1. This paper, although mathematical in content, is motivated by quantumtheoretical considerations. The states of a quantum-mechanical system are usually described by vectors f of norm 1 in some Hilbert space A, and we assume explicitly that to every unit vector f corresponds a state of the system. This correspondence, however, is not one-to-one. In fact, the vectors which describe the same state form a ray f (in Weyl's terminology, cf. [13], p. 4 and p. 20),1 i.e. a set consisting of all vectors f = Tfo where fo is a fixed unit vector in & and r any complex number of modulus 1. (Every vector f in f will be called a representative of the ray f.) We have therefore a one-to-one correspondence between quantum states and rays, and every significant statement in Quantum Theory is a statement about rays. The transition probability from a state f to a state g equals (f, I)'2 where f, g are representatives of the rays f, g respectively. This suggests the introduction of the inner product of two rays by the definition
This paper, although mathematical in content, is motivated by quantumtheoretical considerations. The states of a quantum-mechanical system are usually described by vectors f of norm 1 in some Hilbert space A, and we assume explicitly that to every unit vector f corresponds a state of the system. This correspondence, however, is not one-to-one. In fact, the vectors which describe the same state form a ray f (in Weyl's terminology, cf. [13], p. 4 and p. 20),1 i.e. a set consisting of all vectors f = Tfo where fo is a fixed unit vector in & and r any complex number of modulus 1. (Every vector f in f will be called a representative of the ray f.) We have therefore a one-to-one correspondence between quantum states and rays, and every significant statement in Quantum Theory is a statement about rays. The transition probability from a state f to a state g equals (f, I)'2 where f, g are representatives of the rays f, g respectively. This suggests the introduction of the inner product of two rays by the definition
In the usual formulation of quantum mechanics, groups of automorphisms of quantum states have ray representations by unitary and antiunitary operators on complex Hilbert space, in accordance with Wigner's theorem. In the phase-space formulation, they have real, true unitary representations in the space of square-integrable functions on phase space. Each such phase-space representation is a Weyl–Wigner product of the corresponding Hilbert space representation with its contragredient, and these can be recovered by 'factorizing' the Weyl–Wigner product. However, not every real, unitary representation on phase space corresponds to a group of automorphisms, so not every such representation is in the form of a Weyl–Wigner product and can be factorized. The conditions under which this is possible are examined. Examples are presented.
There is a notational discrepancy in how mathematicians and physicists treat Hilbert spaces. In physics, one often works with finite-dimensional Hilbert spaces treated as ℂ n , and hence wants to write the Hermitian inner product as (x, y) = x ∗ y. This definition dictates the use of a Hermitian inner product that is antilinear in the first argument, the convention followed, for example, in Reed & Simon [246]. This is also the choice adopted in Dirac’s bracket notation, whose usage in quantum mechanics is very widespread. Our notation is chosen to extend-as closely as possible-the traditional notation of standard finite-dimensional matrix algebra to arbitrary complex inner product spaces and associated linear operators. In matrix algebra, column vectors and the corresponding matrices with one column are identical objects, row vectors are the linear functionals, and the adjoint is the conjugate transpose. For example, ℍ = ℍ × = ℂ n is the space of column vectors of size n, the dual space ℍ ∗ is the space of row vectors of size n, and the operator product ϕ ∗ ψ of a row vector ϕ ∗ and a column vector ψ is the standard Hermitian inner product of the column vectors ϕ and ψ. We use Greek lower case letters to write vectors, thus emphasizing their intended use as quantum state vectors in quantum mechanics. On the other hand, mathematicians working on reproducing kernel Hilbert spaces use an inner product (x, y) antilinear in the second argument, related to the physicist’s inner product (x, y) by (x, y) =( ). This is the convention followed, for example, in Rudin [252]. Although the two ways of defining the inner product lead to fully equivalent theories, all details look a bit different, a fact that has to be taken into account when reading the literature on the subject. For example, in the description based on the physical tradition, it is preferable to work with the antidual space in place of the dual space used in the mathematical tradition. In functional analysis, linear operators in Hilbert spaces are usually considered each with their own domain. But many computations in quantum mechanics require the consideration of algebras of operators with a common domain. The latter is a Euclidean space, a dense subspace of a Hilbert space. This space and its antidual play in many respects a more basic role in quantum physics than the Hilbert space itself. Therefore, and to avoid possible confusion caused by the different traditions, we give in the present chapter a self-contained introduction to Euclidean spaces and their associated spaces. All proofs are carried out in detail.
The most general continuous time-dependent evolution of a physical system is represented by a continuous one-parameter semi-group of linear mappings of density operators to density operators. It is shown that if these dynamical mappings form a group they can be represented by a group of unitary operators on the Hilbert space of state vectors. This proof does not assume that the absolute values of inner products of state vectors or ``transition probabilities'' are preserved but deduces this fact from the requirement that density operators are mapped linearly to density operators. An example is given of a continuous one-parameter semi-group of dynamical mappings which is not a group.
It is perhaps the most fundamental principle of Quantum Mechanics that the system of states forms a linear manifold,1 in which a unitary scalar product is defined.2 The states are generally represented by wave functions3 in such a way that φ and constant multiples of φ represent the same physical state. It is possible, therefore, to normalize the wave function, i.e., to multiply it by a constant factor such that its scalar product with itself becomes 1. Then, only a constant factor of modulus 1, the so-called phase, will be left undetermined in the wave function. The linear character of the wave function is called the superposition principle. The square of the modulus of the unitary scalar product (ψ,Φ) of two normalized wave functions ψ and Φ is called the transition probability from the state ψ into Φ, or conversely. This is supposed to give the probability that an experiment performed on a system in the state Φ, to see whether or not the state is ψ, gives the result that it is ψ. If there are two or more different experiments to decide this (e.g., essentially the same experiment, performed at different times) they are all supposed to give the same result, i.e., the transition probability has an invariant physical sense.
1. Introduction.In quantum field theory we must consider the Hilbert space having non-countable bases which corresponds to a sequence of non-negative integers (n, n.,...).Since we can construct one-to-one mapping from the set of the sequences (n, n,...) onto the points in 0, 1 interval [8, we can identify these bases to [0, 1 interval.Let 7 denote a point in 0, 1 interval and let r be the element of the Hilbert space which corresponds to y.The element of this Hilbert space is usually represented by the formulae flCrrdt( 7) and Cr, in [3], E4] and [6], where C, Cr are constants, and d/(') is a measure on E0, 1].By single dz(r), however, we cannot represent every element of this Hilbert space.That is to say, by a continuous measure dz('), we cannot represent the element of the second form.On the other hand by the second form, we cannot represent the element of the first form.
Come forth into the light of things, Let nature be your teacher. William Wordsworth , The Tables Turned The main focus of this book is many-particle systems such as electrons in a crystal. Such systems are studied within the framework of quantum mechanics, with which the reader is assumed to be familiar. Nevertheless, a brief review of this subject will provide an opportunity to establish notation and collect results that will be used later on. The postulates Quantummechanics is based on five postulates, listed below with some explanatory comments. The quantum state The quantum state of a particle, at time t , is described by a continuous, singlevalued, square-integrable wave function Ψ( r , t ), where r is the position of the particle. In Dirac notation, the state is represented by a state vector, or ket, ∣Ψ( t )⟩, which is an element of a vector space V. We define a dual vector space V* whose elements, called bras, are in one-to-one correspondence with the elements of V: ket ∣α⟩ ∈ V ⟷ bra ⟨α∣ ∈ V*, as illustrated in Figure 1.1. The bra corresponding to ket c ∣α⟩ is c * ⟨∣, where c *. is the complex conjugate of c . The inner product of kets ∣α⟩ and ∣β⟩ is denoted by ⟨β∣α⟩, and it is a complex number (c-number). Note that the inner product is obtained by combining a bra and a ket. By definition, ⟨β∣α⟩ = ⟨α∣β⟩*.