3,931 research outputs found
Quantum Variance and Ergodicity for the baker's map
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
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
We show how to do gauge theory on the octonions and other nonassociative
algebras such as `fuzzy ' 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 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
We show that the -product for , 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 to . 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
momentum and to interpret the Duflo map as noncommutative compression. Our
methods also provide a generalised twist operator for the -product.Comment: 53 pages latex, no figures; extended the intro for this final versio
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