3,931 research outputs found

    Quantum Variance and Ergodicity for the baker's map

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    We prove a Egorov theorem, or quantum-classical correspondence, for the quantised baker's map, valid up to the Ehrenfest time. This yields a logarithmic upper bound for the decay of the quantum variance, and, as a corollary, a quantum ergodic theorem for this map

    Light-Front Quantisation as an Initial-Boundary Value Problem

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    In the light front quantisation scheme initial conditions are usually provided on a single lightlike hyperplane. This, however, is insufficient to yield a unique solution of the field equations. We investigate under which additional conditions the problem of solving the field equations becomes well posed. The consequences for quantisation are studied within a Hamiltonian formulation by using the method of Faddeev and Jackiw for dealing with first-order Lagrangians. For the prototype field theory of massive scalar fields in 1+1 dimensions, we find that initial conditions for fixed light cone time {\sl and} boundary conditions in the spatial variable are sufficient to yield a consistent commutator algebra. Data on a second lightlike hyperplane are not necessary. Hamiltonian and Euler-Lagrange equations of motion become equivalent; the description of the dynamics remains canonical and simple. In this way we justify the approach of discretised light cone quantisation.Comment: 26 pages (including figure), tex, figure in latex, TPR 93-

    Gauge theory on nonassociative spaces

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    We show how to do gauge theory on the octonions and other nonassociative algebras such as `fuzzy R4R^4' models proposed in string theory. We use the theory of quasialgebras obtained by cochain twist introduced previously. The gauge theory in this case is twisting-equivalent to usual gauge theory on the underlying classical space. We give a general U(1)-Yang-Mills example for any quasi-algebra and a full description of the moduli space of flat connections in this theory for the cube Z23Z_2^3 and hence for the octonions. We also obtain further results about the octonions themselves; an explicit Moyal-product description of them as a nonassociative quantisation of functions on the cube, and a characterisation of their cochain twist as invariant under Fourier transform.Comment: 24 pages latex, two .eps figure

    Noncommutative Harmonic Analysis, Sampling Theory and the Duflo Map in 2+1 Quantum Gravity

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    We show that the ⋆\star-product for U(su2)U(su_2), group Fourier transform and effective action arising in [1] in an effective theory for the integer spin Ponzano-Regge quantum gravity model are compatible with the noncommutative bicovariant differential calculus, quantum group Fourier transform and noncommutative scalar field theory previously proposed for 2+1 Euclidean quantum gravity using quantum group methods in [2]. The two are related by a classicalisation map which we introduce. We show, however, that noncommutative spacetime has a richer structure which already sees the half-integer spin information. We argue that the anomalous extra `time' dimension seen in the noncommutative geometry should be viewed as the renormalisation group flow visible in the coarse-graining in going from SU2SU_2 to SO3SO_3. Combining our methods we develop practical tools for noncommutative harmonic analysis for the model including radial quantum delta-functions and Gaussians, the Duflo map and elements of `noncommutative sampling theory'. This allows us to understand the bandwidth limitation in 2+1 quantum gravity arising from the bounded SU2SU_2 momentum and to interpret the Duflo map as noncommutative compression. Our methods also provide a generalised twist operator for the ⋆\star-product.Comment: 53 pages latex, no figures; extended the intro for this final versio
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