How to prove that different squeezed vacua are the ground states of inequivalent CCR representations? one can find on wikipedia articles on squeeze operators and squeeze coherent states
these squeezed coherent states depend on a squeezed parameter r. the usual coherent states have r = 0
i have to show that for r not equal to r' the corresponding ground states correspond to ccr inequivalent represetations.
i read in a bood that the scalar product of $\Omega_r$ and $\Omega_{r'}$ is equal to
$1/\sqrt{(cosh r. cosh r' - sinh r.sinh r')} = 1/\sqrt{cosh (r-r')}$
Bogolubov propose this exercise:
Prove that for 0 and r non null the associated representations Wo (the Fock representation) and W(r) are not unitarily equivalent. [Hint: Argue by the method of contradiction. show that for any unit vector f, the projection of the ground state $\Omega_r$ onto the subspace M(f) enerated by the vectors $(a^\dagger(f))^n |O \rangle (n = 1,2, ... )$, has non-zero length which is independent of f.
Finally, choose an infinite orthonormal sequence f1, f2, ...  and use the orthogonality of the
subspaces M(f1, M(f2), ... in order to obtain the required contradiction.
here $a(f) = \int a(x) f(x) dx$ is a smeared annihilation operator. 
for any f if r is not equal to 0 i see that $\langle 0 | \Omega_r \rangle$ is not equal to 1
so its projection to all the M(f) is not null and does not depend on each f. is it correct to say that an infinite number of orthogonal f leads to a contradiction (a null projection on each M(f)?
 A: Let H the Hilbert space associatel to a single boson and 
F(H) its  Fock space 
There are several ordered sets having the same cardinal dim(H). It can be a finite number, the cardinal of N or of R. 
1) A set of vectors of H $ \{e_k \}$ a chosen orthonormal basis of H.
2) Sets of entire numbers $ \{n_k \}$ . Such a sequence is associated to a system of $\Sigma_k n_k $ particles occupying the chosen $e_k$. They belong to the occupancy number basis of F(H).
3) A set of annihilation operators $ \{a_k \}$ (and of their adjoint) obeying the CCR relations:
$$[a_m,a_n] = [a^\dagger_m,a^\dagger_n] = 0$$
$$[a_m,a^\dagger_n] = \delta_{m,n}$$
They modify the  occupancy number on $e_k$ 
As seen in the link above F(H) is closed for the norm. This means that for any set of $ \{n_k \}_j$ vectors  in the occupancy number basis $\sum_j c_j \{n_k \}_j$ belongs to F iff  $\sum_j |c_j|^2$ is finite.
Another set of $b_k$ can obey these Canonical Relations.It was the case with the bosonic displacement $a_k -> a_k + \theta_k $. See the inequivalence problem and we study here the squeezed operators
We say that there is a unitarily equivalence to the Fock repersentation iff there is a unitary map S on F(H) such that
$$\forall k ,b_k S = S a_k$$   and $ \langle0 S|S |0 \rangle = 1 \\$
Wikipedia says that when dim(H) = 1, S is unitary and for a real positive number r,
$$[a_j(r_j),a_k(r_k)] = [a^\dagger_j(r_j),a^\dagger_k(r_k)] = 0$$
$$[a_j(r_j),a^\dagger_k(r_k)] = \delta_{jk}$$
$$S(r)^\dagger a S(r) = a(r) = a .cosh r - a^\dagger sinh r$$
$$S(r)^\dagger a = (a(r)S(r)^\dagger = a .cosh r - a^\dagger sinh r)S(r)^\dagger$$
$$ (a(r)S(r)^\dagger |0 \rangle =( a .cosh r - a^\dagger sinh r)S(r)^\dagger |0 \rangle = 0$$ 
Can $ S(r)^\dagger |0 \rangle$ be normalized? in other worlds is it in F(H)?
Suppose that there is a unit vector $|0_r \rangle $ such
$$0 = a(r)|0_r \rangle =  (a.cosh r - a^\dagger sinh r)(c_0 |0 \rangle + c_1 |1 \rangle +c_2 |2 \rangle + ....$$ 
All these components in the occupancy hilbert basis being null. this gives relations between the   $c_k$.
$c_0$ remains free but cannot be null (the recurrence relations would give $c_k = 0 $ for all k). 
$$0 = c_1 cosh(r)$$
$$0 = -c_0 sinh(r) \sqrt 1 + c_2 cosh (r) \sqrt 2 $$
$$0 = -c_1 sinh(r) \sqrt 2 + c_3 cosh (r) \sqrt 3 $$
$$0 = -c_2 sinh(r) \sqrt 3 + c_4 cosh (r) \sqrt 4 $$
$$0 = -c_3 sinh(r) \sqrt 4 + c_5 cosh (r) \sqrt 5 $$
$$0 = -c_4 sinh(r) \sqrt 5 + c_6 cosh (r) \sqrt 6 $$
so we have 
$$c_1 = c_3 = c_5 .... = 0$$
$$c_2 = c_0 tanh (r) \frac{\sqrt 1}{\sqrt 2} $$ $$\\$$
$$c_4 = c_0 .tanh|^2 (r) \frac{\sqrt 1 \sqrt 3}{\sqrt 2 \sqrt 4}$$
$$c_4 = c_0 .tanh|^3 (r) \frac{\sqrt 1 \sqrt 3 \sqrt 5}{\sqrt 2 \sqrt 4 \sqrt 6} .....$$
$\langle 0_r |0_r  \rangle\ = \Sigma_k |c_k|^2$ is equal to 1
if we take $c_0 = 1/\sqrt{cosh (r) }$. Knight and Buzek wrote that
$$|0(r) = \frac{1}{\sqrt{cosh (r) }} \Sigma_n \frac{\sqrt{(2n)!} }{2^n n!} tanh^n (r) |2n \rangle$$ It belongs to F.
Let us take now the case of 2 annihilation squeezing operators.
$S(r_1)  $ and $  S(r_2)  $ operating on the tensor product of the occupancy basis $|n_1 \rangle \otimes |n_2 \rangle   $
We are looking for a unit vector $|0,0 \rangle _{12}$ which is the proper vector of these opérators 
we have to calculate in the same way $c_{0 0},c_{2 0},c_{0 2}$ and so on so that  $\Sigma_{m n} |c_{m n}|^2 < \infty$ 
Similar formulas appear
$$c_{2 0} = c_{0 0} tanh(r_1)\frac{1}{\sqrt 2}$$
$$c_{0 2} = c_{0 0} tanh(r_2)\frac{1}{\sqrt 2}$$
and they lead to the computation of the unit desired ground state.
$$|0(r_1)0(r_2) \rangle = \frac{1}{\sqrt{cosh (r_1)cosh (r_2) }} \Sigma_m \frac{\sqrt{(2m)!} }{2^m m!} tanh^m (r_1)|2m \rangle \otimes \Sigma_n \frac{\sqrt{(2n)!} }{2^n n!} tanh^n (r_2) |2n \rangle$$
So $$\langle 0,0|0(r_1)0(r_2) \rangle = \frac{1}{\sqrt{cosh (r_1)cosh (r_2) }}$$
In the enumerable infinite limit limit an infinite product of cosh ($r_k$) appears. if it tends to a finite limit, there is no problem about the unitarily equivalence with Fock representation. Look at the similar bosonic displacement question (we had to consider the infinite sum of $\theta^2_k$)
A null scalar product with the Fock vacuum $|0,0....\rangle $ is not a problem
all of the members of the Fock occupancy basis have this property.
But we have still the recurrence relations showing that in this case  it implies a null scalar product with all of the vector basis of F(H) (a contradiction with a unit norm).    
