44 research outputs found
Spacetime-Free Approach to Quantum Theory and Effective Spacetime Structure
Motivated by hints of the effective emergent nature of spacetime structure,
we formulate a spacetime-free algebraic framework for quantum theory, in which
no a priori background geometric structure is required. Such a framework is
necessary in order to study the emergence of effective spacetime structure in a
consistent manner, without assuming a background geometry from the outset.
Instead, the background geometry is conjectured to arise as an effective
structure of the algebraic and dynamical relations between observables that are
imposed by the background statistics of the system. Namely, we suggest that
quantum reference states on an extended observable algebra, the free algebra
generated by the observables, may give rise to effective spacetime structures.
Accordingly, perturbations of the reference state lead to perturbations of the
induced effective spacetime geometry. We initiate the study of these
perturbations, and their relation to gravitational phenomena
Group Fourier transform and the phase space path integral for finite dimensional Lie groups
We formulate a notion of group Fourier transform for a finite dimensional Lie
group. The transform provides a unitary map from square integrable functions on
the group to square integrable functions on a non-commutative dual space. We
then derive the first order phase space path integral for quantum mechanics on
the group by using a non-commutative dual space representation obtained through
the transform. Possible advantages of the formalism include: (1) The transform
provides an alternative to the spectral decomposition via representation theory
of Lie groups and the use of special functions. (2) The non-commutative dual
variables are physically more intuitive, since despite the non-commutativity
they are analogous to the corresponding classical variables. The work is
expected, among other possible applications, to allow for the metric
representation of Lorentzian spin foam models in the context of quantum
gravity.Comment: 7 pages, correction to the applicability of coordinates plus other
minor correction
Quantization maps, algebra representation and non-commutative Fourier transform for Lie groups
The phase space given by the cotangent bundle of a Lie group appears in the
context of several models for physical systems. A representation for the
quantum system in terms of non-commutative functions on the (dual) Lie algebra,
and a generalized notion of (non-commutative) Fourier transform, different from
standard harmonic analysis, has been recently developed, and found several
applications, especially in the quantum gravity literature. We show that this
algebra representation can be defined on the sole basis of a quantization map
of the classical Poisson algebra, and identify the conditions for its
existence. In particular, the corresponding non-commutative star-product
carried by this representation is obtained directly from the quantization map
via deformation quantization. We then clarify under which conditions a unitary
intertwiner between such algebra representation and the usual group
representation can be constructed giving rise to the non-commutative plane
waves and consequently, the non-commutative Fourier transform. The compact
groups U(1) and SU(2) are considered for different choices of quantization
maps, such as the symmetric and the Duflo map, and we exhibit the corresponding
star-products, algebra representations and non-commutative plane waves.Comment: 36 pages; matches published version plus minor correction
Asymptotic Analysis of the Ponzano-Regge Model with Non-Commutative Metric Boundary Data
We apply the non-commutative Fourier transform for Lie groups to formulate
the non-commutative metric representation of the Ponzano-Regge spin foam model
for 3d quantum gravity. The non-commutative representation allows to express
the amplitudes of the model as a first order phase space path integral, whose
properties we consider. In particular, we study the asymptotic behavior of the
path integral in the semi-classical limit. First, we compare the stationary
phase equations in the classical limit for three different non-commutative
structures corresponding to the symmetric, Duflo and Freidel-Livine-Majid
quantization maps. We find that in order to unambiguously recover discrete
geometric constraints for non-commutative metric boundary data through the
stationary phase method, the deformation structure of the phase space must be
accounted for in the variational calculus. When this is understood, our results
demonstrate that the non-commutative metric representation facilitates a
convenient semi-classical analysis of the Ponzano-Regge model, which yields as
the dominant contribution to the amplitude the cosine of the Regge action in
agreement with previous studies. We also consider the asymptotics of the -symbol using the non-commutative phase space path integral for the
Ponzano-Regge model, and explain the connection of our results to the previous
asymptotic results in terms of coherent states
On UV/IR Mixing via Seiberg-Witten Map for Noncommutative QED
We consider quantum electrodynamics in noncommutative spacetime by deriving a
-exact Seiberg-Witten map with fermions in the fundamental
representation of the gauge group as an expansion in the coupling constant.
Accordingly, we demonstrate the persistence of UV/IR mixing in noncommutative
QED with charged fermions via Seiberg-Witten map, extending the results of
Schupp and You [1].Comment: 16 page
Spacetime-Free Approach to Quantum Theory and Effective Spacetime Structure
Motivated by hints of the effective emergent nature of spacetime structure, we formulate a spacetime-free algebraic framework for quantum theory, in which no a priori background geometric structure is required. Such a framework is necessary in order to study the emergence of effective spacetime structure in a consistent manner, without assuming a background geometry from the outset. Instead, the background geometry is conjectured to arise as an effective structure of the algebraic and dynamical relations between observables that are imposed by the background statistics of the system. Namely, we suggest that quantum reference states on an extended observable algebra, the free algebra generated by the observables, may give rise to effective spacetime structures. Accordingly, perturbations of the reference state lead to perturbations of the induced effective spacetime geometry. We initiate the study of these perturbations, and their relation to gravitational phenomena