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

The Metaplectic Representation and Annihilation and Creation Operators, $d$ = 1

In section 22.4 we saw that annihilation and creation operators quantize complexified coordinate functions $\overline{z},z$ on phase space, giving a representation of the complexified Heisenberg Lie algebra ${\mathfrak{h}}_3\otimes \mathbf{C}$ . In this chapter we’ll see what happens for quadratic combinations of the $\overline{z},z.$ which after quantization give quadratic combinations of the annihilation and creation operators. These provide a Bargmann-Fock realization of the metaplectic representation of $\mathfrak{sl}(2,\mathbf{R})$ the representation which was studied in section 17.1 using the Schr¨odinger realization. Using annihilation and creation operators, the fact that the exponentiated quadratic operators act with a sign ambiguity (requiring the introduction of a double cover of $SL(2,\mathbf{R}))$ is easily seen.

The metaplectic representation gives intertwining operators for the $SL(2,\mathbf{R})$ action by automorphisms of the Heisenberg group. The use of annihilation and creation operators to construct these operators introduces an extra piece of structure, in particular picking out a distinguished subgroup $U(1)\subset SL(2,\mathbf{R})$ Linear transformations of the $a,a^{†}$ preserving the commutation relations (and thus acting as automorphisms of the Heisenberg Lie algebra structure) are known to physicists as “Bogoliubov transformations”. They are naturally described using a diferent, isomorphic, form of the group $SL(2,\mathbf{R})$ , a group of complex matrices denoted $SU(1,1)$

中文目录

Chapter contents

• 24.1 The metaplectic representation for d=1 in terms of a and a ^
• 24.2 Intertwining operators in terms of a and a ^
• 24.3 Implications of the choice of z, z
• 24.4 SU(1, 1) and Bogoliubov transformations
• 24.5 For further reading

Previous chapter: 23 Coherent States and the Propagator for the Harmonic Oscillator

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