Physics
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Item Lorentz Harmonics, Squeeze Harmonics and Their Physical Applications(MDPI, 2011-02-14) Kim, Young S.; Noz, Marilyn E.Among the symmetries in physics, the rotation symmetry is most familiar to us. It is known that the spherical harmonics serve useful purposes when the world is rotated. Squeeze transformations are also becoming more prominent in physics, particularly in optical sciences and in high-energy physics. As can be seen from Dirac’s light-cone coordinate system, Lorentz boosts are squeeze transformations. Thus the squeeze transformation is one of the fundamental transformations in Einstein’s Lorentz-covariant world. It is possible to define a complete set of orthonormal functions defined for one Lorentz frame. It is shown that the same set can be used for other Lorentz frames. Transformation properties are discussed. Physical applications are discussed in both optics and high-energy physics. It is shown that the Lorentz harmonics provide the mathematical basis for squeezed states of light. It is shown also that the same set of harmonics can be used for understanding Lorentz-boosted hadrons in high-energy physics. It is thus possible to transmit physics from one branch of physics to the other branch using the mathematical basis common to them.Item Dirac Matrices and Feynman’s Rest of the Universe(MDPI, 2012-10-30) Kim, Young S.; Noz, Marilyn E.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.Item Symmetries Shared by the Poincaré Group and the Poincaré Sphere(MDPI, 2013-06-27) Kim, Young S.; Noz, Marilyn E.Henri Poincaré formulated the mathematics of Lorentz transformations, known as the Poincaré group. He also formulated the Poincaré 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.Item Wigner’s Space-Time Symmetries Based on the Two-by-Two Matrices of the Damped Harmonic Oscillators and the Poincaré Sphere(MDPI, 2014-06-25) Başkal, Sibel; Kim, Young S.; Noz, Marilyn E.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.Item Entangled Harmonic Oscillators and Space-Time Entanglement(MDPI, 2016-06-28) Başkal, Sibel; Kim, Young S.; Noz, Marilyn E.The mathematical basis for the Gaussian entanglement is discussed in detail, as well as its implications in the internal space-time structure of relativistic extended particles. It is shown that the Gaussian entanglement shares the same set of mathematical formulas with the harmonic oscillator in the Lorentz-covariant world. It is thus possible to transfer the concept of entanglement to the Lorentz-covariant picture of the bound state, which requires both space and time separations between two constituent particles. These space and time variables become entangled as the bound state moves with a relativistic speed. It is shown also that our inability to measure the time-separation variable leads to an entanglement entropy together with a rise in the temperature of the bound state. As was noted by Paul A. M. Dirac in 1963, the system of two oscillators contains the symmetries of the 𝑂(3,2) de Sitter group containing two 𝑂(3,1) Lorentz groups as its subgroups. Dirac noted also that the system contains the symmetry of the 𝑆𝑝(4) group, which serves as the basic language for two-mode squeezed states. Since the 𝑆𝑝(4) symmetry contains both rotations and squeezes, one interesting case is the combination of rotation and squeeze, resulting in a shear. While the current literature is mostly on the entanglement based on squeeze along the normal coordinates, the shear transformation is an interesting future possibility. The mathematical issues on this problem are clarified.Item Loop Representation of Wigner’s Little Groups(MDPI, 2017-06-23) Başkal, Sibel; Kim, Young S.; Noz, Marilyn E.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.Item Einstein’s E = mc2 Derivable from Heisenberg’s Uncertainty Relations(MDPI, 2019-11-09) Başkal, Sibel; Kim, Young S.; Noz, Marilyn E.Heisenberg’s uncertainty relation can be written in terms of the step-up and step-down operators in the harmonic oscillator representation. It is noted that the single-variable Heisenberg commutation relation contains the symmetry of the 𝑆𝑝(2) group which is isomorphic to the Lorentz group applicable to one time-like dimension and two space-like dimensions, known as the 𝑂(2,1) group. This group has three independent generators. The one-dimensional step-up and step-down operators can be combined into one two-by-two Hermitian matrix which contains three independent operators. If we use a two-variable Heisenberg commutation relation, the two pairs of independent step-up, step-down operators can be combined into a four-by-four block-diagonal Hermitian matrix with six independent parameters. It is then possible to add one off-diagonal two-by-two matrix and its Hermitian conjugate to complete the four-by-four Hermitian matrix. This off-diagonal matrix has four independent generators. There are thus ten independent generators. It is then shown that these ten generators can be linearly combined to the ten generators for Dirac’s two oscillator system leading to the group isomorphic to the de Sitter group 𝑂(3,2) , which can then be contracted to the inhomogeneous Lorentz group with four translation generators corresponding to the four-momentum in the Lorentz-covariant world. This Lorentz-covariant four-momentum is known as Einstein’s 𝐸=𝑚𝑐2.Item Integration of Dirac’s Efforts to Construct a Quantum Mechanics Which is Lorentz-Covariant(MDPI, 2020-08-01) Kim, Young S.; Noz, Marilyn E.The lifelong efforts of Paul A. M. Dirac were to construct localized quantum systems in the Lorentz covariant world. In 1927, he noted that the time-energy uncertainty should be included in the Lorentz-covariant picture. In 1945, he attempted to construct a representation of the Lorentz group using a normalizable Gaussian function localized both in the space and time variables. In 1949, he introduced his instant form to exclude time-like oscillations. He also introduced the light-cone coordinate system for Lorentz boosts. Also in 1949, he stated the Lie algebra of the inhomogeneous Lorentz group can serve as the uncertainty relations in the Lorentz-covariant world. It is possible to integrate these three papers to produce the harmonic oscillator wave function which can be Lorentz-transformed. In addition, Dirac, in 1963, considered two coupled oscillators to derive the Lie algebra for the generators of the 𝑂(3,2) de Sitter group, which has ten generators. It is proven possible to contract this group to the inhomogeneous Lorentz group with ten generators, which constitute the fundamental symmetry of quantum mechanics in Einstein’s Lorentz-covariant world.