23.3 The Heisenberg group action on operators

23.3 The Heisenberg group action on operators

Concept links · terms present in this machine draft; source roles are unverified: Lie algebra · adjoint representation · irreducible representation · group action

The representation operators

$$\Gamma (\alpha ,\gamma )=D(\alpha )e^{-i\gamma }$$

act not just on states, but also on operators, by the conjugation action

$$D(\alpha )aD(\alpha {)}^{-1}=a-\alpha ,D(\alpha )a^{†}D(\alpha {)}^{-1}=a^{†}-\overline{\alpha }$$

(on operators the phase factors cancel). These relations follow from the fact that the commutation relations

$$[\alpha a^{†}-\overline{\alpha }a,a]=-\alpha ,[\alpha a^{†}-\overline{\alpha }a,a^{†}]=-\overline{\alpha }$$

are the derivatives with respect to t of

$$D(t\alpha )aD(t\alpha {)}^{-1}=a-t\alpha ,D(t\alpha )a^{†}D(t\alpha {)}^{-1}=a^{†}-t\overline{\alpha }\text{\hspace{1em}(23.8)}$$

At $t=0$ this is just equation 5.1, but it holds for all t since multiple commutators vanish.

We thus see that the Heisenberg group acts on annihilation and creation operators by shifting the operators by a constant. The Heisenberg group acts by automorphisms on its Lie algebra by the adjoint representation (see section 15.5), and one can check that the $\Gamma (\alpha ,\gamma )$ are intertwining operators for this action (see chapter 20). The constructions of this chapter can easily be generalized from $d=1$ to general values of the dimension $d.$ For finite values of d the $\Gamma (\alpha ,\gamma )$ act on states as an irreducible representation, as required by the Stonevon Neumann theorem. We will see in chapter 39 that in infinite dimensions this is no longer necessarily the case.

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