Berezin Quantization of Gaussian Functions Depending by a Quantum and Compression Parameter ()
*These notes on the Berezin quantization have been written when the author was a teacher at Liceo “G. B. Bodoni”.
1. Introduction
This paper is devoted to the study of the quantization of Gaussian states. Let us consider a function
, with
, denoted by
as its Berezin quantization. The choice of the Berezin quantization is due to the fact that we will consider Gaussian functions on
instead
and, for this reason, a good scheme of quantization of
is the Berezin quantization. It is well known that this scheme of quantization in comparison with the Weyl quantization presents “a few problems”: for example, it doesn’t preserve polynomial relations, the product rules are more complicated than Weyl quantization and the equivalent of Eherenfest theorem doesn’t hold. Indeed, this scheme of quantization is rarely used to describe the system dynamics. On the other hand, it leads naturally to the definition of a vacuum state
that is usually a Gaussian function.
In the first two sections we will review basic notions on the Berezin quantization of
and the quantum harmonic oscillator [1] - [7] . The other sections are devoted to the proofs of the following results.
Theorem 1.1 In the setting of the Berezin quantization of
we have that the quantization of the complex Gaussian
, with
, is given by:
(1)
where
is a quantum parameter,
,
and
.
It would be desirable to evaluate the trace of the previous Berezin transform. Unfortunately, following the definition of [8] for the trace of the Berezin transform, we find an infinite value. As corollary, we can consider instead the classical trace another kind of trace deriving from the usual inner product on the space
. In this case we have a “modified” version that is the trace of the squared Berezin symbol.
Corollary 1.2 Let
, then we have that:
(2)
We observe that the selected Gaussian function
depends on a “compression factor”
and that
is a quantum parameter that corresponds to the inverse of the Planck constant h. We observe that when
the
trace tends to 1. Moreover, when
we have
. This can be interpreted as an index of the purity of the state. The next result I will present is inspired by the work of [9] and [10] , it corresponds to a “generalized” version of the Heisenberg principle in one dimension.
Theorem 1.3 In the setting of the Berezin quantization of
, if
then
where
is the quantized Gaussian and
is the inner product of elements of
.
In this theorem the quantum parameter has been fixed to 1 as it is convention with the natural units.
2. The Berezin Quantization
Let
be a domain in
with the usual inner product
. Let
be a weight function on
and
be the space of square integrable functions respect
. Let
be the subspace of square integrable holomorphic functions respect to
. This space is also called the “weighted Bergmann space” and has a reproducing kernel
. Let us assume
for all z, we define the “Berezin transform” of
the following integral operator:
(3)
The Berezin transform is an important tool in the contest of Berezin quantization, especially its asymptotic behaviour with the appropriate weights
. As showed in [2] the construction of the Berezin quantization reduces to constructing a family of weights for which the associated Berezin transform
has an asymptotic expansion:
(4)
where
is the “ formal parameter” that when
we have
,
is the identity operator and
are differential operators with
multi-indices. From
it is possible to define the bidifferential operators
, and the star-product:
(5)
If the condition
it’s valid then the star
product coincides with the Berezin star-product and this provides a Berezin quantization. Here
is the Poisson bracket on
and
are quantum observables.
The proof of this assertion and many details on the Berezin quantization of
can be found in [2] . For the Berezin quantization of general function spaces the reader can consult [3] .
For our purpose we consider the Berezin quantization of
with the weighted Bergmann space as function space. In this case
,
will be the quantum parameter that tends to infinity and
.
3. The Quantum Harmonic Oscillator
We consider a slightly modified version of the quantum harmonic oscillator in
. Let
be the Hamiltonian operator where
and
are the usual quantum operators that satisfy the following commutation relations:
(6)
for every
and where h is the Plank constant. We define the operators:
(7)
for every
. The operators
are called respectively the annihilation and creation operator. In this notation the Hamiltonian assume the following form
. It is possible to prove that the ground state corresponding to the energy level
is given by the Gaussian:
(8)
where
. In general we have eigenvalues of energy in this even form:
(9)
with eigenfunctions given by
.
4. Proof of the Theorem 1.1
Proof. The Berezin transform of
with parameter
is:
(10)
We remember the definition of the complex inner product
and, after an algebraic semplification, we have that:
(11)
This is a complex-Gauss integral depending by a quantum parameter
and the positive parameter
. A simple way to solve the integral (11) consists to transform the complex integral in a real integral using the identification
. This gives the product of two real Gauss integrals:
(12)
where
,
,
,
and
,
are the canonical relations. Adjusting the exponents of the two integrals we get two Gauss integrals that can be evaluated:
(13)
Thus we find that the initial integral is equal to
.
Observation 4.1 We observe that when
the complex quantized Gaussian
tends to the classical complex Gaussian
.
Observation 4.2 We can rewrite the
as
and Taylor expand the square brackets:
If we not consider the term
, this is exactly the heat solution operator
according to [2] , where
is the complex Laplacian on
given by
.
5. Proof of the Corollary 1.2
By the modified version of the trace (“that is a sort of a trace of a square”) we have that:
(14)
Thus we must to evaluate the integral:
(15)
Using the canonical relations, this can be written as:
(16)
After the evaluation the Gauss integrals we find that the initial trace is equal to
. Defining
and repeating calculations we find the result.
6. Proof of Theorem 1.3
Before starting to let us fix some notation, we consider a generic Gaussian state with
where
in dimension
. We denote by
respectively the variance of the observable x and p. We have that:
(17)
where for the evaluation the general result in one dimension is useful:
Now we must evaluate
. In a similar way as before we have:
(18)
In order to prove the Heisenberg principle we must evaluate:
(19)
Now by the ordinary Heisenberg principle:
and substituting our quantities we find that:
this is true for every compression
.