It is shown that if ℍ is a Hilbert space for a representation of a group G, then there are triplets of spaces (FH,H,FH), in which FH is a space of coherent state or vector coherent state wave functions and FH is its dual relative to a conveniently defined measure. It is shown also that there is a sequence of maps FH→H→FH which facilitates the construction of the corresponding inner products. After completion if necessary, the spaces (FH,H,FH) become isomorphic Hilbert spaces. It is shown that the inner product for ℍ is often easier to evaluate in FH than in FH. Thus, we obtain integral expressions for the inner products of coherent state and vector coherent state representations. These expressions are equivalent to the algebraic expressions of K-matrix theory, but they are frequently more efficient to apply. The construction is illustrated by many examples.
It is shown that if ℍ is a Hilbert space for a representation of a group G, then there are triplets of spaces (FH,H,FH), in which FH is a space of coherent state or vector coherent state wave functions and FH is its dual relative to a conveniently defined measure. It is shown also that there is a sequence of maps FH→H→FH which facilitates the construction of the corresponding inner products. After completion if necessary, the spaces (FH,H,FH) become isomorphic Hilbert spaces. It is shown that the inner product for ℍ is often easier to evaluate in FH than in FH. Thus, we obtain integral expressions for the inner products of coherent state and vector coherent state representations. These expressions are equivalent to the algebraic expressions of K-matrix theory, but they are frequently more efficient to apply. The construction is illustrated by many examples.
When solving problems involving quantum mechanical systems, it is frequently desirable to find the matrix elements of a unitary representation T of a real algebraic Lie group G. This requires defining an inner product on the Hilbert space mathbbH that carries the representation T. In the case where the representation is determined by a representation of a subgroup containing the lowest weight vector of T, this can be achieved through the coherent state construction. In both the scalar and vector coherent state methods, the process of finding the overlaps can be simplified by introducing the coherent state triplet (mathfrakFmathbbHD, mathbbHD, mathfrakFmathfrakHD) of Bargmann spaces. Coherent state wave functions -- the elements of mathfrakFmathbbHD and of mathfrakFmathbbHD -- are used to define the inner product on mathbbHD in a way that simplifies the calculation of the overlaps. This inner product and the group action Gamma of G on mathfrakFmathbbHD are used to formulate expressions for the matrix elements of T with coefficients from the given subrepresentation.\n\nThe process of finding an explicit expression for Gamma relies on matrix factorizations in the complexification of G even though the representation T does not extend to the complexification. It will be shown that these factorizations are, in fact, justified, that the overlaps and Gamma action can be expressed in terms of the given subrepresentation, and that it is possible to find numerical values for the inner product in mathbbH. The scalar and vector coherent state methods will both be applied to Sp(n) and Sp(n,mathbbR).
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.
Hilbert space and inner product In Ch. 2 it was noted that quantum wave functions form a linear space in the sense that multiplying a function by a complex number or adding two wave functions together produces another wave function. It was also pointed out that a particular quantum state can be represented either by a wave function ψ( x ) which depends upon the position variable x , or by an alternative function ψ( p ) of the momentum variable p . It is convenient to employ the Dirac symbol |ψ〉, known as a “ket”, to denote a quantum state without referring to the particular function used to represent it. The kets, which we shall also refer to as vectors to distinguish them from scalars , which are complex numbers, are the elements of the quantum Hilbert space H . (The real numbers form a subset of the complex numbers, so that when a scalar is referred to as a “complex number”, this includes the possibility that it might be a real number.) If α is any scalar (complex number), the ket corresponding to the wave function ɑψ( x ) is denoted by α|ψ〉, or sometimes by |ψ〉α, and the ket corresponding to ø( x )+ψ( x ) is denoted by |ø〉+|ψ〉 or |ψ〉+|ø〉, and so forth. This correspondence could equally well be expressed using momentum wave functions, because the Fourier transform, (2.15) or (2.16), is a linear relationship between ψ( x ) and ( p ), so that αø( x )+βψ( x ) and αϕ( p ) + βψ( p ) correspond to the same quantum state α|ψ〉+β|ø〉.
The Clebsch–Gordan problems of the Barut–Girardello, and Perelomov coherent states of SL(2,R) are studied using the associated Hilbert spaces as the respective carrier spaces of the representations of the group. For the Barut–Girardello coherent states this Hilbert space is a subspace of the Bargmann–Segal Hilbert space B(C2) called the ‘‘reduced Bargmann space.’’ The generators of the group in this realization are essentially the boson operators of Holman and Biedenharn which provide a convenient starting point of the problem. For the Perelomov coherent states the associated Hilbert space turns out to be Bargmann’s canonical carrier space for the realization of the discrete series of representations, namely, the Hilbert space of functions analytic inside the open unit disc. The scalar product, the principal vector, and a complete orthonormal set in these Hilbert spaces are constructed and used for the explicit evaluation of the Clebsch–Gordan coefficients. For each of the coherent state systems the product state turns out to be the principal vector and, therefore, the coupled state itself is the Clebsch–Gordan coefficient. For the Barut–Girardello coherent states this is, apart from normalization, the product of a Bessel function and d-function. For the Perelomov coherent states, on the other hand, this closely resembles the Clebsch–Gordan coefficient of the SU(2) coherent states.
Quantum mechanics describes the physical state of a microscopic system using vectors in a complex Hilbert space rather than deterministic numerical variables. This mathematical formulation provides a rigorous framework for understanding quantum superposition, probability, and measurement. The Hilbert space structure allows quantum states to be represented as normalized vectors, while inner products define transition probabilities through the Born rule. Linear operators acting on these vectors correspond to measurable physical observables such as position, momentum, and energy. This paper presents a concise mathematical treatment of quantum states in Hilbert space, emphasizing vector representation, superposition, normalization, and projection-based measurement. In addition, the mathematical formulation underlying the concept of quantum immortality is briefly discussed from the perspective of the Many-Worlds Interpretation. The study demonstrates how linear algebra serves as the fundamental language of modern quantum mechanics and provides the theoretical basis for advanced areas including quantum computation, quantum information theory, and quantum communication.
Within the generalized definition of coherent states as group orbits we study the orbit spaces and the orbit manifolds in the projective spaces constructed from linear representations. Invariant functions are suggested for arbitrary groups. The group SU (2) is studied in particular and the orbit spaces of its j = 1/2 and 1 representations completely determined. The orbits of SU (2) in CP N can be either two- or three-dimensional, the former being either isomorphic to S 2 or to RP 2 and the latter being isomorphic to quotient spaces of RP 3 . We end with a look from the same perspective to the quantum mechanical space of states in particle mechanics.