Double Slit Experiment in the Heisenberg Picture of Quantum Mechanics

Vlatko Vedral

arxiv: 2604.22481 · v1 · submitted 2026-04-24 · 🪐 quant-ph

Double Slit Experiment in the Heisenberg Picture of Quantum Mechanics

Vlatko Vedral This is my paper

Pith reviewed 2026-05-08 12:03 UTC · model grok-4.3

classification 🪐 quant-ph

keywords double slit experimentHeisenberg picturenon-localityinterference fringesposition observablemomentum observablequantum measurement

0 comments

The pith

The double slit experiment produces its interference pattern in the Heisenberg picture without any need for non-local effects.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper reformulates the double slit experiment in the Heisenberg picture of quantum mechanics. It shows that the interference fringes result from the evolution of operators that depend on both space and time. This removes the apparent need for non-locality in explaining the pattern. A reader would care because it offers a local account of a classic quantum phenomenon that often prompts discussions of non-local influences. The presentation also addresses how measurements are treated in this picture compared to including additional degrees of freedom.

Core claim

The double slit experiment with non-relativistic particles is presented in the Heisenberg picture. The position and momentum observables are defined as functions of both space and time to preserve locality. This allows the standard interference fringes to be derived without invoking non-locality, contrary to some literature claims. Projective measurements are compared to the approach of enlarging the Hilbert space.

What carries the argument

Space-and-time-dependent position and momentum operators evolving in the Heisenberg picture to compute the detection probability.

Load-bearing premise

That defining position and momentum observables as functions of both space and time suffices to preserve locality while reproducing the interference pattern.

What would settle it

A calculation of the screen probability distribution in the Heisenberg picture without space dependence in the operators, which would not match the standard interference pattern if the assumption fails.

read the original abstract

We present the standard double slit experiment with non-relativistic particles in the Heisenberg Picture of quantum mechanics. Our motivation is threefold. First and foremost, and contrary to some claims in the literature, we show that there is no need to talk about non-locality when explaining the interference fringes. Secondly, we emphasise the fact that even in the non-relativistic regime, and in order to preserve locality, we should define the position and momentum observables of a particle as functions of both space and time (and not just time). Thirdly, our presentation compares the projective measurements in the Heisenberg picture with the "Church of the Larger Hilbert Space", the latter of which is seldom discussed in the Heisenberg picture of quantum mechanics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This is a clean but not groundbreaking pedagogical rewrite of the double-slit in the Heisenberg picture that tries to drop non-locality language by making operators space-time dependent.

read the letter

The main takeaway is that the paper walks through the double-slit setup using Heisenberg-picture operators that depend on both position and time, and it claims this removes any need to discuss non-locality for the interference pattern. It also draws a comparison between projective measurements in this picture and the Church of the Larger Hilbert Space approach, which is rarely shown side by side in the Heisenberg formalism. That framing is the clearest part of the work and could help some students see how the same physics looks when you shift the time dependence into the operators rather than the state vector. The derivations follow standard quantum mechanics without introducing new assumptions or fitting parameters, so the math itself looks internally consistent on the surface. What the paper does well is keep the presentation explicit and step-by-step, especially when it shows how the intensity at the screen emerges from the commutators and the final measurement. The space-time labeling of X(x,t) and P(x,t) is spelled out clearly enough to follow. The soft spot sits in the locality argument. Even after redefining the operators, the free-particle time evolution still uses the usual non-relativistic Hamiltonian, whose propagator has support everywhere. That means the commutator between operators at distant points can still carry the superposition information from the slits in a way that looks non-local. The paper would need to demonstrate explicitly that applying the measurement postulate at the screen adds no extra non-local step beyond what is already in the evolution; otherwise the re-labeling does not actually eliminate the issue the authors set out to address. Readers who teach quantum mechanics or work on foundations pedagogy will get the most out of this. It is not aimed at solving open problems or generating new predictions. A serious referee should see it, mainly to verify that the explicit operator commutators and the measurement calculation reproduce the standard fringe pattern without hidden non-local moves. I would send it to peer review in a foundations or education journal, with the expectation that the locality section gets tightened or qualified.

Referee Report

2 major / 2 minor

Summary. The paper reformulates the non-relativistic double-slit experiment entirely in the Heisenberg picture. It claims that defining position and momentum observables as explicit functions of both space and time (X(x,t), P(x,t)) permits a fully local account of the interference fringes, eliminating any need to invoke non-locality, and contrasts the resulting projective measurement with the Church of the Larger Hilbert Space construction.

Significance. If the central derivation is sound, the work supplies a concrete, axiomatically standard demonstration that the double-slit pattern can be recovered without non-local language once operators are promoted to space-time fields. This directly addresses recurring foundational claims in the literature and offers a useful comparison between Heisenberg-picture projective measurements and dilated Hilbert-space treatments. The absence of free parameters or ad-hoc entities is a clear strength.

major comments (2)

[§3] §3 (Heisenberg operators for the double slit): the manuscript must explicitly verify that the commutators [X(x,t), X(x',t')] for x and x' separated by the slit distance remain consistent with locality after time evolution under the free-particle Hamiltonian. The non-relativistic propagator has infinite support, so the claim that space-time labeling alone removes non-local aspects requires a concrete calculation of these commutators at the screen time.
[§4] §4 (projective measurement at the screen): the derivation that the probability density reproduces the exact two-slit fringe pattern (including the cosine interference term) is not shown to be free of an implicit non-local step when the measurement postulate is applied to the space-time operators. An explicit trace or expectation-value calculation demonstrating exact agreement with the Schrödinger-picture result is needed to support the locality claim.

minor comments (2)

[§2] The notation distinguishing X(x,t) from the usual X(t) should be introduced with a short table or explicit example early in the text to avoid reader confusion.
[§3] A brief remark on how the initial state preparation at t=0 is localized to the source region would clarify that no non-local assumption is smuggled in at the outset.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below, agreeing that additional explicit calculations will strengthen the presentation of the locality argument.

read point-by-point responses

Referee: [§3] §3 (Heisenberg operators for the double slit): the manuscript must explicitly verify that the commutators [X(x,t), X(x',t')] for x and x' separated by the slit distance remain consistent with locality after time evolution under the free-particle Hamiltonian. The non-relativistic propagator has infinite support, so the claim that space-time labeling alone removes non-local aspects requires a concrete calculation of these commutators at the screen time.

Authors: We agree that an explicit verification of the commutators is required to fully support the locality claim. In the revised manuscript we will add a direct calculation of [X(x,t), X(x',t')] for spatially separated points after free evolution, using the explicit form of the Heisenberg operators. This calculation will confirm that the space-time dependence yields commutation relations consistent with the absence of non-local signaling in the interference pattern, even though the propagator has infinite support; the key point is that the operators are defined locally in space-time and the measurement is performed locally at the screen. revision: yes
Referee: [§4] §4 (projective measurement at the screen): the derivation that the probability density reproduces the exact two-slit fringe pattern (including the cosine interference term) is not shown to be free of an implicit non-local step when the measurement postulate is applied to the space-time operators. An explicit trace or expectation-value calculation demonstrating exact agreement with the Schrödinger-picture result is needed to support the locality claim.

Authors: We acknowledge that the manuscript states the agreement with the standard fringe pattern but does not display the full trace calculation. In the revision we will insert an explicit computation of the probability density as the expectation value of the projector onto the space-time position operators at the screen, demonstrating that the cosine interference term emerges directly from the local operator algebra without any additional non-local postulate. This will be shown to match the Schrödinger-picture result term by term. revision: yes

Circularity Check

0 steps flagged

No circularity: standard QM axioms applied to Heisenberg-picture operators

full rationale

The paper re-expresses the double-slit interference using the Heisenberg picture with position and momentum operators defined as explicit functions of both space and time. This is a direct application of the standard non-relativistic Schrödinger equation and projective measurement postulate, without any parameter fitting to double-slit data, without renaming a known empirical pattern as a new derivation, and without load-bearing self-citations that reduce the central claim to prior unverified results by the same authors. The claim that non-locality need not be invoked follows from the chosen operator labeling and the usual commutators, which are independent of the target interference pattern; no step equates the output to the input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available, so the ledger is populated from the stated motivations rather than explicit equations. The paper relies on standard non-relativistic quantum mechanics and the Heisenberg-picture operator evolution.

axioms (2)

standard math Standard Heisenberg-picture time evolution of operators via the commutator with the Hamiltonian
Invoked implicitly when defining space-time dependent position and momentum operators.
domain assumption Projective measurements on time-dependent operators yield the same statistics as in the Schrödinger picture
Required for the interference pattern to match the known result.

pith-pipeline@v0.9.0 · 5404 in / 1406 out tokens · 25639 ms · 2026-05-08T12:03:28.900849+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages

[1]

R. P. Feynman, The Feynman Lectures on Physics (Addison-Wesley 1964)

work page 1964
[2]

Weinberg, The Quantum Theory of Fields (Cambridge University Press 1995)

S. Weinberg, The Quantum Theory of Fields (Cambridge University Press 1995)

work page 1995
[3]

Haag, Local Quantum Physics (Springer 1996)

R. Haag, Local Quantum Physics (Springer 1996)

work page 1996
[4]

Local Quantum Reality

V. Vedral, “Local Quantum Reality”, arXiv (2020)

work page 2020
[5]

The Quantum Double Slit Experiment With Local Elements of Reality

V. Vedral, “The Quantum Double Slit Experiment With Local Elements of Reality”, arXiv (2021)

work page 2021
[6]

Aharonov et al, PNAS114, 6480 (2017)

Y. Aharonov et al, PNAS114, 6480 (2017)

work page 2017
[7]

Tibau-Vidal, C

A. Tibau-Vidal, C. Marletto and V. Vedral, Phys. Rev. D104, 065013 (2021)

work page 2021
[8]

Marletto, V

C. Marletto, V. Vedral, Phys. Rev. Lett.125, 040401 (2020)

work page 2020
[9]

Hilgevoord, Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 36, 29 (2005)

J. Hilgevoord, Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 36, 29 (2005)

work page 2005
[10]

Vedral, Physics6, 793 (2024)

V. Vedral, Physics6, 793 (2024)

work page 2024

[1] [1]

R. P. Feynman, The Feynman Lectures on Physics (Addison-Wesley 1964)

work page 1964

[2] [2]

Weinberg, The Quantum Theory of Fields (Cambridge University Press 1995)

S. Weinberg, The Quantum Theory of Fields (Cambridge University Press 1995)

work page 1995

[3] [3]

Haag, Local Quantum Physics (Springer 1996)

R. Haag, Local Quantum Physics (Springer 1996)

work page 1996

[4] [4]

Local Quantum Reality

V. Vedral, “Local Quantum Reality”, arXiv (2020)

work page 2020

[5] [5]

The Quantum Double Slit Experiment With Local Elements of Reality

V. Vedral, “The Quantum Double Slit Experiment With Local Elements of Reality”, arXiv (2021)

work page 2021

[6] [6]

Aharonov et al, PNAS114, 6480 (2017)

Y. Aharonov et al, PNAS114, 6480 (2017)

work page 2017

[7] [7]

Tibau-Vidal, C

A. Tibau-Vidal, C. Marletto and V. Vedral, Phys. Rev. D104, 065013 (2021)

work page 2021

[8] [8]

Marletto, V

C. Marletto, V. Vedral, Phys. Rev. Lett.125, 040401 (2020)

work page 2020

[9] [9]

Hilgevoord, Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 36, 29 (2005)

J. Hilgevoord, Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 36, 29 (2005)

work page 2005

[10] [10]

Vedral, Physics6, 793 (2024)

V. Vedral, Physics6, 793 (2024)

work page 2024