Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userDeformed Covariant Quantum Phase Spaces as Hopf Algebroids
135
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
We consider two quantum phase spaces which can be described by two Hopf algebroids linked with the well-known $theta_{mu u }$-deformed $D=4$ Poincare-Hopf algebra $mathbb{H}$. The first algebroid describes $theta_{mu u }$-deformed relativistic phase space with canonical NC space-time (constant $theta_{mu u }$ parameters) and the second one incorporates dual to $mathbb{H}$ quantum $theta_{mu u }$-deformed Poincare-Hopf group algebra $mathbb{G}$, which contains noncommutative space-time translations given by $Lambda $-dependent $Theta_{mu u }$ parameters ($% Lambda $ $equiv Lambda_{mu u }$ parametrize classical Lorentz group). The canonical $theta_{mu u }$-deformed space-time algebra and its quantum phase space extension is covariant under the quantum Poincare transformations described by $mathbb{G}$. We will also comment on the use of Hopf algebroids for the description of multiparticle structures in quantum phase spaces.
We consider new Abelian twists of Poincare algebra describing non-symmetric generalization of the ones given in [1], which lead to the class of Lie-deformed quantum Minkowski spaces. We apply corresponding twist quantization in two ways: as generating quantum Poincare-Hopf algebra providing quantum Poincare symmetries, and by considering the quantization which provides Hopf algebroid describing the class of quantum relativistic phase spaces with built-in quantum Poincare covariance. If we assume that Lorentz generators are orbital i.e.do not describe spin degrees of freedom, one can embed the considered generalized phase spaces into the ones describing the quantum-deformed Heisenberg algebras.
We consider the generalized (10+10)-dimensional D=4 quantum phase spaces containing translational and Lorentz spin sectors associated with the dual pair of twist-quantized Poincare Hopf algebra $mathbb{H}$ and quantum Poincare Hopf group $widehat{mathbb{G}}$. Two Hopf algebroid structures of generalized phase spaces with spin sector will be investigated: first one $% mathcal{H}^{(10,10)}$ describing dynamics on quantum group algebra $% widehat{mathbb{G}}$ provided by the Heisenberg double algebra $mathcal{HD=% }mathbb{H}rtimes widehat{mathbb{G}}$, and second, denoted by $mathcal{% tilde{H}}^{(10,10)}$, describing twisted Hopf algebroid with base space containing twisted noncommutative Minkowski space $hat{x}_{mu }$. We obtain the first explicit example of Hopf algebroid structure of relativistic quantum phase space which contains quantum-deformed Lorentz spin sector.
We consider $kappa$-deformed relativistic quantum phase space and possible implementations of the Lorentz algebra. There are two ways of performing such implementations. One is a simple extension where the Poincare algebra is unaltered, while the other is a general extension where the Poincare algebra is deformed. As an example we fix the Jordanian twist and the corresponding realization of noncommutative coordinates, coproduct of momenta and addition of momenta. An extension with a one-parameter family of realizations of the Lorentz generators, dilatation and momenta closing the Poincare-Weyl algebra is considered. The corresponding physical interpretation depends on the way the Lorentz algebra is implemented in phase space. We show how the spectrum of the relativistic hydrogen atom depends on the realization of the generators of the Poincare-Weyl algebra.
The covariant phase space method of Iyer, Lee, Wald, and Zoupas gives an elegant way to understand the Hamiltonian dynamics of Lagrangian field theories without breaking covariance. The original literature however does not systematically treat total derivatives and boundary terms, which has led to some confusion about how exactly to apply the formalism in the presence of boundaries. In particular the original construction of the canonical Hamiltonian relies on the assumed existence of a certain boundary quantity $B$, whose physical interpretation has not been clear. We here give an algorithmic procedure for applying the covariant phase space formalism to field theories with spatial boundaries, from which the term in the Hamiltonian involving $B$ emerges naturally. Our procedure also produces an additional boundary term, which was not present in the original literature and which so far has only appeared implicitly in specific examples, and which is already nonvanishing even in general relativity with sufficiently permissive boundary conditions. The only requirement we impose is that at solutions of the equations of motion the action is stationary modulo future/past boundary terms under arbitrary variations obeying the spatial boundary conditions; from this the symplectic structure and the Hamiltonian for any diffeomorphism that preserves the theory are unambiguously constructed. We show in examples that the Hamiltonian so constructed agrees with previous results. We also show that the Poisson bracket on covariant phase space directly coincides with the Peierls bracket, without any need for non-covariant intermediate steps, and we discuss possible implications for the entropy of dynamical black hole horizons.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
