Abstract Random sets are used to get a continuous partition of the cardinality of the union of many overlapping sets. The formalism uses Möbius transforms and adapts Shapley’s methodology in cooperative game theory, into the context of set theory. These ideas are subsequently generalized into the context of finite-dimensional Hilbert spaces. Using random projectors into the subspaces spanned by states from a total set, we construct an infinite number of continuous resolutions of the identity, that involve Hermitian positive semi-definite operators. The simplest one is the diagonal continuous resolution of the identity, and it is used to expand an arbitrary vector in terms of a continuum of components. It is also used to define the function on the ‘probabilistic quadrant’ , which is analogous to the Wigner function for the harmonic oscillator, on the phase-space plane. Systems with finite-dimensional Hilbert space (which are naturally described with discrete variables) are described here with continuous probabilistic variables.
Abstract Random sets are used to get a continuous partition of the cardinality of the union of many overlapping sets. The formalism uses Möbius transforms and adapts Shapley’s methodology in cooperative game theory, into the context of set theory. These ideas are subsequently generalized into the context of finite-dimensional Hilbert spaces. Using random projectors into the subspaces spanned by states from a total set, we construct an infinite number of continuous resolutions of the identity, that involve Hermitian positive semi-definite operators. The simplest one is the diagonal continuous resolution of the identity, and it is used to expand an arbitrary vector in terms of a continuum of components. It is also used to define the function on the ‘probabilistic quadrant’ , which is analogous to the Wigner function for the harmonic oscillator, on the phase-space plane. Systems with finite-dimensional Hilbert space (which are naturally described with discrete variables) are described here with continuous probabilistic variables.
Abstract A total set of states for which we have no resolution of the identity (a ‘pre-basis’), is considered in a finite dimensional Hilbert space. A dressing formalism renormalizes them into density matrices which resolve the identity, and makes them a ‘generalized basis’, which is practically useful. The dresssing mechanism is inspired by Shapley’s methodology in cooperative game theory, and it uses Möbius transforms. There is non-independence and redundancy in these generalized bases, which is quantified with a Shannon type of entropy. Due to this redundancy, calculations based on generalized bases are sensitive to physical changes and robust in the presence of noise. For example, the representation of an arbitrary vector in such generalized bases, is robust when noise is inserted in the coefficients. Also in a physical system with a ground state which changes abruptly at some value of the coupling constant, the proposed methodology detects such changes, even when noise is added to the parameters in the Hamiltonian of the system.
We study the Wigner Transform (WT) in the case of an infinite-dimensional continuous Hilbert space and a discrete Hilbert space of finite prime-number dimensionality N. In the discrete case we define a family of WT's as a function of a phase parameter. It is only for a specific value of the parameter that all the properties we have examined, including the matrix elements of the phase-space operator P and the relation between Wigner's function and Kirkwood's distribution, have a parallel in both types of Hilbert space. This close parallel was found constructing the two WT's in terms of a complete set of operators having similar properties, in the sense that the position and momentum operators appear disentangled in the two situations. We give a geometric interpretation of the properties involving the phase-space operator in the discrete case.
In this chapter we take up systems whose state spaces form a continuum. These are, naturally, infinite dimensional vector spaces (often called Hilbert spaces). Unfortunately, we could not do any better than treating such systems by formal analogy with their finite dimensional counterparts. A rigorous treatment would require a background in functional analysis and is beyond the scope of the present work. Thus, we essentially lay out a dictionary for translating the mathematical ingredients and formulae, from discrete to continuum descriptions, motivated by plausibility arguments. In particular, we argue why states are now described by complex square integrable functions (called wave functions) and observables by Hermitian differential operators. We then move on to show how to quantize quantum systems with classical analogues, which comprise the bulk of real life physical systems. Here, we explicitly show how the most nontrivial component of the quantization, i.e., the description of the representation of the momentum operator (in position space) can be deduced from both, the de Broglie hypothesis, and the canonical commutation relation. The “particle in a box” problem is treated as an illustration. The parity operator, Ehrenfest's theorem and continuity conditions have been relegated to the problem section.
Abstract Coherent subspaces spanned by a finite number of coherent states are introduced, in a quantum system with Hilbert space that has odd prime dimension d . The set of all coherent subspaces is partitioned into equivalence classes, with d 2 subspaces in each class. The corresponding coherent projectors within an equivalence class, have the ‘closure under displacements property’ and also resolve the identity. Different equivalence classes provide different granularisation of the Hilbert space, and they form a partial order ‘coarser’ (and ‘finer’). In the case of a two-dimensional coherent subspace spanned by two coherent states, the corresponding projector (of rank 2) is different than the sum of the two projectors to the subspaces related to each of the two coherent states. We quantify this with ‘non-addditivity operators’ which are a measure of quantum interference in phase space, and also of the non-commutativity of the projectors. Generalized Q and P functions of density matrices, which are based on coherent projectors in a given equivalence class, are introduced. Analogues of the Lorenz values and the Gini index (which are popular quantities in mathematical economics) are used here to quantify the inequality in the distribution of the Q function of a quantum state, within the granular structure of the Hilbert space. A comparison is made between Lorenz values and the Gini index for the cases of coarse and also fine granularisation of the Hilbert space. Lorenz values require an ordering of the d 2 values of the Q function of a density matrix, and this leads to the ranking permutation of a density matrix, and to comonotonic density matrices (which have the same ranking permutation). The Lorenz values are a superadditive function and the Gini index is a subadditive function (they are both additive quantities for comonotonic density matrices). Various examples demonstrate these ideas.
In the context of phase-space quantization, matrix elements and observables result from integration of c-number functions over phase space, with Wigner functions serving as the quasi-probability measure. The complete sets of Wigner functions necessary to expand all phase-space functions include off-diagonal Wigner functions, which may appear technically involved. Nevertheless, it is shown here that suitable generating functions of these complete sets can often be constructed, which are relatively simple, and lead to compact evaluations of matrix elements. New features of such generating functions are detailed and explored for integer-indexed sets, such as for the harmonic oscillator, as well as continuously indexed ones, such as for the linear potential and the Liouville potential. The utility of such generating functions is illustrated in the computation of star functions, spectra, and perturbation theory in phase space.
Abstract In a quantum system with d -dimensional Hilbert space, the Q -function of a Hermitian positive semidefinite operator θ , is defined in terms of the d 2 coherent states in this system. The Choquet integral of the Q -function of θ , is introduced using a ranking of the values of the Q -function, and Möbius transforms which remove the overlaps between coherent states. It is a figure of merit of the quantum properties of Hermitian operators, and it provides upper and lower bounds to various physical quantities in terms of the Q -function. Comonotonicity is an important concept in the formalism, which is used to formalize the vague concept of physically similar operators. Comonotonic operators are shown to be bounded, with respect to an order based on Choquet integrals. Applications of the formalism to the study of the ground state of a physical system, are discussed. Bounds for partition functions, are also derived.