1,481 research outputs found
Weak associativity and deformation quantization
Non-commutativity and non-associativity are quite natural in string theory.
For open strings it appear due to the presence of non-vanishing background
two-form in the world volume of Dirichlet brane, while in closed string theory
the flux compactifications with non-vanishing three-form also lead to
non-geometric backgrounds. In this paper, working in the framework of
deformation quantization, we study the violation of associativity imposing the
condition that the associator of three elements should vanish whenever each two
of them are equal. The corresponding star products are called alternative and
satisfy an important for physical applications properties like the Moufang
identities, alternative identities, Artin's theorem, etc. The condition of
alternativity is invariant under the gauge transformations, just like it
happens in the associative case. The price to pay is the restriction on the
non-associative algebra which can be represented by the alternative star
product, it should satisfy the Malcev identity. The example of nontrivial
Malcev algebra is the algebra of imaginary octonions. For this case we
construct an explicit expression of the non-associative and alternative star
product. We also discuss the quantization of Malcev-Poisson algebras of general
form, study its properties and provide the lower order expression for the
alternative star product. To conclude we define the integration on the algebra
of the alternative star products and show that the integrated associator
vanishes.Comment: 24 pages, V2, examples corrected, discussion extended, refferences
adde
Non-commutative deformation of Chern-Simons theory
The problem of the consistent definition of gauge theories living on the
non-commutative (NC) spaces with a non-constant NC parameter is
discussed. Working in the L formalism we specify the undeformed
theory, d abelian Chern-Simons, by setting the initial brackets.
The deformation is introduced by assigning the star commutator to the
bracket. For this initial set up we construct the corresponding L
structure which defines both the NC deformation of the abelian gauge
transformations and the field equations covariant under these transformations.
To compensate the violation of the Leibniz rule one needs the higher brackets
which are proportional to the derivatives of . Proceeding in the slowly
varying field approximation when the star commutator is approximated by the
Poisson bracket we derive the recurrence relations for the definition of these
brackets for arbitrary . For the particular case of -like NC
space we obtain an explicit all orders formulas for both NC gauge
transformations and NC deformation of Chern-Simons equations. The latter are
non-Lagrangian and are satisfied if the NC field strength vanishes everywhere.Comment: 33 pages, published version, exposition improved, new material
regarding the definition of the non-commutative field strength and the
treatment of the non-commutativity of general form adde
Dirac equation on coordinate dependent noncommutative space-time
We consider the consistent deformation of the relativistic quantum mechanics
introducing the noncommutativity of the space-time and preserving the Lorentz
symmetry. The relativistic wave equation describing the spinning particle on
coordinate dependent noncommutative space-time (noncommutative Dirac equation)
is proposed. The fundamental properties of this equation, like the Lorentz
covariance and the continuity equation for the probability density are
verified. To this end using the properties of the star product we derive the
corresponding probability current density and prove its conservation. The
energy-momentum tensor for the free noncommutative spinor field is calculated.
We solve the free noncommutative Dirac equation and show that the standard
energy-momentum dispersion relation remains valid in the noncommutative case.Comment: Published versio
Strongly anharmonic current-phase relation in ballistic graphene Josephson junctions
Motivated by a recent experiment directly measuring the current-phase
relation (CPR) in graphene under the influence of a superconducting proximity
effect, we here study the temperature dependence of the CPR in ballistic
graphene SNS Josephson junctions within the the self-consistent tight-binding
Bogoliubov-de Gennes (BdG) formalism. By comparing these results with the
standard Dirac-BdG method, where rigid boundary conditions are assumed at the
SN interfaces, we show on a crucial importance of both proximity effect and
depairing by current for the CPR. The proximity effect grows with temperature
and reduces the skewness of the CPR towards the harmonic result. In short
junctions () current depairing is also important and gives rise to a
critical phase over a wide range of temperatures and doping
levels.Comment: 7 pages, 4 figures. v2 contains very minor change
Position-dependent noncommutativity in quantum mechanics
The model of the position-dependent noncommutativety in quantum mechanics is
proposed. We start with a given commutation relations between the operators of
coordinates [x^{i},x^{j}]=\omega^{ij}(x), and construct the complete algebra of
commutation relations, including the operators of momenta. The constructed
algebra is a deformation of a standard Heisenberg algebra and obey the Jacobi
identity. The key point of our construction is a proposed first-order
Lagrangian, which after quantization reproduces the desired commutation
relations. Also we study the possibility to localize the noncommutativety.Comment: published version, references adde
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