Can the double slit experiment distinguish between QM interpretations?

[PeterDonis]

TL;DR

A paper just released on arxiv claims that the double slit experiment can be used to distinguish between QM interpretations. Is its claim correct?

This paper claims that the double slit experiment can be used to distinguish between QM interpretations:

https://arxiv.org/abs/2301.02641

IMO, the paper goes astray right at the start, when it points out that time is a parameter in the Schrödinger equation, not an operator, so that equation gives no way to uniquely derive a probability distribution for measurement results as a function of time--but then fails to note that the solution to that problem is to not use the Schrödinger equation in the first place. The correct framework in which to treat time on the same footing as other observables is relativistic quantum field theory. That is never even mentioned in this paper.

To be fair, I have never seen *any* discussion or comparison of QM interpretations that uses QFT as its framework; they all use non-relativistic QM, even though we know that's just an approximation. But I think it's still an issue even if it's an extremely common one. What do other QM experts here think?

 

[Nullstein]

There is also a Schrödinger equation in QFT and time is just a parameter in that equation. It's just more common to formulate QFT in the Heisenberg picture or using path integrals. However, all of these descriptions are equivalent. So if the paper goes astray (it may very well, I didn't read it in detail), it's not because its using the Schrödinger equation.

 

[PeterDonis]

There is also a Schrödinger equation in QFT

In non-relativistic QFT, perhaps. But I specified relativistic QFT in the OP.

 

[Nullstein]

In non-relativistic QFT, perhaps. But I specified relativistic QFT in the OP.

There is a Schrödinger equation in any QFT, relativistic or not. Heuristically, it is given by the quantization of the Hamiltonian, which can be obtained by Legendre transformation of the Lagrangian of the theory (see Hamiltonian field theory for an overview). Rigorously, it follows from applying Stone's theorem to the time evolution operator.

 

[PeterDonis]

There is a Schrödinger equation in any QFT, relativistic or not. Heuristically, it is given by the quantization of the Hamiltonian, which can be obtained by Legendre transformation of the Lagrangian of the theory (see Hamiltonian field theory for an overview). Rigorously, it follows from applying Stone's theorem to the time evolution operator.

All of this either requires a preferred frame, which violates the principle of relativity, or accepting that different frames have different Hamiltonian operators, Schrödinger equations, etc., which, as far as QM interpretation is concerned, undermines the whole point of trying to use those things in the first place. As I said in the OP, the correct relativistic approach for QM interpretation is to treat time and space on the same footing. The approach you describe, however valid it is mathematically, does not do that.

 

[Nullstein]

All of this either requires a preferred frame, which violates the principle of relativity, or accepting that different frames have different Hamiltonian operators, Schrödinger equations, etc., which, as far as QM interpretation is concerned, undermines the whole point of trying to use those things in the first place.

No, none of this applies. The Hamiltonian looks the same in all inertial frames and none of them is preferred. One can pass between them using Lorentz transformations.

As I said in the OP, the correct relativistic approach for QM interpretation is to treat time and space on the same footing. The approach you describe, however valid it is mathematically, does not do that.

Time and space is treated symmetrically in the Hamiltonian QFT as well. The equation is not manifestly covariant, but everything is equivalent to a covariant formulation. It's also commonly taught in QFT courses and explained in standard textbooks such as Peskin & Schröder, so I don't see why it should be avoided. Here's another exposition of the formalism. The first couple of sentences in the paper are certainly correct assertions also about QFT, so if the authors can derive some interesting conclusions from that observation, it should be taken seriously.

 

[Nullstein]

Let me put it differently. Time and space are treated symmetrically in QFT, independent of the concrete formulation. But it's not because time becomes an observable described by an operator. It's because position ceases to be an observable described by an operator. Both time and position are just parameters in QFT. The observables in QFT are given by the fields, which are represented by field operators. So we don't need the Schrödinger equation to conclude that time is just a parameter (in both QM and QFT). Given the introductory sentences, the paper seems to be based on the observation that time is just a parameter and not an observable. In order to challenge that assumption, you would have to provide as a counterexample a quantum theory (particle or field, it doesn't matter), wherein time is in fact an observable described by an operator.

 

[Demystifier]

This paper is about the arrival time probability distribution problem in QM. Since time is not an operator, it looks as if it's not clear what the standard quantum mechanical prediction for a time distribution is. However, my opinion is that standard QM does make well defined predictions for time distributions. After all, the exponential time distribution of decays, and deviations from the exponential distribution, are well understood in standard QM. In my opinion, when standard QM is analyzed correctly, it also makes well defined predictions on the arrival time problem. And when the calculations are done correctly, the end result is that the predictions of standard QM are the same as those of the Bohmian interpretation.

I (with two collaborators) have written 3 papers on this topic; the first two are published, the third is not published yet, but the final result that the two interpretations give the same results is explicit only in the third paper:
https://arxiv.org/abs/2010.07575
https://arxiv.org/abs/2107.08777
https://arxiv.org/abs/2207.09140
In the first paper we formulated the general framework. In the second paper we further developed the formalism and made some approximate calculations. In the third paper we understood all this from a deeper point of view and made exact analytic calculations.

 

[Demystifier]

In non-relativistic QFT, perhaps. But I specified relativistic QFT in the OP.

Relativistic QFT also has a Schrödinger equation
$$H|\psi(t)\rangle = i\hbar\partial_t |\psi(t)\rangle$$
where $H$ is the QFT Hamiltonian.

 

[Demystifier]

The correct framework in which to treat time on the same footing as other observables is relativistic quantum field theory.

Time is not an observable in relativistic QFT.

 

[Demystifier]

To be fair, I have never seen *any* discussion or comparison of QM interpretations that uses QFT as its framework;

Do you mean a comparison of their measurable predictions?

 

[vanhees71]

The interpretational issues like the so-called "measurement problem" or the "wave-particle-dualism problem" are not too much different between relativistic and non-relativistic QT, and indeed time is not an observable in both relativistic and non-relativistic QT. Position can be defined as an observable for all kinds of non-relativistic and all kinds of massive relativistic particles. For massless particles a position operator is definable only for spins 0 and 1/2.

In relativistic QT the localizability of particles is, however, much more restricted than in non-relativistic QT, because of pair production, which is a necessary consequence of microcausality and locality, which is the only known causal description of interacting relativistic quantum systems, i.e., the only way to define a relativistic quantum dynamics, that's compatible with relativistic causality is the use of local quantum fields, which allow for the description of local observables that obey the microcausality condition, which in turn also leads to the Poincare covariance and unitarity of the S-matrix. A particle description is also only possible for (asymptotic) free states.

 

[Morbert]

As an aside re/ QFT and the Schroedinger equation: This previous discussion might be of interest

To be fair, I have never seen *any* discussion or comparison of QM interpretations that uses QFT as its framework; they all use non-relativistic QM, even though we know that's just an approximation.

For posterity:
https://arxiv.org/ftp/arxiv/papers/1805/1805.12246.pdf
https://arxiv.org/pdf/gr-qc/9304006.pdf
Some discussion of decoherent histories as applied to quantum theories with different levels of generality

 

[PeterDonis]

Time and space are treated symmetrically in QFT, independent of the concrete formulation. But it's not because time becomes an observable described by an operator. It's because position ceases to be an observable described by an operator. Both time and position are just parameters in QFT.

Yes, this is a valid way of treating time and space on the same footing; I would put it that the parameter in QFT is "which spacetime point", and labeling spacetime points requires a 4-dimensional coordinate $x^\mu$.

However, one can't then pick out just one piece of the parameter $x^\mu$, "time", and treat it differently; yet that's what the Hamiltonian formulation and the Schrödinger equation do. That is why I object to them as a basis for QM interpretation. Of course I don't object to them as a mathematical framework for doing calculations; but that's not what this thread is about.

 

[PeterDonis]

Time is not an observable in relativistic QFT.

Yes, I misspoke in that part of the OP. See post #14.

 

[PeterDonis]

Do you mean a comparison of their measurable predictions?

I mean any discussion of QM interpretations using QFT as a framework.

 

[Demystifier]

I mean any discussion of QM interpretations using QFT as a framework.

But you have seen one of my papers on that, I know because once you correctly summarized the main ideas of it.

 

[Demystifier]

However, one can't then pick out just one piece of the parameter $x^\mu$, "time", and treat it differently; yet that's what the Hamiltonian formulation and the Schrödinger equation do. That is why I object to them as a basis for QM interpretation.

But what remains then, path integral formulation? How do you define the notion of state in the Hilbert space within the path integral approach?

 

[vanhees71]

Yes, this is a valid way of treating time and space on the same footing; I would put it that the parameter in QFT is "which spacetime point", and labeling spacetime points requires a 4-dimensional coordinate $x^\mu$.

However, one can't then pick out just one piece of the parameter $x^\mu$, "time", and treat it differently; yet that's what the Hamiltonian formulation and the Schrödinger equation do. That is why I object to them as a basis for QM interpretation. Of course I don't object to them as a mathematical framework for doing calculations; but that's not what this thread is about.

Well, that's the problem with the operator formulation of relativistic QFT. If you want to use the canonical formalism, you have to use the Hamiltonian description, and this is done in an arbitrary but then fixed inertial frame of reference (I'm talking about standard QFT in SRT, not in curved spacetimes of GRT of course). Then you first get a formulation that is not manifestly Poincare covariant.

A way to stay Poincare covariant at all steps of the calculation is to use the path integral in the Lagrangian formulation, but there you must be very careful, whether you can do this in a naive way. That's only the case if the Hamiltonian is quadratic in the canonical field momenta with field-independent coefficients! For a detailed discussion of the most simple example (free spin-0 field), see Sect. 4.5.1 of

https://itp.uni-frankfurt.de/~hees/publ/lect.pdf

For a non-trivial example see the discussion of (scalar) QED in 6.3, when going from the axial-gauge (partially) fixed Hamiltonian (canonical) path-integral formalism to a manifest Lagrangian path-integral formalism using a covariant gauge, showing that both are equivalent.

 

[PeterDonis]

But what remains then, path integral formulation?

That would treat time and space on a equal footing, yes.

How do you define the notion of state in the Hilbert space within the path integral approach?

Why do you need to do that?

 

[PeterDonis]

But you have seen one of my papers on that, I know because once you correctly summarized the main ideas of it.

Was that the paper that treated QFT as emergent from a non-Lorentz invariant underlying theory?

 

[PeterDonis]

As an aside re/ QFT and the Schroedinger equation: This previous discussion might be of interestFor posterity:
https://arxiv.org/ftp/arxiv/papers/1805/1805.12246.pdf
https://arxiv.org/pdf/gr-qc/9304006.pdf
Some discussion of decoherent histories as applied to quantum theories with different levels of generality

Thanks for the references! They look interesting.

 

[Demystifier]

Was that the paper that treated QFT as emergent from a non-Lorentz invariant underlying theory?

Yes.

 

[Demystifier]

Why do you need to do that?

To discuss the (interpretations of) entanglement, for instance. How would you define entanglement without state in the Hilbert space?

 

[PeterDonis]

How would you define entanglement without state in the Hilbert space?

Nonzero entanglement entropy. I believe that can be computed in the path integral approach.

 

[Demystifier]

Nonzero entanglement entropy. I believe that can be computed in the path integral approach.

Entropy in which state? The path integral approach is developed to compute things (including entanglement entropy) in the vacuum state, but how to define other states with path integrals? Bell state, for instance?

 

[vanhees71]

The path-integral method, together with the Schwinger-Keldysh real-time contour or, for equilibrium, the Matsubara imaginary-time contour, can be applied to any state:

https://itp.uni-frankfurt.de/~hees/publ/off-eq-qft.pdf

 

[Demystifier]

The path-integral method, together with the Schwinger-Keldysh real-time contour or, for equilibrium, the Matsubara imaginary-time contour, can be applied to any state:

https://itp.uni-frankfurt.de/~hees/publ/off-eq-qft.pdf

Of course it can be applied, as long as one defines states as objects in the Hilbert space. But to define the Hilbert space, one needs canonical quantization.

The question is, can one formulate (not merely apply) QFT without ever mentioning Hilbert space and/or canonical quantization? I don't think that path-integral quantization can do that. Perhaps algebraic quantization goes in that direction, which indeed can be made manifestly Lorentz invariant, but with this formalism it's hard to get concrete measurable results.

 

[vanhees71]

I think, it's impossible to avoid Hilbert space, because that's the mathematical framework QT is formulated in. Canonical quantization is a rather heuristic method to guess the possible form of the algebra of observables, Hamiltonians, etc. Here I'd say a more reliable approach is the group-theoretical methods exploiting symmetries.

 

[Demystifier]

I think, it's impossible to avoid Hilbert space, because that's the mathematical framework QT is formulated in. Canonical quantization is a rather heuristic method to guess the possible form of the algebra of observables, Hamiltonians, etc. Here I'd say a more reliable approach is the group-theoretical methods exploiting symmetries.

Yes. And for discussing the resulting interpretations, a good starting point is
http://philsci-archive.pitt.edu/8890/