1673726
Symmetries Shared by the Poincaré Group and the Poincaré Sphere
Henri Poincare formulated the mathematics of Lorentz transformations, known as the Poincare group. He also formulated the Poincare sphere for polarization optics. It is shown that these two mathematical instruments can be derived from the two-by-two representations of the Lorentz group. Wigner’s little groups for internal space-time symmetries are studied in detail. While the particle mass is a Lorentz-invariant quantity, it is shown to be possible to address its variations in terms of the decoherence mechanism in polarization optics.
117540013
Theory and applications of the Poincaré group
I: Elements of Group Theory.- 1. Definition of a Group.- 2. Subgroups, Cosets, and Invariant Subgroups.- 3. Equivalence Classes, Orbits, and Little Groups.- 4. Representations and Representation Spaces.- 5. Properties of Matrices.- 6. Schur's Lemma.- 7. Exercises and Problems.- II: Lie Groups and Lie Algebras.- 1. Basic Concepts of Lie Groups.- 2. Basic Theorems Concerning Lie Groups.- 3. Properties of Lie Algebras.- 4. Properties of Lie Groups.- 5. Further Theorems of Lie Groups.- 6. Exercises and Problems.- III: Theory of the Poincare Group.- 1. Group of Lorentz Transformations.- 2. Orbits and Little Groups of the Proper Lorentz Group.- 3. Representations of the Poincare Group.- 4. Lorentz Transformations of Wave Functions.- 5. Lorentz Transformations of Free Fields.- 6. Discrete Symmetry Operations.- 7. Exercises and Problems.- IV: Theory of Spinors.- 1. SL(2, c) as the Covering Group of the Lorentz Group.- 2. Subgroups of SL(2, c).- 3. SU (2).- 4. 5L(2, c) Spinors and Four-Vectors.- 5. Symmetries of the Dirac Equation.- 6. Exercises and Problems.- V: Covariant Harmonic Oscillator Formalism.- 1. Covariant Harmonic Oscillator Differential Equations.- 2. Normalizable Solutions of the Relativistic Oscillator Equation.- 3. Irreducible Unitary Representations of the Poincare Group.- 4. Transformation Properties of Harmonic Oscillator Wave Functions.- 5. Harmonic Oscillators in the Four-Dimensional Euclidean Space.- 6. Moving O(4) Coordinate System.- 7. Exercises and Problems.- VI: Dirac's Form of Relativistic Quantum Mechanics.- 1. C-Number Time-Energy Uncertainty Relation.- 2. Dirac's Form of Relativistic Theory of "Atom ".- 3. Dirac's Light-Cone Coordinate System.- 4. Harmonic Oscillators in the Light-Cone Coordinate System.- 5. Lorentz-Invariant Uncertainty Relations.- 6. Exercises and Problems.- VII: Massless Particles.- 1. What is the E(2) Group?.- 2. E(2)-like Little Group for Photons.- 3. Transformation Properties of Photon Polarization Vectors.- 4. Unitary Transformation of Photon Polarization Vectors.- 5. Massless Particles with Spin 1/2.- 6. Harmonic Oscillator Wave Functions for Massless Composite Particles.- 7. Exercises and Problems.- VIII: Group Contractions.- 1. SE(2) Group as a Contraction of SO(3).- 2. E(2)-like Little Group as an Infinite-momentum/zero-mass Limit of the O(3)-like Little Group for Massive Particles.- 3. Large-momentum/zero-mass Limit of the Dirac Equation.- 4. Finite-dimensional Non-unitary Representations of the SE(2) Group.- 5. Polarization Vectors for Massless Particles with Integer Spin.- 6. Lorentz and Galilei Transformations.- 7. Group Contractions and Unitary Representations of SE(2).- 8. Exercises and Problems.- IX: SO(2, 1) and SU(1, 1).- 1. Geometry of SL(2, r) and Sp(2).- 2. Finite-dimensional Representations of SO(2, 1).- 3. Complex Angular Momentum.- 4. Unitary Representations of SU(1, 1).- 5. Exercises and Problems.- X: Homogeneous Lorentz Group.- 1. Statement of the Problem.- 2. Finite-dimensional Representations of the Homogeneous Lorentz Group.- 3. Transformation Properties of Electric and Magnetic Fields.- 4. Pseudo-unitary Representations for Dirac Spinors.- 5. Harmonic Oscillator Wave Functions in the Lorentz Coordinate System.- 6. Further Properties of the Homogeneous Lorentz Group.- 7. Concluding Remarks.- XI: Hadronic Mass Spectra.- 1. Quark Model.- 2. Three-particle Symmetry Classifications According to the Method of Dirac.- 3. Construction of Symmetrized Wave Functions.- 4. Symmetrized Products of Symmetrized Wave Functions.- 5. Spin Wave Functions for the Three-Quark System.- 6. Three-quark Unitary Spin and SU(6) Wave Functions.- 7. Three-body Spatial Wave Functions.- 8. Totally Symmetric Baryonic Wave Functions.- 9. Baryonic Mass Spectra.- 10. Mesons.- 11. Exercises and Problems.- XII: Lorentz-Dirac Deformation in High-Energy Physics.- 1. Lorentz-Dirac Deformation of Hadronic Wave Functions.- 2. Form Factors of Nucleons.- 3. Calculation of the Form Factors.- 4. Scaling Phenomenon and the Parton Picture.- 5. Covariant Harmonic Oscillators and the Parton Picture.- 6. Calculation of the Parton Distribution Function for the Proton.- 7. Jet Phenomenon.- 8. Exercises and Problems.- References.
16669427
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.
668406
Quantization of electromagnetic fields in a circular cylindrical cavity
We present a quantization procedure for the electromagnetic field in a circular cylindrical cavity with perfectly conducting walls, which is based on the decomposition of the field. A new decomposition procedure is proposed; all vector mode functions satisfying the boundary conditions are obtained with the help of this decomposition. After expanding the quantized field in terms of the vector mode functions, it is possible to derive the Hamiltonian for this quantized system.
120854919
Covariant harmonic oscillators and the parton picture
It is shown that the covariant-harmonic-oscillator wave function exhibits the peculiarities of the Feynman parton picture in the infinite-momentum frame.
73571003
Time separation as a hidden variable to the Copenhagen school of quantum mechanics
The Bohr radius is a space‐like separation between the proton and electron in the hydrogen atom. According to the Copenhagen school of quantum mechanics, the proton is sitting in the absolute Lorentz frame. If this hydrogen atom is observed from a different Lorentz frame, there is a time‐like separation linearly mixed with the Bohr radius. Indeed, the time‐separation is one of the essential variables in high‐energy hadronic physics where the hadron is a bound state of the quarks, while thoroughly hidden in the present form of quantum mechanics. It will be concluded that this variable is hidden in Feynman’s rest of the universe. It is noted first that Feynman’s Lorentz‐invariant differential equation for the bound‐state quarks has a set of solutions which describe all essential features of hadronic physics. These solutions explicitly depend on the time separation between the quarks. This set also forms the mathematical basis for two‐mode squeezed states in quantum optics, where both photons are observable,...
118479937
Comment on `A collider signature of the supersymmetric golden region'
We show that Z polarization measurement is useful for distinguishing the signal decay chain of the `golden region' [1] in the minimal supersymmetric standard model (MSSM) parameter space, which involves the stop decay of 2 → 1+Z, from other decay chains involving the neutralino decay of 20 → 10+Z in different MSSM scenarios. Within a given decay chain, Z polarization can also serve as an indicator of the mass difference of stops or neutralinos.
119291756
Gluino Stransverse Mass
We introduce a new observable, ’gluino stransverse mass’, which is an application of the Cambridge mT2 variable to the process where gluinos are pair produced in proton-proton collision and each gluino subsequently decays into two quarks and one LSP, i.e. g̃g̃ → qqχ̃01 qqχ̃ 0 1. We show that the gluino stransverse mass can be utilized to measure the gluino mass and the lightest neutralino mass separately, and also the 1st and 2nd generation squark masses if squarks are lighter than gluino, thereby providing a good first look at the pattern of sparticle masses experimentally.
118380480
Majorana zero modes in semiconductor nanowires in contact with higher- T c superconductors
We analyze the prospects for stabilizing Majorana zero modes in semiconductor nanowires that are proximity coupled to higher-temperature superconductors. We begin with the case of iron pnictides which, though they are $s$-wave superconductors, are believed to have superconducting gaps that change sign. We then consider the case of cuprate superconductors. We show that a nanowire on a steplike surface, especially in an orthorhombic material such as YBCO, can support Majorana zero modes at an elevated temperature.
3194304
Kaon Condensation and Enforced Charge Neutrality of the Color-Flavor Locked Phase
We investigate the influence of charged kaon condensation in hadronic sector on the formation of the electron-free charge neutral quark matter of equipartition of three light flavors, i.e., a phenomenon called ``enforced electrical neutrality" of the color-flavor locked (CFL) phase. We employ the chiral quark model which is assumed to be applicable, "bottom-up," toward chiral restoration while accessible to kaon condensation in the hadronic sector. Approaching "bottom-up" the high density regime is more appropriate in addressing the properties of compact stars than approaching "top-down" from asymptotic QCD. It turns out that the presence of hadronic kaon condensation makes the electron-free charge-neutral CFL phase energetically more favorable with a relatively small color-superconducting gap $\Delta_0\sim 20$ MeV compared to $\Delta_0\sim 70$ MeV required in the absence of such kaon condensation.
23657326
AC Power Supply for Wobbler Magnet of the MC-50 Cyclotron
A wobbler magnet and an AC power supply for the MC-50 cyclotron (k=50) at Korea Institute of Radiological and Medical Sciences (KIRAMS) were designed and constructed to increase the area of beam irradiation. The power supply has two independently operating channels for the vertical and horizontal coils of the wobbler magnet. The frequency of the AC power supply for both coils is programmable from 1 to 20 Hz in a step of 1 Hz, and the maximum output current is 12 A rms. With the wobbler magnet, the uniform irradiation area of the proton beam increased ∼ 4 times larger than the one without the magnet, and various properties of the magnet and power supply system indicate that the system is well suited for many ion beam experiments.