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Papers by Richard Shurtleff
arXiv (Cornell University), Jun 30, 2016
For those finite-matrix representations of the Lorentz group of rotations/boosts with spin (A, B)... more For those finite-matrix representations of the Lorentz group of rotations/boosts with spin (A, B) ⊕ (C, D) that can also represent translations, two possible translation subgroups qualify. Of these two, one must be selected, and one discarded, to represent the Poincaré group of rotations/boosts with translations in spacetime. Instead, let us discard the requirement that there be just one translation subgroup. With dual-translations, one gives up agreement with simple macroscopic observations of spacetime. Now the transformations of both possible translation subgroups combine with those of the Lorentz group. The resulting commutation relations require new transformations and generators to satisfy the linearity requirement of a Lie algebra. Special cases of spins are sought to restrict the influx of new transformations. One finds that the Dirac 4-spinor formalism is the only viable solution. The slightly expanded group it represents is the conformal group with just one new transformation, scale change. It follows as a corollary that the Dirac 4-spinor formalism is the only matrix representation of the conformal group with spin (A, B) ⊕ (C, D).
arXiv (Cornell University), 2008
The momentum of a free massive particle, invariant under translation, thereby realizes a trivial ... more The momentum of a free massive particle, invariant under translation, thereby realizes a trivial representation of the translation group. By allowing nontrivial reps of translations, momentum changes with translation, a recipe for force. Here the procedure is applied to the conventional construction of a free quantum field using spacetime symmetries, yielding a more general field with the free field as a special case. It is shown that a particle described by the quantum field follows the classical trajectories of a massive charged particle in electromagnetic and gravitational fields.
arXiv (Cornell University), Sep 25, 2010
Rotations, boosts and translations in 8 + 1 spacetime are developed based on the commutation and ... more Rotations, boosts and translations in 8 + 1 spacetime are developed based on the commutation and anticommutation relations of SU(3). The process follows a process that gives 3 + 1 spacetime from SU(2).
Mass is proportional to phase gain per unit time; for e, π, and p the quantum frequencies are 0.1... more Mass is proportional to phase gain per unit time; for e, π, and p the quantum frequencies are 0.124, 32.6, and 227 Zhz, respectively. By explaining how these particles acquire phase at different rates, we explain why these particles have different masses. Any free particle spin 1/2 wave function is a sum of plane waves with spin parallel to velocity. Each plane wave, a pair of 2-component rotation eigenvectors, can be associated with a 2 × 2 matrix representation of rotations in a Euclidean space without disturbing the plane wave's space-time properties. In a space with more than four dimensions, only rotations in a 4d subspace can be represented. So far all is well known. Now consider that unrepresented rotations do not have eigenvectors, do not make plane waves, and do not contribute phase. The particles e, π, and p are assigned rotations in a 4d subspace of 16d, rotations in an 8d subspace of 12d, and rotations in a 12d subspace of 12d, respectively. The electron 4d subspace, assumed to be as likely to align with any one 4d subspace as with any other, produces phase when aligned with the represented 4d subspace in 16d. Similarly, we calculate the likelihood that a 4d subspace of the pion's 8d space aligns with the represented 4d subspace in 12d. The represented 4d subspace is contained in the proton's 12d space, so the proton always acquires phase. By the relationship between mass and phase, the resulting particle phase ratios are the particle mass ratios and these are coincident with the measured mass ratios, within about one percent.
arXiv (Cornell University), Jun 30, 2016
For those finite-matrix representations of the Lorentz group of rotations/boosts with spin (A, B)... more For those finite-matrix representations of the Lorentz group of rotations/boosts with spin (A, B) ⊕ (C, D) that can also represent translations, two possible translation subgroups qualify. Of these two, one must be selected, and one discarded, to represent the Poincaré group of rotations/boosts with translations in spacetime. Instead, let us discard the requirement that there be just one translation subgroup. With dual-translations, one gives up agreement with simple macroscopic observations of spacetime. Now the transformations of both possible translation subgroups combine with those of the Lorentz group. The resulting commutation relations require new transformations and generators to satisfy the linearity requirement of a Lie algebra. Special cases of spins are sought to restrict the influx of new transformations. One finds that the Dirac 4-spinor formalism is the only viable solution. The slightly expanded group it represents is the conformal group with just one new transformation, scale change. It follows as a corollary that the Dirac 4-spinor formalism is the only matrix representation of the conformal group with spin (A, B) ⊕ (C, D).
The Lie algebra of a Lie group is a set of commutation relations, equations satisfied by the grou... more The Lie algebra of a Lie group is a set of commutation relations, equations satisfied by the group's generators. For SU(2) and many other Lie groups, the equations have been solved and matrix generators are realized as algebraic expressions suitable for further investigation or numerical evaluation. This article presents formulas that give a set of matrix generators for any irreducible representation of the group SU(3), the group of unimodular unitary threedimensional complex matrices with matrix multiplication. To assist in calculating the matrix generators, a Mathematica computer program and a Fortran 90 program are included.
By adding generalizations involving translations, the machinery of the quantum theory of free fie... more By adding generalizations involving translations, the machinery of the quantum theory of free fields leads to the semiclassical equations of motion for a charged massive particle in electromagnetic and gravitational fields. With the particle field translated along one displacement, particle states are translated along a possibly different displacement. Arbitrary phase results. And particle momentum, a spin (1/2,1/2) quantity, is allowed to change when field and states are translated. It is shown that a path of extreme phase obeys a semiclassical equation for force with derived terms that can describe electromagnetism and gravitation.
viXra, 2019
Electric charges may have mass in part or in full because they charged. Supplying details is the ... more Electric charges may have mass in part or in full because they charged. Supplying details is the electromagnetic mass problem. Here, the charge's mass is associated with intrinsic quantum mechanical quantities so that the classical problems with extended charge distributions, for example, are irrelevant. An intrinsic vector potential is defined, based on intrinsic linear momentum. The charge-electromagnetic field interaction energy is gauge-dependent and the needed mass term is placed with the interaction energy in the intrinsic gauge. Traditional electromagnetism retains its gauge invariance. The field equations make no new predictions since all dynamic dependence on intrinsic quantities can be gauged away. The field equations describe a massive, charged 4-spinor Dirac electron-like particle and an uncharged, massless neutrino-like particle, formulas that have been a part of physics for nearly a century.
By adding generalizations involving translations, the machinery of the quantum theory of free fie... more By adding generalizations involving translations, the machinery of the quantum theory of free fields leads to the semiclassical equations of motion for a charged massive particle in electromagnetic and gravitational fields. With the particle field translated along one displacement, particle states are translated along a possibly different displacement. Arbitrary phase results. And particle momentum, a spin (1/2,1/2) quantity, is allowed to change when field and states are translated. It is shown that a path of extreme phase obeys a semiclassical equation for force with derived terms that can describe electromagnetism and gravitation.
arXiv (Cornell University), Jun 30, 2016
For those finite-matrix representations of the Lorentz group of rotations/boosts with spin (A, B)... more For those finite-matrix representations of the Lorentz group of rotations/boosts with spin (A, B) ⊕ (C, D) that can also represent translations, two possible translation subgroups qualify. Of these two, one must be selected, and one discarded, to represent the Poincaré group of rotations/boosts with translations in spacetime. Instead, let us discard the requirement that there be just one translation subgroup. With dual-translations, one gives up agreement with simple macroscopic observations of spacetime. Now the transformations of both possible translation subgroups combine with those of the Lorentz group. The resulting commutation relations require new transformations and generators to satisfy the linearity requirement of a Lie algebra. Special cases of spins are sought to restrict the influx of new transformations. One finds that the Dirac 4-spinor formalism is the only viable solution. The slightly expanded group it represents is the conformal group with just one new transformation, scale change. It follows as a corollary that the Dirac 4-spinor formalism is the only matrix representation of the conformal group with spin (A, B) ⊕ (C, D).
arXiv (Cornell University), 2008
The momentum of a free massive particle, invariant under translation, thereby realizes a trivial ... more The momentum of a free massive particle, invariant under translation, thereby realizes a trivial representation of the translation group. By allowing nontrivial reps of translations, momentum changes with translation, a recipe for force. Here the procedure is applied to the conventional construction of a free quantum field using spacetime symmetries, yielding a more general field with the free field as a special case. It is shown that a particle described by the quantum field follows the classical trajectories of a massive charged particle in electromagnetic and gravitational fields.
arXiv (Cornell University), Sep 25, 2010
Rotations, boosts and translations in 8 + 1 spacetime are developed based on the commutation and ... more Rotations, boosts and translations in 8 + 1 spacetime are developed based on the commutation and anticommutation relations of SU(3). The process follows a process that gives 3 + 1 spacetime from SU(2).
Mass is proportional to phase gain per unit time; for e, π, and p the quantum frequencies are 0.1... more Mass is proportional to phase gain per unit time; for e, π, and p the quantum frequencies are 0.124, 32.6, and 227 Zhz, respectively. By explaining how these particles acquire phase at different rates, we explain why these particles have different masses. Any free particle spin 1/2 wave function is a sum of plane waves with spin parallel to velocity. Each plane wave, a pair of 2-component rotation eigenvectors, can be associated with a 2 × 2 matrix representation of rotations in a Euclidean space without disturbing the plane wave's space-time properties. In a space with more than four dimensions, only rotations in a 4d subspace can be represented. So far all is well known. Now consider that unrepresented rotations do not have eigenvectors, do not make plane waves, and do not contribute phase. The particles e, π, and p are assigned rotations in a 4d subspace of 16d, rotations in an 8d subspace of 12d, and rotations in a 12d subspace of 12d, respectively. The electron 4d subspace, assumed to be as likely to align with any one 4d subspace as with any other, produces phase when aligned with the represented 4d subspace in 16d. Similarly, we calculate the likelihood that a 4d subspace of the pion's 8d space aligns with the represented 4d subspace in 12d. The represented 4d subspace is contained in the proton's 12d space, so the proton always acquires phase. By the relationship between mass and phase, the resulting particle phase ratios are the particle mass ratios and these are coincident with the measured mass ratios, within about one percent.
arXiv (Cornell University), Jun 30, 2016
For those finite-matrix representations of the Lorentz group of rotations/boosts with spin (A, B)... more For those finite-matrix representations of the Lorentz group of rotations/boosts with spin (A, B) ⊕ (C, D) that can also represent translations, two possible translation subgroups qualify. Of these two, one must be selected, and one discarded, to represent the Poincaré group of rotations/boosts with translations in spacetime. Instead, let us discard the requirement that there be just one translation subgroup. With dual-translations, one gives up agreement with simple macroscopic observations of spacetime. Now the transformations of both possible translation subgroups combine with those of the Lorentz group. The resulting commutation relations require new transformations and generators to satisfy the linearity requirement of a Lie algebra. Special cases of spins are sought to restrict the influx of new transformations. One finds that the Dirac 4-spinor formalism is the only viable solution. The slightly expanded group it represents is the conformal group with just one new transformation, scale change. It follows as a corollary that the Dirac 4-spinor formalism is the only matrix representation of the conformal group with spin (A, B) ⊕ (C, D).
The Lie algebra of a Lie group is a set of commutation relations, equations satisfied by the grou... more The Lie algebra of a Lie group is a set of commutation relations, equations satisfied by the group's generators. For SU(2) and many other Lie groups, the equations have been solved and matrix generators are realized as algebraic expressions suitable for further investigation or numerical evaluation. This article presents formulas that give a set of matrix generators for any irreducible representation of the group SU(3), the group of unimodular unitary threedimensional complex matrices with matrix multiplication. To assist in calculating the matrix generators, a Mathematica computer program and a Fortran 90 program are included.
By adding generalizations involving translations, the machinery of the quantum theory of free fie... more By adding generalizations involving translations, the machinery of the quantum theory of free fields leads to the semiclassical equations of motion for a charged massive particle in electromagnetic and gravitational fields. With the particle field translated along one displacement, particle states are translated along a possibly different displacement. Arbitrary phase results. And particle momentum, a spin (1/2,1/2) quantity, is allowed to change when field and states are translated. It is shown that a path of extreme phase obeys a semiclassical equation for force with derived terms that can describe electromagnetism and gravitation.
viXra, 2019
Electric charges may have mass in part or in full because they charged. Supplying details is the ... more Electric charges may have mass in part or in full because they charged. Supplying details is the electromagnetic mass problem. Here, the charge's mass is associated with intrinsic quantum mechanical quantities so that the classical problems with extended charge distributions, for example, are irrelevant. An intrinsic vector potential is defined, based on intrinsic linear momentum. The charge-electromagnetic field interaction energy is gauge-dependent and the needed mass term is placed with the interaction energy in the intrinsic gauge. Traditional electromagnetism retains its gauge invariance. The field equations make no new predictions since all dynamic dependence on intrinsic quantities can be gauged away. The field equations describe a massive, charged 4-spinor Dirac electron-like particle and an uncharged, massless neutrino-like particle, formulas that have been a part of physics for nearly a century.
By adding generalizations involving translations, the machinery of the quantum theory of free fie... more By adding generalizations involving translations, the machinery of the quantum theory of free fields leads to the semiclassical equations of motion for a charged massive particle in electromagnetic and gravitational fields. With the particle field translated along one displacement, particle states are translated along a possibly different displacement. Arbitrary phase results. And particle momentum, a spin (1/2,1/2) quantity, is allowed to change when field and states are translated. It is shown that a path of extreme phase obeys a semiclassical equation for force with derived terms that can describe electromagnetism and gravitation.