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Journal of Mathematical Physics

Limitable Dynamical Groups in Quantum Mechanics. II. A Model Including Arbitrary Spin

H. D. Doebner, O. Melsheimer

Published Apr 1, 1970• 6 citations

Abstract

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.

DOI: 10.1063/1.1665280

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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.

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.

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.

We show that the group of linear canonical transformations in a 2N-dimensional phase space is the real symplectic group Sp(2N), and discuss its unitary representation in quantum mechanics when the N coordinates are diagonal. We show that this Sp(2N) group is the well-known dynamical group of the N-dimensional harmonic oscillator. Finally, we study the case of n particles in a q-dimensional oscillator potential, for which N = nq, and discuss the chain of groups Sp(2nq)⊃Sp(2n)×O(q). An application to the calculation of matrix elements is given in a following paper.

Abstract The representation of an N‐electron Schrödinger Hamiltonian on an orthonormal, spin‐free (freon) orbital product space is exactly modeled by a second‐degree polynomial in the infinitesimal generators of the unitary group. Symmetry adaptation of the orbital product space with respect to the symmetric group yields Gel'fand states which provide base for irreducible represntations of the unitary group. These exist in closed form as, in consequence, does the representation of the model Hamiltonian in this same basis. We retain as physical only those states characterized by tableaux with no more than two columns for which the spin labelling is one‐half the difference in the lengths of the two columns. The unitary group formulation is equivalent to standard, number‐conserving, second‐quantized, many‐body theory.

We use the mathematical structure of group algebras and

H^{+}

-algebras for describing certain problems concerning the quantum dynamics of systems of angular momenta, including also the spin systems. The underlying groups are

{\rm SU}(2)

and its quotient

{\rm SO}(3,\mathbb{R})

. The scheme developed is applied in two different contexts. Firstly, the purely group-algebraic framework is applied to the system of angular momenta of arbitrary origin, e.g., orbital angular momenta of electrons and nucleons, systems of quantized angular momenta of rotating extended objects like molecules. The other promising area of applications is Schrödinger quantum mechanics of rigid body with its often rather unexpected and very interesting features. Even within this Schrödinger framework the algebras of operators related to group algebras are a very useful tool. We investigate some problems of composed systems and the quasiclassical limit obtained as the asymptotics of "large" quantum numbers, i.e., "quickly oscillating" functions on groups. They are related in an interesting way to geometry of the coadjoint orbits of

{\rm SU}(2)

.