25 The Metaplectic Representation and Annihilation and Creation Operators, arbitrary d

In this chapter we’ll turn from the $d=1$ case of chapter 24 to the general case of arbitrary d. The choice of d annihilation and creation operators picks out a distinguished subgroup $U(d)\subset Sp(2d,\mathbf{R})$ of transformations that do not mix annihilation and creation operators, and the metaplectic representation gives one a representation of a double cover of this group. We will see that normal ordering the products of annihilation and creation operators turns this into a representation of $U(d)$ itself (rather than the double cover). In this way, a $U(d)$ action on the finite dimensional phase space gives operators that provide an infinite dimensional representation of $U(d)$ on the state space of the d dimensional harmonic oscillator.

This method for turning unitary symmetries of the classical phase space into unitary representations of the symmetry group on a quantum state space is elaborated in great detail here not just because of its application to simple quantum systems like the d dimensional harmonic oscillator, but because it will turn out to be fundamental in our later study of quantum field theories. In such theories the observables of interest will be operators of a Lie algebra representation, built out of quadratic combinations of annihilation and creation operators. These arise from the construction in this chapter, applied to a unitary group action on phase space (which in the quantum field theory case will be infinite dimensional).

Studying the d dimensional quantum harmonic oscillator using these methods, we will see in detail how in the case $d=2$ the group $U(2)\subset Sp(4,\mathbf{R})$ commutes with the Hamiltonian, so acts as symmetries preserving energy eigenspaces on the harmonic oscillator state space. This gives the same construction of all $SU(2)\subset U(2)$ irreducible representations that we studied in chapter 8. The case $d=3$ corresponds to the physical example of an isotropic quadratic central potential in three dimensions, with the rotation group acting on the state space as an $SO(3)$ subgroup of the subgroup $U(3)\subset Sp(6,\mathbf{R})$ of symmetries commuting with the Hamiltonian. This gives a construction of angular momentum operators in terms of annihilation and creation operators.

中文目录

Chapter contents

• 25.1 Multiple degrees of freedom
• 25.2 Complex coordinates on phase space and U(d) ⊂ Sp(2d, R)
• 25.3 The metaplectic representation and U(d) Sp(2d, R)
• 25.4 Examples in d=2 and 3
  • 25.4.1 Two degrees of freedom and $SU(2)$
  • 25.4.2 Three degrees of freedom and $SO(3)$
• 25.5 Normal ordering and the anomaly in finite dimensions
• 25.6 For further reading

Previous chapter: 24 The Metaplectic Representation and Annihilation and Creation Operators, d=1

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