What is the correct explanation of the Casimir force?

[A. Neumaier]

It should be possible to express any correlation function in both the Schrödinger and the Heisenberg picture.

It can be expressed but it looks very unnatural (one has to distinguish one of the times) and has no longer an obvious interpretation as a time correlation.

The meaning is visible only in the Heisenberg picture. In this sense, the pictures are not equivalent but the Heisenberg picture is superior. Lack of covariance is another reason why the Schroedinger picture is inferior in relativistic quantum field theory.

 

[DarMM]

It should be possible to express any correlation function in both the Schrödinger and the Heisenberg picture. The one is just a time-dependent unitary transformation of the other.

In some QFTs they are not unitary transformations of each other, see Luscher's paper I cited above.

 

[rubi]

Could you give me a pointer to some place that explains this? I haven't come across this before.

The example I gave is discussed in Claus Kiefer's book on quantum gravity for example (chapter 3). It's an application of Dirac's constraint algorithm, which is described in:
- Dirac, "Lectures on QM"
- Henneaux, Teitlboim "Quantization of Gauge Systems"
- Tyutin, "Quantization of Fields with Constraints"

Do you know a reference where Heisenberg picture for canonical quantum gravity is formulated explicitly?

I'm not sure what you mean by explicitly, but for instance in canonical LQG, the quantization programme has been carried out up to the point where you need to solve the Hamiltonian constraint. (This is a very difficult problem, since it is the quantum version of finding the general solution to Einstein's field equations, which isn't possible on the classical side either.) Unfortunately, we don't know many observables that could be studied. The only known Dirac observables are the 10 asymptotic Poincare generators in the asymptotically flat case. In the quantum theory, they would constitute Heisenberg operators. Thiemann's book "Modern canonical quantum general relativity" is a nice reference for this stuff.

 

[Demystifier]

I'm not sure what you mean by explicitly, but for instance in canonical LQG, the quantization programme has been carried out up to the point where you need to solve the Hamiltonian constraint. (This is a very difficult problem, since it is the quantum version of finding the general solution to Einstein's field equations, which isn't possible on the classical side either.) Unfortunately, we don't know many observables that could be studied. The only known Dirac observables are the 10 asymptotic Poincare generators in the asymptotically flat case. In the quantum theory, they would constitute Heisenberg operators. Thiemann's book "Modern canonical quantum general relativity" is a nice reference for this stuff.

My problem is that I usually think of LQG, Wheeler-DeWitt, and similar approaches as theories in the Schrodinger picture. But given the Hamiltonian constraint, I guess this is actually the same as Heisenberg picture.

 

[rubi]

My problem is that I usually think of LQG, Wheeler-DeWitt, and similar approaches as theories in the Schrodinger picture. But given the Hamiltonian constraint, I guess this is actually the same as Heisenberg picture.

Since the Hamiltonian vanishes on the constraint surface, it doesn't generate physical time evolution, but rather just gauge transformations (off-shell). Physical time is a relational concept. One usually uses material reference systems to define the concept of physical time and defines relational observables (see chapter 2 in Thiemann's book). These are to be interpreted as Heisenberg observables and they don't arise from the time evolution given by the Hamiltonian.

 

[Demystifier]

Since the Hamiltonian vanishes on the constraint surface, it doesn't generate physical time evolution, but rather just gauge transformations (off-shell). Physical time is a relational concept. One usually uses material reference systems to define the concept of physical time and defines relational observables (see chapter 2 in Thiemann's book). These are to be interpreted as Heisenberg observables and they don't arise from the time evolution given by the Hamiltonian.

How about the Heisenberg equation of motion of the form
$$\dot{g}=i[H,g]$$
where $H$ is the Hamiltonian and $g$ is some gravitational observable? Does it make sense when Hamiltonian vanishes on-shell?

My current opinion (which I have been changing several times) is that time-diff symmetry should not be taken seriously and gauge should be fixed before quantization. In this way one obtains a $H$ which does not vanish on-shell, so there is no problem of time. The problem, of course, is that such a procedure is far from being unique, but still ...

 

[A. Neumaier]

when Hamiltonian vanishes on-shell?

In that case H=0on the physical Hilbert space and all $g$'s are constant in the Hamiltonian time defined by your equation. The physical time is coordinate-dependent, hence not given by your Heisenberg dynamics. Except if you redefine $H$ to be something frame-dependent.

 

[rubi]

How about the Heisenberg equation of motion of the form
$$\dot{g}=i[H,g]$$
where $H$ is the Hamiltonian and $g$ is some gravitational observable? Does it make sense when Hamiltonian vanishes on-shell?

I'm using the phrase "Heisenberg picture" in the sense that the time dependence is in the observables, not in the state. If you have a set of observables $g(\tau)$, parametrized by some material reference system, then it is not clear that there exists an operator $H_{phys}$ such that $\dot g = i[H_{phys},g]$ and if it exists, then it is not given by the canonical Hamiltonian $H$ (which vanishes).

(However, there are some deparametrized models, in which a physical Hamiltonian exists.)

 

[Demystifier]

Another point is that a similar problem does not appear in classical gravity, when commutators are replaced by Poisson brackets. A long time ago that motivated me to propose a more "classical" method of quantization
http://arxiv.org/abs/gr-qc/0312063
The main weakness of this proposal was that one has to give up linearity (and hence superposition principle) of quantum mechanics.

 

[A. Neumaier]

a similar problem does not appear in classical gravity, when commutators are replaced by Poisson brackets.

In classical gravity, the problem reappears as singularity theorems. The quantum analogue of this would be that it may be impossible to define (in a givenn coordinate system defining an observer frame) a self-adjoint Hamiltonian - since this implies existence at all times.

 

[rubi]

Another point is that a similar problem does not appear in classical gravity, when commutators are replaced by Poisson brackets.

The situation is completely analogous in classical GR. You can introduce matter reference frames already on the classical side, see for example http://arxiv.org/abs/gr-qc/9409001 . The classical canonical Hamiltonian also vanishes on the constraint surface. In a time parametrization invariant system, the canonical Hamiltonian is also not expected to generate physical time evolution. You can also reformulate ordinary QM as a constraint system and then find the physical Hilbert space and recover the physical Hamiltonian. In that case, the canonical Hamiltonian doesn't generate physical time evolution either. (The book by Kiefer that I quoted earlier discusses this for example.)

 

[Demystifier]

In classical gravity, the problem reappears as singularity theorems. The quantum analogue of this would be that it may be impossible to define (in a givenn coordinate system defining an observer frame) a self-adjoint Hamiltonian - since this implies existence at all times.

I don't think that these problems are related. You can have various time-reparametrization invariant theories with vanishing Hamiltonian, the solutions of which do not lead to singularities.

 

[A. Neumaier]

I don't think that these problems are related. You can have various time-reparametrization invariant theories with vanishing Hamiltonian, the solutions of which do not lead to singularities.

But yours is a different Hamilonian - the canonical one, which vanishes on physical states hence generates no evolution at all. The time generating Hamiltonian is necessarily frame dependent since the notion of time is.

 

[Demystifier]

I think you missed my point.

 

[Demystifier]

If I had any doubts when I wrote the post above, now I don't. The argument in
http://lanl.arxiv.org/abs/1605.04143
convinced me that, at the fundamental level, the Casimir force is not created by vacuum energy.

If someone is interested, now a revised version accepted for publication in Phys. Lett. B is available
http://arxiv.org/abs/1605.04143