D. Dimitrijević, G. S. Djordjević and Lj. Nešić. Abstract

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1 Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: htt:// Filomat 21:2 2007), ON GREEN FUNCTION FOR THE FREE PARTICLE D. Dimitrijević, G. S. Djordjević and Lj. Nešić Abstract Alication of the -adic analysis and Green function in relativistic quantum theory is considered. Integrability of the free relativistic article model is discussed. A convenient gauge is fixed. The Green function, as a corresonding roagator, are calculated in -dimensional -adic Minkowski sace. For this model, detailed calculation is erformed for all three different cases: 1mod), 3mod) and = 2. 1 Introduction -Adic numbers, introduced by Kurt Hansel at the end of 19 th century, and -adic analysis, have a quite long history and reach bibliograhy devoted to investigation and alication of this field of mathematics. During the last decades this, at least at the beginning retty exotic, art of ure number theory has found its lace in surrisingly many different fields of science and technology. Maybe the best guide to a reader interested in this toic would be the monograhy [1], even more than ten years old one. In this aer we are mostly interested in alication of -adic mathematics in theoretical hysics. More recisely, our goal is to calculate Green function for -adic free relativistic article FRP), motivated by, so called, Hartle-Hawking HH) aroach to quantum cosmology. Namely, most of the -adic hysicists believe that the most imortant alications of -adic analysis concerns the Planck scale hysics and non-archimedean structure at very small sace-time distances Mathematics Subject Classification. 12H25, 3B27, 11E95, 83F05. Key words and hrases. -Adic Nubers, Free Relativistic Particle, Green Function. Received: July 10, 2007

2 252 D. Dimitrijević, G. S. Djordjević and Lj. Nešić Foundations of -adic quantum mechanics [2] and quantum cosmology [3] were worthwhile achievements in attemt of deeer understanding the fundamental rocesses at fundamental-planck distance. The adelic connection between ordinary and -adic quantum theory has been roosed for strings [], quantum mechanics [5] and quantum cosmology [6]. Dynamics of some simle non-relativistic models is considered in a few aers: free article and article in constant external field [7], harmonic oscillator with constant [5] and time-deendent frequency [8]. Also, de Sitter and other interesting cosmological models have been considered [6]. In all that cases, functional integral aroach and the corresonding Gauss integrals have been used [9]. As usually, while develoing a theory, as -adic) quantum cosmology, it is very convenient to study some technically simler cases. The FRP is formally used as one very instructive and integrable model. Although usual action for the relativistic article is nonlinear, as in the real case [10], that system may be treated like a system with quadratic constraint. This aer is concerned about the -adic asects of such system. For a bit different and interesting aroaches to -adic Green function and -adic free article see [11]. The aer is organized as follows. After the Introduction we briefly recaitulate basic facts about -adics. Section 3 is devoted to classical and quantum FRP over real numbers. In Section -adic roagator Green function) for the FRP is introduced and studied in details. The main results are formulated in the form of three roositions. 2 -Adic Mathematics In considering -adic numbers it is suitable to begin with the field of rational numbers Q, as the simlest field of numbers of characteristics 0. Any rational number can be exanded into one of two forms of infinite series [1] or i + i=m a i 10 i, a i = 0,..., 9, 1) b i i, b i = 0,..., 1, 2) where n and m are some integers, and is a rime number. The above series 1) and 2) are convergent in resect to standard absolute value and -adic norm, resectively. From hysical oint of view, it is imortant that Q contains all results of hysical measurements.

3 On Green function for the free article 253 The comletion of the field of rational numbers Q with resect to the standard norm leads to the field of real numbers R Q exression 1)). According to the Ostrowski theorem, besides absolute value and -adic norms there are no other non-equivalent and nontrivial norms on Q. The comletion of Q with resect to the -adic norm leads to the -adic number field Q exression 2)). The -adic norm is nonarchimedean ultrametric) one, i.e. x + y max x, y ). Generally seaking, there are two analyses over Q, ϕ : Q Q and ψ : Q C. In the case of maing ψ : Q C, there is no standard derivative, and some tyes of seudodifferential oerators have been introduced [1, 12]. However, it turns out that there is a well defined integral with the Haar measure. In the following, we will use the integral formula χ ξy)dy = S γ γ 1 1), if ξ γ γ 1, if ξ = γ+1 0, if ξ γ+2. where χ u) = ex2πiu ) is a -adic additive character. Here, u denotes the fractional art of u Q. Recall that in the real case one has χ x) = ex 2πix). Also, let us state the notation for the ring of -adic integers, -adic circle and disc, resectively: 3) Z = {x Q : x 1} ) S γ a) = {x Q : x a = γ }, 5) B γ a) = {x Q : x a γ }. 6) Even very interesting toics, adeles [13] and their alication to hysics [6, 9] will not be considered here. 3 Classical and Quantum Free Relativistic Particle Usual action for the FRP, in the flat configuration sace, is S = mc τ 2 τ 1 dτ ẋµẋ ν η µν 7) a dot denotes a derivative of -vector x µ with resect to τ, η µν is the usual Minkowski metric with signature, +, +, +) and µ, ν {0, 1, 2, 3}).

4 25 D. Dimitrijević, G. S. Djordjević and Lj. Nešić Seed of light in vacuum is denoted by c and m denotes article mass. The quantity τ arameterizes the wordline, with the boundary condition xτ 1 ) = x 1, xτ 2 ) = x 2. 8) In using action 7) for the ath integral quantization we meet with the roblems [10]. First of all, there is the ractical roblem of evaluating a ath integral which is not Gaussian. However, FRP may be treated as a system with constraint [10] H = η µν k µ k ν + m 2 c 2 = k 2 + m 2 c 2 = 0, k µ is the -momentum, k µ = E/c, k), E is a energy of article and k is a 3-momentum). This Hamiltonian after introduction Lagrange multilier N, is getting the canonical form H c = Nk 2 + m 2 c 2 ). 9) The canonical Hamiltonian 9) leads to the Lagrangian where ẋ α = H c k = 2Nη α µα k µ, k α = gain L x S = L = ẋ α k α H c, 10) τ 2 τ 1 dτ ẋµη µα 2N, k α = ẋα 2N ). For the action we [ ] ẋ2 N m2 c 2 N. 11) The classical trajectory, as a solution of Euler-Lagrange equation d L dt = 0, with the boundary condition 8) leads to the classical action ẋ x = x 2 x 1 τ 2 τ 1 τ + x 1τ 2 x 2 τ 1 τ 2 τ 1, 12) S = x 2 x 1 ) 2 Nτ 2 τ 1 ) m2 c 2 Nτ 2 τ 1 ). 13) In quantization by the functional Feynman) aroach, first of all, we have to calculate the corresonding kernel of the oerator of evolution U for the considered system. For the class of rearametrization-invariant systems, in the standard case, it can be resented as a functional art [10] of the Gx j x i ) = dnτ j τ i ) DkDxe i τ j τ i τ[kẋ NH]. 1)

5 On Green function for the free article 255 After the usual redefinition T = Nτ j τ i ) we gain Gx j x i ) = dt Kx j, T x i, 0) 15) where Gx j x i ) is the roagator or Green function and Kx j, T x i, 0) is above mentioned kernel. Let us note that the Green function Gx j x i ) for the x j = x and x i = 0, in the quantum cosmology is the HH wave function of the universe see [3]). The main results concerning the T integration in the standard case are resented in the Ref Adic Proagator Green function) In the -adic case, the above exressions from 7) to 15) are valid too. Since classical action 13) may be resented in the form S = x0 2 x0 1 )2 T m2 c 2 T + x1 2 x1 1 )2 T m2 c 2 T + x2 2 x2 1 )2 T m2 c 2 T + x3 2 x3 1 )2 T corresonding -adic kernel is [9] m2 c 2 T = S 0 + S 1 + S 2 + S 3 16) K x j, T j x i, T i ) = λ2 ht j T i )) 2hT j T i ) 2 χ 1 h x j x i ) 2 T j T i ) + m2 c 2 ) h T j T i ) 17) where λ u) is an arithmetic comlex valued function [1]. Remind that K x j, T j x i, T i ) is the kernel of the -adic evolution oerator which is defined by its action on the wave function ψx j ) U T )ψ x j ) = K x j, T j x i, T i )ψ x i )dx i. Q 18) Fixing the gauge Ṅ = 0, we will calculate -adic roagator as G x j x i ) = ht 1 dt K x j, T x i, 0), 19),

6 256 D. Dimitrijević, G. S. Djordjević and Lj. Nešić i.e., G x j x i ) = ht 1 dt λ2 ht ) 2hT 2 χ x j x i ) 2 ht + m2 c 2 ) h T. 20) The integration is erformed over the -adic ball, ht 1. The nature of integration force us to divide integration in three different cases: 1) 1mod), λ 2 a) = 1, 3mod), λ 2 a) = ±1, 3) = 2, λ 2 a) = 1) a 1 i. PROPOSITION 1. For 1mod), i.e. λ 2 a) = 1 corresonding -adic Green function reads G x j x i ) = { 1 h, if x j x i ) 2 0, if x j x i ) 2 >. 21) To rove the above roosition, we introduce following changes ht = z dz = h dt ; z = 1 y dz = dy y 2 dt = 1 dy h y 2. 22) Then, the Green function can be evaluated as with q 2 = x j x i ) 2 ) G x j x i ) = 1 h S γ χ q 2 y)dy. 23) Let q 2 2, than q 2 γ+2, γ 0, γ N + {0}. In resect to the third line in 3), it follows S γ χ q 2 y)dy = 0 G x j x i ) = 0, for x j x i ) ) Let now q 2 = δ, δ Z \N δ χ q 2 y)dy = γ 1 1 ) δ = 1. 25) S γ

7 On Green function for the free article 257 That means G x j x i ) = 1 1, h for x j x i ) 2 δ, δ Z \N. 26) On the light cone e.g. q 2 = 0 1dy = γ 1 1 ) = ) + 2 1)... ) = 1. S γ 27) All these results, in fact, are reresented in the comact form 21). PROPOSITION 2. For 3mod), i.e, λ 2 a) = ±1, corresonding -adic Green function reads G x j x i )= 1 h , if x j x i ) 2 2, 1, if x j x i ) 2 =, 2 1 ), ) if x j x i ) 2 = 1, + 1)δ 2 δ +1, if x j x i ) 2 = δ,δ <0,, if x j x i ) 2 = 0. 28) 1 The most imortant difference in resect to 1mod) is λ 2 ht ) = λ 2 z) = λ 2 y) = 1) γ, y = γ. 29) Therefore, we calculate Green function as sum of two arts G x j x i ) = h 1 + 1) 2γ dyχ q 2 y ) + + S 2γ Let q 2 = δ, for δ 2. Follows 1) 2γ+1 S 2γ+1 dyχ q 2 y ). 30) G x j x i ) = 0 for x j x i. 31) For δ = 1, G x j x i ) = 1 h, and for δ = 0, G x j x i ) = 1 h 2 1 ). For odd δ, δ < 0 G x j x i ) = h 1 + 1) γ S γ dyχ q 2 y ) ) 1 = h 1 + 1) 1 2 δ )

8 258 D. Dimitrijević, G. S. Djordjević and Lj. Nešić For even δ, δ < 0, one has G x j x i ) = h 1 δ 1) γ γ 1 1 ) δ 1) δ+1 33) ) 1 = h 1 + 1) δ +1. 3) Finally, on the light cone we find 1)2 + G x j x i ) = h 1 2γ =. 35) Collecting all above results we get form given in the PROPOSITION 2. PROPOSITION 3. In the case = 2 unique even rime number), i.e λ 2 a) = 1) a 1 i, Green function for FRP has the following form G 2 x j x i ) = { 0, if q 2 = 0 or x j x i ) 2 2 1, 1) q2 1 hx j x i ) 2 2, if q 2 0 and x j x i ) ) To rove above roosition we start with G 2 x j x i ) = 2h 1 2 dyλ 2 2y)χ q 2 y ). 37) For y = 2 γ 1 + 2y ), γ N, G 2 x j x i ) = 2h 1 2 i γ=1 S γ,y1 =0 y 2 2 λ 2 2 y) = 1)y 1 i, and dyχ q 2 y) i By the introducing I 0 and I 1 [1] I 0 = dyχ q 2 y), and I 1 = S γ,y1 =1 dyχ q 2 y). 38) dyχ q 2 y). 39) S γ,y1 =0 S γ,y1 =1 we finaly calculate 1)If q 2 = 0, I 0 = I 1 G 2 x j x i ) = 0.

9 On Green function for the free article 259 2)If q 2 2 = 2 δ, δ Z, and y 2 2 δ, I 0 = I 1. 3)If q 2 2 = 2 δ, and y 2 = 2 δ+1, {q 2 y} 2 = 1 2, I 0 = I 1. )If q 2 2 = 2 δ, and y 2 = 1, {q 2 y} 2 δ 2 2 = y 1+q1 2), I 0 = i 1)q 2 1 = I 2 δ 1. 5)If q 2 2 = 2 δ, and y 2 2 δ+3, I 0 = I 1. We have to kee in mind that the range of integration is γ 1, so for unique nonsingular case ) we have δ + 2 1, i.e. δ 1. It corresonds q 2 2 = x j x i ) x j x i ) Comaring 1) to 5) with 36) we finish the roof of the PROPOSITION 3. Acknowledgement The research of authors was artially suorted by the Serbian Ministry of Science and Environment Protection, Projects No. 101 and We are thankful to B. Dragovich for useful discussion. A art of this work was comleted during a stay of G. Djordjevic at LMU-Munich suorted by DFG, and a stay at University of Freiburg suorted by DAAD. References [1] V. S. Vladimirov, I. V. Volovich and E. I. Zelenov, -Adic Analysis and Mathematical Physics World Scientific, Singaore, 199). [2] V. S. Vladimirov and I. V. Volovich, A vacuum state in -adic quantum mechanics, Phys. Lett. B217, ). [3] I. Ya. Aref eva, B. Dragovich, P. Framton and I. V. Volovich, Wave function of the universe and -adic gravity, Int. J. Mod. Phys. A6, ). [] P. G. O. Freund and E. Witten, Adelic string amlitudes, Phys. Lett. B199, ). [5] B. Dragovich, Adelic harmonic oscillator, Int. J. Mod. Phys. A10, ). [6] G. S. Djordjevic, B. Dragovich, Lj. Nesic, I.V.Volovich, -Adic and adelic minisuersace quantum cosmology, Int. J. Mod. Phys. A17, ).

10 260 D. Dimitrijević, G. S. Djordjević and Lj. Nešić [7] G. S. Djordjevic and B. Dragovich, On -adic functional integration, Proc. of the II Mathematical Conference, Priština, Yugoslavia, ); G. Djordjević, On -Adic and Adelic Quantum Mechanics, University of Belgrade 1999) PhD Thesis in Serbian). [8] G. S. Djordjevic and B. Dragovich, -Adic and adelic harmonic oscillator with time-deendent frequency, Theor. Mat. Phys. 12, ). [9] G. Djordjevic, B. Dragovich and Lj. Nesic, Adelic ath integrals for quadratic actions, Infinite Dimensional Analysis, Quantum Probability and Related Toics, 6, ). [10] J. J. Halliwell, Derivation of the Wheeler-DeWitt equation from a ath integral for minisuersace models, Phys. Rev. D, ). [11] A. Kh. Bikulov, Investigation of -adic Green s function, Theor. Math. Phys. 3 Vol. 87, ); N. M. Chuong and N. Van Co, The Multidimensional -adic Green function, Proc. of the American Math. Society 3 Vol. 127, ); A. N. Kochubei, On -adic Green s functions, Theor. Math. Phys. 1 Vol. 96, ); A. Yu. Khrennikov, S. V. Kozyrev, Localization in sace for free article in ultrametric quantum mechanics, quant-h/ [12] D. Dimitrijevic, G. S. Djordjevic and B. Dragovich, On Schroedingertye equation on -adic saces, Bul. J. of Phys. 3 Vol. 27, ). [13] I. M. Gel fand, M. I. Graev and I. I. Piatetskii-Shairo, Reresentation Theory and Automorhic Functions Saunders, London, 1966). Deartment of Physics, Faculty of Science, P.O. Box 22, Niš Serbia gorandj@junis.ni.ac.yu