preferred Lorentz frame in which the apparatus is at rest.

To conclude, it is not correct to say that position operator does not exist for zero mass and spin larger than $1/2$. The correct statement is that position operator satisfying Newton-Wigner axioms does not exist for zero mass and spin larger than $1/2$. What I propose here is to give up one of those axioms (axiom (d)), in which case the position operator exists, does not depend on mass and spin, and is basically the same as in non-relativistic QM.

but don’t you have a problem when two relativistic detectors are “competing” to detect (operate) the position of some propagator? In other words the same old problem reconciling QM and GR. What is it your proposal buys?“The position operator defined as being proportional to the momentum derivative” but I kind of see how that would be utterly circular.

 

[Demystifier]

Newton and Wignenr's paper is here. Do you use their labeling of the axioms, from p.401?

Yes.

Is it uniquely defined by (a)-(c)?

No, it depends also on the Lorentz frame with respect to which the apparatus is at rest.

... why should Nature pick that particular one?

Because an experimentalist (who is a part of Nature) have chosen to use an apparatus which is at rest with respect to that particular Lorentz frame.

 

[Demystifier]

but don’t you have a problem when two relativistic detectors are “competing” to detect (operate) the position of some propagator?

Yes, in that case the position operator does not make sense. Recall that the position operator (as well as other observables) is just an instrumental tool to describe probabilities of outcomes for certain well defined experimental procedures.

“The position operator defined as being proportional to the momentum derivative” but I kind of see how that would be utterly circular.

Recall that the momentum observable is generally considered to be well defined in relativistic QM and QFT. Hence I don't think it's circular.

 

[Jimster41]

Yes, in that case the position operator does not make sense. Recall that the position operator (as well as other observables) is just an instrumental tool to describe probabilities of outcomes for certain well defined experimental procedures.Recall that the momentum observable is generally considered to be well defined in relativistic QM and QFT. Hence I don't think it's circular.

Okay well on the first one I am glad it makes no sense because I am totally confused. But I will try to think of a way of carefully asking the question of how multiple observers in a curved space-time can work how, how microscopically space-time adjusts space and time at the point their light-cones meet or whatever to ...reconcile their respective laws of physics.

On momentum, fair enough. I was reacting to the way it sounds keeping in mind the basic definition of momentum as the product of mass and velocity, velocity being change in x over time. I mean how doesn’t the idea of the back-bootstrapping position more or less from its own temporal derivative - feel suspicious.

Aaaand as I go googling the QM def of momentum space and position space operators and regular old p=mv time seems to be what they all rest on... and the problem for our intuition in trying to imagine a machine that can run space-time itself.

I just don’t see how any microscopic QMGR theory can work unless it can construct and adjust time - out of something more fundamental i.e. figure out how time figures out what to...do, how it gets smoothed out.

If, at the point some Schwarzschild observers QM experiments meet we need a way of keeping time smooth, is it’s at the expense of velocity through length contraction? But we don’t experience contraction do we? We do experience time dilation but it’s smooth physics. So how specifically did time get... bent smoothly, microscopically? Mass maybe, aka energy. To which...

Started a book of QFT of many body systems. Trying really hard. It’s good. No, really. But I’ve already had to dig out my book on “e” and now maybe more interestingly Winfree’s epic on “The Geometry of Biological Time”. Because isn’t “Berry Phase” related in a way to his “resetting types”.

One of the coolest things I’ve read so far in that book (Winfree’s): Did you know that the circadian rhythms of some human beings who spent huge amounts of time in caves severely deprived of normal diurnal forcing turned out to be most correlated to of all things body temperature. And the experiments show this odd type of de-syncing with actual diurnal time - and responses that go like weird phase resetting - non-linear thresh-holds that have got to be - many body?

 

[Elias1960]

Does the "problem of time in quantum mechanics" go for Lorentz-invariant quantum mechanical theories like QED?

The problem of time is a problem of quantum gravity, not of quantum mechanics. Quantum mechanics is simply using good old Newtonian absolute space and time.

It does not follow that there are no incompatibility problems related with time between RQFT and QT. In particular, there is no time measurement in QT. Every clock has even a probability to go backward in QT time (which is the parameter used in the Schrödinger equation). In SR, time is already what clocks measure. That means it is something which has the probability of going backward in QT time.

But these problems are solvable in a quite simple way: Mark everything which has some compatibility problem as bad, and avoid even mentioning it. If what remains is sufficient to compute all observables, then everything is fine.

One thing thrown away is, in particular, the Schrödinger picture. The state (the wave functional) would have to be defined on a particular slice, the Schrödinger equation would have to evolve in that parameter t, maybe there would have to be even a collapse of that wave functional. In the Heisenberg picture, the situation looks much better. The operators one will measure in field theories are local field values, o(x), in the Heisenberg picture they gain the dependence on t so that we have now operators O(x,t) measuring o(x,t). Already much better. One can ignore now that measurements collapse the wave function at a given moment of time, thus, are global (in space) events, they are now simply local measurements measuring o(x,t). All that matters is that if two operators "localized" at space-like separated events commute. If they do, one can prove that no signals faster than light can be sent, and this is sufficient for preserving relativity.

(Moreover, one should not forget that not even these operators O(x,t) are really interesting. What matters are only limits in infinity because this is what is necessary for computing scattering amplitudes, which is what can be compared with observations in particle colliders. I have at one time tried to understand how gauge invariance is handled for experiments with finite distances. I was unable to identify any source which considered this - all that I have seen was happy to prove something for the scattering matrix and did not care about finite distances.)

So, my impression is that there are compatibility problems already for RQFT, but they are usually ignored. And, from a purely pragmatical point of view, this can be justified. It was difficult enough to get a reasonable theory able to compute the numbers for what can be measured in particle colliders. The incompatibility problems related to time can be, instead, left to quantum gravity researchers, where the problems are really much more serious.

 

[PGaccount]

The usual manifestly unitary quantum theory can be described either in the schrodinger or Heisenberg pictures. Both of these pictures are 'non-relativistic', because either the operators or the states evolve in time depending on the picture. The path integral formulation is manifestly relativistic, but in its present form is incomplete in the sense that unitarity is not manifest.

It is possible to interpret canonical quantum mechanics in a relativistic way if we consider the creation operators to be the 'string field theory' limit of quantum gravity.

a⁺ = x + ip creates a particle of momentum p. This state |p⟩ evolves in time. The corresponding relativistic notion would be a creation operator that creates a 'spacetime history', rather than a state that evolves in time. An example of a spacetime history is exp i(px - Et), which is characterized not by an x and a p, but by E and p. The creation operators satisfy

a⁺a^- = N

where N is the number of particles. This means that in a sense a ~ √N. This property of the creation/annihiliation operators must be preserved by a covariant version of these operators for the following reason. The difference between time and space means that particles should be considered to be 'packets of energy', but there is no notion of a 'packet of momentum'. Momentum is the flux of energy. Covariant theories treat time and space on an 'equal footing', but this is very subtle for the reason mentioned.

For example, in loop quantum gravity, the 'Wheeler-de Witt equation'

H|ψ⟩ = 0

Has de Sitter spacetime as a solution (ψ = e^{i(dA^A + A^A^A)}.) Where the term in the exponent is the action for Chern-Simons theory. This equation presumably corresponds to

(p² - m²)φ = 0

Where now p² - m² is in some sense a Hamiltonian, and φ is a spacetime history rather than a state that evolves in time.

Please feel free to ask any questions, as there are many details to include in one post.

 

[PGaccount]

One more thing to mention:

1/p² - m² = ∫ exp τ(p² - m²)

where the limits are 0 and ∞.

In this sense p² - m² is a Hamiltonian conjugate to the 'time' τ. (in string theory there is no difference between time and space.)

 

[Demystifier]

in string theory there is no difference between time and space.

What is that supposed to mean?

 

[PGaccount]

I think it is related to the fact that at the plank scale, the Casimir forces due to vacuum fluctuations and the interactions by virtual exchanges or electrostatic forces become of the same order of magnitude, and you cannot really distinguish between them.

The energy levels of a harmonic oscillator are given by (n + 1/2)hω where n is the number of particles. The vacuum energy or ground state energy 1/2 hω is presumably not visible in minkowski space because it is flat, but probably there should be some way to see Minkowski space as de Sitter space with a cosmological constant Λ, because the minkowski vacuum appears to be excited in rindler or accelerated coordinates. The cosmological constant is in a sense a global concept.

 

[weirdoguy]

I think it is related to the fact that at the plank scale, the Casimir forces due to vacuum fluctuations and the interactions by virtual exchanges or electrostatic forces become of the same order of magnitude, and you cannot really distinguish between them.

Sorry, but I don't think that any of that makes sense at all.

The question was: what does it mean that "in string theory there is no difference between time and space".

 

[PGaccount]

The open strings and closed strings are in some sense equivalent because of the equivalence between gauge theory and gravity. since the boundary and the bulk are of different dimension, space dimensions in the bulk could be equivalent to time dimensions on the boundary.

 

[Demystifier]

The open strings and closed strings are in some sense equivalent because of the equivalence between gauge theory and gravity. since the boundary and the bulk are of different dimension, space dimensions in the bulk could be equivalent to time dimensions on the boundary.

No, the bold part is not what the gauge/gravity correspondence tells.

 

[PGaccount]

The diffeomorphism constraint equation mentioned before is about gravity in 4 dimensional spacetime, but its solution in terms of Chern-Simons theory is 3 dimensional. So the time dimension is somehow present in the 3 dimensional solution representing de Sitter space.

 

[Auto-Didact]

^Anti-de Sitter, if anything. The extrapolation to de Sitter is still as far as I know an unsubstantiated conjecture, unless there have been serious results demonstrating otherwise.

 

[PGaccount]

The Hamiltonian 1/2 (p² - m²) is conjugate to the proper time τ. If instead we use an arbitrary parameter λ, then we must include a factor e(λ) called the 'einbein', and the action becomes

∫ dλ e/2 (p² - m²)

where e(λ) dλ = dτ. The inclusion of this factor makes the theory invariant under reparametrizations τ → λ. In general relativity, the motion of a particle in a gravitational field is independent of its mass. So the value of m is arbitrary in the lagrangian. This is analogous to the arbitrary character of e. In a sense, m = 1/e. In spacetime, we can introduce an einbein field j⁺. When we go from minkowski to general coordinates,

η → j⁺ j^-η

This j is to be considered as the canonical conjugate of a self-dual vector potential A⁺. The j or the A can be considered to be the gravitational field. In terms of these 'new variables', the Hamiltonain of general relativity is

C^- = ϵ j⁺ j^- F

This is called the Hamiltonian constraint. There is another quantity called the diffeomorphism constraint C````<sup>+</sup> = j<sup>-</sup> F. The equations C````⁺ = C````^- = 0 are satisfied by so-called 'physical' states, and belong to H_phys.