On Dynamical Quantization

On Dynamical Quantization Héctor Calisto† and C. A. Utreras-Dı́az‡ arXiv:quant-ph/0603246v1 27 Mar 2006 † Departamento de Fı́sica, Facultad de Ciencias, Universidad de Tarapacá, Casilla 7-D Arica, Chile ‡ Instituto de Fı́sica, Facultad de Ciencias, Universidad Austral de Chile, Casilla 567, Valdivia, Chile E-mail: [email protected] E-mail: [email protected] Abstract. In this article we review some results obtained from a generalization of quantum mechanics obtained from modification of the canonical commutation relation [q, p] = ih̄. We present some new results concerning relativistic generalizations of previous works, and we calculate the energy spectrum of some simple quantum systems, using the position and momentum operators of this new formalism. Submitted to: Eur. J. Phys. PACS numbers: 03.65.w, 01.55+b On Dynamical Quantization 2 1. Historical introduction The validity of any physical theory depends on the experimental data set from which it was originally abstracted. Although a theory may be well established, is not completely unexpected that it fails, or gives unsatisfactory results, when it is applied under experimental sufficiently different conditions that those which originated it. Newtonian mechanics, for example, contains three basic postulates: (i) The existence of an absolute space time (ii) The particles move throughout well defined trajectories. (iii) Space time is continuous. The first two postulates were abandoned at the beginning of last century. As far as the continuity postulate, one may ask whether or not it is a logical necessity, or if it must be accepted for some fundamental reason. The development of relativity and quantum mechanics showed that nature can impose constraints to our measurements; these constraints are related to the existence of two fundamental constants: the speed of light c and Plank’s constant h. Within this same context, we notice that: • Heisenberg quantum mechanics, based upon the canonical commutation relation: [q, p] = i h̄ (1) was formulated more than seventy years ago, originating from experimental data on atomic physics, that is to say, from phenomena whose characteristic energies range from a few eV to about 100 eV. In the particle physics experiments, the involved energies are in the range from 109 eV to 1012 eV. On the base of this observation alone, the question arises on whether the same commutation relations are still valid, or if some suitable modification or generalization is required. This question has been reinforced by experimental observations which suggest that, at high energies, completely new phenomena are observed, which are very difficult, or perhaps impossible, to explain within the framework of the usual quantum mechanics. One of these phenomena is the confinement of quarks; that is to say, the fact that there are quarks within the hadrons, which cannot be observed as free particles. • Relativistic quantum mechanics takes into account the two experimental constraints previously mentioned, but uses differential equations for fields. Since all the physical laws must be verifiable, at least in principle, these equations imply that it has to be possible to measure small space and time intervals without any finite limit. The validity of this assumption has been verified for distances of up to the order of 10−19 cm [1]. To be realistic then, we should consider the value ℓ of the smallest measurable distance, like an empirical parameter to be determined by experiments, instead of assuming a priori that ℓ = 0. On Dynamical Quantization 3 Later in this work, we will assume that ℓ is a universal constant in all the inertial reference frames, just as c and h, and we shall prove that it is possible to construct a quantum theory with the constants c, h and ℓ without falling into logical inconsistencies when ℓ → 0. The continuity assumption enters in the Euclidean geometry through the postulate of infinite divisibility of any line segment, together with the famous postulate on parallel lines. Although both postulates are based on uncontrollable physical extrapolations, they were seen like evident truths, in the sense that alternative assumptions seemed unacceptable. One researcher maintained that the Euclidean postulate on parallel lines could be demonstrated by reduction to an absurd proposition. Nevertheless, Lobachevsky, Bolyai and Riemann, between 1829 and 1854, discovered that it is possible to construct logically consistent non-Euclidean geometries. It is natural to ask, then, what would it happen if one abandoned continuity postulate. According to the literature [2], around 1870, Clifford considered a modification of the Newton laws of the movement, but without changing the other postulates of classical mechanics. He simply provided the absolute space time with a discrete structure, and assumed that the particles may only appear and exist at the points of the resulting network, they would act like lights that can be ignited and extinguished one after another. This concept of discontinuous motion reappeared [2] after the development of relativity, but now combined with the idea of a maximum velocity c. Nevertheless, Einstein had modified the physics in a much deeper form. He recognized that ideally precise measurements of space and time intervals are subject to a universal constraints, namely, we may only obtain results that are related one to the other in such a way that the speed of light in vacuum has the same value c, for any direction and in all the inertial reference frames. Heisenberg modified this idea, requiring that the motion of atomic electrons could be described in terms of all the possible values of a measurement. Using spectroscopy data, he constructed the matrix mechanics, where the concept of coordinates was generalized to be in agreement with the old quantization rules. After the development of the wave mechanics, Heisenberg formulated his famous incertitude relationships, demonstrating very clearly the existence of other universal restriction. A particle may only be localized in space-time with a precision that depends on the incertitude accepted between the momentum and energy of the particle, when these observables are defined in a given inertial reference frame of reference, by their wave properties, and Planck’s constant. Perhaps the single event that gave form to the physics of the twentieth century, was the very surprising discovery that nature can impose constraints to our measurements, a fact that also modified the status of the physical laws. Instead of beginning directly with statements about reality, we make statements about the knowledge that we can obtain from the reality. This knowledge is the result of measurements that are subject to universal constraints, which must be included in the formulation of the physical laws. Relativistic quantum mechanics combines the effects of c and h, but evidence exists today that nature could impose a third universal constraint on physical laws. On Dynamical Quantization 4 Pauli, in a review of the basic principles of quantum mechanics [3] stated that only the relativistic quantum mechanics is logically complete, and expressed vigorously his belief that new limitations in the possibilities of measuring would have to be expressed more directly in a future theory, and that these would be associated with an essential and deep modification of the basic concepts of the formalism of present quantum theory. Pauli also held that the concepts of space and time on very small scales need fundamental modification. The origin of this fact lies in that the calculated values of some physical observables become infinity when the continuous theories are extrapolated to very small distances, although the measured values are in fact finite. This difficulty first appeared in the classic theory of electromagnetism. Quantum electrodynamics attenuated these divergences, but it did not remove them. With the purpose of controlling these divergences, around 1930, Heisenberg [4, 5], proposed to replace the continuous space time by a discrete structure. However, at first sigth, discrete structures break relativistic invariance, a fundamental requirement of any theory. Later Snyder [6], suggested the idea of using a non commutative structure, and showed that this necessarily implies the existence of a length scale below which the notion of physical points ceases to exist. Remarkably, in the Snyder method the space time remains invariant under Lorentz transformations, and it is becomes possible that when this method is used in a field theory it would provide an effective cut-off, that is to say, a minimum length scale in space-time to which the theory is sensible, eliminating therefore the infinities. Unfortunately, the theory of Snyder is not invariant with respect to translations [7], and after some initial developments [8], this idea fell into oblivion, mainly because the renormalization program was revealed appropriate to consistently yield finite numerical values for the observable magnitudes in quantum electrodynamics, without resorting to non commutativity. Some time later, in the fifties, von Newman introduced the term noncommutative geometry when discussing about a geometry in which an algebra of functions is replaced by a noncommutative algebra [9, 10]. Nevertheless, the first example of a noncommutative space that clearly was recognized as so it is the quantum phase space. In fact the first considerations on their quantified differential geometry were developed by Dirac in 1926 [11, 12]. In these works, Dirac discovered the algebraic structure of the quantum phase space, postulating his celebrated quantization method for classical theory, consisting in replacing the Poisson bracket of the classic observable by i h̄ times the commutator of the associated quantum operators. In this way, the coordinates of the phase space p and q become non commutative operators, whose commutator is equal to i h̄. Since these operators do not commute, they cannot be simultaneously diagonalized and the notion of space disappears. In other words, the non commutativity of the operators p and q imply an incertitude relationship between their observed eigenvalues, which replaces the notion of individual points phase space; the closestly related idea remaining in quantum theory being that of the the Bohr cell. In the limit h̄ → 0 recovers the ordinary phase space. This particular algebra of operators was the one that inspired the more radical On Dynamical Quantization 5 idea of replacing the coordinates xµ of the space time by non commutative operators. As it happens in the previous case, the relationship [xµ , xν ] 6= 0 implies an incertitude principle between different coordinates in space time that destroys the idea of points at short distances. One can argue that as the Bohr cell replaces the points of the classic phase space, the appropriate intuitive notion to replace a point is one Planck cell of dimensions given by the Planck area. More recently, the French mathematician Alain Connes developed one more formal definition of the notion of non commutativity from a mathematical point of view [13, 14]. For some time the formalism of Connes was applied to some physical systems, but with very little success, and was subsequently abandoned due to this; however, it generated a renewed interest in the ideas of Snyder about non commutative space time. Further motivation for non commutative theories comes from the idea that, in a quantum theory that includes gravity, the nature of space time must change at distances of the order of the Planck length. The momentum and the energy required to make a measurement at these distances, would by itself modify the geometry of space time [15]. A way to formulate mathematically this is to postulate that, on a scale smaller than the Planck length, the space time is not a differentiable variety, but it has the structure of non commutative space time. Then, a quantum theory of gravity which contains or predicts non commutative coordinates, seems to have good possibilities of being intrinsecally regulated. The string theories have already suggested from the eighties the possibility of a non commutative space time [16] and appears as the main candidate for a quantum theory of the gravity. Also, non commutative field theories play an important role in the area of the condensed matter, which provides not only specific examples of mathematical models used to explore the properties of space time in the physics of high energies and quantum theory of gravity, but that represents specific applications in an area of increasing interest and impact. A classic example is the electron theory in an external magnetic field, projected on the lower Landau level, that can be treated like a non commutative theory. Clear examples of these applications arise in the study of the quantum Hall effect [17, 18]. An recent and convincing examnple of a non commutative theory, in the area of condensed matter, is the quantum theory of mesoscopic electrical circuits developed by Li and Chen [19, 20], that takes explicit account of the discretization of electric charge, leading to a new commutation relationship between the charge and current operators, similarly to those studied in the physics of high energies and quantum theory of gravity. Several advances and applications in the context of the mesoscopic circuits with discrete charge can be found [21, 22]. In what concerns the purely mathematical aspect, the traditional framework of geometry and topology is the set of points with some particular structure that we call space. Nevertheless, as it was discovered very early, fundamental objects such as elliptical curves are better, not in terms of the set of points, but in examining the continuous functions that can be defined on them. Weierstrass opened up a whole new way in geometry when studying directly the set of complex functions that satisfy an On Dynamical Quantization 6 algebra with particular addition rule, and to derive the set of points from these. In non commutative geometry, the general concept to replace sets of points by an algebra of functions is extended. In many cases, the set of points is completely determined by the algebra of functions, then, the set of points can be left, and all the information may be obtained from the funtions alone. On the other hand, in many cases, the set of points is very complex and a direct examination does not provide useful information. In such cases, when the problem is studied from the algebraic point of view, it is common to find that it contains by itself all the necessary information. Nevertheless, this algebra is in general noncommutative. Then the process consists of discovering first how algebras of functions determine the structure of a set of points, and then to determine which are the relevant properties of these algebras that do not depend on the commutativity. After doing this, one can to study noncommutative geometry, generated by an arbitrary noncommutative algebra. Von Newman was the first in trying to describe such quantum spaces rigorously calling to its study geometry without points. The ideas of noncommutative geometries were retaken in the eighties by the mathematicians Connes and Woronowicz [23], who generalized the notion of a differential structure to the noncommutative case, that is to say, to arbitrary algebras. This was completed by a definition of a generalized integration, which provide a more fuller description of the noncommutative space time, and allowed the definition of field theories in such spaces. Summarizing, noncommutative theories have been revealed as useful tools in theoretical physics; they appear as much in the physics of high energies, for the description from a fundamental level of the space time on small scale, as in the area of condensed matter to describe the quantum Hall effect and in the quantum theory of the mesoscopic electrical circuits. The enormous activity around these theories mainly is closely related to the appearance of the noncommutativenes in the limit of low energies of the string theory mentioned previously. Since the string theory is the only well-known theory that could unify all the fundamental interactions, it is possible that the problems of control of divergences in the quantum theory of fields, and the quantization of gravity are in last term intimately related by means of some type of noncommutative algebra. In this work, we begin with a brief review of the generalization of quantum mechanics via the of canonical commutation relation, which was proposed in the eighties by Professor Igor Saavedra [24, 25, 26, 27, 28]. In several dimensions, this theory provides a noncommutative algebra between the space coordinates and in its relativistic version it predicts a space time in which the time is a continuous variable, while at the same time the space has a discrete structure. With few changes we will use this formalism to calculate energy spectrua of some simple quantum systems. 2. Dynamical Quantization The purpose of this section is to review the generalization of quantum mechanics through the canonical commutation relations, proposed by Professor Igor Saavedra On Dynamical Quantization 7 and his collaborators in [24, 25, 26, 27, 28]. In professor Saavedra’s own words, this generalization was inspired by the following reasons: (i) Aesthetic: In general theory of relativity, space is not given a priori; instead, it is given by the energy, the space is curved in the proximity of a massive star. On the contrary, in the usual quantum mechanics, the space, represented here by the position variable, is known beforehand, that is to say, it is independent of the physical phenomena. This reveals an evident and unsatisfactory asymmetry between the macrouniverse, described by general relativity, and the microuniverse, described by the laws of the usual quantum mechanics. (ii) Curiosity: Only from an intuitive point of view, it is a very surprising fact that the same commutation relations extracted from atomic physics could continue to be valid for energies that are twelve orders of magnitude greater. (iii) Phenomenology: It is possible that some phenomena of the physics of high energies, such as the confinement of quarks and certain regularities exhibited by the so-called heavy photons, are connected with the geometry of space. Then, with the purpose of investigating these questions, Profesor Saavedra proposed a generalization of the canonical commutation relationship of the form: iℓ [q, p] = ih̄ + F (q, p), (2) c where ℓ is a constant with dimensions of length and c is the speed of the light in vacuum. In the course of these investigations we assumed that the momentum operator p is well- known, so that the commutation relation (2) determines the position operator q when the function F (q, p) is given. In the low energy limit, therefore, q is the usual position operator: q = ih̄ d/dp. The function F = F (q, p), in general, depends on the dynamics of the problem, and therefore, also the operator q and its eigenvalues; that is to say, physical space is not given here a priori, but it is determined by the physics of the problem represented by the choice of function F . This is the origin of dynamical quantization. In addition, it is assumed that a Hamilton function H = H(q, p) exists and that the Heisenberg equations of motion: dΩ i = [H, Ω] , (3) dt h̄ for any dynamic variable Ω, continues to be valid. 2.1. Non relativistic problems The simplest choice for F that takes into account a possible dependence of the canonical commutation relation with the energy is: F = H(q, p), which leads to the new commutation relation: iℓ [q, p] = ih̄ + H(q, p) (4) c On Dynamical Quantization 8 and to a new uncertainty principle:    h̄ ℓE  ∆p∆q ≥  + , (5) 2 2c  from which we see that the product ∆p∆q grows linearly with the energy in this approach. This result was verified experimentally, except by logarithmic corrections by M. Giffon and E. Predazi [29], using data from the physics of high energies. These authors obtained an approximate value for the parameter ℓ of this theory, that is to say: ℓ = 2.3 × 10−6 fm. For the case of a free particle in one dimension: iℓ 2   [q, p] = ih̄ + p = ih̄ 1 + δ 2 p2 (6) 2mc ℓ δ2 = (7) 2 m h̄ c The corresponding problem eigenvalue problem for the position operator: q ψ(p) = λ ψ(p) (8) lead to the physical space; S. Montecinos, I. Saavedra and O. Kunstmann [26], found that the spectrum is discrete: λn = 2 n h̄ δ, 0, ±1, ±2, . . . . (9) Then, the physical space generated by the hamiltonian H = p2 /2 m, in one dimension is a lattice in which the minimum length interval is: s 2 h̄ ℓ ∆qmin = 2 h̄ δ = (10) mc Once the particle has been located in an arbitrary point of the one dimensional space, the rest of the space feels it, that is to say, the lattice appears; in this sense, geometry acts like a constant force: a linear potential. In addition, if ∆qmin is the space extension of an extended object, it does not make sense to ask for his constituents since no test particle can go ’within’ it. 3. Relativistic generalization Our starting point constitutes the observation that, for a non relativistic free particle p2 1 F= = p · p, 2m 2m which suggests a relativistic generalization of the form: pµ pν F= (11) m Then, the equations (2) and (11) provide the following relativistic generalization:   [qµ , pν ] = −ih̄ gµ ν − δ 2 pµ pν (12) On Dynamical Quantization 9 where g0 0 = −gk k = 1 for k = 1, 2, 3, gµ ν = 0 for µ 6= ν and from now on: ℓ δ2 = (13) m h̄ c As before, we will assume that the operators pµ are known, so that the equation (12) determines the position operators qµ in the four dimensional Minkowski space. The pµ act as multiplicative operators which commute among them: [pµ , pν ] = 0 (14) In this generalization, these not necessarily represent momentum operators, although in the limit δ → 0, we will require that they recover their usual meaning in quantum mechanics. In order to determine position operators qµ who satisfy (12), we postulate the general form: ∂ qµ = Fµ α (p0 , p1 , p2 , p3 ) + ih̄ κ pµ (15) ∂pα where Fµ α is a function to be determined, and we use the Einstein convention (repeated indices are implicitly summed over), κ is a real constant. Later we shall see that the choice of κ determines a weight function in the definition of the internal product. Computing the commutator between pν and (15) we obtain: [qµ , pν ] = Fµ α (p0 , p1 , p2 , p3 )δν α = Fµ ν (p0 , p1 , p2 , p3 ) (16) Here δν α is the Kronecker delta: Comparing (16) with (12) we find:   Fµ ν (p0 , p1 , p2 , p3 ) = −ih̄ gµ ν − δ 2 pµ pν (17) Then:  ∂  qµ = −ih̄ gµ ν − δ 2 pµ pν + ih̄ κ pµ (18) ∂pν Now, in order that the position operators have physical sense, they should represent physical observables, that is to say, they must be hermitian operators. It is not difficult to verify that the operators qµ , are not hermitian with the internal product usually employed in quantum mechanics: Z (ψ, φ) = dτ ψ ∗ φ (19) There are two possible alternatives that they are exactly equivalent to each other: (i) Construct an internal product in which our position operators qµ are hermitian with κ arbitrary. (ii) Choose the constant κ in such a way that these operators are hermitian with the usual internal product. The second possibility is quite simple since it is enough to impose that the operators qµ are hermitian with the internal product (19); the result is: N +1 2 κ= δ (20) 2 On Dynamical Quantization 10 where N = 1, 2, 3, 4 is the dimension of the space time. In this work, however, we will explore the first possibility. Consequently, we needed to construct an internal product in which the operators qµ are self adjoint. We postulate the general form: ψ∗ φ Z (ψ, φ) = dτ (21) W (p · p) where dτ = dp0 dp1 dp2 dp3 , W (p · p) is a weight function to be determined, and p · p = gµ ν pµ pν . Imposing condition of hermeticity of the operators qµ with this new internal product: (qµ ψ, φ) = (ψ, qµ φ) (22) and requiring which the functions ψ and φ vanish suitably fast at infinity, a partial integration shows that, to insure the fulfillment of condition (22), the weight function W satisfy the following differential equation:   ∂W h i g µ ν + δ 2 pµ pν + (N + 1) δ 2 − 2 κ pµ W = 0 (23) ∂ pν where N is the number of dimensions of the space time. The most general solution for W is:  1−β W = 1 − δ 2 g µ ν pµ pν (24) where a multiplicative integration constant has been chosen equal to unity, since any constant of this type may always be included in the normalization of the wave functions ψ, φ. The constant β is given by: κ 1 β = 2 − (N − 1) (25) δ 2 Finally, the sought-after internal product is: ψ∗ φ Z (ψ, φ) = dτ (26) (1 − δ 2 gµ ν pµ pν )1−β from which we can define the probability amplitude: ψ(p) Ψ(p) = 1−β (27) (1 − δ 2 gµ ν pµ pν ) 2 and the probability density: |ψ(p)|2 ρ(p) = Ψ∗ (p)Ψ(p) = (28) (1 − δ 2 gµ ν pµ pν )1−β 3.1. Algebraic properties of space time ¿From equation (18) we have: " !# ∂ ∂ qk = ih̄ + δ 2 pk pν + ih̄ κ pk k = 1, 2, 3 (29) ∂pk ∂pν " !# ∂ 2 ∂ q0 = − ih̄ − δ p0 pν + ih̄ κ p0 (30) ∂p0 ∂pν On Dynamical Quantization 11 These operators look much like the operators introduced by Snyder in [8], the differences are in the additional terms κ pk and κ p0 respectively, in addition, (30) has a global minus sign in the first term. Now we define the operators: Lj = ǫj k ℓ qk pℓ , Mk = qk p0 − q0 pk (31) where ǫj k ℓ is the usual Levi-Civita symbol. Making some simple algebraic manipulations, we find that Lj and Mk have the same explicit expression as in the usual quantum mechanics: ! ∂ ∂ ∂ Lj = −ih̄ ǫj k ℓ pk Mk = ih̄ p0 + pk (32) ∂pℓ ∂pk ∂p0 and they are the infinitesimal generators of the Lorentz group: [Lj , Lk ] = ih̄ǫj k ℓ Lℓ (33) [Mj , Mk ] = − ih̄ǫj k ℓ Mℓ (34) [Lj , Mk ] = ih̄ǫj k ℓ Mℓ (35) Evidently Lj has the usual properties of the angular momentum in quantum mechanics. In addition, other direct calculations allow us to show that: [qj , qk ] = ih̄ δ 2 ǫj k ℓ Lℓ (36) 2 [q0 , qk ] = ih̄δ Mk (37) [qj , Lk ] = ih̄ ǫj k ℓ qℓ (38) [pj , Lk ] = ih̄ ǫj k ℓ pℓ (39) The position operators do not commute, their commutators are proportional to the infinitesimal generators of the Lorentz group. Evidently: ∆qj ∆qk 6= 0, j, k = 1, 2, 3 (40) ∆q0 ∆qk 6= 0, k = 1, 2, 3 (41) That is to say, in this theory, it is not possible to measure two coordinates simultaneously. The algebra obtained when constructing the commutation relations (33), (34), and (35) is identical to the one of Snyder [6]. Consequently, the proposed theory has the important property of being relativistically invariant. This is not accidental, since the commutation relation (12), from which the theory is deduced, are evidently covariant. We will see below that the space time in this theory is discrete, however, we have already seen that Lorentz invariance is included from the begining, it is a fundamental requirement here. 3.2. Structure of the space time In order to find the nature of physical space, we must solve the eigenvalues equation: qµ ψ = λ ψ (42) On Dynamical Quantization 12 with the operators qµ given by (18). In one dimension the equation (42), with κ = 0, is: " !# d 2 d ih̄ +δ p p ψ(p) = λ ψ(p) (43) dp dp this equation can be solved immediately. Nevertheless, is interesting to study it in an auxiliary space, to which we will call background space. In this space the commutation relationships are the usual in quantum mechanics, that is to say: [x̂j , p̂k ] = ih̄ δj k , j, k = 1, 2, 3 (44) [x̂0 , p̂0 ] = − ih̄ (45) We notice that the operator q in (43) becomes the operator x̂ when δ → 0, hence the name given to the space. Interpreting equation (42) like a quantum equation in the momentum representation: h i x̂ + δ 2 p̂ (p̂x̂) ψ(p) = λ ψ(p) (46) we can write, in background representation : " !# 2 2 d d x − h̄ δ x ψ̃(x) = λ ψ̃(x) (47) dx dx where ψ̃(x) is the Fourier transform of ψ(p). This shows that we can solve the eigenvalue problem of the position operator, either in the momentum representation, or in the coordinate representation provided by the background space . This this last one is obtained by the following transformations: ∂ ∂ i ∂ ∂ i pk → −ih̄ , → − xk , p0 → ih̄ , → x0 , ψ → ψ̃ (48) ∂xk ∂pk h̄ ∂x0 ∂p0 h̄ As an example, writing: 2x η η= , ψ̃ = A e− 2 φ(η) (49) h̄ δ equation (47) is reduced to: d2 φ ! dφ λ η + (1 + 1 − η) + −1 φ= 0 (50) d η2 dη 2 h̄ δ the solution, therefore, are the usual associate Laguerre polynomials: φ(η) = L1n−1 (η), n≥1 (51) with eigenvalues: λn = 2 n h̄ δ (52) This result, is the same as the one obtained, by a different procedure, in reference [26]. Using the rules given in equation (48), it may be easily verified that the eigenvalues λn of the position operator q do not depend on the choice of the constant κ. On Dynamical Quantization 13 When the previous procedure is applied to the operators qk and q0 gives by the equations (29) and (30) we obtain the following set of operators in the background space: ! ∂ 2 2 ∂   ∂ ζk = xk + h̄ δ xν + h̄2 κ − N δ 2 (53) ∂xk ∂xν ∂xk ! 2 2 ∂ ∂ ∂   ζ0 = x0 + h̄ δ xν + h̄2 κ − N δ 2 (54) ∂x0 ∂xν ∂x0 which have properties similar to those of the operators qk and q0 , and satisfy an identical algebra. This is correct, of course, since both operators sets are related by a unitary transformation. Using the fact that κ is an arbitrary parameter we can establish relationships with another result existing in the literature. In fact, setting κ = N δ 2 , the operators ζk , ζ0 given in (53) and (54) are reduced to the set of operators introduced by Hellund and Tanaka [30] to describe a quantized space time. These authors assume that time is a continuous variable and demonstrate that the operators ζk have discrete spectrum. In our formalism it is possible to demonstrate explicitly that the operator q0 has continuous spectrum, that is to say, in this theory, the time is a continuous variable. The eigenvalue equation for the operator q0 given in (30) is: q0 ψ = λψ (55) and its solution is: h i iλ exp h̄δ tanh−1 (δ p0 ) ψ=A κ (56) (1 − δ 2 gµ ν pµ pν ) 2 δ2 where A it is a normalization constant. Integrating completely the spatial part we obtain: 4π d p0 Z 2 2 |ψ| = |A| (57) 3δ 3 1 − δ 2 p20 Evidently, the integrand is singular **in one and minus one**. Now, to insure that the theory has physical meaning, ℓ must be the smaller length that it may appears in the problem. In particular, ℓ must be smaller than the Compton wavelength, λc , of the particle of mass m, that is to say, ℓ/λc < 1, and therefore −1 ≤ δ p0 ≤ 1. In addition, with an appropriate choice of the constant A it is possible to prove directly that: (ψ1 , ψ2 ) = δ (λ1 − λ2 ) (58) where δ (λ1 − λ2 ) is the Dirac delta function and ψ this given by (56). The spectrum of q0 is continuous. The generalization of the commutation relation proposed in equation (12) leads to a relativistic model where the space is discrete, but time is continuous, allowing in this way the validity of the Heisenberg equation of motion, as we assumed from the beginning. On Dynamical Quantization 14 4. Energy spectra We have seen already that the commutation  relation (12) determines the position ∂ operators in the form: qµ = qµ p, ∂p . The position operators of this theory depend only on the momentum. In principle, this allows us to solve the eigenvalue problem for the Hamiltonian of any quantum system, in the momentum representation. In this section we shall calculate the energy spectrum and the eigenfunctions of the one dimensional harmonic oscillator and the three-dimensional isotropic harmonic oscillator. That is to say, we will solve in both cases the eigenvalues equation: HΨ=EΨ (59) introducing in H = H(q, p) the position operators (29), which in the one dimensional case become:   ∂ q = ih̄ 1 + δ 2 p2 + ih̄ κ p (60) ∂p In the three-dimensional case we will use: " !# ∂ 2 ∂ qk = ih̄ + δ pk pj + ih̄ κ pk j, k = 1, 2, 3 (61) ∂pk ∂pj We remark that the examples that we present here are clearly non relativistic. However, they are valid examples because in the non relativistic case our generalization becomes rotationally invariant. 4.1. The Harmonic Oscillator in one Dimension The Hamiltonian of the system is: p2 1 H= + m ω2 q2 (62) 2m 2 Replacing q by (60) and rearranging, the equation of eigenvalues (59) becomes: 1 2 d2 Ψ 2 2 dΨ     − mh̄2 ω 2 1 + δ 2 p2 − mh̄2 2 ω κ + δ 2 1 + δ p 2 dp2 dp 2 " # p 1   − E− + m ω 2 1 + (κ + δ 2 ) p2 Ψ = 0 (63) 2m 2 Introducing the changes: √ p = m h̄ ω P, Ψ(p) → Ψ(P ) (64) and using the given definition of δ (13) explicitly we obtain: !2 ω ℓ 2 d2 Ψ ! ! ℓ ωℓ 2 dΨ 1+ P + 2 m h̄ω κ + 1 + P P c dP 2 m h̄ c c dP " ! # 2E ωℓ + + κmh̄ω + κmh̄ω + κ2 m2 h̄2 ω 2 − 1 P 2 Ψ = 0 (65) h̄ω c Now we will assume that the solutions of this differential equation have the form:  σ Ψ = 1 + ξ2 Φ(ξ) (66) On Dynamical Quantization 15 where σ is a unknown parameter that we will fix by imposing that the resulting differential equation for Φ(ξ) reduces to the Hermite equation when we take the limit ℓ → 0. The detailed calculations are actually quite simple, and two possible values for σ are obtained: s 1 mh̄c 1 2c 2   σ1 = − − κ + 1+ (67) 4 2ℓ 4 ωℓ s 1 mh̄c 1 2c 2   σ2 = − − κ − 1+ (68) 4 2ℓ 4 ωℓ For reasons of consistency with the known results of the usual quantum mechanics, we will choose σ = σ2 . The resulting equation for Φ is: !2  s  2 d2 Φ ωℓ  ! ωℓ 2 2c ωℓ 2 dΦ  1+ P 2 + 1− 1+  1+ P P c dP c ωℓ c dP   s  2 2E ωℓ  2c  + − 1+ 1+  Φ =0 (69) h̄ω 2 c ωℓ Finally, making the changes: c η r P = √ , −1 ≤ η ≤ 1 Φ(P ) → Φ(η) (70) ωℓ 1 − η 2 the following differential equation is obtained:  s  2 d2 Φ  2c dΦ    1 − η2 2 − 2+ 1+ η dη ωℓ dη   s  2 2c E 1 2c  + − 1+ 1+  Φ =0 (71) ωℓ h̄ω 2 ωℓ Requiring that: s 2 1 1 2c  + 1+ =a (72) 2 2 ωℓ s 2c E 1 1 2c 2   − − 1+ = n(n + 2 a) (73) ωℓ h̄ω 2 2 ωℓ where n is an integer, the equation (71) can be made to agree with the differential equation for the Gegenbauer polynomials [31]:   d2 Φ dΦ 1 − η2 2 − (1 + 2 a)η + n(n + 2 a)Φ = 0 (74) dη dη Using (72) and (73), the eigenvalues for the energy are obtained:  v  u !2 1 ωℓ 1 ωℓ   u   En = h̄ω  n + 1+ + n2 + n + (75)  t 2 2c 2 2c  Making the inverse change of variables, it is possible to demonstrate very easily that the normalized eigenfunctions, in terms of the original variables, are given by [32]: v ! n!(n + a)δ  δp u σ a 1 + δ 2 p2 Cna √ u Ψn (p) = 2 Γ(a)t (76) 2πΓ(n + 2a) 1 + δ 2 p2 On Dynamical Quantization 16 where Cna is the standard notation for the Gegenbauer polynomials. 4.2. The three-dimensional isotropic harmonic oscillator The Hamiltonian of this system: 1 1 H= pk pk + m ω 2 qk qk , k = 1, 2, 1 (77) 2m 2 is rotationally invariant. Then, using spherical coordinates (p, ϑ, ϕ), in the momentum space and replacing qk by (61) we find that the eigenvalue equation (59, in this case, takes the form: p2 1  2 ∂ 2 Ψ Ψ − m h̄2 ω 2 1 + δ 2 p2 2m 2 ∂p2 1  h   i 2 ∂Ψ − m h̄2 ω 2 1 + δ 2 p3 1 + κ + δ 0 p2 2 p ∂p 2 1 L 1 h   i + m ω 2 2 Ψ − κ m h̄2 ω 2 3 + κ + δ 2 p2 Ψ = E Ψ (78) 2 p 2 where L is the usual angular momentum operator in quantum mechanics. Then, writing the eigenfunctions of the Hamiltonian H like a product of the spherical harmonic Ys m (ϑ, ϕ) and one radial function Π(p) [33], we obtain:‡  d2 Π 2 h   i 2 dΠ 1 + δ 2 p2 + 1 + δ 2 2 p 1 + κ + δ 2 p2 dp2 p dp 2 " # 2E p s(s+1) h   i + 2 2− 2 2− +κ 3+ κ+δ 2 p2 Π = 0 (79) mh̄ ω 2 m h̄ ω p2 As it is usual, to remove the singularity when p → 0 we do: Π(p) = ps Φ(p) (80) In the resulting equation for Φ(p) we make a new change:  γ Φ(p) = 1 + δ 2 p2 Ξ(p) (81) and we pick γ so that the singularity when p → ∞ disappears. Like in the one dimensional problem this provides two possible choices for γ. We chose the one that gives us the correct limits fot the spectrum and the eigenfunctions when we let ℓ → 0: s 1 s m h̄ c 1 2c 2   γ = − − −κ − 1+ (82) 4 2 2ℓ 4 ωℓ Finally, in the resulting equation for Ξ(p), we make following changes: s 1 1+η p= , −1 ≤ η ≤ 1 , Ξ(p) → Ξ(η) (83) δ 1−η ‡ L2 Ys m (ϑ, ϕ) = h̄2 s(s + 1) Ys m (ϑ, ϕ), s = |m| On Dynamical Quantization 17 The result of all these operations is the following differential equation for Ξ(η):   d2 Ξ 1 − η2 dη 2  s  s   2 2 1 1 2c 5 1 2c dΞ   + s + − 1+ − +s+ 1+  η 2 2 ωℓ 2 2 ωℓ dη  s  2 c E 3 s 3 s 2c    + − − − + 1+ Ξ =0 (84) 2ωℓ h̄ω 8 2 8 4 ωℓ where we have used explicitly the given definition of δ (13). Let s 2 1 2c  1+ =a (85) 2 ωℓ 1 s+ =b (86) 2 s c E 3 s 3 s 2c 2     − − − + 1+ = m′ (m′ +a+b+1) (87) 2ωℓ h̄ω 8 2 8 4 ωℓ The equation (84) can be made to agree with the differential equation for the Jacobi polynomials [32, 34]:   d2 Ξ dΞ 1−η 2 2 + [b−a+(a+b+2)η] + m′ (m′ +a+b+1)Ξ = 0 (88) dη dη Using (85-87) we can obtain the eigen-energies of the three-dimensional isotropic harmonic oscillator v u !2 3 u t1+ ωℓ 3 ωℓ     En = h̄ω n+ +h̄ω n2 +3 n − s(s + 1)+ (89) 2 2c 2 2c where n = s + 2 m′ . The solution of (88) are the so called Jacobi polynomials, Ξ(η) = Pn(a,b) (η). The eigenfunctions of Hamiltonian (77) are: Ψsmn (p, ϑ, ϕ) = Ys m (ϑ, ϕ) A Πn (p) (90) where A is a normalization constant. Choosing the spherical harmonical properly normalized: v u 2 s + 1 (s − m)! u Ys m (ϑ, ϕ) = t Psm (cos θ)eiϕ (91) 4π (s + m)! where Psm (cos θ) are the associated Legendre polynomials, the constant A is completely determined by the normalization condition: p2 dp Z ∞ 2 |A| Πn (p)Π∗n (p) =1 (92) 0 W where W is the weight function (24). Replacing Πn (p) and using the change of variables (83) we obtain exactly the normalization integral for the Jacobi polynomials: 22 γ+β−1 1 2 Z  2 |A| (1 − η)a (1 + η)b Pn(a,b) (η) dη = 1 (93) δ 2 s+3 −1 On Dynamical Quantization 18 where all the symbols used have been previously defined. The value of the integral is very well known [34], and we obtain A. Returning to the original variables, the normalized radial function is: v 2 (2n + a + b + 1) n! Γ(n + a + b + 1) 3/2 u u Πn (p) = t δ Γ(n + a + 1)Γ(n + b + 1) δ 2 p2 − 1¨ ! 2 γ   s 2 × (δ p) 1+δ p Pn(a,b) (94) δ 2 p2 + 1 where Pn(a,b) is the standard notation for the Jacobi polynomials. 5. Summary and Conclusions In this work we have presented a relativistic generalization of the canonical commutation relation (1). The main consequence is that, within this theory, the physical space becomes a discrete set of points, whereas the time variable is continuous, as in the usual theory. This result is in agreement with the initial hypothesis, that within this generalization, the Heisenberg equation of motion is valid. The position operators qµ satisfy an algebra that is formally identical to algebra of Snyder [6]. Their commutators are proportional to the infinitesimal generators of the Lorentz group: ∆qj ∆qk 6= 0, ∆q0 ∆qk 6= 0, j, k = 1, 2, 3. Due to this, it is not possible to measure simultaneously two coordinates within this theory. In addition, within this formalism, the product of the incertitudes of the position and momentum operators ∆qµ ∆pν 6= 0, ∀ µ, ν. In particular, for the one dimensional case, ∆q ∆p ∝ p2 . In the low energy regime, when the mass is great compared with the momentum, this result reduces to ∆q ∆p ∝ E. We also showed that an intermediate representation of the position operators exists, in the sense that it is not the representation of momentum nor the representation of coordinates; we named it background space instead. In this representation, the eigenvalue problem for the position operators may be solved in terms of differential equations very well known in quantum mechanics. Finally, we have calculated the energy spectrum of two simple quantum systems: the one dimensional harmonic oscillator and the three-dimensional isotropic harmonic oscillator. In both systems the energy levels depend on n2 . This fact without a doubt is a reflection of our modified commutation relation. The results (75) and (89) show that although the non-dimensional parameter: ω ℓ/2 c may be small, the deviation of the usual dependency in n will be pronounced for sufficiently great values of n. Acknowledgments It is a great pleasure to thank Professor Dr. Igor Saavedra, who suggested this problem a long time ago. H. Calisto acknoledges finantial support from Universidad de Tarapacá On Dynamical Quantization 19 ( UTA Grant # 4723-05). C.A. Utreras-Díaz acknowledges support from Universidad Austral de Chile (DID Grant # S-2004-43), and FONDECYT (Grant #1040311). References [1] M. Dineykhan and Kh. Namsrai, Quantum space-time and estimations on the value of the fundamental length, Mod. Phys. Let. A1, 183 (1986). [2] A. Meessen, Spacetime Quantization, Elementary Particles and Cosmology, Foundation of Physics, 29, 281, (2000). [3] W. Pauli, Die allgemeinen Prinzipien der Wellenmechanik in Handb.d.Physik (Springer, Heidelberg, 1933), 24/1, p. 246, 247, 271, 272. [4] W. Heisenberg, Die Leobachtbaren Grossen in der Theorie der Elemntarteilchen, Z Phys 120, 513 (1943) [5] W. Heisenberg, Quantum Theory of Fields and Elementary Particles, Rev Mod Phys,29, 269 (1957) [6] H. S. 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