Statistic-probability theory and linear algebra are both used. So to avoid confusion with the use of ISSN:

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1 International Journal of Latest Research in Science and Technology Volume 2,Issue 2 :Page No.26-35, March - April (2013) ISSN (Online): STUDY ON A PHASE SPACE REPRESENTATION OF QUANTUM THEORY 1 Tokiniaina RANAIVOSON, 2 RAOELINA ANDRIAMBOLOLONA, 3 Rakotoson HANITRIARIVO, 4 Roland Raboanary 1 Theoretical Physics Dept, Institut National des Sciences et Techniques Nucléaires (INSTN-Madagascar) 2 Theoretical Physics Dept, INSTN-Madagascar 3 Theoretical Physics Dept, INSTN-Madagascar 4 University of Antananarivo, Faculty of Science, Department of Physics Abstract - A study on a method to establish phase space representation of quantum theories is presented. The approach utilizes properties of Gaussian distribution, properties of Hermite polynomials, Fourier analysis and linear operators theory. Phase space representation of quantum state and wave functions in phase space are introduced using properties of a set of functions that we call harmonic Gaussian function. Then, new operators that we call dispersion operators are defined as the operators which admit as eigenstates the states associated to the phase space representation. Generalization of the approach for multidimensional case is shown. Examples of application of the formalism that we introduce are given too. Keywords - Phase space, Quantum theory, Quantum state, Wave function, Operator, Hermite polynomials, Gaussian distribution I. INTRODUCTION According to the uncertainty relation, it is well known that in general the exact values of coordinate and momentum of a particle can not be measured and known simultaneously. Let us consider a particle which move along a one dimensional axis: let be his coordinate and his momentum. Let be the quantum vector state of the particle and let and be its wave functions respectively in coordinate and momentum representations. We have Let and be the mean values of and The standard deviations and respectively of and is defined by the relation Using properties of Fourier transform and Cauchy-Schwarz inequality, it may be shown that we have the inequality The inequality (1.8) is the uncertainty relation. Because of this uncertainty relation, it seems not easy to talk about phase space in quantum mechanics. However, some authors has already study the possibility of formulating quantum theory in phase space. A well known approach is based on the use of Wigner distribution [1],[2],[3],[4] instead of the wave function. It permits to obtain interesting results [5],[6],[7] but some difficulties remain about the physical interpretation of this distribution which is not positive definite [8] [9]. In this work, we adopt another approach to tackle the problem of formulation of quantum theory in phase space. We utilize properties of Gaussian distribution, Hermite polynomials, Fourier analysis and linear operators. Phase space states are introduced to build a basis in states space which permits to define the phase space representation. And wave functions in phase space are defined as wave functions corresponding to this representation. Then, it is shown that positive definite probability density function in phase space may be defined using these phase space wave functions. Important insights in our approach are the introduction of new operators that we call dispersion operators. Our approach may be applied to nonrelativstic or relativistic cases. Statistic-probability theory and linear algebra are both used. So to avoid confusion with the use of ISSN:

2 the same words variance and covariance in statistical meaning and in linear algebra meaning (covariance of a tensor), we will use the word dispersion and codispersion to designate statistical variance and covariance. II. HARMONIC GAUSSIAN FUNCTIONS An For positive integers, let us consider the set of orthonormalized functions defined by the relations is Hermite polynomial of degree. Properties of Hermite polynomial may be found in [10]. We may establish International Journal of Latest Research in Science and Technology. According to the relation (2.8) and (2.9), is the momentum mean value and the momentum dispersion corresponding to the function. For further utilization, we designate the momentum dispersion corresponding to the ground momentum dispersion. Because of the relation, it is sufficient to use only one of the parameters or. So from now on we will use in all expressions. For instance, we will utilize the notation instead of for the functions. The set of functions is an orthonormal basis in the vectorial space of Lebesgue square integrable functions. Let and two wave functions, corresponding to a state, respectively in coordinate and momentum representations. The functions are elements of ; we have the expansions: According to the relations (2.3) and (2.4), is the coordinate mean value and the coordinate dispersion corresponding to. For further utilization, we call the coordinate dispersion corresponding to the ground coordinate dispersion. As in our work [11], we call a harmonic Gaussian function. According to the relation (2.2), we have in particular From the relation (2.5) and the properties of orhonormality of the basis and we may deduce the relations Thus a function may be considered as a wave function of a particle in coordinate representation, the corresponding wave function in momentum representation is We may show, that between the functions and, we have also the relations In which As for the case of the functions properties for, we have the following III. PHASE SPACE REPRESENTATION 3.1 Phase space states and phase space representation According to the results obtained in the section 2, the functions and may be considered as wave functions respectively in coordinate and momentum representations. If we denote the state which corresponds to these wave functions, we have According to the relations is a state characterized by the coordinate mean value, the state ISSN:

3 the momentum mean value the coordinate dispersion the momentum dispersion a state may be then considered as a state in phase space i.e. a phase space state. From the relations and we may have From the relations we may establish also According to the relations and, we can deduce that the set is a basis in the states space. This basis may be used to define a phase space representation. According to the relation, a function is a wave function corresponding to a state in this representation. The results in the next section give more justification to this interpretation. According to the relations and, the phase space representation that we have defined has a particularity: the expansion of the state in the basis may be obtained by making a summation on the index or by making an integration on the plane. The first expansion needs the knowledge of the values of all the functions at a point for a given values of. The second one needs the knowledge of the expressions of one function for a given value of. 3.2 Wave functions in the phase space representation The functions may be considered as a phase space wave function corresponding to a state. In this section, we will justify this assumption. Let and be the following functions The analysis of these relations suggest the following interpretations The function is a representation of the probability density corresponding to the coordinate for a given value of the ground momentum dispersion. The function is a representation of probability density corresponding to the momentum for a given value of the ground momentum dispersion. The function is a representation of the probability density on the plane for a given value of the momentum ground dispersion. These interpretations justify the consideration of the functions as a wave function in phase space representation. According to the relation, for a given values of and, may be also interpreted as the probability to find in a state a particle which is in the state. 3.3 Particular properties of phase space states and wave functions in phase space representation We may enumerate and establish some useful properties of the phase space states and wave functions in phase space representation. Property 1: Scalar product of two phase space basis states Let be a phase space state. According to the relation and the corresponding wave functions in coordinate and momentum representations are respectively Using the properties of the functions and, we may prove the following relations ISSN:

4 . From the relation we may also deduce the relations According to the relation, we have the expansion Property 2: Relations between two values at two points of two phase space wave functions is the wave function corresponding to the state in phase space representation. Let us look for the general expression of this wave function Let be a state, according to the relation we can expand in the basis or in the basis. To perform the calculation of this integral, we have to use the following property of Hermite polynomials Then, taking into account the relation the relation we may deduce We find IV. DISPERSION OPERATORS For a given values of the coordinate mean, the momentum mean and the momentum ground dispersion, The state, uses for the definition of the phase space representation is a state characterized by the knowledge of in which and is the complex polynomial the coordinate dispersion the momentum dispersion It follows that a state may be considered as an eigenstate of coordinate dispersion operator with eigenvalue equal to momentum dispersion operator with eigenvalue equal to If and, we have We may obtain the expression of these operators in coordinate representation by using properties of Harmonic Gaussian functions. In fact, because of the relation, in this representation we may write These functions may be used to perform a basis change between the basis and ISSN:

5 Using directly the expressions (2.1) of Harmonic Gaussian functions, we can verify that in coordinate representation, the exact expressions of the operators and which satisfy exactly the relations (4.3) and (4.4) are and are respectively the ground coordinate dispersion and ground momentum dispersion. They are related by the relation (2.6). and, are the related: International Journal of Latest Research in Science and Technology. We may generalize our results to the cases of higher dimension than one.. We study the case of uncorrelated and correlated variables one after the others. Calculation related to linear algebra are based on formulation given in [12] 5.1 Case of uncorrelated variables Let us consider a quantum system which may be described with a position vector belonging to a dimensional vector space,. is a basis in the space. We introduce the momentum associated to as a covector on that we denote, in which is the cobasis of the basis i.e. is the basis in the dual of verifying [12]. Let and be the basic states for the coordinate and momentum representation. Any state of the system may be expanded in the basis and The relations (4.5) and (4.6) suggest that if we denote and the operators associated respectively to the coordinate and momentum in the ordinary formulation of quantum mechanics, we have in general (in any representation) the relation: and are respectively the wave functions in coordinate and momentum representations. The variables are uncorrelated if we have the relations. To find a particular expression in a given representation, we have to replace the operators and by their expression in the considered representation. For instance, to have the expressions in coordinate representation we have to replace in the relations and to obtain their expression in momentum representation, we have to perform the replacement With the mean and dispersion operators, we may also define mean quadratic operators and in which, is a one dimensional coordinate state and is one dimensional momentum state. and are the wave functions corresponding to the variable respectively in coordinate and momentum representations. We may introduce the phase space state corresponding to the variable and phase space wave functions such as And we have the eigenvalue equations V. MULTIDIMENSIONAL GENERALIZATION ISSN:

6 We may define the state the expression of takes the form and the wave functions From the properties (2.2), (2.3) and (2.4) of the one dimensional harmonic function, we may deduce for the From the relations (2.7), (2.8) and (2.9), we may establish analogous properties for the functions In these relations, is the -uplet. The summation in is to be performed for all possible values of -uplet.. is the diagonal matrix. is the function Then we call the position mean vector and the diagonal matrix the dispersion matrix corresponding to. We call the dispersion matrix We may call a uncorrelated multidimensional harmonic Gaussian functions. We may introduce the matrix corresponding to We call matrix the ground position dispersion matrix. the momentum mean covector and the the momentum dispersion matrix corresponding to call the momentum dispersion matrix. We Then with the notations corresponding to matrix. the ground momentum dispersion From the result obtained for one dimensional case, we may deduce that for uncorrelated variables, coordinates and momentum dispersion operators may be associated to each doublet. According to the relations (4.8) and (4.9), if ISSN:

7 we denote and the operators associated respectively to and in ordinary formulation of quantum mechanics, we may associate to them the dispersion operators such as International Journal of Latest Research in Science and Technology. and Let us introduce the two order contravariant tensor and a two order covariant tensor such as The eigenstates of these operators are the states in the relations (5.12), the eigenvalue equations are There is a summation over the index in the last term of these equalities. The inverses of the relations (5.38) and (5.39) are and we may also establish we may also define quadratic mean operators If we choose the linear transformation such as. We may establish from the relation (5.42) 5.2 Case of Correlated variables To study correlated variables, we begin with a more general definition of multidimensional harmonic Gaussian functions with correlated variables. We consider a linear transformation in the space which mix the noncorelated variables considered as components of the vector. Let be a linear transformation in the space and let be its matrix elements in the basis. We introduce a new position vector with its mean. and the correspondent momentum covector by the relation with its mean By introducing the expressions of and in the relations (5.36) and (5.37), we may deduce As the tensors and are by definitions bilinear forms respectively on and on [12], we may write (5.43) and (5.44) in more condensed forms In the relation ( ), We have introduced the inverse of and are the elements of the matrix of in the basis and. We have If we define the functions The expression (5.24) of the correlated variables considered now as a function of become ISSN:

8 We may define the phase space state such as The relation (5.48a) and (5.48b) may be used to describe basis change from the basis and to the basis. This basis change corresponds to the change from coordinate and momentum representations to the phase space representation and then defines this latest one. Using the relations (5.34), we may find the expressions of dispersion operators for the case of corellated variables. Using the relations (5.17), (5.38) and (5.39), we may establish VI. EXAMPLE OF APPLICATION IN NONRELATIVISTIC QUANTUM MECHANICS In nonrelativistic mechanics, the classical expression of the energy of a free particle may be written as in which is the momentum covector. If we denote the position vector of the particle in the three dimensional Euclidian space and the time, the equation of motion of the particle is If we consider quantum mechanics, the wave function of the free particle, of momentum and energy, respectively in coordinate and momentum representation are considered as the functions We have the eigenvalues equations But these functions don t fulfill the normalization relation We remark that this difficulty is a consequence of the fact that the limit is intrinsically assumed. This difficulty may be resolved by introducing our approach in which this limit is not assumed even for a free particle. For the application of our approach, we assume the following hypothesis the states are eigenstates of the operators and respectively with the eigenvalues Hypothesis 1 We associate to the system dispersion operators, and quadratic mean operators. The variables are considered as uncorellated -The means values and associated to each variables and are identified to the classical values of these quantities. So this hypothesis means that we have a mean trajectory of the particle defined by the equation For the case, by taking into account the relation (5.38) and (5.39), we may obtained -To the square of the modules of the momentum is associated the quadratic mean operators : in which the expression of a quadratic mean operator may be deduced from the general expression (5.33) ISSN:

9 The expression of the momentum dispersion operator be deduced from the general expression (5.29) may is a one dimensional harmonic Gaussian function. the operators and is the operator associated to the quantity and in ordinary quantum mechanics. We have for instance in coordinate representation Hypothesis 2 There is an Hamiltonian operator which admit as eigenvalues the values of the energy. The expression of the Hamiltonian may be deduced from the classical expression of the energy by replacing by the quadratic mean operator. According to this hypothesis, we have for the free particle the expression of the Hamiltonian operator in which From the relation (6.10), we may deduce that the eigen states of the Hamiltonian are equal to the eigenstates of the momentum (and coordinate) dispersion operators with the corresponding eigenvalues which may be deduced from the relations (5.30) and (5.31) So the eigenstates of the Hamiltonian operator and the corresponding eigenvalues are is the triplet, are the momentum ground dispersion.the matrix is the ground momentum dispersion matrix. According to this results, the eigenstates of the Hamiltonian operators are phase space states. The corresponding wave functions, respectively in coordinate and momentum representations are are Contratry to the functions in the relations and, the functions in and fulfill the normalization relation (6.4). The relation (6.12) show that the energy of the free particle is equal to the sum of a classical kinetic term and a quantified term which is a linear function of the square of the momentum ground dispersions According to the hypothesis 1, the particle has a mean trajectory in coordinate space or in the phase space. The equation of this trajectory is given by the relation (6.5). The first term in the expression of the energy which is equal to the classical kinetic energy may be associated with the mean motion associated with this mean trajectory. And the second term may be associated to the quantum effect which results from the dispersion of the values of momentum and coordinates around their mean values. This results obtained for the case of a free particle may be generalized: a general system may have a mean trajectory which is a classical trajectory in phase space and quantum effect appear in the dispersion of the values of coordinates and momentum around this mean trajectory : In our method, these facts are described by the introduction of mean and dispersion operators. The study about the relation between uncorrelated and correlated variables that we have considered give a possibility to include in the analysis the study of linear change in the coordinate, for instance a rotation of the coordinates axis. ISSN:

10 VII. EXAMPLE OF APPLICATION IN RELATIVISTIC QUANTUM THEORY Let us give another example of application of our approach in relativistic theory. We may establish a Klein-Gordon like equation which may be utilized to study a formulation of VIII. CONCLUSION scalar field theory in phase space. In the theory of special relativity, the relation between momentum and energy is If we make the replacement Gordon equation for a function, we obtain the Klein From our approach, we may consider an analogous method to establish a Klein Gordon like equation. Let us write the relation in the form may also describe Lorentz Transformation by the relations (5.36). The results obtained in section 2, 3 and 4 show that properties of harmonic Gaussian functions may be exploited to introduce phase space representation in quantum mechanics for the case of one dimensional motion. According to the relation (4.1) and (4.2), the states vector introduced for this representation may be considered as eigenstates of dispersion operators. It was shown in the section 5 that the results obtained for the case of one dimension may be generalized to multidimensional cases. The example of application described in the section 6 and 7 show that our approach may be applied both in formulation of nonrelativstic and relativistic theory and may lead to the obtention of interesting results. Results thus obtained show that our approach may be considered as a possible method to establish a framework for formulation of quantum theory in phase space. if we consider a set of uncorrelated spacetime coordinates. According to the general relation (5.35) we may associate to the square of a component of the momentum the quadratic mean operator is the mean operator and The expression of (5.29) is the dispersion operator. may be deduced from the relation The operator and are the operator associated to and in the ordinary formulation of quantum mechanics. For instance in the coordinate representations, we have Then, we have in coordinate representation REFERENCES [1] E.P. Wigner, "On the quantum correction for thermodynamic equilibrium", Phys. Rev. 40, , 1932 [2] H.J. Groenewold, "On the Principles of elementary quantum mechanics",physica, 12, 1946 [3] J.E. Moyal, "Quantum mechanics as a statistical theory", Proceedings of the Cambridge Philosophical Society, 45, , 1949 [4] T.L Curtright,C.K Zachos, Quantum Mechanics in Phase Space, arxiv: v2 [physics.hist-ph], [5] D.Dragoman, Phase space formulation of quantum mechanics, Insight into the measurement problem, Physica Scripta, 72, , 2005 [6] A. Nassimi, Quantum Mechanics in Phase Space, ArXiv: [quant-ph], 2008 [7] H.-W. Lee, Theory and application of the quantum phase-space distribution functions, Phys.Rep Volume 79, Issue 3, , 1995 [8] A. Kenfack, K. Zyczkowski, Negativity of the Wigner function as an indicator of non-classicality, Journal of optics B: Quantum Semiclass. Opt. 6, , [9] D. I. Bondar, R.Cabrera, D. V. Zhdanov, H. A. Rabitz, Wigner function's negativity reinterpreted: Non-conservation as quantum efficiency indicator, arxiv: v3 [quant-ph], 2013 [10] Raoelina Andriambololona, Mécanique quantique, Collection LIRA, INSTN Madagascar,1990. [11] Tokiniaina Ranaivoson, Raoelina Andriambololona, Hanitriarivo Rakotoson, Time frequency analysis and harmonic Gaussian functions, arxiv: v1 [math.gm], 2013 [12] Raoelina Andriambololona, Algèbre linéaire et multilinéaire, Collection LIRA, INSTN Madagascar, And If we replace in the relations (7.3) the quantity by the above expression of, we may establish a Klein-Gordon like equation which may be used for the study of scalar field theory in phase space. Extension to the theory of spinorial field and vectorial field may be considered too. We remark that in the framework of the analogy that we have made, we ISSN: