A Generalization of the Bargmann's Theory of Ray Representations

The paper contains a complete theory of factors for ray representations acting in a Hilbert bundle, which is a generalization of the known Bargmann's theory. With the help of it we have reformulated the standard quantum theory such that the gauge freedom emerges naturally from the very nature of quantum laws. The theory is of primary importance in the investigations of covariance (in contradistinction to symmetry) of a quantum theory which possesses a nontrivial gauge freedom. In that case the group in question is not any symmetry group but it is a covariance group only - that case which has not been deeply investigated. It is shown on the paper that the factor of its representation depends on space and time when the system in question possesses a gauge freedom. In the nonrelativistic theories the factor depends on the time only. In the relativistic theory the Hilbert bundle is over the spacetime in the nonrelativistic one it is over the time. We explain two applications of this generalization: in a theory of a quantum particle in the nonrelativistic limit and in the quantum electrodynamics.Comment: 37 pages, LateX, revised version, submitted to Comm. Math. Phy

Parametric statistics of zeros of Husimi representations of quantum chaotic eigenstates and random polynomials

Local parametric statistics of zeros of Husimi representations of quantum eigenstates are introduced. It is conjectured that for a classically fully chaotic systems one should use the model of parametric statistics of complex roots of Gaussian random polynomials which is exactly solvable as demonstrated below. For example, the velocities (derivatives of zeros of Husimi function with respect to an external parameter) are predicted to obey a universal (non-Maxwellian) distribution ${d P(v)}/{dv^2} = 2/(\pi\sigma^2)(1 + |v|^2/\sigma^2)^{-3},$ where $\sigma^2$ is the mean square velocity. The conjecture is demonstrated numerically in a generic chaotic system with two degrees of freedom. Dynamical formulation of the ``zero-flow'' in terms of an integrable many-body dynamical system is given as well.Comment: 13 pages in plain Latex (1 figure available upon request

Parity violating vertices for spin-3 gauge fields

The problem of constructing consistent parity-violating interactions for spin-3 gauge fields is considered in Minkowski space. Under the assumptions of locality, Poincar\'e invariance and parity non-invariance, we classify all the nontrivial perturbative deformations of the abelian gauge algebra. In space-time dimensions $n=3$ and $n=5$, deformations of the free theory are obtained which make the gauge algebra non-abelian and give rise to nontrivial cubic vertices in the Lagrangian, at first order in the deformation parameter $g$. At second order in $g$, consistency conditions are obtained which the five-dimensional vertex obeys, but which rule out the $n=3$ candidate. Moreover, in the five-dimensional first order deformation case, the gauge transformations are modified by a new term which involves the second de Wit--Freedman connection in a simple and suggestive way.Comment: 27 pages, 1 table, revtex4, typos correcte

Why odd-space and odd-time dimensions in even-dimesional spaces?

We are answering the question why 4-dimensional space has the metric 1+3 by making a general argument from a certain type of equations of motion linear in momentum for any spin (except spin zero) in any even dimension d. All known free equations for non-zero spin for massless fields belong to this type of equations. Requiring Hermiticity(This is a generalization of an earlier work which shows that without assuming the Lorentz invariance -which in the present work is assumed- the Weyl equation follows using Hermiticity.) of the equations of motion operator as well as irreducibility with respect to the Lorentz group representation, we prove that only metrics with the signature corresponding to q time + (d - q) space dimensions with q being odd exist. Correspondingly, in four dimensional space, Nature could only make the realization of 1+3 dimensional space.Comment: (only small corrections made

Fidelity preserving maps on density operators

We prove that any bijective fidelity preserving transformation on the set of all density operators on a Hilbert space is implemented by an either unitary or antiunitary operator on the underlying Hilbert space.Comment: This is corrected version of the paper math.OA/0108060. The paper has already appeared in ROMP (vol. 48 (2001), 299-303

Spin transport, spin diffusion and Bloch equations in electron storage rings

We show how, beginning with the Fokker--Planck equation for electrons emitting synchrotron radiation in a storage ring, the corresponding equation for spin motion can be constructed. This is an equation of the Bloch type for the polarisation density.Comment: 7 pages. No figures. Latex: Minor corrections in the tex

The exotic Galilei group and the "Peierls substitution"

Taking advantage of the two-parameter central extension of the planar Galilei group, we construct a non relativistic particle model in the plane. Owing to the extra structure, the coordinates do not commute. Our model can be viewed as the non-relativistic counterpart of the relativistic anyon considered before by Jackiw and Nair. For a particle moving in a magnetic field perpendicular to the plane, the two parameters combine with the magnetic field to provide an effective mass. For vanishing effective mass the phase space admits a two-dimensional reduction, which represents the condensation to collective ``Hall'' motions and justifies the rule called ``Peierls substitution''. Quantization yields the wave functions proposed by Laughlin to describe the Fractional Quantum Hall Effect.Comment: Revised version, to appear in Phys. Lett. B. Souriau's scheme and its relation of with the Faddeev-Jackiw hamiltonian reduction is explained. 11 pages, LaTex, no figure