Double Slit Experiment in the Heisenberg Picture of Quantum Mechanics

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.

• The paper demonstrates that interference arises from the local evolution of observables, eliminating the need for non-local wavefunction collapse.
• It uses time-evolved projective measurement operators and operator-valued spacetime fields to detail the propagation from slits to detection.
• The analysis integrates non-relativistic quantum mechanics with local quantum field theory, offering valuable insights for quantum control and measurement design.

Locality in the Double Slit Experiment: A Heisenberg Picture Analysis

Overview

This paper rigorously formulates the double slit experiment within the Heisenberg picture for non-relativistic quantum particles, challenging frequently cited claims regarding quantum non-locality in such setups. The author demonstrates that all observable phenomena, including interference fringes, can be comprehensively described using local observables—where “local” is operationalized as operator-valued fields dependent explicitly on both space and time. The presentation also juxtaposes projective measurements in the Heisenberg formalism with the “Church of the Larger Hilbert Space” account, offering clarification on the proper locus of quantum locality.

Formal Treatment of Successive Measurements

The author opens by reviewing the probability structure for two consecutive measurements separated by unitary evolution, contrasting the Schrödinger and Heisenberg pictures. In the Heisenberg representation, the state remains stationary while operators evolve, such that measurement projectors must be appropriately "time-evolved" (i.e., Heisenberg evolved) backward in time via the relevant unitary transformations. These manipulations exploit the cyclic property of the trace and the unitarity of quantum evolution, allowing various equivalent forms for the joint probability of outcomes.

Applying this scheme to the double slit experiment, the initial stage is described as a general pure state, evolved freely to the slit-screen, which is represented by a spatially projective measurement operator acting only at the slit positions. This approach establishes the post-slit state as a spatial superposition over the two slit locations. The post-slit propagation to the detection screen is another unitary time translation, concluded with local projective detection.

Calculation of Interference and Locality

The explicit calculation of detection probabilities reproduces the canonical result for interference fringes. The critical detection probability, $p(2/1)$, emerges as the modulus-squared of the transition amplitude between spatially localized eigenstates at the two distinct times—first at the slits, then at a final detection point. For infinitesimal slit width, this leads to an interference term proportional to $\cos\left(\frac{msd}{2\hbar (t_2-t_1)}\right)$, as per standard analysis. The time translation $(t_2 - t_1)$ is set by the geometry and kinematics in the $y$-direction, which the author emphasizes can serve as an intrinsic “clock," making time an internal variable rather than an external parameter.

The formalism is developed so that observables, particularly position and momentum, are operator-valued distributions—i.e., fields—depending on spacetime coordinates, not merely parameterized by time. This feature aligns non-relativistic quantum mechanics with the axiomatic structure of local quantum field theory. In this framework, $\hat{x}(x, t)$ and $\hat{x}(x', t')$ are intrinsically distinct operators; critically, their commutator vanishes for $x \neq x'$. For a free particle, the absence of mediators (i.e., interactions) ensures that these commutators vanish for all time differences except at coincident points, which is a formal manifestation of locality.

Projective Measurements and the Larger Hilbert Space

To further clarify the operational meaning of locality, the paper invokes the “Church of the Larger Hilbert Space”—an expanded setup in which all measurement interactions are modeled as entanglement with additional ancillary/measurement degrees of freedom. The double slit screen is treated as a quantum system; the interaction Hamiltonian couples the particle’s local position operator to the screen only at the corresponding position. The measurement’s effect manifests as a local entanglement structure, with each spatial sector interacting solely with the screen at that coordinate. The post-selection on the screen’s unperturbed state implements the physical projection onto the “went through the slits” sector of the particle Hilbert space, justifying the locality of the projective operation.

Theoretical and Practical Implications

The analysis posits unambiguously that there is no operational or formal requirement for non-locality to explain interference in the canonical double slit experiment within non-relativistic quantum mechanics. Locality is a structural property of the operator algebra—observables are local fields, and interactions occur only at coincident points in spacetime. The author critiques prevailing misconceptions that ascribe “instantaneous action at a distance” to quantum phenomena, attributing these errors to an overreliance on the Schrödinger picture’s dynamics where wavefunction delocalization can obscure the locality underlying operator fields.

The discussion extends beyond the specific experimental context: the treatment and conclusions generalize to arbitrary measurement protocols and experimental arrangements described by standard quantum mechanics. Even entanglement and related quantum correlations are treated as arising from local phase evolution, not from any underlying non-local triggers.

Practically, this perspective highlights the importance of modeling measurement interactions as local, both in conventional quantum mechanics and in engineered quantum technologies where measurement and interaction fields are physically delineated. The formal clarification may inform foundational debates and modeling strategies in contexts ranging from quantum information theory to quantum simulation protocols.

Future Directions

The field-theoretic formalism for quantum observables suggested by this analysis raises natural lines of inquiry for both theoretical physics and quantum information. It motivates further examination of the role of operator locality in more elaborate many-body, interacting, or relativistic systems—particularly in assessing the boundary conditions under which non-local effective observables emerge. The conceptual framework may also interact productively with quantum control, quantum sensing, and device engineering, wherein the specification of local measurement and manipulation is operationally critical. On a foundational level, understanding the interplay between locality in operator algebra and operational definitions of causality in quantum protocols could guide new analyses of quantum non-locality and related paradoxes.

Conclusion

This work provides a comprehensive Heisenberg picture account of the double slit experiment, demonstrating the sufficiency of spacetime-localized observables in capturing all observed quantum phenomena, including interference, within non-relativistic quantum mechanics (2604.22481). The emphasis on operator-valued fields unifies quantum measurement, dynamics, and locality in a manner consistent with quantum field theory, dispelling misconceptions regarding the necessity of non-local phenomena in standard quantum physics. This framework invites continued analysis of locality in broader quantum mechanical settings and encourages further integration of Heisenberg-picture treatments in both foundational and applied quantum research.