The second-order differential equation for a damped harmonic oscillator can be converted to two c... more 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...
Einstein's photo-electric effect allows us to regard electromagnetic waves as massless partic... more Einstein's photo-electric effect allows us to regard electromagnetic waves as massless particles. Then, how is the photon helicity translated into the electric and magnetic fields perpendicular to the direction of propagation? This is an issue of the internal space-time symmetries defined by Wigner's little group for massless particles. It is noted that there are three generators for the rotation group defining the spin of a particle at rest. The closed set of commutation relations is a direct consequence of Heisenberg's uncertainty relations. The rotation group can be generated by three two-by-two Pauli matrices for spin-half particles. This group of two-by-two matrices is called SU(2), with two-component spinors. The direct product of two spinors leads to four states leading to one spin-0 state and one spin-1 state with three sub-states. The SU(2) group can be expanded to another group of two-by-two matrices called SL(2,c), which serves as the covering group for the gr...
The two-by-two scattering matrix for one-dimensional scattering processes is a three-parameter Sp... more The two-by-two scattering matrix for one-dimensional scattering processes is a three-parameter Sp(2) matrix or its unitary equivalent. For one-dimensional crystals, it would be repeated applications of this matrix. The problem is how to calculate N repeated multiplications of this matrix. It is shown that the original Sp(2) matrix can be written as a similarity transformation of Wigner's little group matrix which can be diagonalized. It is then possible to calculate the repeated applications of the original Sp(2) matrix.
It is noted that the single-variable Heisenberg commutation relation contains the symmetry of the... more 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.
Heisenberg’s uncertainty relation can be written in terms of the step-up and step-down operators ... more 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 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 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 generat...
The second-order differential equation for a damped harmonic oscillator can be converted to two c... more 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...
Einstein's photo-electric effect allows us to regard electromagnetic waves as massless partic... more Einstein's photo-electric effect allows us to regard electromagnetic waves as massless particles. Then, how is the photon helicity translated into the electric and magnetic fields perpendicular to the direction of propagation? This is an issue of the internal space-time symmetries defined by Wigner's little group for massless particles. It is noted that there are three generators for the rotation group defining the spin of a particle at rest. The closed set of commutation relations is a direct consequence of Heisenberg's uncertainty relations. The rotation group can be generated by three two-by-two Pauli matrices for spin-half particles. This group of two-by-two matrices is called SU(2), with two-component spinors. The direct product of two spinors leads to four states leading to one spin-0 state and one spin-1 state with three sub-states. The SU(2) group can be expanded to another group of two-by-two matrices called SL(2,c), which serves as the covering group for the gr...
The two-by-two scattering matrix for one-dimensional scattering processes is a three-parameter Sp... more The two-by-two scattering matrix for one-dimensional scattering processes is a three-parameter Sp(2) matrix or its unitary equivalent. For one-dimensional crystals, it would be repeated applications of this matrix. The problem is how to calculate N repeated multiplications of this matrix. It is shown that the original Sp(2) matrix can be written as a similarity transformation of Wigner's little group matrix which can be diagonalized. It is then possible to calculate the repeated applications of the original Sp(2) matrix.
It is noted that the single-variable Heisenberg commutation relation contains the symmetry of the... more 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.
Heisenberg’s uncertainty relation can be written in terms of the step-up and step-down operators ... more 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 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 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 generat...
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