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arXiv:1008.5215 · Few-Body Systems

Mini Review of Poincaré Invariant Quantum Theory

Wayne Nicholas Polyzou, Ch. Elster, W. Glöckle, J. Golak, Y. Huang, Hiroyuki Kamada, R. Skibiński, Henryk Witała

Published Dec 2, 2010• 34 citations
arXivHTMLPDF
DOI
10.1007/s00601-010-0149-x

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Abstract In this paper, we develop the quantum theory that has discrete Poincaré symmetry on the one-dimensional Bravais lattice. We review the recently discovered discrete Lorentz symmetry which coexists with the discrete space translational symmetry on a Bravais lattice. The discrete Lorentz transformations and spacetime translations form the discrete Poincaré group, which are represented by unitary operators in a quantum theory. We find the conditions for the existence of representation, which are expressed as the congruence relation between quasi-momentum and quasi-energy. We then build the Lorentz-invariant many-body theory of indistinguishable particles by expressing both the unitary operators and Floquet Hamiltonians in terms of the field operators. Some typical Hamiltonians include the long-range hopping which fluctuates as the distance between sites increases. We calculate the Green’s function of the theory. The spacetime points where the Green’s function is nonzero display a lattice structure. During the propagation, the particle stays localized on a single or a few sites to preserve the Lorentz symmetry.
We consider how the reduced dynamics of an open quantum system coupled to an environment is realized in the Poincar\'e symmetry. The reduced dynamics is described by a dynamical map, which is a quantum channel (a completely positive and trace-preserving linear map) given by tracing out the environment from the total unitary evolution without initial correlations. We investigate the dynamical map invariant under the Poincar\'e transformations and discuss how the invariance constrains the form of the map. Based on the unitary representation theory of the Poincar\'e group, we develop a systematic way to construct the dynamical map with the Poincar\'e invariance. Using this method, we derive such a dynamical map for a spinless massive particle, and the conservation of the Poincar\'e generators is discussed. We then find the map with the Poincar\'e invariance and the four-momentum conservation. Further, we show that the conservation of the angular momentum and the boost operator makes the map of a spinless massive particle unitary.
We give a brief review of the theory of quantum and optical universal invariants, i.e., certain combinations of the second- and higher-order moments (variances) of quantum-mechanical operators or the transverse phase-space coordinates of optical paraxial beams that are preserved in time (or along the axis of the beam) independently of the concrete form of the coefficients of the Hamiltonian or the parameters of the optical system, provided that the effective Hamiltonian is either a generic quadratic form of the generalized coordinate-momenta operators or a linear combination of generators of certain finite-dimensional algebras. Using the phase-space representation of quantum mechanics (paraxial optics) in terms of the Wigner function, we elucidate the relation between the quantum invariants and the classical universal integral invariants of Poincaré and Cartan. The specific features of the Gaussian beams are discussed as examples.