Applied not toz,zbut to their quantizationsa,a†, suchSU(1,1) trans-
formations are known to physicists as “Bogoliubov transformations”. One can
easily see that replacing the annihilation operatoraby
a′=αa+βa†leads to operators with the same commutation relations when|α|^2 −|β|^2 = 1,
since
[a′,(a′)†] = [αa+βa†,αa†+βa] = (|α|^2 −|β|^2 ) 1
By equation 24.11 theSO(2)⊂SL(2,R) subgroup of equation 24.1 appears
in the isomorphicSU(1,1) group as the special caseα=eiθ,β= 0, so matrices
of the form (
eiθ 0
0 e−iθ
)
Acting with this subgroup on the annihilation and creation operators just changes
aby a phase (anda†by the conjugate phase).
The subgroup 24.7 provides more non-trivial Bogoliubov transformations,
with conjugation byUgr giving (see equation 24.9) annihilation and creation
operators
ar=acoshr+a†sinhr, a†r=asinhr+a†coshr
Forr 6 = 0, the state
| 0 〉r=er 2 (a^2 −(a†)^2 )
| 0 〉will be an eigenstate of neitherHnor the number operatorN, and describes
a state without a definite number of quanta. It will be the ground state for a
quantum system with Hamiltonian operator
Hr=a†rar+1
2
=(cosh^2 r+ sinh^2 r)a†a+ coshrsinhr(a^2 −(a†)^2 )−1
2
=(cosh 2r)a†a+1
2
sinh 2r(a^2 −(a†)^2 )−1
2
Such quadratic Hamiltonians that do not commute with the number operator
have lowest energy states| 0 〉rwith indefinite number eigenvalue. Examples of
this kind occur for instance in the theory of superfluidity.
24.5 For further reading
The metaplectic representation is not usually mentioned in the physics litera-
ture, and the discussions in the mathematical literature tend to be aimed at
an advanced audience. Two good examples of such detailed discussions can be
found in [26] and chapters 1 and 11 of [94]. To see how Bogoliubov transfor-
mations appear in the theory of superfluidity, see for instance chapter 10.3 of
[86].