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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- Accuracy, Applicability, and Tarskian Semantics
- Overheard at New Directions in Foundations of Physics ’10