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Journal of Mathematical Physics

O(3,3)-like symmetries of coupled harmonic oscillators

D. Han, Y. S. Kim, Marilyn E. Noz

Published Aug 1, 1995• 38 citations
Publisher

Abstract

In classical mechanics, the system of two coupled harmonic oscillators is shown to possess the symmetry of the Lorentz group O(3,3) or SL(4,r) in the four-dimensional phase space. In quantum mechanics, the symmetry is reduced to that of O(3,2) or Sp(4), which is a subgroup of O(3,3) or SL(4,r), respectively. It is shown that among the six Sp(4)-like subgroups, only one possesses the symmetry which can be translated into the group of unitary transformations in quantum mechanics.

DOI
10.1063/1.530940

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In classical mechanics, the system of two coupled harmonic oscillators is shown to possess the symmetry of the Lorentz group O(3,3) or SL(4,r) in the four-dimensional phase space. In quantum mechanics, the symmetry is reduced to that of O(3,2) or Sp(4), which is a subgroup of O(3,3) or SL(4,r), respectively. It is shown that among the six Sp(4)-like subgroups, only one possesses the symmetry which can be translated into the group of unitary transformations in quantum mechanics.
In classical mechanics, the system of two coupled harmonic oscillators is shown to possess the symmetry of the Lorentz group O(3,3) applicable to a six-dimensional space consisting of three space-like and three time-like coordinates, or SL(4,r) in the four-dimensional phase space consisting of two position and two momentum variables. In quantum mechanics, the symmetry is reduced to that of O(3,2) or Sp(4), which is a subgroup of O(3,3) or SL(4,r) respectively. It is shown that among the six Sp(4)-like subgroups, only one possesses the symmetry which can be translated into the group of unitary transformations in quantum mechanics. In quantum mechanics, there is the lower bound in the size of phase space for each mode determined by the uncertainty principle while there are no restriction on the phase-space size in classical mechanics. This is the reason why the symmetry is smaller in quantum mechanics.
It is shown that the basic symmetry of two-mode squeezed states is governed by the group Sp(4) in the Wigner phase space which is locally isomorphic to the (3+2)-dimensional Lorentz group. This symmetry, in the Schr\"odinger picture, appears as Dirac's two-oscillator representation of O(3,2). It is shown that the SU(2) and SU(1,1) interferometers exhibit the symmetry of this higher-dimensional Lorentz group. The mathematics of two-mode squeezed states is shown to be applicable to other branches of physics including thermally excited states in statistical mechanics and relativistic extended hadrons in the quark model.
It is known that two coupled harmonic oscillators can support the symmetry group as rich as O(3,3) which corresponds to the Lorentz group applicable to three space-like and three time-like coordinates. This group contains many subgroups, including O(3), O(3,2), O(2,1) which are already familiar to us. In this report, we discuss the symmetry of O(1,1) which plays pivotal roles in quantum optics and particle physics. For this one-parameter group, a full-fledged group theory is not necessary, and we can start the discussion from the Hamiltonian of the coupled oscillator system. It is shown first that, from the group theoretical point of view, the squeeze state of light is a representation of this O(1,1) group. It is then shown that the same mathematical device supports three seemingly different ideas of Feynman, namely the parton model, the relativistic quark model for hadrons, and the ``rest of the universe'' in connection with the density matrix. If these three theories are combined, they produce a covariant picture of Feynman's parton model with a built-in decoherence mechanism.
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.
Other than scattering problems where perturbation theory is applicable, there are basically two ways to solve problems in physics. One is to reduce the problem to harmonic oscillators, and the other is to formulate the problem in terms of two-by-two matrices. If two oscillators are coupled, the problem combines both two-by-two matrices and harmonic oscillators. This method then becomes a powerful research tool to cover many different branches of physics. Indeed, the concept and methodology in one branch of physics can be translated into another through the common mathematical formalism. Coupled oscillators provide clear illustrative examples for some of the current issues in physics, including entanglement and Feynman's rest of the universe. In addition, it is noted that the present form of quantum mechanics is largely a physics of harmonic oscillators. Special relativity is the physics of the Lorentz group which can be represented by the group of two-by-two matrices commonly called SL(2,c)SL(2,c)SL(2,c). Thus the coupled harmonic oscillators can play the role of combining quantum mechanics with special relativity. It is therefore possible to relate the current issues of physics to the Lorentz-covariant formulation of quantum mechanics.
Other than scattering problems where perturbation theory is applicable, there are basically two ways to solve problems in physics. One is to reduce the problem to harmonic oscillators, and the other is to formulate the problem in terms of two‐by‐two matrices. If two oscillators are coupled, the problem combines both two‐by‐two matrices and harmonic oscillators. This method then becomes a powerful research tool to cover many different branches of physics. Indeed, the concept and methodology in one branch of physics can be translated into another through the common mathematical formalism. It is noted that the present form of quantum mechanics is largely a physics of harmonic oscillators. Special relativity is the physics of the Lorentz group which can be represented by the group of by two‐by‐two matrices commonly called SL(2, c). Thus the coupled harmonic oscillators can therefore play the role of combining quantum mechanics with special relativity. Both Paul A. M. Dirac and Richard P. Feynman were fond of harmonic oscillators, while they used different approaches to physical problems. Both were also keenly interested in making quantum mechanics compatible with special relativity. It is shown that the coupled harmonic oscillators can bridge these two different approaches to physics.