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Conformal group $Yields$ Schroedinger group $Yields$ dynamical group: the maximal kinematical group of the massive Schroedinger particle

A. O. Barut

Published Jan 1, 1973• 3 citations
OpenAlex

Abstract

The 15-parameter nonrelativistic limit of the conformal group and its linear realization in the six-dimensional space are determined. It is shown that the massive free Schroedinger equation is invariant under a 15-parameter set of transformations enlarging the 12-parameter Schroedinger group. It is shown how this set arises from the conformal group, and the quantum-mechanica1 representation of the generators is determined. (auth)

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

H. D. Doebner, O. Melsheimer

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.

Journal of Mathematical Physics•Apr 1, 1970•6 citations
Conformal group YieldsYieldsYields Schroedinger group YieldsYieldsYields dynamical group: the maximal kinematical group of the massive Schroedinger particle
The 15-parameter nonrelativistic limit of the conformal group and its linear realization in the six-dimensional space are determined. It is shown that the massive free Schroedinger equation is invariant under a 15-parameter set of transformations enlarging the 12-parameter Schroedinger group. It is shown how this set arises from the conformal group, and the quantum-mechanica1 representation of the generators is determined. (auth)
The largest group of coordinate transformations leaving invariant the Schroedinger equation of the n-dimensional harmonic oscillator is determined and shown to be isomorphic to the corresponding group of the free-particle equation. It can be described as a Galilei group in which the time translations have been replaced by the group SL(2,R) of projective transformations. The relation between the oscillator group and the spectrumgenenating algebra of the harmonic oscillator is investigated. The relevance of the oscillator group and the group SL(2,R) for general quantum systems is discussed. (auth)
The dynamics of a system of particles acting on one another at a distance can be relativistically invariant if the assumption of invariant world-lines is given up. This is shown by constructing a particular dynamics in which invariance over the homogeneous Lorentz group is trivial, but space as well as time displacement requires the solution of equations of motion or of a Schroedinger equation. This particular dynamics reduces in the nonrelativistic limit to the most general dynamics of a system of interacting particles admitting the Newtonian group.
The Galilean symmetry of a free particle in one-dimensional space is examined under the scope of non-Abelian quantum kinematics. Within the Hilbert space that carries the regular ray representation of the Galilei group the Schrödinger operator appears as one of the three fundamental invariants of the extended kinematic algebra. By means of a superselection rule the physical Hilbert subspaces of the system are identified, in which a complementary ray representation of the Galilean transformation produces the time-dependent Schrödinger equation, and the Feynman space-time propagator. The quantization approach used in this paper is purely group theoretic and relativistic.
Momentum space Ward identities are derived for the amputated n-point Green’s functions in 3 + 1 dimensional non-relativistic conformal field theory. For n = 4 and 6 the implications for scattering amplitudes (i.e. on-shell amputated Green’s functions) are considered. Any scale invariant 2-to-2 scattering amplitude is also conformally invariant. However, conformal invariance imposes constraints on off-shell Green’s functions and the three particle scattering amplitude which are not automatically satisfied if they are scale invariant. As an explicit example of a conformally invariant theory we consider non-relativistic particles in the infinite scattering length limit. 1 Poincaré invariant theories that are scale invariant usually have a larger symmetry group called the conformal group1. A similar phenomena happens for 3+1 dimensional nonrelativistic systems. These are invariant under the extended Galilean group, which consists of 10 generators: translations (4), rotations (3), and Galilean boosts (3). The largest space-time symmetry group of the free Schrödinger equation is called the Schrödinger or nonrelativistic
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.