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Texts and readings in mathematics

Systems with a Configuration Under a Group Action

Kalyanapuram Rangachari Parthasarathy

Published Jan 1, 2005• 0 citations
Publisher

Abstract

We start with a heuristic description of a quantum mechanical system which admits ‘position’ as an observable in the sense of Definition 1.2.1. Suppose the position of the system can be identified with a point x in an abstract space X with the Borel structure B $$ \mathfrak{B} $$ . We can then speak of the event ‘position x ϵ E’ or ‘x ϵ E’ for any E ϵ B $$ \mathfrak{B} $$ . Suppose that ‘all’ the events concerning the system are described by the lattice P $$ \mathcal{P} $$ (H) $$ (\mathcal{H}) $$ of all orthogonal projections on a Hilbert space H $$ \mathcal{H} $$ . Then the event ‘x ϵ E’ can be identified with an element P(E) ϵ P $$ \mathcal{P} $$ (H) $$ (\mathcal{H}) $$ . Then the existence of a position observable is equivalent to the fact that the map E → P(E) from B $$ \mathfrak{B} $$ to P $$ \mathcal{P} $$ (H) $$ (\mathcal{H}) $$ is a spectral measure.

DOI
10.1007/978-93-86279-28-6_2

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We start with a heuristic description of a quantum mechanical system which admits ‘position’ as an observable in the sense of Definition 1.2.1. Suppose the position of the system can be identified with a point x in an abstract space X with the Borel structure B B\mathfrak{B}B . We can then speak of the event ‘position x ϵ E’ or ‘x ϵ E’ for any E ϵ B B\mathfrak{B}B . Suppose that ‘all’ the events concerning the system are described by the lattice P P\mathcal{P}P (H) (H)(\mathcal{H})(H) of all orthogonal projections on a Hilbert space H H\mathcal{H}H . Then the event ‘x ϵ E’ can be identified with an element P(E) ϵ P P\mathcal{P}P (H) (H)(\mathcal{H})(H) . Then the existence of a position observable is equivalent to the fact that the map E → P(E) from B B\mathfrak{B}B to P P\mathcal{P}P (H) (H)(\mathcal{H})(H) is a spectral measure.
It is proposed that an unconstrained quantum system is completely specified by its configuration space CCC and its group of symmetries GGG. It is then shown that the theory of projective unitary representations of GGG on a certain Hilbert space determined by CCC and GGG leads to a definitive resolution of supposed ambiguities relating to superselection rules, choice of wave functions, topological properties of GGG and CCC, anomalies, etc.
The quantities in physics that can be measured are conventionally denoted observables : entities that can be observed (directly or indirectly) and to which a value in appropriate units can be assigned. Examples include position, momentum, and energy. Unlike in classical physics, the concept of an observable in quantum theory possesses a significance beyond that of the measurable: to each observable there corresponds a unique operator in an abstract, linear vector space. These operators (and others) govern all aspects of quantum systems. Quantum operators act on abstract vectors in the linear space. Especially important are the eigenvectors and eigenvalues of these operators, in particular because some of the eigenvectors are the quantum states mentioned in the preceding chapters. Furthermore, for any operator that is the mathematical image of an observable, its eigenvalues constitute the entire set of values that measurement of the observable can yield . Finally, from the quantum states one can obtain the probabilities for the occurrence of events in the microscopic world, for example, the most likely value that measurement of an observable will yield. The concepts of operators and eigenvalue problems, as well as those of quantal probabilities, of Hermiticity, and of complete sets, were encountered and exemplified in the preceding chapter using the language of differential equations. While much of quantum mechanics could be presented in terms of this language alone, certain fundamental concepts (spin, for instance) cannot be, and it eventually becomes necessary to consider quantum mechanics as a theoretical framework embedded in that abstract linear vector space known as a Hilbert space.
Suppose (Ω i , F i ), 1 ≤ i ≤ n are sample spaces describing the elementary outcomes and events concerning n different statistical systems in classical probability. To integrate them into a unified picture under the umbrella of a single sample space one takes their cartesian product (Ω, F) where Ω = Ω1 x … x Ω n , F = F1 x … x F n , the smallest σ-algebra containing all rectangles of the form F1 x F1 x … F n , F j ∈ F j for each j. Now we wish to search for an analogue of this description in quantum probability when we have n systems where the events concerning the j-th system are described by the set P(ℋ j ) of all projections in a Hubert space ℋ j j = 1, 2,…, n. Such an attempt leads us to consider tensor products of Hilbert spaces. We shall present a somewhat statistically oriented approach to the definition of tensor products which is at the same time coordinate free in character. To this end we introduce the notion of a positive definite kernel.
This paper presents a generalization of quantum mechanics from conventional Hilbert space formalism to Banach space one. We construct quantum theory starting with any complex Banach space beyond a complex Hilbert space, through using a basic fact that a complex Banach space always admits a semi-inner product. Precisely, in a complex Banach space X\mathbb{X}X with a given semi-inner product, a pure state is defined by Lumer \cite{Lumer1961} to be a bounded linear functional on the space of bounded operators determined by a normalized element of X\mathbb{X}X under the semi-inner product, and then the state space S(X)\mathcal{S} (\mathbb{X})S(X) of the system is the weakly closed convex set spanned by all pure states. Based on Lumer's notion of the state, we associate a quantum system with a complex Banach space X\mathbb{X}X equipped with a fixed semi-inner product, and then define a physical event at a quantum state ω∈S(X)ω\in \mathcal{S}(\mathbb{X})ω∈S(X) to be a projection PPP (bounded operator such that P2=PP^2 =PP2=P) in X\mathbb{X}X satisfying the positivity condition 0≤ω(P)≤1,0 \le ω(P) \le 1,0≤ω(P)≤1, and a physical quantity at a quantum state ωωω to be a spectral operator of scalar type with real spectrum so that the associated spectral projections are all physical events at ω.ω.ω. The Born formula for measurement of a physical quantity is the natural pairing of operators with linear functionals satisfying the probability conservation law. A time evolution of the system is governed by a one-parameter group of invertible spectral operators determined by a scalar type operator with the real spectrum, which satisfies the Schrödinger equation. Our formulation is just a generalization of the Dirac-von Neumann formalism of quantum mechanics to the Banach space setting. We include some examples for illustration.