Internal Space-time Symmetries Research Articles

Loop Representation of Wigner’s Little Groups

Wigner’s little groups are the subgroups of the Lorentz group whose transformations leave the momentum of a given particle invariant. They thus define the internal space-time symmetries of relativistic particles. These symmetries take different mathematical forms for massive and for massless particles. However, it is shown possible to construct one unified representation using a graphical description. This graphical approach allows us to describe vividly parity, time reversal, and charge conjugation of the internal symmetry groups. As for the language of group theory, the two-by-two representation is used throughout the paper. While this two-by-two representation is for spin-1/2 particles, it is shown possible to construct the representations for spin-0 particles, spin-1 particles, as well as for higher-spin particles, for both massive and massless cases. It is shown also that the four-by-four Dirac matrices constitute a two-by-two representation of Wigner’s little group.

Wigner’s Space-Time Symmetries Based on the Two-by-Two Matrices of the Damped Harmonic Oscillators and the Poincaré Sphere

The second-order differential equation for a damped harmonic oscillator can be converted to two coupled first-order equations, with two two-by-two matrices leading to the group Sp(2). It is shown that this oscillator system contains the essential features of Wigner’s little groups dictating the internal space-time symmetries of particles in the Lorentz-covariant world. The little groups are the subgroups of the Lorentz group whose transformations leave the four-momentum of a given particle invariant. It is shown that the damping modes of the oscillator correspond to the little groups for massive and imaginary-mass particles respectively. When the system makes the transition from the oscillation to damping mode, it corresponds to the little group for massless particles. Rotations around the momentum leave the four-momentum invariant. This degree of freedom extends the Sp(2) symmetry to that of SL(2, c) corresponding to the Lorentz group applicable to the four-dimensional Minkowski space. The Poincaré sphere contains the SL(2, c) symmetry. In addition, it has a non-Lorentzian parameter allowing us to reduce the mass continuously to zero. It is thus possible to construct the little group for massless particles from that of the massive particle by reducing its mass to zero. Spin-1/2 particles and spin-1 particles are discussed in detail.

Poincaré Sphere and a Unified Picture of Wigner’s Little Groups

It is noted that the Poincar\'e 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 Poincar\'e sphere, in addition, has a symmetry parameter corresponding to reducing the particle mass from a positive value to zero. The Poincar\'e sphere thus the gives one unified picture of Wigner's little groups for massive and massless particles.

Internal space-time symmetries of particles derivable from periodic systems in optics

While modern optics is largely a physics of harmonic oscillators and two-by-two matrices, it is possible to learn about some hidden properties of the two-by-two matrix from optical systems. Since two-by-two matrices can be divided into three conjugate classes depending on their traces, optical systems force us to establish continuity from one class to another. It is noted that those three classes are equivalent to three different branches of Wigner’s little groups dictating the internal space-time symmetries massive, massless, and imaginary-mass particles. It is shown that the periodic systems in optics can also be described by the same class-based matrix algebra. The optical system allow us to make continuous, but not analytic, transitions from massiv to massless, and massless to imaginary-mass cases.

ABCD matrices as similarity transformations of Wigner matrices and periodic systems in optics

It is shown that every ray transfer matrix, often called the ABCD matrix, can be written as a similarity transformation of one of the Wigner matrices that dictate the internal space-time symmetries of relativistic particles, while the transformation matrix is a rotation preceded by a squeeze. The implementation of this mathematical procedure is described, and how it facilitates the calculations for scattering processes in periodic systems is explained. Multilayer optics and resonators such as laser cavities are discussed in detail. For both cases, the one-cycle transfer matrix is written as a similarity transformation of one of the Wigner matrices, rendering the computation of the ABCD matrix for an arbitrary number of cycles tractable.

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 $ABCD$ 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.

Slide-rule-like property of Wigner's little groups and cyclic S matrices for multilayer optics.

It is noted that 2x2 "S" matrices in multilayer optics can be represented by the Sp(2) group whose algebraic property is the same as the group of Lorentz transformations applicable to two spacelike and one timelike dimensions. It is also noted that Wigner's little groups have a slide-rule-like property that allows us to perform multiplications by additions. It is shown that these two mathematical properties lead to a cyclic representation of the S matrix for multilayer optics, as in the case of ABCD matrices for laser cavities. It is therefore possible to write the N-layer S matrix as a multiplication of the N single-layer S matrices resulting in the same mathematical expression with one of the parameters multiplied by N. In addition, it is noted, as in the case of lens optics, that multilayer optics can serve as an analog computer for the contraction of Wigner's little groups for internal space-time symmetries of relativistic particles.

Lens optics as an optical computer for group contractions.

It is shown that the one-lens system in para-axial optics can serve as an optical computer for contraction of Wigner's little groups and an analog computer that transforms analytically computations on a spherical surface to those on a hyperbolic surface. It is shown possible to construct a set of Lorentz transformations which leads to a 2x2 matrix whose expression is the same as those in the para-axial lens optics. It is shown that the lens focal condition corresponds to the contraction of the O(3)-like little group for a massive particle to the E(2)-like little group for a massless particle, and also to the contraction of the O(2,1)-like little group for a spacelike particle to the same E(2)-like little group. The lens-focusing transformations presented in this paper allow us to continue analytically the spherical O(3) world to the hyperbolic O(2,1) world, and vice versa. Since the traditional role of Wigner's little groups has been to dictate the internal space-time symmetries of massive, massless, and imaginary-mass particles, the one-lens system provides a unification of those symmetries.

Neutrino polarization as a consequence of gauge invariance

There are gauge-transformation operators applicable to massless spin-1/2 particles within the little-group framework of internal space-time symmetries of massive and massless particles. It is shown that two of the SL(2, c) spinors are invariant under gauge transformations while the remaining two are not. The Dirac equation contains only the gauge-invariant spinors leading to polarized neutrinos. It is shown that the gauge-dependent SL(2, c) spinor is the origin of the gauge dependence of electromagnetic four-potentials. It is noted also that, for spin-1/2 particles, the symmetry group for massless particles is an infinite-momentum/zero-mass limit of the symmetry group for massive particles.

Internal space-time symmetries of massive and massless particles and their unification

It is noted that the internal space-time symmetries of relativistic particles are dictated by Wigner's little groups. The symmetry of massive particles is like the three-dimensional rotation group, while the symmetry of massless particles is locally isomorphic to the two-dimensional Euclidean group. It is noted also that, while the rotational degree of freedom for a massless particle leads to its helicity, the two translational degrees of freedom correspond to its gauge degrees of freedom. It is shown that the E(2)-like symmetry of of massless particles can be obtained as an infinite-momentum and/or zero-mass limit of the O(3)-like symmetry of massive particles. This mechanism is illustrated in terms of a sphere elongating into a cylinder. In this way, the helicity degree of freedom remains invariant under the Lorentz boost, but the transverse rotational degrees of freedom become contracted into the gauge degree of freedom.

Wigner rotations and Iwasawa decompositions in polarization optics.

Wigner rotations and Iwasawa decompositions are manifestations of the internal space-time symmetries of massive and massless particles, respectively. It is shown to be possible to produce combinations of optical filters which exhibit transformations corresponding to Wigner rotations and Iwasawa decompositions. This is possible because the combined effects of rotation, phase-shift, and attenuation filters lead to transformation matrices of the six-parameter Lorentz group applicable to Jones vectors and Stokes parameters for polarized light waves. The symmetry transformations in special relativity lead to a set of experiments which can be performed in optics laboratories.

Photons, neutrinos, and gauge transformations

A unified description is presented for internal space-time symmetries of photons and neutrinos. Lorentz-boosted Dirac equation for a massive spin-1/2 particle is discussed in the Weyl representation. In the large-momentum/small-mass limit, the resulting large component of the Dirac spinor becomes invariant under transformations of the E(2)-like little group, and the small component is not invariant under the E(2)-like transformation. The gauge dependence of the photon four-potential can thus be derived from a direct product of the gauge-independent large component and the gauge-dependent small component.

Internal space-time symmetries of massive and massless particles

A unified description is given for internal space-time symmetries for massive and massless particles. It is noted that the little groups governing the internal symmetries of massive and massless particles are locally isomorphic to the three-dimensional rotation group or O(3) and the two-dimensional rotation group or E(2), respectively. It is noted also that the E(2) group can be obtained from a large-radius/flat-surface approximation of O(3). This procedure is then shown to be directly applicable to that of obtaining the E(2)-like little group for massless particles from the O(3)-like little group for massive particles in the infinite-momentum/zero-mass limit.

C-number time-energy uncertainty relation in the quark model

The time-energy uncertainty relation is studied for the time- and energy-separation variables between two constitutent quarks inside a hadron bounded by a harmonic-oscillator force. The O(3,1) and O(3)-like internal space-time symmetries discussed in a previous paper are formulated in terms of covariant commutators. It is shown that the resulting commutation relations constitute a covariant realization of Dirac's $c$-number time-energy uncertainty relation.

Charge conjugation and internal space-time symmetries

Charge conjugation and internal space-time symmetries

Charge conjugation and internal space-time symmetries

We adopt the relativistic framework in which fundamental particles are regarded as extended objects. Then, we show that the (geometrical) operation which reflects theinternal space-time of a particle is equivalent to the operationC which invertes the sign of all its additive charges.