Dirac field and gravity in NC $$SO(2,3)_\star $$ model
Eur. Phys. J. C
Dirac field and gravity in NC S O (2, 3)
Dragoljub Gocˇanin 0
Voja Radovanovic´ 0
0 Faculty of Physics, University of Belgrade , Studentski trg 12, 11000 Beograd , Serbia
Action for the Dirac spinor field coupled to gravity on noncommutative (NC) Moyal-Weyl spacetime is obtained without prior knowledge of the metric tensor. We emphasize gauge origins of gravity and its interaction with fermions by demonstrating that a classical action invariant under S O(2, 3) gauge transformations can be exactly reduced to the Dirac action in curved spacetime after breaking the original symmetry down to the local Lorentz S O(1, 3) symmetry. The commutative S O(2, 3) invariant action can be straightforwardly deformed via Moyal-Weyl -product to its NC S O(2, 3) invariant version which can be expanded perturbatively in powers of the deformation parameter using the Seiberg-Witten map. The NC gravity-matter couplings in the expansion arise as an effect of the gauge symmetry breaking. We calculate in detail the first order NC correction to the classical Dirac action in curved spacetime and show that it does not vanish. Moreover, linear NC effects are apparent even in flat spacetime. We analyse NC deformation of the Dirac equation, Feynman propagator and dispersion relation for electrons in Minkowski spacetime and conclude that constant NC background acts as a birefringent medium for electrons propagating in it.
1 Introduction
Quantum Field Theory (QFT) and General Relativity (GR)
are two cornerstones of modern theoretical physics. Although
these theories have been tested to an excellent degree of
accuracy in their respective areas of applicability, occurrence of
singularities in both of them strongly indicates that they are
incomplete. GR, as a classical theory of gravity, describes
large-scale geometric structure of spacetime and its relation
to the distribution of matter. On the other hand, QFT,
standing on the principles of Quantum Mechanics and Special
Relativity, provides us with the Standard Model of
elementary particles which successfully utilizes the idea of local
symmetry to describe the fundamental particle interactions.
Understanding quantum nature of spacetime and reconciling
gravity with other fundamental interactions is considered to
be one of the main goals of contemporary physics.
In order to obtain a consistent unified theory, certain
modifications of the basic concepts of QFT and GR are necessary.
Various approaches have been proposed so far, stemming
from String Theory, Loop Quantum Gravity,
Noncommutative (NC) Field Theory, etc. and all of them, in some radical
way, change the notion of point particle and/or that of
spacetime.
In the last twenty years, Noncommutative Field Theory
has become a very important direction of investigation in
theoretical high energy physics and gravity. Its basic insight
is that the quantum nature of spacetime, at the microscopic
level, should mean that even the spacetime coordinates are to
be treated as mutually incompatible observables, satisfying
some non trivial commutation relations. The simplest choice
of noncommutativity is the so called canonical
noncommutativity, defined by
where θ μν are components of a constant antisymmetric
matrix.
To establish canonical noncommutativity, instead of using
abstract algebra of coordinates, i.e. noncommutative
spacetime, one can equivalently introduce the noncommutative
Moyal-Weyl -product,
f (x ) g(x ) = e 2i θαβ ∂x∂α ∂y∂β f (x )g(y)|y→x ,
as a multiplication of functions (fields) defined on the usual,
commutative (undeformed) spacetime. The quantity θ μν is
considered to be a small deformation parameter that has
dimensions of (length)2 (in natural units). It is a fundamental
constant, like the Planck length or the speed of light.
(1.1)
(1.2)
Recently, a lot of attention has been devoted to NC
gravity, and many different approaches to this problem have been
developed. In [1–3] a deformation of pure Einstein
gravity based on the Seiberg-Witten map is proposed. Twist
approach to noncommutative gravity was explored in [4–
7]. Lorentz symmetry in NC field theories was considered
in [8,9]. Some other proposals are given in [10–22]. The
connection to supergravity was established in [23,24]. The
extension of NC gauge theories to orthogonal and
symplectic algebra was considered in [25,26]. Finally, in the
previous papers of one of the authors [27–30] an approach
based on the deformed Anti de Sitter (AdS) symmetry group,
i.e. S O(
2, 3
) group, and canonical noncommutativity was
established. In this approach NC gravity is treated as a gauge
theory. It becomes manifest only after the suitable
symmetry breaking. The action was constructed without previous
introduction of the metric tensor and the second order NC
correction to the Einstein-Hilbert action was found
explicitly. Special attention has been devoted to the meaning of
the coordinates used. Namely, it was shown that
coordinates in which we postulate canonical nonco (...truncated)