On quantization of nondispersive wave packets

On quantization of nondispersive wave packets arXiv:1208.4356v1 [math-ph] 21 Aug 2012 M.V.Altaisky∗, N.E.Kaputkina† Aug 21, 2012 Abstract Canonical commutation relations for the Bateman-Hillion type nondispersive wave packets are constructed 1 Introduction The packet-like asymptotic solutions of the wave equation (∂x22 + ∂y22 + ∂z22 − ∂t22 )u(x, y, z, t) = 0. (1) trace their origin from the Bateman’s works on conformal symmetry [1, 2]. Historically such solutions were first derived approximately in terms of parabolic equation of the diffraction and are related to paraxial optical beams, see [3] for a review. The interest to the localized solutions of wave equation is encouraged by the progress in generation of ultra-short laser pulses [4], in view of the fact that large variety of optical problems can be solved in terms of scalar equation. The packet-like localized solutions of equation (1) have a general form [5]: f (θ) √ x2 + y 2 u= √ , θ= 2ξ− + √ , (2) 2ξ+ − ıε 2ξ+ − ıε ∗ Space Research Institute RAS, Profsoyuznaya 84/32, Moscow, 117997, Russia; and Joint Institute for Nuclear Research, Joliot-Curie 6, Dubna, 141980, Russia; e-mail: al- [email protected] † National University of Science and Technology “MISiS”, Leninsky prospect 4, Moscow, 119049, Russia; e-mail: [email protected] 1 where f is an arbitrary localized function, z±t ξ± = √ , (3) 2 are the light-cone coordinates, describing a particle moving at a speed of light (c = 1) along z axis, ε > 0. Their particular form with f (θ) = eıqθ was called ’quasiphotons’ by Babich and Ulin [6], or generally ’focus wave modes’ [7], for the waves which remain focused in ξ− coordinate. Although the plane-wave solutions exp(ı(kx − ωt)) of equation (1) have laid the basis of quantum electrodynamics [8, 9], the packet-like solutions of type (2) were considered only as a special case of classical electrodynam- ics. Their importance has raised after the seminal paper of Brittingham [7], who has proved the (2)-type solutions – the focus wave modes – (i) satisfy the homogeneous Maxwell’s equations, (ii) are continuous and non-singular, (iii) have a three-dimensional pulse structure, (iv) be non-dispersive for all time, (v) move at light velocity in straight lines. This stimulated a series of work indented for practical application of localized electromagnetic pulses for long-range energy transfer without dispersion, electromagnetic and acoustic bullets, Gaussian wave beams in optics and geophysics, optical information processing in microcavities [10, 11, 12, 13]. The application of the Bateman-Hillion-like solutions or weakly localized wave packets to quantum field theory was restricted to the formal studies of the localized solutions of the Klein-Gordon and the Dirac equations [14, 15]. In contrast to the solitons, the solutions localized due to nonlinearity, which have long been studied in quantum field theory, see e.g.[16], neither the Bateman-Hillion solutions of wave equations nor the Moses-Prosser wave bullets have been ever considered as operator-valued functions and have been ever subjected to canonical commutation relations. In the present paper we consider the nondispersive solutions of wave equa- tion, as particle-like solutions of the field equations, describing a quantum particles subjected to canonical commutation relations. This quantization condition results in certain restrictions on the amplitude and the width of the pulse-wave which makes it into quantum particle. In the next sections we will consider this problem for free scalar field theory models in 1 + 1 and 3 + 1 dimensions. 2 2 d=2 The nondispersive localized wave solutions in three plus one dimensions are generalizations of the traveling wave solutions of the wave equation in one plus one dimension: (∂z22 − ∂t22 )u(z, t) = 0, (4) which is equivalent to the equation ∂ξ2+ ξ− u(ξ+ , ξ− ) = 0. The solution of the equation (4) is a superposition of two independent solutions traveling right and left along the z axis: u = f (ξ− ) + g(ξ+ ), where f and g are arbitrary functions. In quantum field theory the field φ, which satisfies the massless field equation (4), is considered as an operator-valued function φ = φ(z, t). Using the Fourier transform the field φ can be casted as a sum of the positive and the negative frequency components Z ± φ (x) = e±ıkx δ(k 2 − m2 )φ(±k)dk. k0 >0 In case of massless field in two dimensions, x = (z, t), this gives dk  −ıωk t+ıkz Z û(k) + eıωk t−ıkz û† (k) ,  φ(z, t) = e (5) 2π2ωk dkdω where the integration over (2π) 2 is made into one dimensional integration dk 2π2ωk using the mass-shell delta-function δ(ω 2 −k 2 ), which results in ωk = |k|. The operators û(k) and û† (k) are referred to as the annihilation and the cre- ation operators for the quanta with momentum k. They satisfy commutation relations [û(k), û† (k ′ )] = 2π2ωk δ(k − k ′ ). (6) The equation (5) is the basis of field quantization. However, its physical interpretation leads to counterintuitive result: if a photon is described by plane wave (5), then the absorption of photon by photographic plate should expose the whole plate: for the plane wave is present everywhere. In reality the exposure is very local. This prompts us to use some localized function, travelling at a speed of light, instead of plane waves. Same as for classical 3 c-valued fields the localization is achieved by substituting plane wave by the Fourier image of some localized function. This leads to the operator-valued solution of the massless field equation dk h −ıωk t+ıkz Z i ıωk t−ıkz ∗ † φ(z, t) = e c(k)û(k) + e c (k)û (k) , (7) 2π2ωk here we assume c∗ (k) = c(−k). The canonical momentum, conjugated to the field density φ(z, t) is π(z, t) = ∂φ ∂t : ı dk h −ıωk t+ıkz Z i π(z, t) = − e c(k)û(k) − eıωk t−ıkz c∗ (k)û† (k) . (8) 2 2π Since we consider the wave packet φ as a quantum particle we can intro- duce the operator of the ”mean field coordinate” Q̂(t) and the total momen- tum P̂ (t) of the wave packet 1 Z Z Q̂(t) = φ(z, t)dz, P̂ (t) = π(z ′ , t)dz ′ , V where V is the volume occupied by the field. Using the commutation relations (6) for the Fourier modes, we get the commutator ı dk Z ′ [Q̂, P̂ ] = dzdz ′ eık(z−z ) |c(k)|2 . V 2π Thus to ensure canonical commutation relations for the wave packet [Q̂, P̂ ] = ı (9) we need to fulfil the constraint dzdz ′ Z Λ(z − z ′ ) = 1, V where dk Z Λ(z) = eıkz |c(k)|2 . 2π 2 2 Using the symmetry |c(k)| = |c(−k)| the constraint that ensures canonical commutation relations (9) for the operator-valued wave packet (7) can be written in the form Z ∞ Z ∞ dk 2 dz cos(kz)|c(k)|2 = 1. (10) 0 −∞ 2π 2 σ 2 /2 For the Gaussian wave packet with c(k) = Ae−k this leads to the con- straint A2 = 1 independent on σ. 4 3 d=4 Without loss of generality we can consider the Klein-Gordon equation in 3+1 dimensions (∂x22 + ∂y22 + ∂z22 − ∂t22 − m2 )u(x, y, z, t) = 0 (11) with m set to zero for the case of massless field. The Fourier image of the localized solution of the Klein-Gordon equation [15] can be written as √   B ˜ k+ kz + ωk g( 2k+ ) = δ f √ δ(k 2 − m2 ), k+ = √ , (12) k+ 2 2 where B is a constant, δ = 0, 12 – for massless, and massive field, respectively, see B, C. This leads to the operator-valued solution of the field equation d3 k h −ıωk t+ıkx Z φ(x, t) = e g(kz + ωk )û(k) (2π)3 2ωk i + eıωk t−ıkx g(−kz − ωk )û† (k) , (13) where the annihilation and the creation operators satisfy the commutation relations √ [û(k), û† (k′ )] = (2π)3 · 2ωk · δ(k − k′ ), ωk = k2 + m2 . (14) The canonical momentum, conjugated to the field density φ(x, t), is π(x, t) = ∂φ ∂t : ı d3 k h −ıωk t+ıkx Z π(x, t) = − e g(kz + ωk )û(k) 2 (2π)3 i ıωk t−ıkx † − e g(−kz − ωk )û (k) . (15) The ”mean field coordinate” Q̂(t) and the total momentum P̂ (t) of the wave packet are: 1 Z Z Q̂(t) = φ(x, t)d x, P̂ (t) = π(x′ , t)d3 x′ . 3 V The constraint (9) results in d3 k Z 3 ıkx 1 = d xe g(kz + ωk )g(−kz − ωk ) (2π)3 = g(kz + ωk )g(−kz − ωk )|k=0 . (16) 5  k2 σ 2 For Gaussian packet g(k) = Ae− 2 this gives the normalization constraint 2 m2 A2 e−σ = 1. (17) 4 Conclusions The idea of construction of nondispersive wave packets, which follow a tra- jectory of classical particle, from the coherent superposition of harmonic oscillators can be traced back to Schrödinger [17]. In classical scales such packets of different form can be created by combination of electromagnetic pulses of equispaced frequencies. In quantum physics, where the energy lev- els of atoms, used for laser pulse generation, are not equispaced, the known way to suppress wave packet dispersion is to use external electromagnetic fields [18, 19]. The most intensive experimental studies have been performed in creation Trojan wave packets – nondispersive wave packets of electron density on circular orbits [20]. The creation of nondispersive packet on a circular orbit is achieved by using circularly polarized laser beams and is a technically sophisticated, however it seems that a simpler problem of creating a nondispersive wave packet moving linearly, which was studied in classical electrodynamics since Brittingham [7] and is implemented experimentally, was not implemented in quantum case, regardless an evident advantage of no need of the external field for suppressing dispersion. In present paper, using a simple scalar model, we constructed the quantization condition for such packets in purely relativistic case. If such mechanism exists in quantum electrodynamics, it may be related to the coherent electromagnetic energy transfer at mesoscopic scales, along with excitons and other nonrelativistic mechanisms, see e.g. [21]. In future we hope to apply the localized wave packets, subjected to quantization constraints described in this paper, to the scattering problems of localized quantum particles described by quan- tum field theory methods [22]. In particular the existence of the localized focus wave modes [7] in classical electrodynamics may be of interest for the quantum theory of gauge fields [23]. Acknowledgement The research was supported in part by the Program of Creation and Devel- opment of the National University of Science and Technology ”MISiS” and 6 by the RFBR project 11-02-00604a. The authors have benifited from the discussions of these results with Drs. A.L.Kataev, A.P.Kiselev, M.V.Perel and O.V.Teryaev. References [1] H. Bateman, The conformal transformations of a space of four dimen- sions and their applications to geometrical optics, Proc. Lond. Math. Soc. 7 (1909) 70–89. [2] H. 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A 786 (2007) 107–130. 8 A The Bateman-Hillion solution of wave equa- tion The Bateman-Hillion solution of the wave equation (1) can be obtained by the change of variable (§4.2.1 of [3]): Z ∞ u= ûeıαa da, (18) −∞ where û = û(x, y, α, β), α = z − t, β = z + t. The new function û satisfies the parabolic equation ∂ 2 û ∂ 2 û ∂ û 2 + 2 + 4ıa = 0, ∂x ∂y ∂β which has the solution û = β −1 exp(ıa(x2 + y 2)/β)fˆ(a), (19) where fˆ(a) is arbitrary function. The Bateman-Hillion solution is obtained by substitution of (19) into (18). B Fourier image of the massless field solution To evaluate the Fourier image of the solution (2) of wave equation (1) we use the light-cone variables z±t kz ± ω ξ± = √ , k± = √ . 2 2 In these variables f (θ) −ı(kx x+ky y+k+ ξ− +k− ξ+ ) Z φ̃(kx , ky , k+ , k− ) = dxdydξ−dξ+ √ e 2ξ+ x2 + y 2
i   1 f (θ) k+ Z h = dθdξ+ dxdy exp −ı kx x + ky y + k− ξ+ + √ θ − √ . 2 ξ+ 2 2ξ+ 9 We have the product of two identical integrals in x and y transversal coor- dinates Z ∞ Z ∞ ıkx2 −ıkx x 2 ıπ k⊥2 e dx eıky −ıky y dy = e−ı 4k (20) −∞ −∞ k k+ with k = 2ξ + . The remaining integration in ξ+ is performed by the change of variable r = ξ+ /k+ . This gives    2  2˜ k+ k⊥ φ̃(kx , ky , k+ , k− ) = 2ıπ f √ δ + k+ k− (21) 2 2 C Fourier image of the massive solution To evaluate the Fourier image of the localized solution for the Klein-Gordon equation we integrate the massless solution in 5d over the extra variable z ′ [15]: Z ′ φ̃m (kx , ky , k+ , k− ) = e−ı(kx x+ky y+mz +k+ ξ− +k− ξ+ ) × f (θ) Z ′ × √ dxdydz dξ− dξ+ = e−ı(kx x+ky y+mz +k− ξ+ ) × ′ ( 2ξ+ )3/2   k 2 +y 2 +z ′ 2 ı √+ θ− x √ f (θ) dθ ×e 2 2ξ+ √ √ dξ+ dxdydz ′ . ( 2ξ+ )3/2 2 The product of three identical integrals in x, y, z ′ , cf. Eq. 20, is equal to 3/2 (2π)3/2  ξ+ − 2ı (k⊥ ξ 2 +m2 ) + √ e k + . ı k+ ξ+ Introducing the new variable r = k+ and integrating over it we get   f˜ k √+(2π)5/2  k 2 + m2 2  ⊥ φ̃m (kx , ky , k+ , k− ) = p δ + k+ k− . (22) 2ık+ 23/4 2 10