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Journal of Physics Conference Series

Wasserstein space as state space of quantum mechanics and optimal transport

Muhammad Farchani Rosyid, K Wahyuningsih

Published Mar 1, 2019• 0 citations
Publisher

Abstract

In this work, we are in the position to view a measurement of a physical observable as an experiment in the sense of probability theory. To every physical observable, a sample space called the spectrum of the observable is therefore available. We have investigated the Wasserstein spaces over the spectrums of a quantum observables, i.e. the space of all probability measures defined on the suitable σ -algebra of subsets of the spectrum equipped with certain metric. It has been shown here the existence of one-to-one correspondence between the set of all quantum states and the set of all distribution functions defined on the spectrum of an observable. Thanks to the correspondence, every quantum state can be represented by a probability measure associated with the corresponding distribution function. It has been shown that the Wasserstein space over the spectrum of an observable is homeomorphic to the Wasserstein space over the spectrum of another observable. Such homeomorphism is referred to as generalized Fourier transform. By using the work of von Renesse, it has been shown that the Newtonian dynamics of the system can be formulated in the Wasserstein space over the spectrum of each observable which is equivalent to the Schroedinger dynamics, starting from the solution of Newton equation of motion in the Wasserstein space over the spectrum of a certain observable.

DOI
10.1088/1742-6596/1170/1/012006

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In this work, we are in the position to view a measurement of a physical observable as an experiment in the sense of probability theory. To every physical observable, a sample space called the spectrum of the observable is therefore available. We have investigated the Wasserstein spaces over the spectrums of a quantum observables, i.e. the space of all probability measures defined on the suitable σ -algebra of subsets of the spectrum equipped with certain metric. It has been shown here the existence of one-to-one correspondence between the set of all quantum states and the set of all distribution functions defined on the spectrum of an observable. Thanks to the correspondence, every quantum state can be represented by a probability measure associated with the corresponding distribution function. It has been shown that the Wasserstein space over the spectrum of an observable is homeomorphic to the Wasserstein space over the spectrum of another observable. Such homeomorphism is referred to as generalized Fourier transform. By using the work of von Renesse, it has been shown that the Newtonian dynamics of the system can be formulated in the Wasserstein space over the spectrum of each observable which is equivalent to the Schroedinger dynamics, starting from the solution of Newton equation of motion in the Wasserstein space over the spectrum of a certain observable.
In this article we define and build the one method of description of stochastic evolution of a physical quantum system.For each quantum state U E ω ∈ we construct the probability measure ω µ in the space ( ) , U P S , where U P is the space of the pure states of the quantum system, S the Borel σ -algebra in U P .Farther, for any Hermit's positive element with norm 1 u = , in the C * -algebra of observables U , we define the probability measure u µ on the set of states U E .If strongly conti- nuous group { } t α of * automorphisms on U describes the evolution of struc- ture of observables, according to this, we have a picture of evolution of distribution of states of quantum system relatively to each observable u .
Newtonian and Schrodinger dynamics can be formulated in a physically meaningful way within the same Hilbert space framework. This fact was recently used to discover an unexpected relation between classical and quantum motions that goes beyond the results provided by the Ehrenfest theorem. The Newtonian dynamics was shown to be the Schrodinger dynamics of states constrained to a submanifold of the space of states, identified with the classical phase space of the system. Quantum observables are identified with vector fields on the space of states. The commutators of observables are expressed through the curvature of the space. The resulting embedding of the Newtonian and Schrodinger dynamics into a unified geometric framework is rigid in the sense that the Schrodinger dynamics is a unique extension of the Newtonian one. Under the embedding, the normal distribution of measurement results associated with a classical measurement implies the Born rule for the probability of transition of quantum states. The mathematics of the discovered relationship between the classical and the quantum is reviewed and investigated here in detail, and applied to the process of measurement of spin and position observables.
Newtonian and Schrödinger dynamics can be formulated in a physically meaningful way within the same Hilbert space framework. This fact was recently used to discover an unexpected relation between classical and quantum motions that goes beyond the results provided by the Ehrenfest theorem. The Newtonian dynamics was shown to be the Schrödinger dynamics of states constrained to a submanifold of the space of states, identified with the classical phase space of the system. Quantum observables are identified with vector fields on the space of states. The commutators of observables are expressed through the curvature of the space. The resulting embedding of the Newtonian and Schrödinger dynamics into a unified geometric framework is rigid in the sense that the Schrödinger dynamics is a unique extension of the Newtonian one. Under the embedding, the normal distribution of measurement results associated with a classical measurement implies the Born rule for the probability of transition of quantum states. The mathematics of the discovered relationship between the classical and the quantum is reviewed and investigated here in detail and applied to the process of measurement of spin and position observables.
In this work we discuss the notion of observable - both quantum and classical - from a new point of view. In classical mechanics, an observable is represented as a function (measurable, continuous or smooth), whereas in (von Neumann's approach to) quantum physics, an observable is represented as a bonded selfadjoint operator on Hilbert space. We will show in the present part II and the forthcoming part III of this work that there is a common structure behind these two different concepts. If R\mathcal{R}R is a von Neumann algebra, a selfadjoint element A∈RA \in \mathcal{R}A∈R induces a continuous function fA:Q(P(R))→Rf_{A} : \mathcal{Q}(\mathcal{P(R)}) \to \mathbb{R}fA​:Q(P(R))→R defined on the \emph{Stone spectrum} Q(P(R))\mathcal{Q}(\mathcal{P(R)})Q(P(R)) (\cite{deg3}) of the lattice P(R)\mathcal{P(R)}P(R) of projections in R\mathcal{R}R. fAf_{A}fA​ is called the observable function corresponding to AAA. The aim of this part is to study observable functions and its various characterizations.
A generalized probability theory arises naturally from an attempt to formulate a statistical model for quantum mechanics.One soon finds that the probability theory of Kolmogorov is inadequate for the description of a quantum mechanical system mainly because the quantum mechanical events fail to form a er-algebra of subsets of a set.Instead, they have a much less richly endowed algebraic structure which we shall a "logic".We therefore look upon generalized probability as a stochastic theory in which the events form a logic.Besides being useful in the study of quantum mechanics, this theory has a purely mathematical interest since it constructs a stochastic formalism on an algebraic structure and at the same time gives an abstract generalization of Hubert space theory.The first formulation of this theory is due to G. Bodiou [1], [2].More recently, V. S. Varadarajan [8] has emphasized the stochastic properties of an axiomatic formulation of quantum mechanics due to G. W. Mackey [5] and has proved some important results in this theory.The author heartily recommends these references for the background and motivation of this study.In this paper, the main interest centers on the spectral theory of the subject.The spectrum is important physically since it represents the "allowable" values of a quantum mechanical observable.It is also mathematically useful since it introduces topological methods into the formalism.2. Definitions and notation.Let L be a complemented lattice.That is, L is a lattice with first and last elements, 0,1 respectively and a-*a' satisfying the conditions (i) ia'Y = a for all a eL;(ii) a z% b implies b' zi a'; (iii) a V a ' -1 for all a e L.
It has been recently shown that Newtonian dynamics is the Schrodinger dynamics of the system whose state is constrained to a submanifold in the space of states of the system. Thus defined, the submanifold can be identified with the classical phase space of the system. The classical space is then also embedded into the space of states in a physically meaningful way. The resulting unified geometric framework establishes a new connection between classical and quantum physics. The framework is rigid in the sense that the Schrodinger dynamics is a unique extension of the Newtonian one from the classical phase space submanifold to the space of states. Quantum observables in the framework are identified with vector fields on the space of states. The commutators of canonical conjugate observables are expressed through the curvature of the sphere of normalized states. The velocity and acceleration of a particle in Newtonian dynamics are components of the velocity of state under the corresponding Schrodinger evolution. The metric properties of the embedding of the classical space into the space of states result in a relationship between the normal distribution of the position of a particle and the Born rule for the probability of transition of quantum states. In this paper the implications of the obtained mathematical results to the process of measurement in quantum physics are investigated. It is argued that interaction with the environment constrains the state of a macroscopic body to the classical space. The notion of collapse of a quantum state is analyzed. The double-slit, EPR and Schrodinger cat type experiments are reviewed anew. It is shown that, despite reproducing the usual results of quantum theory, the framework is not simply a reformulation of the theory. New experiments to discover the predicted effects are proposed.