Double Slit Experiment in the Heisenberg Picture of Quantum Mechanics

This paper presents a Heisenberg picture formulation of the double-slit experiment to demonstrate that interference fringes can be explained without invoking non-locality, provided that position and momentum observables are defined as functions of both space and time.

The Gist

The Mystery of the "Spooky" Particle: A New Way to Look at the Double Slit Experiment

Imagine you are playing a game of hide-and-seek with a magical ball. You throw this ball toward a wall with two narrow vertical slits. In the weird world of quantum mechanics, something strange happens: instead of the ball going through one slit or the other like a marble, it acts like a wave. It passes through both slits at once, interferes with itself, and creates a pattern of stripes on the far wall.

For decades, many scientists have looked at this and said, "Aha! This proves that particles are 'non-local.' They are in two places at once, and they are communicating with themselves instantly across space!" They call this "spooky action at a distance."

Vlatko Vedral’s paper argues that we don't need the "spookiness" at all. He suggests that the mystery isn't caused by particles teleporting or acting weirdly across distances; it’s caused by the fact that we’ve been looking at the "map" of the universe incorrectly.

Here is the breakdown of his argument using everyday analogies.

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1. The "GPS" Problem (Observables as Fields)

Most people think of a particle like a single, tiny dot moving through space. In this view, if the dot is at Slit A, it cannot be at Slit B. Therefore, if it somehow affects Slit B, it must be "teleporting."

Vedral says we should stop thinking of the particle as a single "dot" and start thinking of its properties (like position) as a weather system.

The Analogy: Think of a storm front. You don't ask, "Where is the storm located?" as if it were a single marble. Instead, the "storm" is a field of wind and pressure that exists across a wide area. Even if the center of the storm is in one place, the effects of the storm (the wind) are felt everywhere at once.

Vedral argues that in quantum mechanics, "position" isn't just a single number attached to a particle; it is a "field" that exists at every point in space and time. When the particle hits the slits, it isn't "teleporting" between them; the "weather system" of the particle is simply interacting with the slits locally, point by point.

2. The "Heisenberg Movie" (The Heisenberg Picture)

In standard physics (the Schrödinger Picture), we usually imagine the particle moving and changing while the "rules of the world" stay still. It’s like watching a movie where the actors move, but the stage and the props are frozen.

Vedral uses the Heisenberg Picture, which flips the script. In this version, the particle stays still, but the rules and the environment change.

The Analogy: Imagine you are watching a movie of a person walking through a door.

• Schrödinger style: You watch the person move across the screen.
• Heisenberg style: The person stands perfectly still, but the door moves, the floor moves, and the camera moves.

By using this "Heisenberg" view, Vedral shows that the interference pattern isn't caused by the particle "doing something weird" across a distance. Instead, it’s caused by how the measurement tools (the slits and the screen) interact with the particle's field at specific moments in time. Everything happens through local, direct contact.

3. The "Church of the Larger Hilbert Space" (The Interaction)

Vedral uses a fancy term called the "Church of the Larger Hilbert Space" to explain how a measurement actually happens. He argues that a "measurement" isn't some magical, instant event that collapses reality. Instead, it is just a very complex handshake between the particle and the object measuring it.

The Analogy: Imagine a person walking through a dark room filled with hanging bells. If the person brushes against a bell, the bell rings. The "measurement" (the ringing bell) isn't a spooky, instant event; it is a local physical interaction. The person only rings the bells they actually touch.

In the double-slit experiment, the particle "brushes against" the slits. The interference pattern is just the result of these local "handshakes" happening at different points.

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The Big Conclusion

Vedral is essentially saying: "Stop looking for ghosts."

He argues that we don't need to invent "non-locality" (the idea that things affect each other instantly across space) to explain quantum mechanics. If we simply accept that a particle's properties are spread out like a field (like wind or temperature) and that measurements are just local "handshakes" between that field and the world, the "spookiness" vanishes.

The universe isn't acting weirdly at a distance; we were just using a map that was too simple to see the full picture.

Technical Summary

Technical Summary: Double Slit Experiment in the Heisenberg Picture of Quantum Mechanics

Author: Vlatko Vedral (University of Oxford)

1. Problem Statement

The paper addresses a fundamental conceptual tension in quantum mechanics: the perceived conflict between the phenomenon of quantum interference (exemplified by the double-slit experiment) and the principle of locality.

The author identifies two specific misconceptions in the existing literature:

• The Non-Locality Myth: The claim that interference fringes imply non-local "action at a distance" or that the particle "knows" about both slits simultaneously in a non-local way.
• The Operator Misconception: The tendency in non-relativistic quantum mechanics to treat position ($\hat{x}$) and momentum ($\hat{p}$) as single entities belonging to a particle as a whole, rather than as local observables defined at specific spacetime coordinates.

2. Methodology

The author employs the Heisenberg Picture of quantum mechanics to re-examine the double-slit experiment. The methodology is built on three technical pillars:

• Heisenberg Evolution of Projectors: Instead of evolving the state (Schrödinger Picture), the author evolves the measurement operators (projectors). The probability of a sequence of measurements is expressed as $p(1, 2) = \text{Tr}\{(U_1^\dagger P_1 U_1)(U_2 U_1)^\dagger P_2 (U_2 U_1)(U_1^\dagger P_1 U_1)\rho\}$, where the projectors evolve "backwards in time" to meet the initial state.
• Spacetime-Dependent Observables: The author treats position and momentum not merely as functions of time, but as fields—observables $\hat{x}(x, t)$ that are functions of both space and time. This allows for the calculation of commutators at different spacetime points.
• The "Church of the Larger Hilbert Space": To formalize locality, the author replaces abstract projective measurements with physical interactions. The measurement is modeled as an entanglement between the particle's position and a quantized screen via a local interaction Hamiltonian: $\hat{H}_{int} = g(x)\hat{x}(x)\hat{h}_{screen}(x)$.

3. Key Contributions

• Demonstration of Locality via Commutators: By defining observables as functions of spacetime, the author shows that for a free particle, the commutator $[\hat{x}(x, t), \hat{x}(x', t')] = \delta(x - x')i\hbar(t - t')/m$. Crucially, if $x \neq x'$, the commutator vanishes for all $t, t'$, proving that an interaction at one point cannot instantaneously affect another point.
• Local Interaction Modeling: The author proves that the "projection" at the slits is actually a local coupling. The particle only affects the screen at the specific point where it is located. The interference pattern arises from the evolution of phases acquired through these local interactions, not from non-local communication.
• Reinterpretation of the Double Slit: The interference pattern is derived using the overlap of position eigenstates at different times, showing that the "mystery" is simply the mathematical result of local unitary evolution and local projective measurements.

4. Results

• Mathematical Derivation: The author successfully derives the standard interference term $p(2/1) \propto \cos\{\frac{msd}{2\hbar(t_2 - t_1)}\}$ using the Heisenberg formalism.
• Resolution of Non-Locality: The paper demonstrates that the double-slit experiment can be fully described using only locally defined observables. The "spooky" nature of the experiment is revealed to be a byproduct of the Schrödinger picture's tendency to focus on a delocalized wavefunction, which can mislead one into thinking the particle acts non-locally.
• Field-Like Nature of Non-Relativistic QM: The results show that even in non-relativistic regimes, position and momentum behave like fields, a property usually reserved for Quantum Field Theory (QFT).

5. Significance

This paper provides a rigorous theoretical framework to defend locality as a universal principle in quantum mechanics. Its significance lies in:

1. Conceptual Clarification: It provides a toolset to resolve debates regarding "spooky action at a distance" by shifting the focus from delocalized states to local operators.
2. Unification of Perspectives: It bridges the gap between non-relativistic quantum mechanics and the local structure of Quantum Field Theory.
3. Pedagogical Correction: It suggests that the Heisenberg picture is a more "transparent" way to view quantum dynamics, as it prevents the mathematical artifacts of the Schrödinger wavefunction from being misinterpreted as physical non-locality.