We consider the general D=4 (10+10)-dimensional kappa-deformed quantum phase space as given by Heisenberg double mathcal{H} of D=4 kappa-deformed Poincare-Hopf algebra H. The standard (4+4) -dimensional kappa - deformed covariant quantum phase space
spanned by kappa - deformed Minkowski coordinates and commuting momenta generators ({x}_{mu },{p}_{mu }) is obtained as the subalgebra of mathcal{H}. We study further the property that Heisenberg double defines particular quantum spaces with Hopf algebroid structure. We calculate by using purely algebraic methods the explicite Hopf algebroid structure of standard kappa - deformed quantum covariant phase space in Majid-Ruegg bicrossproduct basis. The coproducts for Hopf algebroids are not unique, determined modulo the coproduct gauge freedom. Finally we consider the interpretation of the algebraic description of quantum phase spaces as Hopf algebroids.
After short historical overview we describe the difficulties with application of standard QFT methods in quantum gravity (QG). The incompatibility of QG with the use of classical continuous space-time required conceptually new approach. We present br
iefly three proposals: loop quantum gravity (LQG), the field-theoretic framework on noncommutative space-time and QG models formulated on discretized (triangularized) space-time. We evaluate these models as realizing expected important properties of QG: background independence, consistent quantum diffeomorphisms, noncommutative or discrete structure of space-time at very short distances, finite/renormalizable QG corrections. We only briefly outline an important issue of embedding QG into larger geometric and dynamical frameworks (e.g. supergravity, (super)strings, p-branes, M-theory), with the aim to achieve full unification of all fundamental interactions.
We introduce the free quantum noncommutative fields as described by braided tensor products. The multiplication of such fields is decomposed into three operations, describing the multiplication in the algebra M of functions on noncommutative space-ti
me, the product in the algebra H of deformed field oscillators, and the braiding by factor Psi_{M,H} between algebras M and H. For noncommutativity generated by the twist factor we shall employ the star-product realizations of the algebra M in terms of functions on standard Minkowski space. The covariance of single noncommutative quantum fields under deformed Poincare symmetries is described by the algebraic covariance conditions which are equivalent to the deformation of generalized Heisenberg equations on Poincare group manifold. We shall calculate the covariant braided field commutator, which for free quantum noncommutative fields provides the field quantization condition and is given by standard Pauli-Jordan function. For ilustration of our new scheme we present explicit calculations for the well-known case in the literature of canonically deformed free quantum fields.
In order to obtain free kappa-deformed quantum fields (with c-number commutators) we proposed new concept of kappa-deformed oscillator algebra [1] and the modification of kappa-star product [2], implementing in the product of two quantum fields the c
hange of standard kappa-deformed mass-shell conditions. We recall here that the kappa-deformed oscillators recently introduced in [3]-[5] lie on standard kappa-deformed mass-shell. Firstly, we study kappa-deformed fields with the standard kappa-star product, what implies that in the oscillator algebra the corresponding kappa-deformed oscillators lie on standard kappa-deformed mass-shell. We argue that for the kappa-deformed algebra of such field oscillators which carry fourmomenta on kappa-deformed mass-shell it is not possible to obtain the free quantum kappa-deformed fields with the c-number commutators. Further, we study kappa-deformed quantum fields with the modified kappa-star product which implies the modification of kappa-deformed mass-shell. We obtain large class of kappa-deformed statistics depending on six arbitrary functions which provides the c-number field commutator functions. Such general class of kappa-oscillators can be described as the kappa-deformation of standard oscillator algebra obtained by composing general kappa-deformed multiplication with the deformation of the flip operator.
We transform the oscillator algebra with kappa-deformed multiplication rule, proposed in [1],[2], into the oscillator algebra with kappa-deformed flip operator and standard multiplication. We recall that the kappa-multiplication of the kappa-oscillat
ors puts them off-shell. We study the explicit forms of modified mass-shell conditions in both formulations: with kappa-multiplication and with kappa-flip operation. On the example of kappa-deformed 2-particle states we study the clustered nonfactorizable form of the kappa-deformed multiparticle states. We argue that the kappa-deformed star product of two free fields leads in similar way to a nonfactorizable kappa-deformed bilocal field. We conclude with general remarks concerning the kappa-deformed n-particle clusters and kappa-deformed star product of n fields.
We show how some classical r-matrices for the D=4 Poincare algebra can be supersymmetrized by an addition of part depending on odd supercharges. These r-matrices for D=4 super-Poincare algebra can be presented as a sum of the so-called subordinated r
-matrices of super-Abelian and super-Jordanian type. Corresponding twists describing quantum deformations are obtained in an explicit form. These twists are the super-extensions of twists obtained in the paper arXiv:0712.3962.
We describe the deformed E.T. quantization rules for kappa-deformed free quantum fields, and relate these rules with the kappa-deformed algebra of field oscillators.
We describe D=4 twistorial membrane in terms of two twistorial three-dimensional world volume fields. We start with the D-dimensional p-brane generalizations of two phase space string formulations: the one with $p+1$ vectorial fourmomenta, and the se
cond with tensorial momenta of $(p+1)$-th rank. Further we consider tensionful membrane case in D=4. By using the membrane generalization of Cartan-Penrose formula we express the fourmomenta by spinorial fields and obtain the intermediate spinor-space-time formulation. Further by expressing the worldvolume dreibein and the membrane space-time coordinate fields in terms of two twistor fields one obtains the purely twistorial formulation. It appears that the action is generated by a geometric three-form on two-twistor space. Finally we comment on higher-dimensional (D>4) twistorial p-brane models and their superextensions.
