# An argument for hidden variables

Detlef Dürr, Shelly Goldstein, and Nino Zanchí once gave a very interesting argument for hidden variables. I’ll give their argument a careful reconstruction. But first, here’s what they say.

According to the quantum formalism, measurements performed on a quantum system with definite wave function ψ typically yield random results. Moreover, even the specification of the wave function of the composite system including the apparatus for performing the measurement will not generally diminish this randomness. However, the quantum dynamics governing the evolution of the wave function over time, at least when no measurement is being performed, and given, say, by Schrödinger’s equation, is completely deterministic. Thus, insofar as the particular physical processes which we call measurements are governed by the same fundamental physical laws that govern all other processes, one is naturally led to the hypothesis that the origin of the randomness in the results of quantum measurements lies in random initial conditions, in our ignorance of the complete description of the system of interest — including the apparatus — of which we know only the wave function.

I’m going to interpret their “randomness” to mean the lack of a determinate value at a given time. Now, here’s my reconstruction.

- If the laws governing a physical process are deterministic, and its initial conditions are completely specified, then our description of the physical process is guaranteed to have determinate values at any given time.
- The wave-function description of the measurement processes is *not* guaranteed to have determinate values at any given time.
- All physical processes are governed by the same fundamental physical laws (and hence by
*equally deterministic*equations of motion). - So, since Schrödinger evolution is governed by deterministic laws, the measurement process must be governed by deterministic laws as well.
- But since the measurement process does not have determinate values, this implies by (1) that the initial conditions of the measurement process are not completely specified, when given by the wave function alone.

This argument is very different than the kind of locality complaint espoused by Einstein. And it’s part of what leads these authors to adopt Bohmian mechanics, which supplements the “unspecified initial conditions” allowed by quantum mechanics with exact particle positions.

As much as I’d like to be convinced, I just don’t understand the motivation for premise (3). The Schrödinger equation is deterministic. But the authors want to conclude that whatever basic fundamental law governs *both* Schrödinger evolution and measurement must therefore also be deterministic. Why?

The authors don’t say in this article. And the following seems to be a counterexample: Measurement in quantum mechanics is indeterministic. But because of Ehrenfest’s theorem, it still (on average) satisfies a deterministic law. So, a single *indeterministic* law appears able to give rise to *deterministic* law-like behavior. Therefore, deterministic law-like behavior (e.g., Schrödinger evolution) doesn’t imply that *all more fundamental laws* are also deterministic. So premise (3) fails. Right?

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Eh, not entirely convincing, I think. First, they fail to rule out quantum chaos as a contributor, second they fail to rule out the influence of external stochastic variables. In the first case, small imprecisions in the specification of the wave function should be magnified to make for an unpredictable outcome (at least if quantum chaos is like classical chaos, and I confess I have no idea if this is actually the case). In the second case the influences of ‘external’ forces is sufficient to cause indeterminacy the same way thermal noise does classically.

In either case the variables aren’t so much ‘hidden’ as either beyond our reach or ignored out of necessity.

Good — it does seem this argument leads just as naturally to dynamical collapse theories like GRW. Knowledge of the stochastic parameter then becomes the “missing information.”

I don’t quite see how quantum chaos can help though. Care to elaborate? How does this explain the indeterminism observed (say) in a Stern-Gerlach measurement?

It looks like (3) is something you would want at first glance. If you are in the business of finding fundamental physical laws then wouldn’t you want to find some very basic fundamental law that governs all the physical processes?

Sure Noah — but that’s not enough to imply their conclusion. I don’t see how their argument can work unless they assume that if one consequence of the fundamental law is deterministic (e.g., Schrodinger evolution), then every other consequence is as well (e.g., the measurement process). And that assumption seems wrong, for the reasons I outlined above.

The lack of an argument for (3) leads me to believe that they believe it on principle. My only concern is that your counter-example won’t impress someone who holds (3) on principle. Still, this is their problem, not yours.

Excellent point about the SG experiment, I had forgot to consider that. Combined with Bell’s inequalities they show that no amount of variables hidden within the wave function being measured can account for the quantum indeterminacy. That, in my mind, forces the possibility that many worlds works (at least sometimes).

So it would seem that chaos can only serve to magnify the role of the external stochastic variables…