Solving oscillations problems through affine quantization
aa r X i v : . [ qu a n t - ph ] D ec Solving oscillations problems through affine quantization
Isiaka Aremua and Laure Gouba Laboratoire de Physique des Mat´eriaux et des Composants `a Semi-Conducteur,Universit´e de Lom´e (UL), B.P. 1515 Lom´e, Togo.International Chair of Mathematical Physics and Applications,ICMPA-UNESCO Chair, University of Abomey-Calavi,072 B.P. 50 Cotonou, Republic of Benin. Email: [email protected] The Abdus Salam International Centre for Theoretical Physics (ICTP),Strada Costiera 11, I-34151 Trieste Italy. Email: [email protected] (Dated: January 1, 2021)In this paper the benefits of affine quantization method are highlighted through oscillation prob-lems. We show how affine quantization is able to solve oscillation problems where canonical quan-tization fails.
I. INTRODUCTION
Quantization is generally understood as a correspondence between a classical and a quantumtheory and it appears natural in a quantization procedure that classical variables be chosen beforechoosing what quantum operators to employ since classical physics that refers to pre-1900 physicsis seen as the oldest theory. How to formulate a quantum theory if a classical system is given? Theresponse to that question is addressed in some overviews of the better known quantization techniquesfound in the current literature and used both by physicists and mathematicians [1, 2]. However,quantum mechanics, like any other physical theory, classical mechanics, electrodynamics, relativity,thermodynamics, cannot be derived. The laws of quantum mechanics, expressed in mathematicalform, are the results of deep physical intuition, as indeed, are all other physical theories. Theirvalidity can only be checked experimentally. From this point of view, quantization is not a methodfor deriving quantum mechanics, rather it is a way to understand the deeper physical reality whichunderlies the structure of both classical and quantum mechanics. In his most recent paper J. R.Klauder has addressed how to overcome the two separate languages and create a smooth and commonprocedure that provides a clear and continuous passage between the conventional distinction of eithera strictly classical and quantum expressions [3]. A connection between the classical and the quantumrealms is provided by the coherent states [4–6] and in that sense we consider the procedure of coherentstates quantization and integral quantization among the new trends of quantization procedures [7–10]. We have been particularly interested in new developments of quantization techniques, [11–14],bearing in our mind that affine coherent states framework is useful for investigating quantum gravity[20].The main motivation of investigating in affine quantization is that although Dirac’s canonicalquantization works reasonably well and has got success it has some severe shortcoming from atheoretical point of view when it comes to non-trivial spaces. We are interested for instance inexamining quantization procedure on configuration spaces other than R . The simple exampleof a particle restricted to move on the positive real line is a well illustration. In such case, theconfiguration space is Q = R + . At a classical level it is reasonable to use the position x andthe momentum p x as classical observables and they satisfy the usual Poisson brackets relations.Promoting naively the classical variables respectively to operators ˆ x ≡ x and ˆ p x ≡ − i ~ ∂∂ x , it turnsout that the momentum operator ˆ p x is not self adjoint on the Hilbert space H = L ( R + , dx ) and thismeans that a straightforward application of Dirac’s canonical quantization method is impossible. Themethod of affine quantization expands similar procedures of canonical quantization to solve problemsthat canonical quantization can not solve [15–19]. For instance in a recent work, the usefullness ofthe affine quantization has been highlighted in the quantization of the classical Brans-Dicke Theory[18]. Indeed, the authors have used affine quantization rather than canonical quantization, sincethe domain of the variables involved (scale factor and scalar field) is the real half-line and thecorresponding phase space can be identified with the affine group.In order to make affine quantization procedure our favorite to be solving more complex problemsthat canonical quantization cannot solve such as gravity, we start with some simple one-dimensionalmodels. Indeed, the solution of one-dimensional (1D) problems is usually the first approach tothe calculation of quantum effects in physics, because solutions of the Schr¨odinger equation in onedimension are technically easier to find. Although considered as simple, a careful look into one-dimensional problems shows that they cannot always be solved without serious challenges. One ofus has recently examined the affine quantization on the half line where the case of the free particleand the harmonic oscillator have been solved [21]. The spectrum of the half harmonic oscillator isgiven in terms confluent hypergeometric functions φ n ( x ) = p n + 1)( mω ~ ) x / e − mω ~ x F ( − n, , mω ~ x ) , E n = 2( n + 1) ~ ω, n = 1 , , . . . (1)As we can see the energy eigenvalues are equally spaced as the one of the linear harmonic oscillatorand the eigenfunctions are the well known solution of the harmonic oscillators with repulsive potentialproportional to 1 /x . The polynomial confluent hypergeometric functions are actually proportionalto associated Laguerre polynomials with quadratic variable and the latter are proportional to Hermitepolynomial when the parameter is 1 /
2, more details on the links between those special functionscan be found in the literature [22–24]. Solving the case of the half harmonic oscillator brought toour attention the case of a coupled half harmonic oscillators. The motivation to study this specificsystem is that the decoupling gives different scenario than the case of a coupled two full harmonicoscillator and a different spectrum. An other benefit of the affine quantization is the possibility torecover the solution of the full harmonic oscillator by fixing the symmetry point and changing theend point with the aim to recover the solution of the full harmonic oscillator. That scheme is to gofrom affine quantization to canonical quantization, the reverse scheme not being possible. In section(II), we study the interaction of two half harmonic oscillators, in section (III) we examine how torestore the solutions of the full harmonic oscillator from the solutions of the half harmonic oscillator.Concluding remarks are given in section (IV).
II. INTERACTIONS BETWEEN TWO HALF HARMONIC OSCILLATORS
We consider the two harmonic oscillators represented by the classical Hamiltonian on the half line H c ( x , x , p x , p x ) = 12 m p x + 12 m p x + 12 mw x + 12 mw x + gx x , (2)where ( x , x ) ∈ R + × R + and ( p x , p x ) ∈ R × R , g is a coupling constant that we impose to boundedas | g | < mω . The associated nonvanishing Poisson brackets are given by { x , p x } = 1; { x , p x } = 1 . (3)Let’s redefine the model by mean of change of variables as follows y = x + x ; y = x − x ; p y = p x + p x ; p y = p x − p x , (4)where ( y , y ) ∈ R + × R and ( p y , p y ) ∈ R × R . Regarding the change of variables in equation (4),the Hamiltonian in equation (2) is rewritten as˜ H c = 14 m p y + 14 m p y + 14 ( mω + g ) y + 14 ( mω − g ) y . (5)The Hamiltonian in equation (5) is now describing a model of two independent harmonic oscillatorson R + × R , the fundamental variables being y , y , p y , p y from which the associated nonvanishingPoisson brackets are given by { y , p y } = 2; { y , p y } = 2 . (6)We may perform some rescaling to have the kind of usual Poisson brackets structure in case it isnecessary. Since y >
0, a canonical quantization procedure may fail as the associated conjugatemomenta p y fails to be self adjoint. An alternative is to perform affine quantization procedure.In order to do that we consider the dilation variable d y = p y y which together with y formthe fundamental Poisson bracket { y , d y } = 2 y , they are not canonical coordinates but form a Liealgebra and are worthy of consideration as new pair of classical variables. The fundamental variablesare now y , y , d y , p y . The classical hamiltonian (5) and the equation (6) are now substituted by˜ H c = 14 m d y ( y − ) d y + 14 m p y + 14 ( mω + g ) y + 14 ( mω − g ) y , (7)and { y , d y } = 2 y ; { y , p y } = 2 . (8)For the quantization procedure, the classical variables are promoted to operators as follows y → ˆ y ; y → ˆ y ; d y → ˆ d y ; p y → ˆ p y , and˜ H q = 14 m ˆ d y ˆ y − ˆ d y + 14 m ˆ p y + 14 ( mω + g )ˆ y + 14 ( mω − g )ˆ y . (9)The nonvanishing commutators are[ˆ y , ˆ d y ] = 2 i ~ ˆ y ; [ ˆ y , ˆ p y ] = 2 i ~ . (10)The Schr¨odinger representation is as followsˆ y ψ ( y , y ) = y ψ ( y , y ); (11)ˆ y ψ ( y , y ) = y ψ ( y , y ); (12)ˆ d y ψ ( y , y ) = − i ~ ( y ∂ y + 12 ) ψ ( y , y ); (13)ˆ p y ψ ( y , y ) = − i ~ ∂∂ y ψ ( y , y ) . (14)In order to solve the Schr¨odinger equation, i ~ ∂ t ψ ( y , y , t ) = H q ψ ( y , y , t ) , (15)we assume now that the constant of coupling g is bounded as 0 < g < mω . In presence ofautonomous system, we consider the Ansatz ψ ( y , y , t ) = e − itE y ,y / ~ φ ( y , y ) , (16)the corresponding time-independent eigenvalue equation is H q (ˆ y , ˆ y , ˆ d y , ˆ p y ) φ ( y , y ) = E y ,y φ ( y , y ) . (17)From the expression of equation (9), the equation (17) can be decoupled and in that sense we mayview φ ( y , y ) as φ ( y , y ) = φ ( y ) φ ( y ) and φ ( y ) and φ ( y ) being determined separately, associatedrespectively to E y and E y with E y ,y = E y + E y . We have then to solve (cid:20) − d dy + 34 1 y + m ~ ( mω + g ) y (cid:21) φ ( y ) = 4 mE y ~ φ ( y ) , (18)and (cid:20) − d dy + m ~ ( mω − g ) y (cid:21) φ ( y ) = 4 mE y ~ φ ( y ) . (19)It is easy to derive the solutions of (18) from [21] as follows φ n ( y ) = p n + 1)( mω ~ p g/ ( mω )) y / e − mω ~ √ g/ ( mω ) y F (cid:16) − n, , mω ~ p g/ ( mω ) y (cid:17) . (20)Setting α = mω ~ p g/ ( mω ), we have φ n ( y ) = p n + 1) α y / e − α y F (cid:0) − n, , α y (cid:1) , (21) E y ,n = ( n + 1) ~ ω p g/ ( mω ) , n = 1 , , . . . (22)The solutions of (19) is also easy to derive from the one of simple harmonic oscillator as φ n ( y ) = (cid:18) √ α n n ! √ π (cid:19) / e − α y / H n ( √ α y ) , α = mω ~ p − g/ ( mω ) , (23)where H n represents the Hermite polynomial of degree n and the eigenvalues are given by the simpleformula E y ,n = ( n + 1 / ~ ω p − g/ ( mω ) . (24) III. FROM AFFINE QUANTIZATION TO CANONICAL QUANTIZATION
The solution of the half harmonic oscillator solved in ([21]) is displayed in equation (1). We havebeen curious on the possibility of obtaining the solution of the full harmonic oscillator from theequation (1), that is getting back to canonical quantization from affine quantization. The questionis whether it is possible to recover the solutions of the usual harmonic oscillator on the line from thesolutions of the half harmonic oscillator? In order to analyse that question, we consider a positivereal b ≥
0, where − b < x . In the situation of the half line, the end point and the symmetry pointcoincide to 0. Now we imagine the situation in which we save the symmetry point that remains0 while the end point can move toward negative infinity. The situation in the quantum picture isdescribed by the Hamiltonianˆ H a (ˆ x, ˆ d x ) = 12 m (cid:16) ( ˆ d x + b ˆ p x )(ˆ x + b ) − ( ˆ d x + b ˆ p x ) (cid:17) + 12 mω ˆ x , (25)where ˆ d x is the dilation operator defined in the previous sections and[ˆ x, ˆ p x ] = i ~ , [ˆ x, ˆ d x ] = i ~ ˆ x . (26)The corresponding time independent Schr¨odinger equation is given by (cid:20) − d dx + 34 1( x + b ) + m ω ~ x (cid:21) φ ( x ) = 2 mE ~ φ ( x ) , (27)When b = 0, we recover the problem in Section 3 of [21] brieffly introduced in section (I) and when b → + ∞ the equation (27) tends to the one of the harmonic oscillator on the line. Indeed for b sufficiently large, we assume | xb | <
1, and therefore the approximation1( x + b ) ∼ b (cid:18) − xb + 3 x b − x b + 5 x b + . . . (cid:19) (28)If the expansion is considered up to the fourth level, then, inserting the equation (28) in the equation(27), the potential to consider is V ( x ) = 34 b (cid:18) − xb + 3 x b − x b + 5 x b (cid:19) + m ω ~ x . (29)The potential in equation (29) converges strongly as b → ∞ to the one of the full harmonic oscillator.While the equation (27) is not obvious to solve analytically, we guess that the limit b → + ∞ leadsfrom affine to canonical and should lead to usual even and odd about the symmetry point x = 0eigenfunctions. A temptative way to solve quasi-analytically the equation (27) is to consider thepotential in the equation (29) and consider the method of resolution by Christiane Quesne in herpaper titled “ Quasi-exactly solvable polynomial extensions of the quantum harmonic oscillator ”[25]. The study of this problem is another example of how affine quantization deals with harmonicoscillator problems that canonical quantization cannot resolve. IV. CONCLUDING REMARKS
The coupling of two half harmonic oscillator lead to a problem of a decoupled harmonic oscillatoron the full line and on the half line. In such situation, we show that canonical quantization procedureand affine quantization procedure interviened simultaneously in elegant way. We also show thepossibility of getting back to canonical quantization from affine quantization. Our investigationson the oscillations problems on non trivial configurations space provided an insight to investigateaffine coherent states and their applications y include solving more complicated problems like howto quantize gravity using affine quantization techniques.
Acknowledgements : I. Aremua and L. Gouba would like to gratefully thank Professor J.R. Klauder for drawing the consequent oscillators problems to their attention and for the usefulinstructions. [1] Ali S T, 1993, Survey of quantization methods, in
Classical and Quantum Systems - Foundations andSymmetries (Proc. II. Intern. Wigner Symposium) , p. 29[2] Ali S T and Engliˇ s M, 2005 , Quantization methods: A guide for physicists and analysts,
Rev. Math.Phys.
17, 391 - 490[3] Klauder J R, 2020, A unified combination of classical and quantum systems, arXiv:2010.03984 [quant-ph] [4] Klauder J R and Skagerstam B S, 1985, Coherent states. Application in physics and mathematicalphysics.
World Scientific Publishing Co., Singapore (eds) [5] Gazeau J P, 2009, Coherent States in Quantum Physics, (Wiley -VCH, Berlin ) [6] Ali S T, Antoine J P, Gazeau J P, 2014, Coherent states, wavelets and their generalizations, (2nd edition,Theoretical and Mathematical Physics, Springer, New York) [7] Gazeau J P, 2012, Frame quantization or exploring the world in the manner of a starfish,
AIP ConferenceProceedings
Annals of physics , 344pp. 43 - 68[9] Ali S T, Antoine J P, Gazeau J P, 2014, Coherent States, Wavelets, and Their Generalizations,
Springer,New York, NY [10] Gazeau J P and Murenzi R, 2016, Covariant affine integral quantizations,
Journal of MathematicalPhysics
57, 052102[11] Aremua I, Gazeau J P, Hounkonnou M N, 2012, Action-angle coherent states for quantum systems withcylindrical phase space,
J. phys. A: Math. Theor. 45 , 335302 -1-335302-16 (2012)[12] Aremua I, Hounkonnou M N, Baloitcha E, 2015, Coherent states for Landau levels: algebraic andthermodynmaical properties,
Rep. Math. Phys , 76 (2), 247-269[13] Gouba L, 2019, Beyond coherent states quantization,
J. Phys.: Conf. Ser. arXiv:1912.05388 [math-ph] [15] Klauder J R, 2012, Enhanced quantization: a primer,
J. Phys. A: Math. Theor. arXiv: 1611.02107[quant-ph] [17] Klauder J R, 2020, The benefits of affine quantization, Journal of High Energy Physics, Gravitationand Cosmology , 175-185[18] Frion E, Almeida C R, 2019, Affine quantization of the Brans-Dicke theory : smooth bouncing and theequivalence between the Einstein and Jordan frames, Phys. Rev. D (2), 023524[19] Fanuel M, Zonetti S, 2013, Affine Quantization and the Initial Cosmological Singularity, EPL ,10001[20] Klauder J R, 2006, Attractions of affine quantum gravity, In: Sidharth B, Honsell F, De Angelis A.(eds) Frontiers of Fundamental Physics. Springer [21] Gouba L, Affine quantization on the half line, arXiv: 2005.08696 [22] Almeida C R, Bergeron H, Gazeau J P, Scardua A C, 2018, Three examples of quantum dynamics onthe half-line with smooth bouncing,Annals of Physics
Pages 206 -228[23] Bergeron H, Dapor A, Gazeau J P and Malkiewicz P, 2014, Smooth big bounce from affine quantization,Phys. Rev. D Phys. : Conf. Ser.1071