Quantum Mechanics for Mathematicians

This is ford= 1, for arbitrarydone gets states parametrized by a vector
α∈Cd, and
|〈α| 0 〉|^2 =e−

∑d

j=1|αj|^2

In the infinite dimensional case, for any sequence ofαjwith divergent

∑∞

j=1|αj|

2

one will have
〈α| 0 〉= 0

For each such sequenceα, this leads to a different representation of the Heisen-
berg group, spanned by acting with various products of the

aj(α)†=D(α)a†jD−^1 (α)

on|α〉.
These representations will all be unitarily inequivalent. To show that the
representation built on|α〉is inequivalent to the one built on| 0 〉, one shows

that|α〉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)

aj(α)=D(α)ajD−^1 (α) =aj−αj

so

〈 0 |ajn|α〉=〈 0 |anj−^1 (aj(α) +αj)|α〉

=αj〈 0 |anj−^1 |α〉=...

=αnj〈 0 |α〉= 0

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 “con-
dense” into the lowest energy state. This could be a state like|α〉, with

∑d

j=1

〈α|a†jaj|α〉

having a physical interpretation in terms of a non-zero particle density in the
condensate state|α〉.
Other examples of this phenomenon can be constructed by considering changes
in the complex structureJused to define the Bargmann-Fock construction of
the representation. For finited, representations defined using| 0 〉Jfor different
complex structures are all unitarily equivalent, but this can fail in the limit as
dgoes to infinity.
In both the standard oscillator case withSp(2d,R) acting, and the fermionic
oscillator case withSO(2d,R) acting, we found that there were “Bogoliubov
transformations”: elements of the group not in theU(d) subgroup distinguished
by the choice ofJ, which acted non-trivially on| 0 〉J, taking it to a different state.
As in the case of the Heisenberg group action on coherent states above, such
action by Bogoliubov transformations can, in the limit ofd→ ∞, take| 0 〉to