Phase Space Observables Research Articles

Moment operators of the Cartesian margins of the phase space observables

The theory of operator integrals is used to determine the moment operators of the Cartesian margins of the phase space observables generated by the mixtures of the number states. The moments of the x-margin are polynomials of the position operator and those of the y-margin are polynomials of the momentum operator.

The norm-1-property of a quantum observable

A normalized positive operator measure X⟼E(X) has the norm-1-property if ‖E(X)‖=1 whenever E(X)≠O. This property reflects the fact that the measurement outcome probabilities for the values of such observables can be made arbitrarily close to one with suitable state preparations. Some general implications of the norm-1-property are investigated. As case studies, localization observables, phase observables, and phase space observables are considered.

The uniqueness question in the multidimensional moment problem with applications to phase space observables

The theory of holomorphic functions of several complex variables is applied in proving a multidimensional variant of a theorem involving an exponential boundedness criterion for the classical moment problem. A theorem of Petersen concerning the relation between the multidimensional and one-dimensional moment problems is extended for half-lines and compact subsets of the real line. These results are used to solve the moment problem for the quantum phase space observables generated by the number states.

Continuous-time histories: Observables, probabilities, phase space structure and the classical limit

The continuous-time histories program stems from the consistent histories approach to quantum theory and aims to provide a fully covariant formalism for quantum mechanics. In this paper we examine some structural points of the formalism. We demonstrate a general construction of history Hilbert spaces and identify a large class of time-averaged observables. We pay particular attention to the construction of the decoherence functional (the object that encodes probability information) in the continuous-time limit and its relation to the temporal structure of the theory. Phase space observables are introduced, through the study of general representations of the history group, which is the analog of the canonical group in the formalism. We can also define a closed-time-path (CTP) generating functional for each observable, which encodes the information of its correlation functions. The phase space version of the CTP generating functional leads to the implementation of Wigner–Weyl transforms, that gives a description of quantum theory solely in terms of phase space histories. These results allow the identification of an algorithm for going to the classical (stochastic) limit for a generic quantum system.

Phase space observables and isotypic spaces

We give necessary and sufficient conditions for the set of Neumark projections of a countable set of phase space observables to constitute a resolution of the identity, and we give a criteria for a phase space observable to be informationally complete. The results will be applied to the phase space observables arising from an irreducible representation of the Heisenberg group.

Positive operator measures determined by their moment sequences

We solve the moment problem for the polar margins of a physically relevant class of phase space observables.

Covariant phase observables in quantum mechanics

In this paper we characterize all the phase shift covariant normalized positive operator measures, i.e., phase observables, and we investigate some of their examples. We also characterize those phase observables which arise from the phase space observables as their polar coordinate angle margins.

Operator integrals and phase space observables

The operator integral ∫f dE of a complex valued measurable function f with respect to a positive operator measure E is considered. If F is a Neumark dilation of E into a projection measure, then the “projected” operator integral pr(∫ fdF) is a restriction of the operator ∫ f dE. Necessary and sufficient conditions for the equality pr(∫f dF)=∫ f dE are obtained. The results are applied to determine the moment operators of the phase space observables generated by the number states.

The moment operators of phase space observables and their number margins

We show that the moment operators of any of the phase space observables B( C) ) ̄ Z ↦ A ¦s〉(Z)≔ 1 π ∫ Z/LL> D¦s〉〈s¦D z ∗ dλ(z) ¬E L(H) associated with the number states ¦s〉 are the powers of the lowering operator. We also determine all the moment operators of the number margins of these observables. They are integer valued polynomials of the number operator, the degree of the polynomial being the degree of the moment and the integer coefficients depending on ¦s〉. In the Appendix we develop the necessary tools for integrating unbounded functions with respect to operator measures.

Quantum groups and q-lattices in space

Quantum groups lead to an algebraic structure that can be realized on quantum spaces. These are noncommutative spaces that inherit a well defined mathematical structure from the quantum group symmetry. In turn such quantum spaces can be interpreted as noncommutative configuration spaces for physical systems which carry a symmetry like structure. These configuration spaces will be generalized to noncommutative phase space. The definition of the noncommutative phase space will be based on a differential calculus on the configuration space which is compatible with the symmetry. In addition a conjugation operation will be defined which will allow us to define the phase space variables in terms of algebraically selfadjoint operators. An interesting property of the phase space observables will be that they will have a discrete spectrum. These noncommutative phase space puts physics on a lattice structure.

Phase statistics and phase-space measurements.

The statistics of quantum optical phase observables are recovered as patterns from both number as well as phase-space measurement statistics. A model for the measurement of generalized Q functions studied recently by Leonhardt and Paul [Phys. Rev. A. 47, R2460 (1993); 48, 3265 (1993)] is shown to yield measurements of arbitary phase-space observables.