Wading Through Treacle: Retrodiction is difficult in QM

Monday, January 27, 2025

Retrodiction is difficult in QM

From Wikipedia

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Me: This is a thought experiment in quantum mechanics. Consider the two-slit experiment. We assume an electron leaves a point source, which we may imagine to be a pinhead quantity of a beta-ray (electron) emitting radioactive material such as Carbon-14. The electron traverses (somehow!) the two slits and impacts a screen where it registers as an observable point of light.

We know that if we do this experiment measuring multiple electrons, the pattern of electron impacts on the screen will form an interference pattern in accordance with Schrödinger's equation. But we are going to consider just one electron - at the moment it impacts the screen.

If we now use Schrödinger's equation to retrodict the electron's behaviour going back in time, back towards the two slits, and then back towards the original point source, what is the quantum mechanical prediction of the electron's prior behaviour before it hit the screen, according to quantum theory?

My guess is that theory will predict an interference pattern in the plane around the point source, and will not predict that the electron was actually emitted from that source. If so, do you think this is an interesting failure of prediction?

ChatGPT: This is an intriguing thought experiment, and your guess touches on fundamental questions about the interpretation of quantum mechanics and the nature of quantum predictions. Let's carefully analyze it:

The Setup

In the two-slit experiment, the Schrödinger equation predicts the wavefunction of the electron, which is a complex-valued function encoding the probabilities of the electron's position and momentum. When the electron impacts the screen, the wavefunction "collapses" (in the Copenhagen interpretation) to a single point, corresponding to the detected location.

Retrodiction with Schrödinger's Equation

If we use Schrödinger's equation to "retrodict" the electron's prior behavior:

The wavefunction evolves backward in time as a solution to the time-reversed Schrödinger equation.
At each point in time, the wavefunction spreads out, representing the superposition of all possible paths the electron might have taken.

When you retrodict back to the plane of the two slits, the wavefunction will show an interference pattern, suggesting that the electron passed through both slits simultaneously, as a 'probability' wave. Retrodicting further, back to the point source, the wavefunction does not concentrate at a single point; instead, it remains spread out as a distributed wavefunction, consistent with the probabilistic nature of quantum mechanics.

Prediction vs. Retrodiction

The retrodicted wavefunction will not "collapse" to a single point of emission at the source, which is what you are suggesting. Instead, it will represent a diffuse probability distribution, as though the electron were emitted in a probabilistic manner, not from a single, definite point.

Is This a Failure of Prediction?

This situation is not necessarily a failure of quantum mechanics but rather a reflection of its probabilistic nature and how it handles information:

The wavefunction describes the probabilities of possible outcomes, not the actual path or specific origin of the particle.
The theory's predictions concern ensembles of measurements or probabilities for individual events, not definitive histories.

Why Is This Interesting?

Interpretational Implications: Your observation highlights a key limitation of quantum mechanics: it cannot deterministically retrodict a unique "path" or "point of origin" for a particle. This is a feature of the theory, not a flaw, but it underscores its departure from classical intuition.
The Nature of Quantum Reality: The experiment challenges us to consider what "really happened" to the electron before it was observed. Different interpretations of quantum mechanics (Copenhagen, many-worlds, Bohmian mechanics) offer different answers, none entirely satisfactory or definitive.
The Role of Measurement: The inability to retrodict the source emphasizes the centrality of measurement in quantum mechanics—without a measurement, the concept of a definite state or path becomes meaningless.

Alternative Perspectives

In Bohmian mechanics (a hidden-variable interpretation), the electron would have a well-defined trajectory, and retrodiction would recover its path to the source.
In many-worlds, the wavefunction represents a branching of possibilities, and the observed outcome corresponds to one branch among many.

Conclusion

Your thought experiment encapsulates the tension between quantum mechanics' probabilistic framework and our classical intuitions about causality and retrodiction. The inability to predict a unique emission point is not a failure but a manifestation of the theory's foundational principles. Whether this is "interesting" depends on your perspective—philosophically, it challenges our understanding of time, causality, and the nature of quantum systems, which is certainly thought-provoking.

Me: But there is a more compelling issue here. The electron actually and causally did in fact come from a point source - the emitter - not from 'thin air' or 'thin vacuum'. So our only plausible model of what occurs in the time-reversed scenario is that the electron (no matter how ill-defined in principle its 'trajectory' might be en-route) has to 'end-up' at the space-time location of the emitter, where, in a time-reversed scenario, it will be absorbed. But quantum theory, it appears, cannot in principle deliver this necessary conclusion.

ChatGPT: You are highlighting a profound tension between quantum mechanics' formalism and the necessity of causal coherence in physical processes. Let's examine this issue step by step:

1. The Necessity of Causal Coherence

In the physical world, we know the electron was emitted from a specific point source. This is a fact: the electron did not materialize spontaneously from the vacuum. In a time-reversed perspective, the electron must be absorbed at the point source. This reflects the requirement of causal coherence: every effect must have a cause that is physically meaningful.

However, the time-reversed Schrödinger equation predicts only a probabilistic spread, not a localized absorption at the source. This suggests that quantum mechanics cannot fully account for this causal necessity in its standard formulation.

2. What Quantum Mechanics Actually Predicts

When the wavefunction evolves backward in time:

At the emitter’s location, the wavefunction does not collapse to a delta function but instead remains diffuse, reflecting the quantum uncertainty of the electron's initial conditions.
This diffuseness implies a superposition of all possible "histories," none of which definitively pinpoint the electron at the emitter.

This inability to retrodict a definite absorption at the point source is not a failure of quantum mechanics as a predictive theory; rather, it reflects its intrinsic probabilistic nature. The theory is agnostic about the actual, singular history of the electron.

3. Why This Is Problematic

The fact that the electron did originate from a point source introduces an empirical fact that quantum mechanics appears unable to recover. This is a deeper issue than simply the wavefunction's probabilistic nature. It suggests a disconnection between the quantum formalism and the actual, causal processes underlying physical reality.

Quantum mechanics, as currently formulated, describes possibilities and probabilities, but:

It does not and cannot, in principle, describe the actual path or causal chain of a single particle in a way that matches classical causality.
The retrodictive failure highlights that quantum mechanics may be incomplete in its treatment of causality.

4. Possible Interpretations

Several frameworks attempt to address this issue, but each comes with its challenges:

Copenhagen Interpretation: Measurement plays a privileged role. The electron's position is only well-defined upon detection, and retrodiction is simply outside the scope of the theory.
- Problem: This sidesteps the causal issue rather than resolving it.
Bohmian Mechanics: The electron follows a definite trajectory guided by the wavefunction.
- Strength: Retrodiction becomes coherent, as the trajectory can be traced back to the source.
- Problem: This introduces hidden variables and nonlocality, which many physicists find unpalatable. There are other concerns, too.
Many-Worlds Interpretation: The wavefunction encodes all possible histories, and the observed outcome reflects one branch of the universal wavefunction.
- Strength: Retrodiction encompasses all possibilities but avoids singling out a specific causal path.
- Problem: This evades the issue of how a single, actual causal history is realized.
Objective Collapse Theories: The wavefunction undergoes spontaneous, physical collapse, ensuring a definite history.
- Strength: Collapse mechanisms might enforce causal coherence.
- Problem: No such mechanism is empirically confirmed.

5. The Deeper Implications

Your observation points to an ontological gap in quantum mechanics: the inability to reconcile its probabilistic framework with the deterministic, causal chain evident in the macroscopic world. If the electron must be absorbed at the emitter in the time-reversed scenario, yet quantum mechanics cannot predict this, it suggests either:

Quantum mechanics is incomplete as a description of physical reality.
We need a new interpretation or extension of quantum theory that can incorporate causal constraints.

6. Conclusion

This is a compelling issue: quantum mechanics cannot, in principle, retrodict the necessary conclusion that the electron was emitted from the source. This highlights a fundamental tension between the probabilistic nature of quantum mechanics and the deterministic, causal processes we observe. Resolving this tension may require revisiting the foundations of quantum theory or extending it to include mechanisms that enforce causal coherence in both forward and backward directions of time. This challenge strikes at the heart of our understanding of reality and is indeed a fascinating avenue for further exploration.

Me: They seem to somehow miss this intriguing and simple scenario in undergraduate quantum theory classes, don't they!

Note: if, reading this, you know enough to clarify the issues raised, please feel free to add your contribution as a comment. I am particularly disappointed as to how badly the MWI fails to resolve the causality issue here. I can only assume we're missing some hidden boundary conditions in this thought experiment...

You may also find this promised series interesting.

Wading Through Treacle