Classical limits of quantum mechanics on a noncommutative configuration space
Abstract
We consider a model of noncommutative Quantum Mechanics given by two harmonic oscillators over a noncommutative two dimensional configuration space. We study possible ways of removing the noncommutativity based on the classical limit context known as antiWick quantization. We show that removal of noncommutativity from the configuration space and from the canonical operators are not commuting operations.
I Introduction
Since the beginning, the classical limit of quantum mechanics has been of primary interest. The most suitable context where to study it is provided by the notion of coherent states as in klaus .
In this work, we study the classical limit not of a standard quantum system, but of two quantum harmonic oscillators whose spatial coordinates are themselves noncommuting operators with noncommutative parameter laure . For a more physical approach of the problem see euro . One immediately has various possibilities to go to the limit of classical harmonic oscillators on the commutative configuration space . One can first go from the noncommutative configuration space to by letting and then remove the quantumness by letting ; one can remove quantumness first and get to a nonquantum system over a noncommutative configuration space and then remove the residual noncommutativity; finally, one can remove both noncommutativities together.
In order to study these possibilities, we use the quantization/dequantization schemes known as antiWick quantization monique . In such a scheme we first quantize a algebra of continuous functions with identity by means of suitably constructed Weyl operators and corresponding Gaussian states that allow to set up a positive unital map from functions to bounded operators and then dequantize it by getting back to functions via another positive unital map. Combining them together one has a means to first let and then and vice versa: the main result is that the two procedures do not commute.
Further, we study a harmonic like dynamics of the two noncommutative quantum oscillators and show that the asymmetry in the two limits is even stronger; letting first regains the standard quantum mechanics of two independent harmonic oscillators. However, letting first does not leave any dynamics on the nonquantum system over the noncommutative configuration space.
We start by giving a brief review of the antiWick quantization in Section II. Then, in Section III, we briefly recall the model of noncommutative quantum harmonic oscillators in two dimensions; in III.1 we construct the Weyl operators and Gaussian states on which the antiWick quantization is based and in Section IV we study the various classical limits. The time evolution and its classical limits will be discussed in Section V.
Ii AntiWick Quantization
In this section we shall shortly review the classical limit of quantum mechanics in the algebraic setting known as antiWick quantization; this technique is based on the quasiclassical properties of coherent states whose definition and properties we shall also summarize. For later extension to the noncommutative quantum mechanical context, we shall consider the standard setting of a classical system with degrees of freedom described by a phasespace with canonical coordinates and momenta ; and denote vectors in whose components satisfy the canonical Poissonbracket relations . From now, scalar products will be denoted by and by norm of vectors.
Let be the dimensional vector of quantized coordinates and momenta operators acting on the Hilbert space of squaresummable functions over . They satisfy the Heisenberg commutation relations , where is the symplectic matrix
(1) 
A useful algebraic description of the quantized system in terms of bounded operators on makes use is of the unitary Weyl operators
(2) 
where denotes the scalar product over generic dimensional vectors. They satisfy the Weyl algebraic relations
(3) 
whence they linearly span an algebra whose norm closure known as Weyl algebra. One can pass from a real formulation whereby the Weyl operators are labelled by to a complex formulation where they are labelled by a complex vector : this is done by introducing creation and annihilation operators
(4) 
where is a suitable parameter such that is adimensional and . Then, one rewrites
(5) 
Let denote the state annihilated by all operators : . We shall refer to it as to the ground state, which in position representation amounts to the Gaussian state
(6) 
The coherent states
(7) 
are eigenstates of the vector operator with eigenvalue ,
(8) 
whence
(9) 
In order to set a useful algebraic setting for the classical limit, we will consider the algebra of continuous functions over which vanish at infinity to which we add the identity function: we shall denote by this commutative algebra. In this context, a particularly suitable algebraic setting for the classical limit is the socalled antiWick quantization that is based on the overcompleteness of coherent states:
(10) 
where denotes the identity operator on .
Then, one may define two positive maps: a quantization map , given by
(11) 
which represents the quantization of the classical function , and a dequantization map given by
(12) 
which dequantizes the operator mapping it back to a function in .
Remark 1
The quantization and dequantization maps are positive as they send positive functions into positive operators and vice versa; they are unital as they map the identity function in into the identity operator .
Now, one computes
(13)  
whence the classical limit
(14) 
ensues. If the classical system evolves in time according to a Hamiltonian function then the antiWick quantization allows one to recover such an evolution from the quantized one when , the simplest situation occurs when corresponds to a quantized . In such a case, phasespace points evolve into where is an symplectic matrix; namely
(15) 
where , respectively denote the inverse of the matrix , respectively its transposed. Furthermore, exactly the same transformation affects the operators in when subjected to the quantized Hamiltonian while the state does not change. Therefore, Weyl operators are sent into Weyl operators according to
(16)  
Then,
(17)  
so that
(18) 
Then, in such a simple case, the classical limit of the quantum timeevolution amounts to the classical timeevolution:
(19) 
Iii Noncommutative Quantum Mechanics
We shortly review the formalism of noncommutative quantum mechanics, more details being available in laure . We consider the two dimensional noncommutative configuration space, where the coordinates satisfy the commutation relation
(20) 
with a real positive parameter and the completely antisymmetric tensor with . Since, the operators
(21) 
satisfy the commutation relations , one can introduce a Focklike vacuum vector such that and construct a noncommutative configuration space isomorphic to the boson Fock space
(22) 
where the span is taken over the field of complex numbers.
A proper Hilbert space over such noncommutative configuration space is the HilbertSchmidt Banach algebra of bounded operators on such that
(23) 
The denotes the trace over noncommutative configuration space and the set of bounded operators on . This space has a natural inner product and norm
(24) 
Next we introduce the noncommutative Heisenberg algebra
(25) 
where a unitary representation in terms of the operators and acting on the quantum Hilbert space (23) with the inner product (24) is
(26) 
In the above representation, the position acts by left multiplication and the momentum adjointly. We shall also consider the system to be equipped with a harmonic oscillator like Hamiltonian operator
(27) 
and refer to the model as two noninteracting noncommutative quantum oscillators.
One can associate to position and momentum operators creation and annihilationlike operators , that satisfy the algebra
(28) 
The explicit expressions of the are as follows laure
(29)  
(30) 
where
(31) 
Interestingly, the operators can be interpreted as proper annihilation and creation operators as there is a vector in , that is a HilbertSchmidt operator such that
(32) 
given by laure
(33) 
After normalization, the ground state corresponding to is
(34) 
Furthermore, the Hamiltonian (27) becomes
(35) 
Clearly, there are two possible quantization and dequantization schemes playing possibly together in this context: one is passing from a commutative to a noncommutative configuration space and back, another one is to pass from commuting position and momentum operators to noncommuting ones and back. In order to make the antiWick quantization works, we proceed by extending the coherent state construction of the previous section to this noncommutative quantum system with two degrees of freedom.
iii.1 Gaussianlike states of the noncommutative quantum harmonic oscillators
In analogy with what we presented in Section (II), we introduce the coordinate vector and the operator vector . Then, we construct the Weyllike operators
(36) 
where is a parameter with the dimension of an action. Using the commutation relations (25), the Weyl algebraic composition law (3) now read
(37) 
As to the parameter , it will eventually let vanish with and . However, there are three possible ways we can reach the full commutative limit :

by linking and so that one may consider the classical limit ;

by letting first so to get to standard quantum mechanics and then let ;

by letting first so to get to a nonquantum noncommutative system and then let .
In order to explore these three possibilities, we shall choose such that
(38) 
Notice that the latter expression is the only natural constant with the dimensions of an action when in the model. A most natural choice is provided by (31)
(39) 
whereas when .
By inverting the relations (29) and (30),
(40)  
(41)  
(42)  
(43) 
one can rewrite the Weyl operators (36) in the form
(44) 
which is similar to (5), with a two dimensional complex vector whose real and imaginary parts are connected to the real four dimensional vector by
(45) 
By using the ground state (34) and the relations (28), we now introduce the noncommutative analogues of the coherent states (7),
(46) 
where . Exactly as in the case of (8), because of the algebraic relations (28), it follows that
(47) 
These states are not exactly coherent states as they do not satisfy the noncommutative analog of minimal indeterminacy joe ; however, they have a Gaussian character and constitute an overcomplete set.
Lemma 1
Proof: Denote the integral by ; then, one checks whether , where the states
constitute an orthonormal basis in the noncommutative Hilbert space . Then, (47) yields
whence the result follows by Gaussian integration.
Iv The classical limits of the noncommutative harmonic oscillators
Following the prescriptions of the antiWick quantization in Section II, we start by choosing the classical algebra, that we choose as made of continuous functions that vanish at infinity augmented with the identity function. Then, following (11) and (12), we define the quantization map dequantization maps.
Definition 1
Let be the algebra generated by the Weyl operators (36), the quantization of will be given by the positive unital map defined by
(49) 
while the dequantization map by the following positive, unital map
(50) 
In order to study the classical limit of the noncommutative quantum oscillators we shall focus upon the following functions
(51) 
that, after some manipulations reported in the Appendix, explicitly reads
where
(53)  
(54) 
In the following, we compute and discuss various possible limits in terms of and or both going to zero.
iv.1 Classical limit:
If and vanish together with the same speed, that is if , with a suitable constant, then , tend to constants and we get the classical limit
(55) 
iv.2 Commutative configurationspace limit:
In the limit , , from (53) and (54) we get the limit behaviours
so that
(56) 
This is nothing but the map (13) for two independent harmonic oscillators with , in fact by a change of variable that we include in the Appendix, we show that the equation (56) is equivalent to
(57) 
The corresponding Weyl operators are
and the Gaussian ground state
in the position representation. The classical limit then yields
(58) 
exactly as in the previous Section.