14 research outputs found
DIRAC GAMMA MATRICES AS REPRESENTATIONS OF THE ALGEBRA OF LITTLE GROUPS
The algebra of O(3, 3) is contracted to that of E(2)-like groups in the infinite-momentum limit. It is shown through this process that there are 18 different possibilities for the representations of E(2)-like algebra if they are to be expressed in terms of Dirac γ-matrices. It is also pointed out that the generators of this algebra are independent of the particular representation of the γ-matrices. Possible implications of this analysis is discussed
The language of Einstein spoken by optical instruments
Einstein had to learn the mathematics of Lorentz transformations in order to
complete his covariant formulation of Maxwell's equations. The mathematics of
Lorentz transformations, called the Lorentz group, continues playing its
important role in optical sciences. It is the basic mathematical language for
coherent and squeezed states. It is noted that the six-parameter Lorentz group
can be represented by two-by-two matrices. Since the beam transfer matrices in
ray optics is largely based on two-by-two matrices or matrices, the
Lorentz group is bound to be the basic language for ray optics, including
polarization optics, interferometers, lens optics, multilayer optics, and the
Poincar\'e sphere. Because the group of Lorentz transformations and ray optics
are based on the same two-by-two matrix formalism, ray optics can perform
mathematical operations which correspond to transformations in special
relativity. It is shown, in particular, that one-lens optics provides a
mathematical basis for unifying the internal space-time symmetries of massive
and massless particles in the Lorentz-covariant world.Comment: LaTex 8 pages, presented at the 10th International Conference on
Quantum Optics (Minsk, Belarus, May-June 2004), to be published in the
proceeding
Little groups and Maxwell-type tensors for massive and massless particles
The massive and massless representations of four-vectors and
Maxwell-type tensors are constructed as bilinear combinations of
SL(2,C)-spinors and are evaluated on a unified description upon
adopting the group contraction procedure of O(3)-like little group
to E(2)-like little group. Contraction of massive particle
representations leads to gauge-dependent vectors as well as the gauge-independent
"state vectors" constructed by
Weinberg. It is shown that gauge degrees of freedom associated with the translation-like
transformations of the E(2)-like little group can be traced to the
spinors that undergo spin flips in the infinite-momentum/zero-mass limit
Poincaré Symmetry from Heisenberg’s Uncertainty Relations
It is noted that the single-variable Heisenberg commutation relation contains the symmetry of the S p ( 2 ) group which is isomorphic to the Lorentz group applicable to one time-like dimension and two space-like dimensions, known as the S O ( 2 , 1 ) group. According to Paul A. M. Dirac, from the uncertainty commutation relations for two variables, it possible to construct the de Sitter group S O ( 3 , 2 ) , namely the Lorentz group applicable to three space-like variables and two time-like variables. By contracting one of the time-like variables in S O ( 3 , 2 ) , it is possible to construct the inhomogeneous Lorentz group I S O ( 3 , 1 ) which serves as the fundamental symmetry group for quantum mechanics and quantum field theory in the Lorentz-covariant world. This I S O ( 3 , 1 ) group is commonly known as the Poincaré group
Dirac Matrices and Feynman’s Rest of the Universe
There are two sets of four-by-four matrices introduced by Dirac. The first set consists of fifteen Majorana matrices derivable from his four γ matrices. These fifteen matrices can also serve as the generators of the group SL(4, r). The second set consists of ten generators of the Sp(4) group which Dirac derived from two coupled harmonic oscillators. It is shown possible to extend the symmetry of Sp(4) to that of SL(4, r) if the area of the phase space of one of the oscillators is allowed to become smaller without a lower limit. While there are no restrictions on the size of phase space in classical mechanics, Feynman’s rest of the universe makes this Sp(4)-to-SL(4, r) transition possible. The ten generators are for the world where quantum mechanics is valid. The remaining five generators belong to the rest of the universe. It is noted that the groups SL(4, r) and Sp(4) are locally isomorphic to the Lorentz groups O(3, 3) and O(3, 2) respectively. This allows us to interpret Feynman’s rest of the universe in terms of space-time symmetry
Poincaré Sphere and a Unified Picture of Wigner’s Little Groups
It is noted that the Poincaré sphere for polarization optics contains the symmetries of the Lorentz group. The sphere is thus capable of describing the internal space-time symmetries dictated by Wigner’s little groups. For massive particles, the little group is like the three-dimensional rotation group, while it is like the two-dimensional Euclidean group for massless particles. It is shown that the Poincaré sphere, in addition, has a symmetry parameter corresponding to reducing the particle mass from a positive value to zero. The Poincaré sphere thus the gives one unified picture of Wigner’s little groups for massive and massless particles