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arXiv:0802.2297

Predictive approach to some quantum paradoxes

Henryk Gzyl

Published Feb 15, 2008• 1 citation
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Abstract

In classical probability theory, the best predictor of a future observation of a random variable $X,$ is its expected value $E_P[X]$ when no other information is available When information consisting in the observation of another random variable $Y$ is available, then the best predictor of $X$ is another random variable $E_P[X|Y].$ It is the purpose of this note to explore the analogue of this in the case of quantum mechanics. We shall see that exactly as in classical prediction theory, when the result of an observation is taken into account by means of a non-commutative conditional expectation, some of the usual paradoxes cease to be such.

DOI
10.48550/arxiv.0802.2297

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In classical probability theory, the best predictor of a future observation of a random variable X,X,X, is its expected value EP[X]E_P[X]EP​[X] when no other information is available When information consisting in the observation of another random variable YYY is available, then the best predictor of XXX is another random variable EP[X∣Y].E_P[X|Y].EP​[X∣Y]. It is the purpose of this note to explore the analogue of this in the case of quantum mechanics. We shall see that exactly as in classical prediction theory, when the result of an observation is taken into account by means of a non-commutative conditional expectation, some of the usual paradoxes cease to be such.
In classical probability theory, the best predictor of a future observation of a random variable X,X,X, is its expected value EP[X]E_P[X]EP​[X] when no other information is available, whereas if information consisting in the observation of another random variable is available, then the best predictor is is another random variable EP[X∣Y].E_P[X|Y].EP​[X∣Y]. It is the purpose of this note to explore the analogue of this in the case of quantum mechanics. we shall see that when the result of an observation is taken into account by means of a non-commutative conditional expectation, exactly as in classical prediction theory, some of the usual paradoxes cease to be so. The moral of this note is that the mystery of the probabilistic interpretation of quantum mechanics lies in the superposition principle
In this note we discuss a few simple classical (as opposed to quantum) prediction problems. The thrust of this work is to examine the predictive role of probability theory and we shall see that some situations are not re-ally paradoxical. In particular we want to separate the role of probability as a predictive tool from one of the basic sources of strangeness in quan-tum mechanics, namely the principle of superposition of states. For that we shall re-examine some of the quantum paradoxes, recast as a problem about making a prediction about the result of a random experiment in classical physics when some information is known. 1
Given a normalized state-vector ψψψ, we define the conditional expectation $\mathbb{E }_{ψ} (A | B ) $ of a Hermitian operator $A $ with respect to a strongly commuting family of self-adjoint operators $B $ as the best approximation, in the operator mean square norm associated to ψψψ, of $A $ by a real-valued function of $B . $ A fundamental example is the conditional expectation of the momentum operator $P $ given the position operator $X $, which is found to be the Bohm momentum. After developing the Bohm theory from this point of view we treat conditional expectations with respect to general $B $, which we apply to non-relativistic spin 1/2-particles. We derive the dynamics of the conditional expectations of momentum and spin with respect to position and the third spin component. These dynamics can be interpreted in terms of classical continuum mechanics as a two-component fluid whose components carry intrinsic angular momentum. Interpreting the joint spectrum of the conditioning operators as a space of beables, we can introduce a classical-stochastic particle dynamics on this space which is compatible with the time-evolution of the Born probability, by combining the de Broglie-Bohm guidance condition with a Markov jump process, following an idea of J. Bell. This results in a new Bohm-type model for particles with spin. A basic problem is that such auxiliary particle dynamics are far from unique. We finally examine the relation of our conditional expectations with the conditional expectations of the theory of $C^* $-algebras and, as an application, derive a general evolution equation for conditional expectations for operators acting on finite dimensional Hilbert spaces. Two appendices re-interpret the classical Bohm model as an integrable constrained Hamiltonian system, and provide the details of the two-fluid interpretation.
Both classical and respectively quantum observables can be modeled as somewhat similar examples of random variables. In such a model the associated measurements preserve the values spectrum of an observable but change the corresponding probabilistic weights (probability density or respectively the wave function). Such a model ensures theoretical estimations for predicted errors specific to the mean values as well as to the fluctuations of both types of observables. The model stands out of ourdays prevalent opinion that measurement theories must be depicted in entirely different manners in the classical respectively quantum cases.
Abstract Given the state of a quantum system, we define the conditional expectation of an observable with respect to a second observable, or commuting family of observables, as the real-valued function of the latter which best approximates the observable, in the sense of minimizing the quantum mechanical expectation of the difference squared. This corresponds to the notion of best predictor of a random variable by another random variable in non-parametric statistical regression. A basic example is the conditional expectation of the quantum mechanical momentum with respect to position, which can be identified with the Bohm momentum. We re-examine the de Broglie–Bohm theory from this perspective. As a second example we consider the simplest quantum system, that of a spin- 1 / 2 paticle in a magnetic field. We study the conditional expectations of the Pauli spin-operators S 1 , S 2 , S 3 with respect to S 3 , and determine their dynamics if the pure-state wave-function solves the relevant Schrödinger equation. Considering S 3 as a beable we can, following Bell, introduce a generalized guidance law in the form of a Markov jump process on the spectrum of S 3 which is compatible with the time evolution of the Born probabilities of the two eigenstates of S 3 . The conditional expectations of the remaining spin operators are then shown to follow a piece-wise deterministic Markov process, defined by an explicit SDE. We finally examine the relationship between the conditional expectations of this paper, and the von Neumann-algebraic conditional expectation onto the sub-algebra generated by the commuting family of conditioning operators when the joint spectrum of the latter is discrete, and derive a general expression for the conditional expectation dynamics in that context.