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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

New Dynamical Group for the Relativistic Quantum Mechanics of Elementary Particles

J. J. Aghassi, P. Román, Ruggero María Santilli

Published May 15, 1970• 43 citations
Publisher

Abstract

Nonrelativistic Galilean quantum mechanics and the standard transition to relativistic Poincar\'e quantum mechanics is analyzed in terms of group theory. Special emphasis is given to the discussion of the relation between dynamics and geometry. Certain unsatisfactory features are pointed out and a new relativistic group ${\mathcal{G}}_{5}$ is suggested as the symmetry group of dynamics. ${\mathcal{G}}_{5}$ contains both the nonrelativistic Galilei group and the Poincar\'e group as subgroups, and it is a group extension of the restricted Lorentz group. For use in relativistic quantum mechanics, the central extension of ${\mathcal{G}}_{5}$ by a phase group must be employed. The Lie algebra of this relativistic quantum-mechanical Galilei group ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathcal{G}}}_{5}$ contains an acceptable covariant space-time position operator and a nontrivial relativistic mass operator. The latter also serves to describe dynamical development. The irreducible unitary projective representations of ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathcal{G}}}_{5}$ correspond to infinite towers of states with increasing spin.

DOI
10.1103/physrevd.1.2753

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Nonrelativistic Galilean quantum mechanics and the standard transition to relativistic Poincar'e quantum mechanics is analyzed in terms of group theory. Special emphasis is given to the discussion of the relation between dynamics and geometry. Certain unsatisfactory features are pointed out and a new relativistic group G5{\mathcal{G}}_{5}G5​ is suggested as the symmetry group of dynamics. G5{\mathcal{G}}_{5}G5​ contains both the nonrelativistic Galilei group and the Poincar'e group as subgroups, and it is a group extension of the restricted Lorentz group. For use in relativistic quantum mechanics, the central extension of G5{\mathcal{G}}_{5}G5​ by a phase group must be employed. The Lie algebra of this relativistic quantum-mechanical Galilei group G\ifmmode~\else˜\fi5{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathcal{G}}}_{5}G\ifmmode~\else˜\fi​5​ contains an acceptable covariant space-time position operator and a nontrivial relativistic mass operator. The latter also serves to describe dynamical development. The irreducible unitary projective representations of G\ifmmode~\else˜\fi5{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathcal{G}}}_{5}G\ifmmode~\else˜\fi​5​ correspond to infinite towers of states with increasing spin.
This paper is devoted to the study of the Galilei group and its representations. The Galilei group presents a certain number of essential differences with respect to the Poincaré group. As Bargmann showed, its physical representations, here explicitly constructed, are not true representations but only up-to-a-factor ones. Consequently, in nonrelativistic quantum mechanics, the mass has a very special role, and in fact, gives rise to a superselection rule which prevents the existence of unstable particles. The internal energy of a nonrelativistic system is known to be an arbitrary parameter; this is shown to come also from Galilean invariance, because of a nontrivial concept of equivalence between physical representations. On the contrary, the behavior of an elementary system with respect to rotations, is very similar to the relativistic case. We show here, in particular, how the number of polarization states reduces to two for the zero-mass case (though in fact there are no physical zero-mass systems in nonrelativistic mechanics). Finally, we study the two-particle system, where the orbital angular momenta quite naturally introduce themselves through the decomposition of the tensor product of two physical representations.
Covariant four-vector position operators ${X}^{\ensuremath{\mu}}$ are proposed, which form a natural operator generalization of the four-position in relativistic classical mechanics. These ${X}^{\ensuremath{\mu}}$ are defined by specifying commutation relations of the ${X}^{\ensuremath{\mu}}$ with the Poincar\'e generators ${P}^{\ensuremath{\mu}}$ and ${M}^{\ensuremath{\mu}\ensuremath{\nu}}$, and thereby extending the Poincar\'e algebra to a larger algebra whose representations are subsequently found. The ${X}^{\ensuremath{\mu}}$ are shown to be acceptable relativistic position operators within a proper-time dynamical framework. A single Hamiltonian is used for all spins, with a covariant proper-time description. The dynamics is capable of describing the time evolution of states which are not mass eigenstates. An automatic Foldy-Wouthuysen-type diagonalization is achieved for all spin representations, with spin and orbital angular momentum being separately conserved. The connection with the standard theory is made via the specific field equations. In making this connection to the standard theory of half-integral spins, the origin of Zitterbewegung and the nonseparate conservation of spin and angular momentum are clarified. The connection to Maxwell's equations provides an interesting statement of those equations and of gauge invariance. The unphysical representations of negative and imaginary mass and continuous spin are not present in this formalism. Other features are discussed.
We present the formulation of a version of Lorentz covariant quantum mechanics based on a group theoretical construction from a Heisenberg–Weyl symmetry with position and momentum operators transforming as Minkowski four-vectors. The basic representation is identified as a coherent state representation, essentially an irreducible component of the regular representation, with the matching representation of an extension of the group C*-algebra giving the algebra of observables. The key feature is that it is not unitary but pseudo-unitary, exactly in the same sense as the Minkowski spacetime representation. The language of pseudo-Hermitian quantum mechanics is adopted for a clear illustration of the aspect, with a metric operator obtained as really the manifestation of the Minkowski metric on the space of the state vectors. Explicit wavefunction description is given without any restriction of the variable domains, yet with a finite integral inner product. The associated covariant harmonic oscillator Fock state basis has all the standard properties in exact analog to those of a harmonic oscillator with Euclidean position and momentum operators. Galilean limit and the classical limit are retrieved rigorously through appropriate symmetry contractions of the algebra and its representation, including the dynamics described through the symmetry of the phase space.
Noninertial transformations on time–position–momentum–energy space { t , q , p , e } with invariant Born–Green metric and the symplectic metric −d e ∧ d t + d p ∧ d q are studied. This group of transformations contains the Lorentz group as the inertial special case and, in the limit of small forces and velocities, reduces to the expected Hamilton transformations leaving invariant the symplectic metric and the nonrelativistic line element d s 2 = −d t 2 . The transformations bound relative velocities by c and relative forces by b . Spacetime is no longer an invariant subspace but is relative to noninertial observer frames. In the limit of b → ∞, spacetime is invariant. Born was lead to the metric by a concept of reciprocity between position and momentum degrees of freedom and for this reason we call this reciprocal relativity. For large b , such effects will almost certainly only manifest in a quantum regime. Wigner showed that special relativistic quantum mechanics follows from the projective representations of the inhomogeneous Lorentz group. Projective representations of a Lie group are equivalent to the unitary representations of its central extension. The same method of projective representations for the inhomogeneous group is used to define the quantum theory in the noninertial case. The central extension of the inhomogeneous group is the cover of the quaplectic group . is the Weyl–Heisenberg group. The group, and the associated Heisenberg commutation relations central to quantum mechanics, results directly from requiring projective representations. A set of second-order wave equations result from the representations of the Casimir operators.
In the framework of limitable dynamical groups, a group Dst=Dt⊗SO(N)I, Dt=GE0(N) [multiplication in left half circle]Sp(2N, R),is discussed. It is shown that the physical representation U(Dst) describes interacting nonrelativistic particles of arbitrary spin in N dimensions, with linear combinations of second-order polynomials in momentum and position operators Pi, Qi with i = 1, ⋯, N, and spin-flip operators as Hamiltonians. GE0(N) is the central extension of the inhomogeneous, pure Galilei group, and SO(N)I a rotation group in N dimensions. Sp(2N, R) is the noncompact form of the symplectic group. It is proven that there exists no irreducible unitary representation of Dt describing particles with spin s > 0. Hence it was necessary to enlarge Dt to Dst. The physical free-particle group D0⊂Dst possesses projection parts in Sp(2N, R) as well as in SO(N)I. To calculate them, we decompose the physical representation U(D0) by group-theoretical methods such that the angular momentum splits into a spin and an orbital part. D0 is isomorphic to the central extension of the inhomogeneous Galilei group. The physical representation U(Dst) is calculated by the postulate that U(Dst) and U(D0) can be limited into each other, and that mass and spin conservation hold. The limitation is realized as a group contraction leading to a nonfaithful representation of the contracted group. We prove that U(Dst) is uniquely determined by mass and spin of the free particle which is obtained if the interaction in Dst is turned off. Furthermore, the generators are identified as functions of Pi, Qi, and spin-flip operators. Hence the limitation postulate leads to the physical representation of the abstract group Dst and to the identification of the described physical system.