This is the second part of an article about q-deformed analogs of spinor calculus. The considerations refer to quantum spaces of physical interest, i.e. q-deformed Euclidean space in three or four dimensions as well as q-deformed Minkowski space. The Clifford algebras corresponding to these quantum spaces are treated. Especially, their commutation relations and their Hopf structures are written down. Bases of the four-dimensional Clifford algebras are constructed and their properties are discussed. Matrix representations of the Clifford algebras lead to q-deformed Dirac-matrices for the four-dimensional quantum spaces. Moreover, q-analogs of the four-dimensional spin matrices are presented. A very complete set of trace relations and rearrangement formulae concerning spin and Dirac-matrices is given. Dirac spinors together with their bilinear covariants are defined. Their behavior under q-deformed Lorentz transformation is discussed in detail.
Using an elementary perturbative open string field theory solution involving a twistor-like parameter, we study the cohomology of new nilpotent BRST charge corresponding to the space-time background defined by this solution. The BRST cohomology of the deformed background automatically cuts off the GSO-odd spectrum in RNS superstring theory and keeps the GSO-even spectrum intact, without a need of GSO-projection. The on-shell constraints in the GSO-even sector get deformed in the new background, corresponding to BRST type transformations of the low-energy effective action with the ghost-like co…
The relativistic Lorentz-covariant quantum space-times obtained by Snyder can be described by the coset generators of (anti) de-Sitter algebras. Similarly, the Lorentz-covariant quantum phase spaces introduced by Yang, which contain additionally quantum curved fourmomenta and quantum-deformed relativistic Heisenberg algebra, can be defined by suitably chosen coset generators of conformal algebras. We extend such algebraic construction to the respective superalgebras, which provide quantum Lorentz-covariant superspaces (SUSY Snyder model) and indicate also how to obtain the quantum relativistic…
In this note we give an explicit formula for the preserved Killing spinors in deformed string theory backgrounds corresponding to integrable Yang--Baxter deformations realized via (sequences of) TsT transformations. The Killing spinors can be expressed only in terms of the bi-vector $Θ$ which encodes the deformation. This formula is applicable to deformed backgrounds related to $r$-matrices of various ranks, including those that do not satisfy the unimodularity condition and give rise to backgrounds in generalized supergravity. We conjecture that our formula also remains valid for integrable d…
The Nontriviality of Trivial General Covariance: How Electrons Restrict 'Time' Coordinates, Spinors (Almost) Fit into Tensor Calculus, and 7/16 of a Tetrad Is Surplus Structure
In this work we investigate the issue of fermion localization and resonances in $(4,1)$-deformed branes constructed with one scalar field coupled with gravity. Such models provide us branes with internal structures that turns the gravitational interaction more effective for fermions aside the brane, increasing their lifetime. The coupling between the scalar field and spinors is a necessary condition for fermions to be localized on such branes. After performing a chiral decomposition of the five-dimensional spinor we found resonances with both chiralities. The correspondence between the spectra…
We present a supermatrix realisation of q-deformed spinor-spinor and spinor-vector R-matrices. These R-matrices are then used to construct transfer matrices for $U_{q^2}(\mathfrak{so}_{2n+1})$- and $U_{q}(\mathfrak{so}_{2n+2})$-symmetric closed spin chains. Their eigenvectors and eigenvalues are computed.
We study the minimal unitary representation (minrep) of SO(5,2), obtained by quantization of its geometric quasiconformal action, its deformations and supersymmetric extensions. The minrep of SO(5,2) describes a massless conformal scalar field in five dimensions and admits a unique "deformation" which describes a massless conformal spinor. Scalar and spinor minreps of SO(5,2) are the 5d analogs of Dirac's singletons of SO(3,2). We then construct the minimal unitary representation of the unique 5d superconformal algebra F(4) with the even subalgebra SO(5,2) X SU(2). The minrep of F(4) describes…
In this paper, we study the quantisation of Dirac field theory in the $κ$-deformed space-time. We adopt a quantisation method that uses only equations of motion for quantising the field. Starting from $κ$-deformed Dirac equation, valid up to first order in the deformation parameter $a$, we derive deformed unequal time anti-commutation relation between deformed field and its adjoint, leading to undeformed oscillator algebra. Exploiting the freedom of imposing a deformed unequal time anti-commutation relations between $κ$-deformed spinor and its adjoint, we also derive a deformed oscillator alge…
Classification of quantum spinor fields according to quantum bilinear covariants is introduced in a context of quantum Clifford algebras on Minkowski spacetime. Once the bilinear covariants are expressed in terms of algebraic spinor fields, the duality between spinor and quantum spinor fields is thus discussed. Hence, by endowing the underlying spacetime with an arbitrary bilinear form with a antisymmetric part in addition to a symmetric spacetime metric, quantum algebraic spinor fields and deformed bilinear covariants can be constructed. They are therefore compared to the classical (non quant…
We study the thermodynamic properties of massless Dirac fermions in graphene under a uniform magnetic field and Rashba spin-orbit coupling with a $q-$deformed Heisenberg algebra calculus. The thermodynamic functions such as the Helmholtz free energy, total energy, entropy and heat capacity are obtained by using an approach based on the zeta function and Euler-Maclaurain formula. These functions will be numerically examined for different values of $η={1\over i}\ln(q)$. In particular, the heat capacity in the presence of deformation, all curves coincide and reach the fixed value $C=6K_B$ three t…
We investigate the hidden symmetries and existence of Killing spinors of D=5 minimal gauged supergravity solutions which admit a Killing-Maxwell system in the sense of Carter, such as the Chong-Cveti$\check{c}$-Lü-Pope (CCLP) Kerr-dS black hole spacetime and a 5-dimensional minimal gauged supergravity solution endowed with a Sasaki structure deformed by torsion, a limiting case of the generalization of the 5-dimensional toric Sasaki-Einstein $L^{a,b,c}$ spacetime. We note that when an electromagnetic tensor is present and an associated Killing-Maxwell system can be constructed in the sense of …
We introduce the deformed fermionic numbers, corresponding to the skein relations, the main characteristics of knots and links. These fermionic numbers allow one to restore the skein relations. For the Alexander (Jones) skein relation we introduce corresponding Alexander (Jones) fermionic q-numbers, and for the HOMFLY skein relation - the HOMFLY deformed (p,q)-numbers with one fermionic parameter.
