55 research outputs found
Scalar field theory on kappa-Minkowski spacetime and translation and Lorentz invariance
We investigate the properties of kappa-Minkowski spacetime by using
representations of the corresponding deformed algebra in terms of undeformed
Heisenberg-Weyl algebra. The deformed algebra consists of kappa-Poincare
algebra extended with the generators of the deformed Weyl algebra. The part of
deformed algebra, generated by rotation, boost and momentum generators, is
described by the Hopf algebra structure. The approach used in our
considerations is completely Lorentz covariant. We further use an adventages of
this approach to consistently construct a star product which has a property
that under integration sign it can be replaced by a standard pointwise
multiplication, a property that was since known to hold for Moyal, but not also
for kappa-Minkowski spacetime. This star product also has generalized trace and
cyclic properties and the construction alone is accomplished by considering a
classical Dirac operator representation of deformed algebra and by requiring it
to be hermitian. We find that the obtained star product is not translationally
invariant, leading to a conclusion that the classical Dirac operator
representation is the one where translation invariance cannot simultaneously be
implemented along with hermiticity. However, due to the integral property
satisfied by the star product, noncommutative free scalar field theory does not
have a problem with translation symmetry breaking and can be shown to reduce to
an ordinary free scalar field theory without nonlocal features and tachionic
modes and basicaly of the very same form. The issue of Lorentz invariance of
the theory is also discussed.Comment: 22 pages, no figures, revtex4, in new version comments regarding
translation invariance and few references are added, accepted for publication
in Int. J. Mod. Phys.
Tensor Coordinates in Noncommutative Mechanics
A consistent classical mechanics formulation is presented in such a way that,
under quantization, it gives a noncommutative quantum theory with interesting
new features. The Dirac formalism for constrained Hamiltonian systems is
strongly used, and the object of noncommutativity plays
a fundamental rule as an independent quantity. The presented classical theory,
as its quantum counterpart, is naturally invariant under the rotation group
.Comment: 12 pages, Late
Kappa-deformed Snyder spacetime
We present Lie-algebraic deformations of Minkowski space with undeformed
Poincare algebra. These deformations interpolate between Snyder and
kappa-Minkowski space. We find realizations of noncommutative coordinates in
terms of commutative coordinates and derivatives. Deformed Leibniz rule, the
coproduct structure and star product are found. Special cases, particularly
Snyder and kappa-Minkowski in Maggiore-type realizations are discussed. Our
construction leads to a new class of deformed special relativity theories.Comment: 12 pages, no figures, LaTeX2e class file, accepted for publication in
Modern Physics Letters
KMS states on Quantum Grammars
We consider quantum (unitary) continuous time evolution of spins on a lattice
together with quantum evolution of the lattice itself. In physics such
evolution was discussed in connection with quantum gravity. It is also related
to what is called quantum circuits, one of the incarnations of a quantum
computer. We consider simpler models for which one can obtain exact
mathematical results. We prove existence of the dynamics in both Schroedinger
and Heisenberg pictures, construct KMS states on appropriate C*-algebras. We
show (for high temperatures) that for each system where the lattice undergoes
quantum evolution, there is a natural scaling leading to a quantum spin system
on a fixed lattice, defined by a renormalized Hamiltonian.Comment: 22 page
Noncommutative Complex Scalar Field and Casimir Effect
A noncommutative complex scalar field, satisfying the deformed canonical
commutation relations proposed by Carmona et al. [27]-[31], is constructed.
Using these noncommutative deformed canonical commutation relations, a model
describing the dynamics of the noncommutative complex scalar field is proposed.
The noncommutative field equations are solved, and the vacuum energy is
calculated to the second order in the parameter of noncommutativity. As an
application to this model, the Casimir effect, due to the zero point
fluctuations of the noncommutative complex scalar field, is considered. It
turns out that in spite of its smallness, the noncommutativity gives rise to a
repulsive force at the microscopic level, leading to a modifed Casimr potential
with a minimum at the point amin= racine(5/84){\pi}{\theta}.Comment: Revtex style, 28 page
Kappa Snyder deformations of Minkowski spacetime, realizations and Hopf algebra
We present Lie-algebraic deformations of Minkowski space with undeformed
Poincar\'{e} algebra. These deformations interpolate between Snyder and
-Minkowski space. We find realizations of noncommutative coordinates in
terms of commutative coordinates and derivatives. By introducing modules, it is
shown that although deformed and undeformed structures are not isomorphic at
the level of vector spaces, they are however isomorphic at the level of Hopf
algebraic action on corresponding modules. Invariants and tensors with respect
to Lorentz algebra are discussed. A general mapping from -deformed
Snyder to Snyder space is constructed. Deformed Leibniz rule, the Hopf
structure and star product are found. Special cases, particularly Snyder and
-Minkowski in Maggiore-type realizations are discussed. The same
generalized Hopf algebraic structures are as well considered in the case of an
arbitrary allowable kind of realisation and results are given perturbatively up
to second order in deformation parameters.Comment: 38 pages, LaTeX2e class fil
Formulation, Interpretation and Application of non-Commutative Quantum Mechanics
In analogy with conventional quantum mechanics, non-commutative quantum
mechanics is formulated as a quantum system on the Hilbert space of
Hilbert-Schmidt operators acting on non-commutative configuration space. It is
argued that the standard quantum mechanical interpretation based on Positive
Operator Valued Measures, provides a sufficient framework for the consistent
interpretation of this quantum system. The implications of this formalism for
rotational and time reversal symmetry are discussed. The formalism is applied
to the free particle and harmonic oscillator in two dimensions and the physical
signatures of non commutativity are identified.Comment: 11 page
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