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 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 under the semi-inner product, and then the state space
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 equipped with a fixed semi-inner product, and then define a physical event at a quantum state
ω∈S(X) to be a projection
P (bounded operator such that
P2=P) in
X satisfying the positivity condition
0≤ω(P)≤1, and a physical quantity at a quantum state
ω to be a spectral operator of scalar type with real spectrum so that the associated spectral projections are all physical events at
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.
A new calculus for functions of noncommuting operators is developed, based on the notion of mapping of functions of operators onto
c-number functions. The class of linear mappings, each member of which is characterized by an entire analytic function of two complex variables, is studied in detail. Closed-form solutions for such mappings and for the inverse mappings are obtained and various properties of these mappings are studied. It is shown that the most commonly occurring rules of association between operators and
c-numbers (the Weyl, the normal, the antinormal, the standard, and the antistandard rules) belong to this class and are, in fact, the simplest ones in a clearly defined sense. It is shown further that the problem of expressing an operator in an ordered form according to some prescribed rule is equivalent to an appropriate mapping of the operator on a
c-number space. The theory provides a systematic technique for the solution of numerous quantum-mechanical problems that were treated in the past by ad hoc methods, and it furnishes a new approach to many others. This is illustrated by a number of examples relating to mappings and ordering of operators.
An overcomplete family of states (OFS) is constructed for a countably infinite linear vector space with an indefinite metric for the case that the metric is diagonal with eigenvalues (−1)n, where n is an integer. A continuous representation is indicated and the properties of a semiclassical description of a quantum mechanical system (the pseudo oscillator whose creation and destruction operators ā and ā+ satisfy [ā, ā+] = −1) defined in this vector space are studied. It is found that a consistent OFS |z> can be constructed if the operator G(z) which generates the state |z> from the vacuum is unitary. Furthermore, with the statistical state of this system specified by a bounded pseudo-Hermitian density matrix ρ̄, the related semiclassical complex function ρA(z) for antinormal ordering of operators in the indefinite metric space is found to be bounded, with ρA(z) and [ρA(z)]2 integrable, continuous, and a boundary value of an entire analytic function of two complex variables. The semiclassical function ρN(z) for normal ordering is associated with a sequence of functions ρN(ν)(z) whose square is integrable and related to a sequence of tempered distributions ρN(ν) such that the corresponding sequence of density matrices ρ̄(ν) converges to ρ̄ in the norm.