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arXiv:2411.08532

Conditional expectations in Quantum Mechanics and causal interpretations: the Bohm momentum as a best predictor

Raymond Brummelhuis

Published Nov 13, 2024• 0 citations
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Abstract

Given a normalized state-vector $ψ$, 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 $ψ$, 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.

DOI
10.48550/arxiv.2411.08532

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Given a normalized state-vector ψψψ, we define the conditional expectation Eψ(A∣B)\mathbb{E }_{ψ} (A | B ) Eψ​(A∣B) of a Hermitian operator AA A with respect to a strongly commuting family of self-adjoint operators BB B as the best approximation, in the operator mean square norm associated to ψψψ, of AA A by a real-valued function of B.B . B. A fundamental example is the conditional expectation of the momentum operator PP P given the position operator XX 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 BB 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∗C^* 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.
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.
de Broglie-Bohm theory (dBBT), treating quantum particles as point objects moving along well defined (Bohmian) trajectories, offers an appealing solution of the measurement problem in quantum mechanics; it has, however, problems relating to spin, relativity and lack of proper integration with the Hilbert space based framework. In this work, we present a consistent formalism which has the traditional state-observable framework integrated with the desirable features of dBBT. We adopt ensemble interpretation for the Schrodinger wave function ψψψ. Given a Schrodinger wave function ψψψ, we use its value ψ0ψ_0ψ0​ at some fixed time (say, t=0t = 0t=0) to define the probability measure ∣ψ0∣2dx|ψ_0|^2 {\rm d}x∣ψ0​∣2dx on the system configuration space MMM (=Rn=\mathbb{R}^n=Rn). On the resulting probability space M0\mathcal{M}_0M0​, we introduce a stochastic process ξ(t)ξ(t)ξ(t) corresponding to the Heisenberg position operator XH(t)X_H(t)XH​(t) such that, in the Heisenberg state ∣ψh⟩|ψ_h\rangle∣ψh​⟩ corresponding to ψ0ψ_0ψ0​, the expectation value of XH(t)X_H(t)XH​(t) equals that of ξ(t)ξ(t)ξ(t) in M0\mathcal{M}_0M0​. This condition leads to the de Broglie-Bohm guidance equation for the sample paths of the process ξ(t)ξ(t)ξ(t) which are, therefore, Bohmian trajectories supposedly representing time-evolutions of individual members of the ψ0ψ_0ψ0​-ensemble. Stochastic processes and Bohmian trajectories corresponding to observables with discrete eigenvalues (in particular spin) are treated by extending the configuration space to the spectral space of the commutative algebra obtained by adding appropriate discrete observables to the position observables. Pauli's equation is treated as an example. A straightforward derivation of von Neumann's projection rule employing the Schrodinger-Bohm evolution of individual systems along their Bohmian trajectories is given. Some comments on the potential application of the formalism developed here to quantum mechanics of the universe are included.
A type of mechanics will be presented that possesses some distinctive properties. On the one hand, its physical description & rules of operation are readily comprehensible & intuitively clear. On the other, it fully satisfies all observable predictions of non-relativistic quantum mechanics. Within it, particles exist at points in space, follow continuous, piecewise differentiable paths, and their linear momentum is equal to their mass times their velocity along their path. Yet the probabilities for position and momentum, conditioned on the state of the particle's environment, follow the rules of quantum theory. Indeed, all observable consequences of quantum theory are satisfied; particles can be entangled, have intrinsic spin, this spin is not local to the particle, particle identity can effect probabilities, and so forth. All the rules of quantum mechanics are obeyed, and all arise in a straightforward fashion. After this is established, connections will be drawn out between this type of mechanics and other types of quantum worlds; those that obey Bohmian mechanics, stochastic mechanics, the many worlds interpretation, and physical collapse. In the final section, a relativistic version of the mechanics will be presented.
The non-relativistic quantum theory has been interpreted causally by de Broglie, David Bohm, and others, where a quantum entity is viewed as a particle with a definite position and momentum. This interpretation contrasts with the Copenhagen orthodox interpretation, which conveys uncertainty through the commutator relationship between position and momentum operators. This is further exacerbated by the de Broglie-Bohm interpretation, which uses no operators corresponding to these observables. We reconcile these opposing viewpoints by introducing mathematical nonlinear transforms in Bohmian mechanics for the momentum observable. While these nonlinear operator-like transforms cannot be easily represented as matrices in the Hilbert space, we show that they can be utilized in the de Broglie-Bohm interpretations of quantum theory.
As mentioned previously, a classical Markov semigroup of transition operators { T t , t ≥ 0} can be generated from a classical Markov process { X t , t ≥ 0}. On the other hand, given a Markov semigroup, a Markov process { X t , t ≥ 0} can be constructed based on the Markov semigroup { T t , t ≥ 0} using the Kolmogorov consistency theorem. In the context of quantum probability space ( A , ρ ), a Markov dilation (i.e., construction of a quantum Markov process from a given quantum Markov semigroup of operators { I t , t ≥ 0}) turns out to be a nontrivial matter. This is partly due to the fact that, although the general concept of conditional expectation of an observable given a sub-von Neumann algebras B ⊂ A can be defined and required properties can be described, an explicit construction of such a conditional expectation is still unavailable in general. The main objective of this chapter is to introduce relevant concepts and to develop some properties of quantum Markov processes in the content of quantum probability explored in previous chapters. There are 2 major components in this chapter: (i) introduction of concepts and derivation of properties of a quantum Markov processes based on some assumed and/or derived properties of conditional expectation E ρ [· | A t ] ] given a filtration of sub-von Neumann algebras { A t ] , t ≥ 0} of A , and (ii) Markov dilation or construction of a weak quantum Markov flow (WQMF) from a given quantum Markov semigroup { I t , t ≥ 0} using the weak conditional expectation E [· | F t ], where { F t , t ≥ 0} is a filtration of orthogonal projection operators defined on the complex Hilbert space ℍ. The main objective of the first component is to introduce the relevant concept and develop some properties of quantum Markov processes in the content of quantum probability explored in previous chapters. Although the theory of (classical) Markov processes is well known in the probability community (see, for example, Dynkin [Dyn62], Ethier and Kurtz [EK85]), we outline some basic concept and results from the classical theory of homogeneous Markov processes below for comparison purpose.
We begin with a study of operations and the effects they measure. We define the probability that an effect aaa occurs when the system is in a state ρ\rhoρ by Pρ(a)=Tr(ρa)P_{\rho}(a)=\textrm{Tr}(\rho a)Pρ​(a)=Tr(ρa). If Pρ(a)≠0P_{\rho}(a)\ne0Pρ​(a)=0 and I\mathcal{I}I is an operation that measures aaa, we define the conditional probability of an effect bbb given aaa relative to I\mathcal{I}I by Pρ(b∣a)=Tr[I(ρ)b]/Pρ(a)P_{\rho}(b\mid a)=\textrm{Tr}[\mathcal{I}(\rho)b]/P_{\rho}(a)Pρ​(b∣a)=Tr[I(ρ)b]/Pρ​(a). We characterize when Bayes' quantum second rule Pρ(b∣a)=Pρ(b)Pρ(a) Pρ(a∣b)P_{\rho}(b\mid a)=\frac{P_{\rho}(b)}{P_{\rho}(a)}\,P_{\rho}(a\mid b)Pρ​(b∣a)=Pρ​(a)Pρ​(b)​Pρ​(a∣b) holds. We then consider Lüders and Holevo operations. We next discuss instruments and the observables they measure. If AAA and BBB are observables and an instrument I\mathcal{I}I measures AAA, we define the observable BBB conditioned on AAA relative to I\mathcal{I}I and denote it by (B∣A)(B\mid A)(B∣A). Using these concepts, we introduce Bayes' quantum first rule. We observe that this is the same as the classical Bayes' first rule, except it depends on the instrument used to measure AAA. We then extend this to Bayes' quantum first rule for expectations. We show that two observables BBB and CCC are jointly commuting if and only if there exists an atomic observable AAA such that B=(B∣A)B=(B\mid A)B=(B∣A) and C=(C∣A)C=(C\mid A)C=(C∣A). We next obtain a general uncertainty principle for conditioned observables. Finally, we discuss observable conditioned quantum entropies. The theory is illustrated with many examples. Quanta 2024; 13: 1–10.