Localization and diffusion in polymer quantum field theory

Abstract

Polymer quantization is a non-standard approach to quantizing a classical system inspired by background independent approaches to quantum gravity such as loop quantum gravity. When applied to field theory it introduces a characteristic polymer scale at the level of the fields classical configuration space. Compared with models with space-time discreteness or non-commutativity this is an alternative way in which a characteristic scale can be introduced in a field theoretic context. Motivated by this comparison we study here localization and diffusion properties associated with polymer field observables and dispersion relation in order to shed some light on the novel physical features introduced by polymer quantization. While localization processes seems to be only mildly affected by polymer effects, we find that polymer diffusion differs significantly from the "dimensional reduction" picture emerging in other Planck-scale models beyond local quantum field theory.

Decoherence and discrete symmetries in deformed relativistic kinematics

“…According to this picture one would intuitively expect that the general three-momentum of a moving particle will be given by another S L(2, R) group element obtained by "boosting" the rest momentum h α . This is indeed the case and it can be shown [12,13] that the three-momentum of a point particle of mass m is given by the group element obtained by boosting with a generic g ∈ S L(2, R), via group conjugation, the rest three-momentum h α :…”

Signal propagation on κ -Minkowski spacetime and nonlocal two-point functions

“…From the action of the derivatives onê k one can straightforwardly derive the action of the operators∂ a on the generic function of non-commuting coordinatesf (x). This can be done by resorting to the following Fourier expansion [32,[52][53][54] in terms of right-ordered non-commutative plane wavesê k…”

Deformed phase spaces with group valued momenta

“…a group element belonging to SL(2, R), the (double cover) of the group of isometries of threedimensional Minkowski space. A description of a moving defect can be obtained by boosting the conical metric, in this case the three momentum of the particle will be a general element of SL(2, R) [3,51]. Various treatments exist for the description of the phase space of point particles coupled to gravity in three dimensions [3,52,53] and its symmetries [54,55].…”

“…We consider the "exponential coordinates" [51] for which g ∈ SL(2, R) is obtained as g = e −pµX µ , µ = 0, 1, 2. The mass parameter ℓ = 1/4πG is determined by the three-dimensional Newton's constant [2] and in the limit G → 0 one recovers the usual flat momentum space R 2,1 .…”

“…We can consider cartesian coordinates on the group manifold by transforming the general group element in exponential coordinates (109) through the relations [51] …”