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.
We discuss the geometry of states of quantum systems in an n-dimensional Hilbert space in terms of an explicit parameterization of all such systems. The geometry of the space of pure as well as mixed states for n -state systems is discussed. The parameterization is particularly useful since it allows for the simple construction and isolation of various physically meaningful subspaces of the space of all density matrices. This is used to describe possible geometric phases, their calculation, and analyze entropy and purity (or linear entropy) functions. In particular we provide conditions under which nontrivial abelian and/or nonabelian geometric phases aris in these subspaces in terms of the given parameterization, an explicit example is given, and multidimensional ientopic surfaces are discused.
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).
Abstract This paper studies coherent quantization, the way operators in the quantum space of a coherent space—defined in the recent book ’Coherent Quantum Mechanics’ by the first author—can be studied in terms of objects defined directly on the coherent space. The results may be viewed as a generalization of geometric quantization, including the non-unitary case. Care has been taken to work with the weakest meaningful topology and to assume as little as possible about the spaces and groups involved. Unlike in geometric quantization, the groups are not assumed to be compact, locally compact, or finite-dimensional. This implies that the setting can be successfully applied to quantum field theory, where the groups involved satisfy none of these properties. The paper characterizes linear operators acting on the quantum space of a coherent space in terms of their coherent matrix elements. Coherent maps and associated symmetry groups for coherent spaces are introduced, and formulas are derived for the quantization of coherent maps. The importance of coherent maps for quantum mechanics is due to the fact that there is a quantization map that associates homomorphically with every coherent map a linear operator from the quantum space into itself. The quantization map generalizes the second quantization procedure for free classical fields to symmetry groups of general coherent spaces. Field quantization is obtained by specialization to Klauder spaces, whose quantum spaces are the bosonic Fock spaces. Implied by the new approach is a short, coordinate-free derivation of all basic properties of creation and annihilation operators in Fock spaces.
We present a survey of the theory of coherent states (CS) and some of their generalizations, with emphasis on the mathematical structure, rather than on physical applications. Starting from the standard theory of CS over Lie groups, we develop a general formalism, in which CS are associated to group representations which are square integrable over a homogeneous space. A further step allows us to dispense with the group context altogether, and thus obtain the so-called reproducing triples and continuous frames introduced in some earlier work. We discuss in detail a number of concrete examples, namely semisimple Lie groups, the relativity groups and various types of wavelets. Finally we turn to some physical applications, centering on quantum measurement and the quantization/dequantization problem, that is, the transition from the classical to the quantum level and vice versa.
Understanding the geometry of state space is of fundamental interest in the field of quantum information, in particular to study entanglement. The states of a d dimensional system form a convex set in the d2−1 dimensional Euclidean space Rd2−1. For a two dimensional system this convex set is nothing but a solid unit sphere (Bloch sphere) with centre at the origin. The surface S of this unit sphere consists of all the pure states of the system. Every unitary transformation U ∈ SU(2) of the Hilbert space corresponds to an associated SO(3) rotation of this space. This kind of correspondence becomes much richer when we go to higher dimension. For example, in the d = 3 case the corresponding unitary group is the eight parameter group SU(3) whereas the rotation group in R is the 28 parameter group SO(8). Therefore, not every point on or inside the sphere S in R will correspond to a state of the system. So, not every rotation in R will correspond to a valid unitary transformation of the 3 dimensional Hilbert space. This gives us a fairly complex convex object as the set of all states. In order to gain insight into the structure of this state space, we study its two dimensional and three dimensional cross-sections. In particular, we describe in detail the discrete symmetries of a non-trivial 3-section of the convex set of states (density matrices) for three dimensional systems. This group turns out to be the same as that of a tetrahedron.