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Classical and quantum q-deformed physical systems

On the basis of the non-commutative q-calculus, we investigate a q-deformation of the classical Poisson bracket in order to formulate a generalized q-deformed dynamics in the classical regime. The obtained q-deformed Poisson bracket appears invariant under the action of the q-symplectic group of transformations. In this framework we introduce the q-deformed Hamilton's equations and we derive the evolution equation for some simple q-deformed mechanical systems governed by a scalar potential dependent only on the coordinate variable. It appears that the q-deformed Hamiltonian, which is the generator of the equation of motion, is generally not conserved in time but, in correspondence, a new constant of motion is generated. Finally, by following the standard canonical quantization rule, we compare the well known q-deformed Heisenberg algebra with the algebra generated by the q-deformed Poisson bracket.


Introduction
Quantum algebra and quantum groups arise as the underlying mathematical structure in several physical phenomena. It has been shown that such a formalism can play an important role in conformal field theory, exact soluble models in statistical physics [1] and in a wide range of applications, from cosmic strings and black holes to solid state physics problems [2,3,4].
Many physical applications have been investigated on the basis of the q-deformation of the Heisenberg algebra [5,6,7,8,9]. For instance, q-deformed Schrödinger equations have been proposed in the literature [10,11] and applications to the study of q-deformed version of the hydrogen atom and of the quantum harmonic oscillator [12,13,14] have been presented. In Ref. [15] the Weyl-Heisenberg algebra has been studied in the framework of the Fock-Bargmann representation allowing a rigorous treatment of the squeezed states, lattice quantum mechanics and Bloch functions.
The theory of the q-deformed harmonic oscillator, based on the construction of SU q (2) algebra of q-deformed commutation or anticommutation relations between creation and annihilation operators [16,17,18], has opened the possibility of studying intermediate q-boson and qfermion statistical behavior [19,20]. A kinetic approach of this problem, within a semiclassical treatment, was presented in Ref. [21]. Moreover, it has recently lead to the Correspondence to: antonio.scarfone@polito.it application of q-calculus in the construction of generalized statistical mechanics where nonextensivity properties can arise from the q-deformed theory [22]. Along these lines, a generalized thermostatistics based on the formalism of qcalculus has been formulated in Ref. [23], whereas in Ref. [24] a q-deformed entropy was applied to study a gas of q-deformed bosons.
In the recent past, some tentative approaches to construct a q-deformed version of classical mechanics have been investigated. The primary motivation for such a study is to understand the origin of the q-deformed Heisenberg algebra which forms the basis of the deformed quantum mechanics. In other words, we can ask: how does one introduce a q-deformed algebraic structure for the quantum plane coordinates, such that, after canonical quantization, the q-deformed Heisenberg algebra follows? This question was investigated, for instance, in Ref. [25] where the quasi-classical limit of the q-oscillator has been discussed and a q-deformed version of the Poisson bracket (PB) was derived in terms of variables of the quantum plane. A related question is how one can describe the dynamical evolution of a classical object existing in such a q-deformed quantum plane. In Ref. [26] the author has presented a tentative formulation to construct q-deformed classical mechanics based on the introduction of a q-Lagrangian and a q-Hamiltonian where the equations of motion are derived from the q-deformed analog of the Euler-Lagrange equation. A similar formulation was presented in Ref. [27] by introducing a deformed phase-space based on the elliptic algebra with different deformed parameters for the space coordinates and the momenta.
Finally, among the many motivations for the study of a q-deformed generalization of classical mechanics, it is important to mention the relevance of symmetries in physics [28]. In this respect, quantum groups give a new symmetric aspect for the resolution of classical problems. Along these lines, Ref. [29] deals with the investigation of a set of Poisson algebra structures obtained from the elliptic quantum algebra. It has been shown that the resulting Poisson structure contains the q-deformed Virasaro algebra which plays a central role in the resolution of several integrable systems both in quantum mechanics and statistical mechanics.
Recently, in Ref. [30], a possible definition of q-deformed PB has been derived by requiring that it be invariant under the action of the q-deformed symplectic group Sp q (1), in analogy with the classical (undeformed) case, where PB are invariant under the action of the symplectic group Sp(1). In this paper we start by considering a q-deformed version of PB previously introduced but following another approach, and in some sense, a more systematic derivation instead of the one adopted in Ref. [30].
In the commutative classical mechanics, the PB between two functions f (x, p) and g(x, p) can be defined through the contraction of the corresponding Hamiltonian fields X f and X g with the canonical symplectic form ω = dx ∧ dp. In the same manner, we can define a q-deformed version of PB between two q-functionsf (x,p) andĝ(x,p), defined on the noncommutative q-plane, through the contraction of the corresponding q-Hamiltonian fieldsXf and Xĝ with the q-deformed canonical symplectic formω = dx ∧ dp (throughout this paper we denote by a hat the elements belonging to the q-algebra to distinguish them from the element belonging to the ordinary commutative algebra). We then attempt to formulate, by means of the q-deformed Hamilton's equations, a q-deformed classical mechanics describing the time evolution of a mechanical system. Finally, in analogy with standard canonical quantization method, where the (undeformed) PB between canonically conjugate variables x and p are replaced by the (undeformed) commutator of the corresponding quantum operators x and p, we discuss the scenario of a possible canonical quantization in the q-deformed framework, putting in correspondence the algebra generated by the q-deformed PB betweenx andp with the well known algebra generated by the q-deformed commutator of the corresponding q-deformed operatorsx andp.
Our paper is organized as follows. After a brief review of the derivation of the standard PB in the formalism of the exterior calculus, presented in Sec. 2, we introduce, in Sec. 3, the q-commutative phase-space and recall the definition of its q-calculus. On the basis of the previous sections, we will be able to obtain the most original result of our paper: the introduction of a q-deformed PB in Sec. 4 and the formulation of a possible q-deformed mechanics by means of q-Hamilton's equations is presented in Sec. 5. In Sec. 6, starting from the q-deformed PB we explain a possible derivation of the q-deformed Heisenberg algebra. Finally the conclusions are presented in Sec. 7.

Poisson bracket in the commutative phase space
We begin by recalling briefly the derivation of the standard PB in the formalism of the exterior calculus, referring to the relevant literature for the details [31,32].
Let us consider the real plane IR 2 generated by the commutative coordinates x 1 ≡ x and x 2 ≡ p and introduce the associative algebra A = Fun(IR 2 ) of the functions on IR 2 freely generated by the elements x and p. The tangent space T A on A is generated by the vectors where f (x, p) and g(x, p) are smooth functions on A and λ, µ are ordinary commutative numbers (C-numbers). Any vector v can be spanned on the base of In the same way we introduce the cotangent space (one-forms) T * A, generated by the elements dx and dp. Any one-form ω can be spanned on this base as ω = dx g 1 + dp g 2 , where g i ∈ A.
Higher order forms are constructed by means of the differential operator d which takes k-forms into (k + 1)forms. In particular starting from a function f ∈ A (0form), its differential is a 1-form and, starting from a 1-form ω = dx g 1 + dp g 2 we obtain the 2-form where we have introduced the exterior product between two 1-forms, that is linear The d operator fulfills the following main properties: where k and p are the degree of ω (k) and ω (p) , respectively. Finally, we introduce the contraction operator i v (ω), in the axiomatic way, through its main propriety listed below, In order to derive the Poisson bracket and their algebra within this formalism we begin by introducing the symplectic form ω = dx ∧ dp , and define the Hamiltonian vector field X f , associated with a function f ∈ A, through the relation ¿From Eqs. (4) and (5), taking into account the properties of the contraction operator (3), it follows that, for any function f , the corresponding Hamiltonian vector field X f assumes the expression As a consequence, we can write the Poisson bracket between the two functions f, g ∈ A through the relation It is easy to show, accounting for Eq. (6), that Eq. (7) can be written in the usual form We recall the main properties of the PB defined through Eq. (7) which are: bi-linearity We remark that Eq. (8) can also be expressed in the form (repeated indexes are summed over), where we have introduced the structure functions which can be arranged in 2 × 2 matrix J. Taking into account the expression of the symplectic form (4), we easily recognize that J is the symplectic unity, with entries with f ′ = f (x ′ , p ′ ) and g ′ = g(x ′ , p ′ ) and where T i j , the entries of a matrix T ∈ Sp(1), satisfies the symplectic relation 3 q-commutative differential calculus In order to generalized the PB in the framework of a qdeformed theory, we review the basic properties of the q-commutative differential calculus.
The real quantum plane IR 2 is generated by the qcommutative elementx 1 ≡x andx 2 ≡p, obeying the relation [33] which is invariant under the action of Gl q (2) transformations and q is the real deformation parameter. We denote by A = Fun( IR 2 ) the associative algebras freely generated by the elementsx andp.
In analogy with the commutative case, we define the qtangent space T A [34], generated by the q-derivatives∂ 1 ≡ ∂ x and∂ 2 ≡∂ p , whose action on the generatorsx i is defined as [5] They are linear operators satisfying∂ i (λf +µĝ) = λ∂ if + µ∂ iĝ which fulfill the q-Leibniz rulê Let us now introduce the q-cotangent space T * A, generated by the elements dx and dp. Any q-deformed 1-form ω can be spanned on this basis asω = dxĝ 1 + dpĝ 2 wherê g i ∈ A. We observe that, dx if =f dx i , due to the qcommutative structure of the q-calculus. Nevertheless, accounting for the following relationŝ p dp = q 2 dpp , x dp = q dpx , any q-deformed 1-form is well defined and admits a unique, left or right, expansion: can be obtained each from the other by means of Eqs. (18).
Through the action of the operator d we can construct higher order q-deformed forms. In particular, the differential of a q-functionf ∈ A is given by which, by means of Eqs. (18), can be written equivalently in the (left) form We recall that in the framework of the q-calculus, the operator d fulfills the same formal properties as in standard calculus. The elements dx and dp satisfy the relations dp ∧ dp = 0 , dp ∧ dx = −q −1 dx ∧ dp , where, the exterior product is still linear and associative. Finally, we introduce the contraction operator i v (ω) between q-deformed vectors and q-deformed forms, in the axiomatic way, through its main properties listed below which are formally identical to Eq. (3) with the difference that now, the ordering is important. A rigorous derivation of these properties can be found in Ref. [34]. A realization of the above q-algebra and its q-calculus can be accomplished by the replacements [35] x → x , where is the dilatation operator along the x direction (reducing to the identity operator for q → 1), whereas are the Jackson derivatives (JD) with respect to x and p [36]. Their action on an arbitrary function f (x, p) is which reduce to the ordinary derivatives when q goes to unity. Therefore, as a consequence of the non-commutative structure of the q-plane, in this realization thex coordinate becomes a C-number and its derivative is the JD whereas thep coordinate and its derivative are realized in terms of the dilatation operator and JD.
4 Poisson bracket in the q-commutative phase space On the basis of the q-commutative differential calculus, in this section we derive the expression for the q-deformed PB by following, in analogy, the same formal steps used in the classical derivation reviewed in Sec. 2. To begin with, let us introduce the q-deformed symplectic form and we define the q-Hamiltonian fieldX f , associated with the functionf ∈ A, through the relation According to q-calculus, the expression forX f is given bŷ which reduces to the standard Hamiltonian field (5) in the q → 1 limit. We introduce the q-Poisson bracket between the qdeformed functionsf andĝ by means of the relation, Accounting for Eq. (20) and the properties (22) we obtain the general expression for the q-PB (36) Properties of the q-PB (35) are consequences of the properties of the q-deformed contraction operator (22). In particular they are bi-linear, but in general they are not skew-symmetric. This can be seen, for instance, if we consider the q-deformed generator functions which can be constructed by settingf ≡x and g ≡p. The canonical q-Hamiltonian fields are respectivelŷ and, after observing that, according to Eq. (15), (∂ ix j ) L ≡ (∂ ix j ) R , from Eq. (36) we immediately derive the q-deformed structure functions for the q-Poisson algebra Their explicit values are obtained as follows p,x q = −q −1/2 , so that we easily obtain Finally, the Jacobi identity is trivially satisfied for the qstructure functions In Ref. [30] a q-deformed Poisson bracket has been obtained by requiring that, in analogy with the classical case, where standard PB is invariant with respect to a symplectic transformation, the q-PB should be invariant with respect to a q-symplectic transformation. As a result we obtained a q-PB formally equivalent to Eq. (36) but with the replacement of q → 1/q 2 . It is also easy to see that the q-PB given in Eq. (36) is preserved under the action of a q-symplectic transformation. In order to show such a property, we recall that, in the fundamental representation of Sp q (1), any elementT is given by a 2 × 2 matrix, with entriesT j i , satisfying the following equation where C ij q = ǫ ij q −ǫ ij . Observing that Eq. (36) can be expressed as where J q = −C q ′ , with q ′ = √ q, according to the property (45), it is easy to see that under the action of quantum group Sp q ′ (1), Eq. (46) transforms as and we assumed the commutations between the group elements and the plane elements. It may be observed that Eq. (45) can be rewritten aŝ which mimics, in this way, the classical expression (13), where the matrix J q plays the role of the symplectic unit J introduced in Eq. (10) and recovered in the q → 1 limit. The q-deformed generator functions are then related to the entries of the matrix J q as which also shows their invariance under the action of the q-symplectic group Sp q ′ (1).

q-deformed Hamilton's equations
As a preliminary application of the q-PB derived in the previous section, it is natural to investigate the effect due to the q-commutativity of the coordinates of the phase space on the time evolution of a classical object existing in this space. We postulate that the dynamics in the deformed phase space is described, in analogy with classical mechanics, by means of the following q-deformed evolution equations written in the forṁ It is assumed that time enters in the q-generators as a normal parameter. The time derivative, indicated by a dot, meansẋ = dx/dt where dt is a C-number. In this way, the q-commutative algebra ofẋ(t) andṗ(t) and its qcalculus are the same as that of dx(t) and dp(t).Ĥ(x,p) is the q-Hamiltonian function which is assumed to not depend explicitly on time.
According to Eq. (36) the evolution Eqs. (51) and (52) can be written down in the forṁ which are the q-deformed Hamilton's canonical equations. It can be shown that, as a consequence of Eqs. (51) and (52), the time evolution of any dynamical function f (x(t),p(t); t), depending on the generatorsx(t) andp(t), can be described by means of the following evolution equation where, the last term in the right hand side takes into account the explicit dependence off from t. In fact, we recall readily that the most general form of a functionf ∈Â can be written as a polynomial in the q-variablesx(t) andp(t) where c nm (t) are C-numbers which may be time dependent and we have assumed thex-p ordering prescription which can be always accomplished by means of Eq. (14). Let us consider the generic term in Eq. (56). Its time derivative becomes where Eq. (18) has been employed and where we have introduced the q-basic number By employing the equations of motion (51) and (52) in the form dx/dt = iX H (dx) and dp/dt = iX H (dp), and accounting for the properties of the operators d and iv, we obtain which, by linearity, implies Eq. (55). Finally, we recall that, in standard mechanics, the canonical transformations are a kind of coordinate transformations mixing x and p in a way that preserves the Hamilton structure of the dynamical system. An important property of the canonical transformations is that they preserve the Poisson bracket. Thus, this family of transformations, also called symplectic transformations [31], are realized by the symplectic group transformations. In the q-deformed case, the situation is very similar because, as shown in the previous section, the q-PB are invariant under the action the q-deformed symplectic group. In this sense, the quantum group Sp q ′ (1) is realized as a version of q-deformed canonical transformation. It is outside the scope of this work to discuss this important subject of the theory and its implications which will be discussed in a future investigation.
In the following we are going to consider some non relativistic systems described by the q-Hamiltonian where m is an ordinary C-number andV (x) is the external potential which we assume to be an arbitrary polynomial in the generatorx, with C-numbers coefficients. Within this formalism let us discuss some examples.

The q-free particle
As a first simple example, we chooseV (x) = 0, a free particle. The q-Hamiltonian field is readily computed from Eq. (34) and is given bŷ These equations show that the effect of the deformation is to rescale the mass m of the particle as the effective mass m q . We remark that the q-Hamiltonian of the free particlesĤ 0 is a constant of motion of the system. In fact, by using Eq. (55), it can be verified that {Ĥ 0 ,Ĥ 0 } q = 0 so that the HamiltonianĤ 0 could represent the energy of the free particle which is conserved in time. This fact, in the undeformed canonical theory, is merely a consequence of the skew-symmetry of the PB: In the q-deformed case, because the q-PB is no longer skew-symmetric, the q-PB evaluated between the same functionf in general does not vanish. An immediate consequence of this is that, contrary to standard classical mechanics, in the q-deformed theory, the Hamiltonian function in general is not conserved in time. Such a situation has been encountered also in other proposed q-deformed classical systems [27]. Let us investigate the consequence in the next example.

The q-harmonic oscillator
We consider the harmonic oscillator with a potential where the angular frequency ω is a C-number.
The q-Hamiltonian field is evaluated aŝ whereĤ h =Ĥ 0 +V h and ω q = ω [2] q /2 q 2 while the equations of motion becomė The effect of the deformation is thus taken into account only through a rescaling of both the mass m → m q and the angular frequency ω → ω q of the harmonic oscillator. The time evolution of HamiltonianĤ h can be determined by employing Eq. (55) and we obtaiṅ which vanishes only in the q → 1 limit. As a consequence, the HamiltonianĤ h cannot be associated with the total energy of the conservative system. Nevertheless, the following function is time conserved,Ė q = 0, and reduces, in the q → 1 limit, to the Hamiltonian of the harmonic oscillator. Hence, the non-conservation ofĤ is not necessarily fatal for the theory due to the existence of other constants of motions that could be identified with the energy of the system instead ofĤ. In other words, in the q-deformed theory,Ĥ is the generator of the equation of motion which now does not necessarily coincide with the total energy of the system, but in correspondence, a new constant of motion may be generated.

The general case
The previous result can be extended to the general case by considering a q-deformed mechanical system governed by a potentialV (x) defined by a polynomial series ofx with C-number coefficients. The q-Hamiltonian field is given byX where (∂ xV ) L dx = dx ∂ xV R , whereas the equation of motion can be expressed in the form withF q = −q −1/2 (∂ xV ) L the q-deformed external force. Again, the Hamiltonian is not conserved in time, whereas the function with d n = c n q 4−3 n , reduces to the Hamiltonian function of the system in the q → 1 limit and fulfills the relatioṅ E q = 0.

Canonical quantization
Let us now compare, by means of canonical quantization, the well-known q-deformed Heisenberg algebra with the algebra (42) generated by the q-deformed PB. To start with we recall that the q-deformed Heisenberg algebra reads [6,7,8,10] x,p where is the q-deformed commutator and we have denoted by a tilde the q-deformed quantum generatorsx andp. The extra generatorΛ q in Eqs. (72) and (73) plays the role of a dilatator and, in the q → 1 limit, reduces to the identity operator [7,37].
As it is known, in the undeformed case, the simplest canonical quantization prescription is given by identify the position variable x with the corresponding multiplicative operator x and the canonically conjugate variables p is assumed to be proportional to the space derivative according to the rule At the same time, the undeformed PB is replaced with the undeformed commutator where [x, p] = x p − p x. In analogy with this scenario, it appears natural to impose, in the q-deformed framework, the following quantization rule on the q-variableŝ and to replace the q-PB with the q-deformed commutator (74). It should be observed that, contrary to the q-deformed classical theory developed starting from the 2-dimensional q-deformed calculus, the q-deformed quantum mechanics is spanned in the 1-dimensional configuration space. This requires us to consider the 1-dimensional q-deformed calculus generated by the elementsx,∂ x and dx which differs from the corresponding one in 2-dimensions and introduced in Section 3. In particular, the Leibnitz rule now reads [5]∂ which differs from the last of Eq. (16). Let us observe that, due to Eq. (79), assumingX is a Hermitian quantity, the prescription (78) deals with a non Hermitian definition of momentum. In accordance with current literature, we can define a physical momentum where the Hermitian conjugation of∂ X is defined bŷ After the redefinitionx it is easy to verify that the definitions (80), (82) and (83) fulfils Eqs. (72) and Eq. (73). Thus, by posing formallŷ we can state the following correspondence among the q-PB between the classical space phase variables and the q-commutator between the corresponding quantum operators. Finally, by taking the Hermitian conjugate of Eq. (72), accounting for the unitariety ofΛ q , we obtain We can verify that Eqs. (85) and (86) which follows directly from Eqs. (42), is replaced by as can be verified by a direct calculation.
In light of the above prescriptions, starting from the classical evolution equation (55), it appears natural to postulate the q-deformed Heisenberg equation for an operatorÕ q as follows

Conclusion
In this paper, we have developed a q-deformed version of the Poisson bracket by generalizing, in the quantum groups framework, the method based on the exterior calculus for the definition of the classical PB. We have derived, in a systematic way, an expression of the q-PB which substantially equivalent to the one recently obtained by us in Ref. [30] by following a different and more consistent method. The two approaches differ in the following sense.
In the previous one [30], the expression for the q-PB was conjectured by requiring the invariance of the q-PB under the action of the symplectic group Sp q (1), leading to the q-deformation of the phase space, whereas in the present investigation, only the non-commutative q-deformation of the phase space has been imposed and the q-PB has been obtained as a consequence of the q-calculus. It has been shown that the new version of the deformed bracket is still invariant under the action of the q-symplectic group of transformations Sl q ′ (2) with q ′ = √ q.
We have discussed some properties of the q-deformed PB and we have presented some elementary examples to illustrate how a possible q-deformed classical mechanics can be introduced. It has been shown that, in contrast to the undeformed case, the q-Hamiltonian, which is the generator of the evolution equation, is in general not conserved in time and cannot be identified with the total energy for conservative systems. However, a suitable function, reducing to the Hamiltonian function in the q → 1 limit and remaining constant during the evolution of the system, has been obtained for the examples studied. These properties, related to derivation of the q-PB given in Eq. (36), are the most relevant results of this paper.
Finally, we have discussed a possible quantization method in the q-deformed picture. Based on the standard method consisting of the replacement of the (undeformed) PB for canonically conjugate variables with the (undeformed) commutator for the corresponding quantum operator, we have postulated a similar scheme by replacing the q-PB for quantum conjugate variables with the q-deformed commutator of the corresponding q-deformed quantum operators.
In conclusion we would like to mention the possible applications of the q-deformed classical mechanics that we have developed to the study of some relevant physical phenomenologies and in particular to the framework of thermostatistics [23] where it could lead to a generalization of the theory in a manner similar to what the classical Tsallis' thermostatistics does with respect to the Boltzmann-Gibbs theory [22].