Group Of Linear Canonical Transformations Research Articles

Unitary rotation and gyration of pixelated images on rectangular screens.

In the two space dimensions of screens in optical systems, rotations, gyrations, and fractional Fourier transformations form the Fourier subgroup of the symplectic group of linear canonical transformations: U(2)F⊂Sp(4,R). Here, we study the action of this Fourier group on pixelated images within generic rectangular Nx×Ny screens; its elements here compose properly and act unitarily, i.e., without loss of information.

The Fourier U(2)F group on square and round pixellated arrays

The Fourier group U(2)F is the maximal compact group of the group of linear canonical transformations on configuration space. When we finitely quantize the two-dimensional harmonic oscillator we turn this space into an array of N2 points arranged along either Cartesian or polar coordinates, as a square or a circle, that model pixellated screens. We investigate the action of U(2)F on these two N2-dimensional spaces, which become a homogeneous space for this group action.

Geometry and dynamics of squeezing in finite systems

Squeezing and its inverse magnification form a one-parameter group of linear canonical transformations of continuous signals in paraxial optics. We search for corresponding unitary matrices to apply on signal vectors in N-point finite Hamiltonian systems. The analysis is extended to the phase space representation by means of Wigner quasi-probability distribution functions on the discrete torus and on the sphere. Together with two previous studies of the fractional Fourier and Fresnel transforms, we complete the finite counterparts of the group of linear canonical transformations.

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.

Linear canonical transformations of coherent and squeezed states in the Wigner phase space.

It is shown that classical linear canonical transformations are possible in the Wigner phase space. Coherent and squeezed states are shown to be linear canonical transforms of the ground-state harmonic oscillator. It is therefore possible to evaluate the Wigner functions for coherent and squeezed states from that for the harmonic oscillator. Since the group of linear canonical transformations has a subgroup whose algebraic property is the same as that of the (2+1)-dimensional Lorentz group, it may be possible to test certain properties of the Lorentz group using optical devices. A possible experiment to measure the Wigner rotation angle is discussed.

Normal forms for classical and boson systems

The group of Bogoliubov transformations of annihilation and creation operators is a subgroup of U( n,n) where n is the number of distinct pairs of annihilation and creation operators. Here, it is established that this subgroup of U( n,n) is isomorphic to Sp(2 n, R ), which appears in classical dynamics as the group of linear canonical transformations on a 2 n-dimensional phase space. Well-known results in classical dynamics are then to used to deduce the full set of normal forms for Boson Hamiltonians. These are classified using a para-eigenvalue notation applicable to both classical and Bose field systems. A simple sufficient condition is given for the non-removability of pairs of creation operators. Explicit normal forms have not previously been given for Hamiltonians with this pathology, which may occur even when the corresponding classical Hamiltonian can be diagonalized.

Stable states of a relativistic harmonic oscillator imbedded in a random stochastic thermostat

An analysis in phase space of the behavior of a relativistic four-dimensional harmonic oscillator undergoing stochastic interactions shows that the group of linear canonical transformations in phase space which leaves invariant the Poisson brackets is anSp(12,4) group, withSp(12,4)⊃U(6,2)⊃SU(1,1)⊗SO(6,2). The application of Cartan's treatment to its behavior implies a classification of its stable states characterized by a set of discrete numbers.

Linear Canonical Transformations and Their Unitary Representations

We show that the group of linear canonical transformations in a 2N-dimensional phase space is the real symplectic group Sp(2N), and discuss its unitary representation in quantum mechanics when the N coordinates are diagonal. We show that this Sp(2N) group is the well-known dynamical group of the N-dimensional harmonic oscillator. Finally, we study the case of n particles in a q-dimensional oscillator potential, for which N = nq, and discuss the chain of groups Sp(2nq)⊃Sp(2n)×O(q). An application to the calculation of matrix elements is given in a following paper.