330 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
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
Nonassociative Weyl star products
Deformation quantization is a formal deformation of the algebra of smooth
functions on some manifold. In the classical setting, the Poisson bracket
serves as an initial conditions, while the associativity allows to proceed to
higher orders. Some applications to string theory require deformation in the
direction of a quasi-Poisson bracket (that does not satisfy the Jacobi
identity). This initial condition is incompatible with associativity, it is
quite unclear which restrictions can be imposed on the deformation. We show
that for any quasi-Poisson bracket the deformation quantization exists and is
essentially unique if one requires (weak) hermiticity and the Weyl condition.
We also propose an iterative procedure that allows to compute the star product
up to any desired order.Comment: discussion extended, tipos corrected, published versio
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
Noncommutative via closed star product
We consider linear star products on of Lie algebra type. First we
derive the closed formula for the polydifferential representation of the
corresponding Lie algebra generators. Using this representation we define the
Weyl star product on the dual of the Lie algebra. Then we construct a gauge
operator relating the Weyl star product with the one which is closed with
respect to some trace functional, . We introduce
the derivative operator on the algebra of the closed star product and show that
the corresponding Leibnitz rule holds true up to a total derivative. As a
particular example we study the space with type
noncommutativity and show that in this case the closed star product is the one
obtained from the Duflo quantization map. As a result a Laplacian can be
defined such that its commutative limit reproduces the ordinary commutative
one. The deformed Leibnitz rule is applied to scalar field theory to derive
conservation laws and the corresponding noncommutative currents.Comment: published versio
Gauge invariance and classical dynamics of noncommutative particle theory
We consider a model of classical noncommutative particle in an external
electromagnetic field. For this model, we prove the existence of generalized
gauge transformations. Classical dynamics in Hamiltonian and Lagrangian form is
discussed, in particular, the motion in the constant magnetic field is studied
in detail.Comment: 10 page
Noncommutativity due to spin
Using the Berezin-Marinov pseudoclassical formulation of spin particle we
propose a classical model of spin noncommutativity. In the nonrelativistic
case, the Poisson brackets between the coordinates are proportional to the spin
angular momentum. The quantization of the model leads to the noncommutativity
with mixed spacial and spin degrees of freedom. A modified Pauli equation,
describing a spin half particle in an external e.m. field is obtained. We show
that nonlocality caused by the spin noncommutativity depends on the spin of the
particle; for spin zero, nonlocality does not appear, for spin half, , etc. In the relativistic case the noncommutative
Dirac equation was derived. For that we introduce a new star product. The
advantage of our model is that in spite of the presence of noncommutativity and
nonlocality, it is Lorentz invariant. Also, in the quasiclassical approximation
it gives noncommutativity with a nilpotent parameter.Comment: 11 pages, references adda
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