April 27, 2002
The lattice of integral points of 4-dimensional Minkowski space, together with the inherited indefinite distance function, is considered as a model for discrete space-time. The Lorentz and Poincare groups of this discrete space-time are identified as subgroups of the corresponding Lie groups. The lattice Lorentz group has irreducible projective (including linear) representations which are restrictions of (all) finite-dimensional irreducible projective representations of the Lorentz Lie group and hence can be used to describe all integral and half-odd-integral helicity. The (4-torus) momentum space has a well-defined ``light cone'' of null points and there are orbits of the lattice Lorentz group lying entirely in the torus light cone and having the lattice euclidean group of the plane as little group. Wigner's method for the Poincare Lie group can then be adapted to show, in the first instance, that the lattice Poincare group has unitary representations describing lattice free fields of zero mass and an arbitrary Lorentz helicity, in particular chiral fermions. There are no representations with a nonzero invariant mass.
Similar papers 1
April 6, 2024
The unitary representations of the Poincare group of a discrete space-time are constructed, following the Wigner method in continuum relativity. They can be interpreted as elementary particles with one significant new feature: the momentum space being the 4-torus is identified as the Brillouin zone of space-time where all physical phenomena occur. Consequently 4-momentum is defined and conserved only modulo a reciprocal lattice vector of the order of the Planck mass, implying...
January 5, 2004
Following standard methods we explore the construction of the discrete Poincare group, the semidirect product of discrete translations and integral Lorentz transformations, using the Wigner-Mackey construction restricted to the momentum and position space on the lattice. The orbit condition, irreducibility and assimptotic limit are discussed.
January 29, 2024
In this review, we have reached from the most basic definitions in the theory of groups, group structures, etc. to representation theory and irreducible representations of the Poincar'e group. Also, we tried to get a more comprehensible understanding of group theory by presenting examples from the nature around us to examples in mathematics and physics and using them to examine more important groups in physics such as the Lorentz group and Poincar'e group and representations ...
December 29, 2003
We explore the mathematical consequences of the assumption of a discrete space-time. The fundamental laws of physics have to be translated into the language of discrete mathematics. We find integral transformations that leave the lattice of any dimension invariant and apply these transformations to field equations.
January 3, 2004
We continue the program, presented in previous Symposia, of discretizing physical models. In particular we calculate the integral Lorentz transformations with the help of discrete reflection groups, and use them for the covariance of Klein-Gordon and Dirac wave equation on the lattice. Finally we define the unitary representation of Poincar group on discrete momentum and configuration space, induced by integral representations of its closed subgroup.
November 24, 2006
An extensive group-theoretical treatment of linear relativistic field equations on Minkowski spacetime of arbitrary dimension D>2 is presented in these lecture notes. To start with, the one-to-one correspondence between linear relativistic field equations and unitary representations of the isometry group is reviewed. In turn, the method of induced representations reduces the problem of classifying the representations of the Poincare group ISO(D-1,1) to the classication of the...
January 9, 2004
We deduce the structure of the Dirac field on the lattice from the discrete version of differential geometry and from the representation of the integral Lorentz transformations. The analysis of the induced representations of the Poincare group on the lattice reveals that they are reducible, a result that can be considered a group theoretical approach to the problem of fermion doubling.
January 26, 2019
The following work demonstrates the viability of Poincar\'e symmetry in a discrete universe. We develop the technology of the discrete principal Poincar\'e bundle to describe the pairing of (1) a hypercubic lattice `base manifold' labeled by integer vertices-denoted $\{\mathbf{n}\}=\{(n_t,n_x,n_y,n_z)\}$-with (2) a Poincar\'e structure group. We develop lattice 5-vector theory, which describes a non-unitary representation of the Poincar\'e group whose dynamics and gauge trans...
August 4, 1998
A survey of results on quantum Poincare groups and quantum Minkowski spaces is presented.
September 20, 2013
The formulation of quantum mechanics with a complex Hilbert space is equivalent to a formulation with a real Hilbert space and particular density matrix and observables. We study the real representations of the Poincare group, motivated by the fact that the localization of complex unitary representations of the Poincare group is incompatible with causality, Poincare covariance and energy positivity. We review the map from the complex to the real irreducible representations-...