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Applied Optics

Skew projectors and generalized observables in polarization optics: a biorthogonal analyses [Invited]

Tiberiu S. Tudor

Published Feb 3, 2014• 2 citations
Publisher

Abstract

This paper constitutes an application of the polarization optics in the problem of quantum measurement. The non-Hermitian operators of the nonorthogonal multilayer optical polarizers represent observables in the sense of the generalized quantum theory of measurement. The intimate spectral structure of these polarizers can be disclosed in the frame of skew-angular vector bases and biorthonormal vector systems. We show that these polarizers correspond to skew projectors; their operators are "generated" by skew projectors in the sense of the spectral theorem of linear operators theory. Thus the common feature of all the polarizers (Hermitian and non-Hermitian) is that their "nuclei" are (orthogonal or skew) projectors--the generating projectors.

DOI
10.1364/ao.53.000b80

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This paper constitutes an application of the polarization optics in the problem of quantum measurement. The non-Hermitian operators of the nonorthogonal multilayer optical polarizers represent observables in the sense of the generalized quantum theory of measurement. The intimate spectral structure of these polarizers can be disclosed in the frame of skew-angular vector bases and biorthonormal vector systems. We show that these polarizers correspond to skew projectors; their operators are "generated" by skew projectors in the sense of the spectral theorem of linear operators theory. Thus the common feature of all the polarizers (Hermitian and non-Hermitian) is that their "nuclei" are (orthogonal or skew) projectors--the generating projectors.
The operatorial theory of optical polarization devices can provide some of the clearest examples in which von Neumann's model of orthogonal measurement is violated. This paper analyses the global and spectral properties of some two- and three-layer optical polarizers of non-orthogonal kind by calculating the dyadic expressions of their operators in a Dirac-algebraic form. These constitute some of the most expressive examples of non-Hermitian operators corresponding to generalized observables and it is expected that the theory of generalized quantum measurement will take advantage of the simplicity of their physical realization.
The polarization vectors of monochromatic electromagnetic fields can be transformed by multiplying them by 2 ? 2 matrices representing, respectively, retardation plates, rotators and partial polarizers. The present description of partial polarizers is related to a two-dimensional Lorentz algebra SU(1,1) leading to nonunitary squeezing as the present SU(1,1) algebra includes non-Hermitian operators in addition to the Hermitian ones. The present analysis exploits the use of similarity transformations which are more general than the unitary transformations used usually in quantum mechanics. By performing a similarity transformation on a set of Hermitian physical operators the transformed operators are Hermitian with respect to new scalar product defined by the use of a new metric. It is shown that such transformations can be implemented experimentally by the use of polarization optics and the metric obtained in such transformations is analysed.
Abstract In this chapter we present the mathematical framework of describing wave functions as vectors in a Hilbert space and observables as Hermitian operators on such a vector space. We describe the mathematical formulation of these ideas for finite and infinite-dimensional cases via the Dirac notation, and discuss the example of the Hilbert space of photon polarization. Important classes of linear operators on Hilbert spaces are discussed, including projectors, Hermitian and self-adjoint operators, as well as unitary operators. As an important centre piece for the mathematical language of quantum mechanics we present the spectral theorem. We discuss the subtleties of this framework in infinite dimensions, giving a brief overview of the distributional aspects of quantum mechanics and the conceptually important problem of finding self-adjoint extensions via the Weyl-von Neumann theorem
In quantum mechanics it is well known that observables are represented by hermitian matrices (or operators). Uncertainty relations are represented as some kinds of trace inequalities satisfied by two observables and one density matrix (or operator). By releasing the hermitian restriction on observables, we obtained non-hermite uncertainty relations. In this paper we give several non-hermitian extensions of Heisenberg or Schrödinger type q uncertainty relations for generalized skew information under some conditions.
The bases used in quantum mechanics, in general, are orthonormal. The relations in orthonormal basis vectors are simpler than those in nonorthogonal basis vectors. Also, the operators that represent different physical quantities are Hermitian and the eigenvectors of these operators belonging to different eigenvalues are orthonormal to each other. But, in many cases, such as sets of screened hydrogenic wavefunctions, nonorthogonal sets of basis vectors occur. It is, therefore, necessary to consider the representation of vectors and operators in nonorthogonal bases. Making use of the set of vectors reciprocal to the given set of nonorthogonal base vectors, the expression for the norms of vectors and for the matrix elements has been found. Also, adjoint and Hermitian matrices in nonorthogonal bases have been worked out using a set of reciprocal basis vectors.