39.1 Inequivalent irreducible representations

39.1 Inequivalent irreducible representations

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In our discussion of quantization, an important part of this story was the Stonevon Neumann theorem, which says that the Heisenberg group has only one interesting irreducible representation, up to unitary equivalence (the Schr¨odinger representation). In infinite dimensions, this is no longer true: there will be an infinite number of inequivalent irreducible representations, with no known complete classification of the possibilities. Before one can even begin to compute things like expectation values of observables, one needs to find an appropriate choice of representation, adding a new layer of dificulty to the problem that goes beyond that of just increasing the number of degrees of freedom.

To get some idea of how the Stone-von Neumann theorem can fail, one can consider the Bargmann-Fock quantization of the harmonic oscillator degrees of freedom and the coherent states (see section 23.2)

$$|\alpha ⟩=D(\alpha )|0⟩=e^{\alpha a^{†}-\overline{\alpha }a}|0⟩$$

where $D(\alpha )$ is a unitary operator. These satisfy

$$|⟨\alpha |0⟩{|}^2=e^{-|\alpha {|}^2}$$

Each choice of $\alpha$ gives a diferent, unitarily equivalent using $D(\alpha )$ , representation of the Heisenberg group. This is on the space spanned by

$$D(\alpha )(a^{†}{)}^n|0⟩=(a(\alpha {)}^{†}{)}^n|\alpha ⟩$$

where

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

This is for $d=1$ , for arbitrary d one gets states parametrized by a vector $\alpha \in {\mathbf{C}}^d$ , and

$$|⟨\alpha |0⟩{|}^2=e^{-{\sum }_{j=1}^d|{\alpha }_j{|}^2}$$

In the infinite dimensional case, for any sequence of ${\alpha }_j$ with divergent ${\sum }_{j=1}^{\infty }|{\alpha }_j{|}^2$ one will have

$$⟨\alpha |0⟩=0$$

For each such sequence $\alpha .$ this leads to a diferent representation of the Heisenberg group, spanned by acting with various products of the

$$a_j(\alpha {)}^{†}=D(\alpha )a_j^{†}{D}^{-1}(\alpha )$$

on $|\alpha ⟩$

These representations will all be unitarily inequivalent. To show that the representation built on $|\alpha ⟩$ is inequivalent to the one built on |0⟩, one shows that $|\alpha ⟩$ is not only orthogonal to $|0⟩$ , but to all the other $(a_j^{†}{)}^n|0⟩$ also. This is true because one has (see equation 23.8)

$$a_j(\alpha )=D(\alpha )a_j{D}^{-1}(\alpha )=a_j-{\alpha }_j$$

so

$$\begin{array}{rl}⟨0|a_j^n|\mathbf{\alpha }⟩& =⟨0|a_j^{n-1}(a_j(\alpha )+{\alpha }_j)|\mathbf{\alpha }⟩\\ & ={\alpha }_j⟨0|a_j^{n-1}|\mathbf{\alpha }⟩=\dots \\ & ={\alpha }_j^n⟨0|\mathbf{\alpha }⟩=0\end{array}$$

Examples of this kind of phenomenon can occur in quantum field theories, in cases where it is energetically favorable for many quanta of the field to “condense” into the lowest energy state. This could be a state like $|\alpha ⟩$ , with

$$\sum _{j=1}^d⟨\alpha |a_j^{†}{a}_j|\alpha ⟩$$

having a physical interpretation in terms of a non-zero particle density in the condensate state |$\alpha$⟩.

Other examples of this phenomenon can be constructed by considering changes in the complex structure $J$ used to define the Bargmann-Fock construction of the representation. For finite $d,$ representations defined using $|0{⟩}_J$ for diferent complex structures are all unitarily equivalent, but this can fail in the limit as d goes to infinity.

In both the standard oscillator case with $Sp(2d,\mathbf{R})$ acting, and the fermionic oscillator case with $SO(2d,\mathbf{R})$ acting, we found that there were “Bogoliubov transformations”: elements of the group not in the $U(d)$ subgroup distinguished by the choice of ${}^{J,}$ which acted non-trivially on $|0{⟩}_J,$ , taking it to a diferent state. $\mathrm{As}$ in the case of the Heisenberg group action on coherent states above, such action by Bogoliubov transformations can, in the limit of $d\to \infty$ , take |0⟩ to an orthogonal state. This introduces the possibility of inequivalent representations of the commutation relations, built by applying operators to orthogonal ground states. The physical interpretation again is that such states correspond to condensates of quanta. For the usual bosonic oscillator case, this phenomenon occurs in the theory of superfluidity, for fermionic oscillators it occurs in the theory of superconductivity. It was in the study of such systems that Bogoliubov discovered the transformations that now bear his name.

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