A Can the double slit experiment distinguish between QM interpretations?

[PeterDonis]

TL;DR Summary

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 Schrodinger 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 Schrodinger 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, Schrodinger 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, Schrodinger 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 Schrodinger 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 Schrodinger 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 Schrodinger 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 Schrodinger 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/

 

[Demystifier]

But to return to the main subject of this thread, quantum theory does not need to be relativistic to study time distributions. For instance, non-relativistic QM has well defined predictions for the time of decay. The question is, does it also has well defined predictions for the time of arrival? I claim that it has, because both can be described by the same theoretical framework (post #8).

 

[vanhees71]

Are you talking about the "tunneling time" and related questions? I think, here the solution can only be to look at specific experiments and (try) to describe them with QT. I think the problem with this is that there's not a clear definition of what "tunneling time" means, and this is not restricted to QT but also within classical theory of waves. E.g., some years ago there was a big debate about faster-than-light signals in electromagnetic wave guides. Of course there's nothing faster than light that's not allowed to be faster than light within Maxwell's theory, which is relativistic of course. In this case the question has been answered already by Sommerfeld and Brillouin in 1907-1913 ;-).

 

[Demystifier]

Are you talking about the "tunneling time" and related questions?

Yes.

I think, here the solution can only be to look at specific experiments and (try) to describe them with QT.

At first I thought that too. But then I developed a general framework of idealized measurements that can be applied to tunneling time as well, see the first paper in #8.

 

[LittleSchwinger]

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.

Algebraic QFT is capable of defining QFTs without the notion of a Hilbert Space. This is increasingly necessary in curved spacetimes, especially for generic spacetimes without the symmetries required to define the notion of particle or where one possibly lacks a global state.

Of course Hilbert spaces are closely tied to this formalism, as they are involved in representations of the abstract observable algebra. Only recently though have researchers found how to compute directly physical quantities using the formalism. See the monograph of Kasia Rejzner.

 

[vanhees71]

The problem with AQFT, however is that for decades nobody was able to describe interacting particles in (1+3) dimensions!

 

[LittleSchwinger]

The problem with AQFT, however is that for decades nobody was able to describe interacting particles in (1+3) dimensions!

I think here you are discussing the lack of a rigorous existence proof, correct me if I am wrong. If so I think one should distinguish between work in rigorous QFT and algebraic field theory, as I don't think what you mention is really a problem for C*-algebra methods.

Algebraic Field Theory involves using C*-algebras to extract physical predictions. This can be done at a non-rigorous level, just as in normal particle physics applications of QFT one uses operator-valued distributions without concerning oneself with their operator domain or smearing class of functions.

Recent papers by Witten on the arxiv about how deSitter space uses a Type $II_{1}$ algebra or many papers on QFT in curved spacetime are examples were C*-algebras are used in a non-rigorous or semi-rigorous way. In these "physicist" approaches to C*-algebras it has been possible to handle particles in 3+1 dimensions for around forty years and in the last ten years extract the same physical information one would from normal Feynman diagram techniques. It's become increasingly necessary to use C*-algebraic methods for discussing issues of quantum information in curved spacetime once we lack the symmetries giving a preferred Hilbert space structure or anything like particle states or global states.

Existence proofs in 3+1D are a separate issue and are equally an issue/non-issue for regular operator or path integral approaches to field theory as they are for C*-algebra methods.

 

[vanhees71]

Indeed, that's the point. We have to use the, unfortunately mathematically non-rigorous "descriptions" of QFT as effective theories using perturbative methods and (regularization and) renormalization. I think in this respect the more abstract and also more general $\text{C}^*$-algebra methods also belong to this class of "physicists' treatment".

I think to discuss the physics content of QFT AQFT is of little use, and one must rely on these non-rigorous but from a physicist's point of view very successful, descriptions.

 

[LittleSchwinger]

I think to discuss the physics content of QFT AQFT is of little use, and one must rely on these non-rigorous but from a physicist's point of view very successful, descriptions.

I take it here you are using AQFT to denote "rigorous use of C*-algebras", for example by people like Haag, as distinct from physicists using C*-algebras in a non-rigorous way as one finds in Witten's papers and the literature on QFT in curved spacetime.

If so, yes certainly it is very difficulty to extract physical results from Haag-Kastler type axiomatic set ups. It's not much different from rigorous non-Relativistic QM where results on operator domains and results like Kato's theorem usually don't give you much physical information.

Here's some sample papers of a physicist usage:
https://arxiv.org/abs/2206.10780
https://arxiv.org/abs/2301.07257

As I mentioned above the reason physicists have to use C*-algebras in such set ups is that the normal Hilbert space approach involves certain assumptions that fail in the general curved spacetime case.

 

[vanhees71]

I don't know much about QFT in curved spacetimes. I was more referring to usual QFT in Minkowski spacetime, but I guess it's even more difficult to find a rigorous formulation than in flat spacetime.

 

[LittleSchwinger]

I don't know much about QFT in curved spacetimes. I was more referring to usual QFT in Minkowski spacetime, but I guess it's even more difficult to find a rigorous formulation than in flat spacetime.

Yes definitely. For a brief explanation if we take the path integral approach, then in the Minkowski case the space of scalar, spinor, tensor fields to integrate over on Euclidean* space have a very nice and well studied linear structure. In the curved spacetime case there are very few such results and one even lacks a single unique space of fields or canonical measures on them and there can be subtleties in defining the Riemannian continuation of a general spacetime.

*Rigorously the path integral has to be in a Riemannian space as you might be aware.

 

[vanhees71]

You mean you study Euclidean QFT in the flat-spacetime case, i.e., the Wick rotated imaginary time to make the metric Euclidean rather than Lorentian. That's of course a simplifying step to formulate the path integral (though also the path-integral formulation is not entirely mathematically rigorous but also relies on regularization-renormalization and perturbation theory for the interacting case) and the trouble then is to do the analytic continuation back to real time quantities, which is everything than simple. I guess in curved spacetimes it's even more complicated, if not impossible. I've to read more about this!

 

[LittleSchwinger]

You mean you study Euclidean QFT in the flat-spacetime case, i.e., the Wick rotated imaginary time to make the metric Euclidean rather than Lorentian

Yes indeed, it's the only way to make the Path Integral rigorous as one can prove the path integral doesn't exist rigorously in the Lorentzian case.
In a rigorous approach regularization-renormalization shows up in the fact that the interacting measure $\mathcal{D}\mu[\phi]$ and the free measure $\mathcal{D}\nu[\phi]$ are relatively singular.
The curved spacetime case is exactly more difficult for the analytic continuation reasons you mentioned.