Linear Canonical Transformations Research Articles

Time observables with projective covariance

We study the positive-operator-valued measures (POVMs) on the projective real line covariant with respect to the projective group. We interpret the projective line as a compactified time axis and we assume that the energy is a positive operator. This formalism may describe a time-of-arrival observable for a free particle covariant with respect to linear canonical transformations. The problem is similar to the more complicated and physically more relevant problem of finding the POVMs on the compactified Minkowski space-time covariant with respect to the conformal group.

Linear canonical transformations and quantum phase: a unified canonical and algebraic approach

The algebra of generalized linear quantum canonical transformations is examined in the prespective of Schwinger's unitary-canonical basis. Formulation of the quantum phase problem within the theory of quantum canonical transformations and in particular with the generalized quantum action-angle phase space formalism is established and it is shown that the conceptual foundation of the quantum phase problem lies within the algebraic properties of the quantum canonical transformations in the quantum phase space. The representations of the Wigner function in the generalized action-angle unitary operator pair for certain Hamiltonian systems with the dynamical symmetry are examined. This generalized canonical formalism is applied to the quantum harmonic oscillator to examine the properties of the unitary quantum phase operator as well as the action-angle Wigner function.

Branching rules for restriction of the Weil representations of Sp(n,R) to its maximal parabolic subgroup CM(n)

The symplectic group Sp(n,R) is the group of linear canonical transformations of a real 2n-dimensional phase space and CM(n)⊂Sp(n,R) is a maximal parabolic subgroup. The symplectic groups are the fundamental dynamical groups of classical and quantal Hamiltonian mechanics. In particular, Sp(3,R) is the dynamical group of the spherical harmonic oscillator and its Weil (harmonic series) representations are important for the microscopic (shell model) description of the collective motions of many-particle systems. The subgroup CM(3)⊂Sp(3,R) also appears in the microscopic theory of nuclear collective motion as the dynamical group of a hydrodynamic model of quadrupole vibrations and rotations of a nucleus. Thus, the Sp(3,R)→CM(3) branching rules are needed in finding the embedding of the hydrodynamic collective model in the microscopic shell model. Some new developments are made in the vector-coherent-state theory of induced representations.

Quantum correspondence for linear canonical transformations on general Hamiltonian systems

General Hamiltonian systems related by linear canonical transformations, which are combinations of the scale and gauge transformations, are considered. Using path integrals, it is proven that each quantum Hamiltonian can be expressed by the classical Hamiltonian whose canonical variables are replaced by their corresponding quantum operators. The relation between the Schr\"odinger solution and the propagator for the transformed (new) and for the original (old) systems are evaluated by applying the unitary operator which describes the linear relationship between their quantum operators as the corresponding relations between their classical forms. It is shown that the uncertainty relations between the canonical position and momentum operator depend on the gauge function chosen, and satisfy Heisenberg's uncertainty principle. The uncertainty relations between the canonical position and kinetic momentum operator (distinguished from the momentum operator) do not depend on the gauge function chosen, and may not satisfy Heisenberg's uncertainty principle. By the gauge transformation, a single system has innumerable Schr\"odinger equations, but the quantum averages of the function of the position and kinetic momentum operators are invariant for all solutions, as for classical cases.

Coherent state path integral for the interaction of spin with external magnetic fields

Linear canonical transformation and time transformation are used within the formalism of path integrals in coherent state representation to solve the problem of an interacting spin with arbitrary time-dependent external magnetic fields. The Factorization technique is used to solve the equation of motion. Examples of the calculation of the transition probability in terms of elementary functions and special functions for solvable systems are presented. For completeness Berry's phase is also derived under adiabatic expansion.

Linear canonical transformations in quantum mechanics

We find explicit unitary operators that implement linear canonical transformations of the quantum mechanical operators for a system with N degrees of freedom. We then relate the operators effecting this transformation to the previous formulation of quantum canonical transformations in terms of an effective generating function introduced by Ghandour.

Optimal filtering with linear canonical transformations

Optimal filtering with linear canonical transformations allows smaller mean-square errors in restoring signals degraded by linear time- or space-variant distortions and non-stationary noise. This reduction in error comes at no additional computational cost. This is made possible by the additional flexibility that comes with the three free parameters of linear canonical transformations, as opposed to the fractional Fourier transform which has only one free parameter, and the ordinary Fourier transform which has none. Application of the method to severely degraded images is shown to be significantly superior to filtering in fractional Fourier domains in certain cases.

Wave emission by resonance crossing

The emission of collective waves by a moving charged particle in a nonuniform medium is discussed. Emission occurs in a nonuniform medium when the local dispersion relation of the collective wave is satisfied. This is a form of resonance crossing. Using the Weyl symbol calculus, a local expansion of the collective wave equation driven by the particle source is derived in the neighborhood of the crossing. The collective wave dispersion manifold and the gyroballistic wave dispersion manifold can be used as a pair of local coordinates in the neighborhood of the resonance crossing, which greatly simplifies the analysis. This change of representation is carried out using a metaplectic transform (a generalization of the fourier transform). The Wigner function of the emitted wave field is then computed in the new coordinates. The Wigner function is a phase space scalar, hence the numerical value is invariant under linear canonical transformations. This invariance is invoked to finally arrive at the Wigner function in the original (physical) coordinates. The wave-action and -energy emission rates are then computed from the Wigner function.

Coherent-mode decomposition of general anisotropic Gaussian Schell-model beams

We present a complete solution to the problem of coherent-mode decomposition of the most general anisotropic Gaussian Schell-model (AGSM) beams, which constitute a ten-parameter family. Our approach is based on symmetry considerations. Concepts and techniques familiar from the context of quantum mechanics in the two-dimensional plane are used to exploit the Sp(4, R) dynamical symmetry underlying the AGSM problem. We take advantage of the fact that the symplectic group of first-order optical system acts unitarily through the metaplectic operators on the Hilbert space of wave amplitudes over the transverse plane, and, using the Iwasawa decomposition for the metaplectic operator and the classic theorem of Williamson on the normal forms of positive definite symmetric matrices under linear canonical transformations, we demonstrate the unitary equivalence of the AGSM problem to a separable problem earlier studied by Li and Wolf [ Opt. Lett.7, 256 ( 1982)] and Gori and Guattari [ Opt. Commun.48, 7 ( 1983)]. This connection enables one to write down, almost by inspection, the coherent-mode decomposition of the general AGSM beam. A universal feature of the eigenvalue spectrum of the AGSM family is noted.

Wigner function and the parity operator

Abstract It is shown that the linear canonical transformation properties of the Wigner function P(q, p) naturally require it to be proportional to the expectation value of the parity operator that performs reflections about the phase-space point (q, p).

Correspondence between the classical and quantum canonical transformation groups from an operator formulation of the wigner function

An explicit expression of the “Wigner operator” is derived, such that the Wigner function of a quantum state is equal to the expectation value of this operator with respect to the same state. This Wigner operator leads to a representation-independent procedure for establishing the correspondence between the inhomogeneous symplectic group applicable to linear canonical transformations in classical mechanics and the Weyl-metaplectic group governing the symmetry of unitary transformations in quantum mechanics.

Group-theoretical derivation of the Wigner distribution function

A derivation of the Wigner distribution function, which is based on the simple postulate that the correspondence between the distribution function and the wave function be invariant under linear canonical transformations which form a Lie group, is given.

Quantum-noise matrix for multimode systems: U(n) invariance, squeezing, and normal forms.

We present a complete analysis of variance matrices and quadrature squeezing for arbitrary states of quantum systems with any finite number of degrees of freedom. Basic to our analysis is the recognition of the crucial role played by the real symplectic group Sp(2n,openR) of linear canonical transformations on n pairs of canonical variables. We exploit the transformation properties of variance (noise) matrices under symplectic transformations to express the uncertainty-principle restrictions on a general variance matrix in several equivalent forms, each of which is manifestly symplectic invariant. These restrictions go beyond the classically adequate reality, symmetry, and positivity conditions. Towards developing a squeezing criterion for n-mode systems, we distinguish between photon-number-conserving passive linear optical systems and active ones. The former correspond to elements in the maximal compact U(n) subgroup of Sp(2n,openR), the latter to noncompact elements outside U(n). Based on this distinction, we motivate and state a U(n)-invariant squeezing criterion applicable to any state of an n-mode system, and explore alternative ways of expressing it. The set of all possible quantum-mechanical variance matrices is shown to contain several interesting subsets or subfamilies, whose definitions are related to the fact that a general variance matrix is not diagonalizable within U(n). Definitions, characterizations, and canonical forms for variance matrices in these subfamilies, as well as general ones, and their squeezing nature, are established. It is shown that all conceivable variance matrices can be generated through squeezed thermal states of the n-mode system and their symplectic transforms. Our formulas are developed in both the real and the complex forms for variance matrices, and ways to pass between them are given.

Metaplectic formulation of linear mode conversion.

The metaplectic formulation of linear mode conversion is presented. We begin by discussing the connection between wave operators in weakly inhomogeneous media, their symbols, and related pseudodifferential operators. A brief summary of WKB theory and Hamiltonian ray dynamics is given. In regions where mode conversion occurs, the WKB approximation breaks down and must be replaced by an appropriate local approximation. This is done by expanding in a Taylor series about the degenerate region and keeping only the leading-order terms. At leading order a linear canonical transformation on the ray phase space can be performed which brings the system into a simpler form. This linear canonical transformation induces a unitary transformation, called a metaplectic transformation, in the wave function's Hilbert space. This is a generalization of the Fourier transformation. The advantage of metaplectic techniques over Fourier techniques lies in the wider range of transformations available to simplify the problem. We show how to construct the S matrix, relating incoming and outgoing waves, and the Wigner tensor. We examine the Wigner function in detail with particular attention to its asymptotic properties.

U(3,1) Transformations with invariant symplectic and orthogonal metrics

The linear canonical transformations leave the symplectic metric on time-position-energy-momentum space invariant. The Lorentz metric defines a non-compact orthogonal metric on time-position space and separately on energy-momentum space. A physical constantb extends the orthogonal metric to time-position-energy-momentum space as defined by Born and Caianiello. The constantb bounds the rate of change of momentum and hence acceleration. TheU(3,1) group of transformations leaving both of these metrics invariant is studied and it is shown to reduce to the expected form in the Newtonian limit.

Phase-space action conservation for non-eikonal wave fields

We derive a local phase-space wave-action conservation law, valid for non-eikonal wave fields for which the medium and/or the wave amplitudes have rapid spatial variation. This six-dimensional conservation law leads to conservation laws on three-dimensional subspaces. The law is covariant under linear canonical transformations of phase-space, and under congruent transformations of the multi-component wave field.

Graph theory and a special class of symplectic transformations: The generalized Jacobi variables

The well-known Jacobi variables in celestial mechanics are generalized to other Hamiltonian systems which include vortex dynamics. A combinatorial algorithm for constructing the generalized Jacobi variables is given; for any binary tree T(N) with N leave, there is a 2N×2N real symplectic matrix MT (T(N),Γ) which completely defines a linear canonical transformation to these relative variables. This algorithm yields a direct proof of the symplectic property for all the generalized Jacobi variables. An application to vortex dynamics is outlined here.

A discretization ofp-adic quantum mechanics

We show that some compact subgroups (ℋn,m) of thep-adic Heisenberg group act irreducibly on corresponding finite dimensional spaces of test-functions (Sm,n). Under certain conditions, a compact group (Am+n) of linear canonical transformations, isomorphic toSL(2,Zp), can be represented unitarily onSm,n as a group of automorphisms of ℋn,m. The restriction toSm,n can be considered as a discretization because an invariant subgroup (In,m) ofAm+n is represented trivially. It is possible to take a limit whereIm,n becomes an arbitrarily small neighborhood of the identity, while the dimension ofSm,n becomes arbitrarily large. This is a possible definition of the “continuum limit” that we relate to other projective limits appearing naturally in the present context.

On the symplectically invariant variational principle and generating functions

The generating function for canonical transformations derived by Marinov has the important property of symplectic invariance (i. e. under linear canonical transformations). However, a more geometric approach to the rederivation of this function from the variational principle reveals that it is not free from caustic singularities after all. These singularities can be avoided without breaking the symplectic invariance by the definition of a complementary generating function bearing an analogous relation to the Woodward ambiguity function in telecommunications theory as that tying Marinov’s function to the Wigner function and the Weyl transform in quantum mechanics. Marinov’s function is specially apt to describe canonical transformations close to the identity, but breaks down for reflections through a point in phase space, easily described by the new generating function.

Linear canonical transformations of coherent and squeezed states in the Wigner phase space. III. Two-mode states.

It is shown that the basic symmetry of two-mode squeezed states is governed by the group Sp(4) in the Wigner phase space which is locally isomorphic to the (3+2)-dimensional Lorentz group. This symmetry, in the Schr\"odinger picture, appears as Dirac's two-oscillator representation of O(3,2). It is shown that the SU(2) and SU(1,1) interferometers exhibit the symmetry of this higher-dimensional Lorentz group. The mathematics of two-mode squeezed states is shown to be applicable to other branches of physics including thermally excited states in statistical mechanics and relativistic extended hadrons in the quark model.