Quantum Field Theory of Particle Orbits with Large Fluctuations

Hagen Kleinert∗ Institut f¨ur Theoretische Physik, Freie Universit¨at Berlin, 14195 Berlin, Germany and ICRANeT Piazzale della Repubblica, 10 -65122, Pescara, Italy

We show that an ensemble of particle orbits with large fluctuations around their classical paths are described by quantum field theory in the strong-coupling limit.

PACS numbers: 95.35+d.04.60,04.20C,04.90

Grand-canonical ensembles of gaussian random walks Thus gaussian random walks are a natural starting can be described by quantum field theory [1–5]. Indeed, point for many stochastic processes. For instance, they the relativistic scalar free-particle propagator of mass m form the basis of the most important tool in the the- in D-dimensional euclidean momentum space p has the ory of financial markets, the Black-Scholes option price form theory [10] (Nobel Prize 1997), by which a portfolio of ∞ assets is intended to grow steadily via hedging. In fact, 1 − 2 − 2 G(p)= = ds e sm e sp . (1) the famous central-limit theorem permits us to prove that p2 + m2 Z0 many independent random movements of finite variance − 2 always pile up to display a gaussian distribution [11]. The Fourier transform of e sp is the distribution of gaussian random walks of length s in D euclidean di- However, since the last crash and the ensuing financial crisis, it has become clear that realistic stochastic distri- mensions butions in nature belong to a more general universality − − 2 P (s, x)=(4πs) D/2e x /4s, (2) class, the so-called L´evy distribution. While gaussian distributions alway arise from a pile up of arbitrary fi- so that the propagator (1) is a superposition of gaussian nite steps, L´evy distribution emerge if these step have an 2 random walks whose lenghts are distributed like e−sm : infinite variance. They describe that rare events, which ∞ initiate crashes, are much more frequent than in gaussian − 2 G(x)= ds e sm P (s, x). (3) distributions. Such tail-events also occur in the distribu- Z0 tion of earthquakes, with catastrophic consequences [12]. 1+α If δ-function-like interaction are added to the random These are events in the so-called power tails ∝ 1/|x| walks, their statistical mechanics is described by the eu- of the distributions, with α ≤ 2, whose description re- clidean action quires a fractional Fokker-Planck equation [13] 2 α/2 gc [∂ + (pˆ ) ]P (s, x)=0. (5) A = d3x φ(x)(pˆ2 + m2)φ(x)+ φ4(x) . (4) s α 4! Z h i In the limit α → 2, the L´evy distributions reduce to gaus- The evaluation of the partition function based on this ac- sian distributions. In general the solution for Pα(s, x) is tion is usually done approximately by perturbation the- [14] ory order by order in the coupling strength g. The results are divergent power series in g from which the physical (1,1);(1,1) 1 |x|α properties must be extracted by renormalization. There x 2,1 Pα(s, )= D/2 D/2 H2,3 α , (6) π |x| 2 s D α α exists ao-called critical dimension, here Dc = 4, where (1,1),( 2 , 2 );(1, 2 )! the action is scale invariant. If the physical dimension 2,1 D lies slightly below Dc, say at D = Dc − ǫ, the the- where H2,3 is a Fox H-function [15]. In the limits γ =0 ory appears to be still scale invariant, with fields ac- and α = 2, this reduces to the standard quantum me- quiring an anomalous dimension [φ] = 1 − η/2, and chanical gaussian expression (2). For γ = 0, α = 1, the the interaction becoming effectively φδ+1(x), where δ = result is (D +2 − η)/(D − 2+ η). This is known as the renormal- − s s2 (1/2 D/2,1) ization group approach to critical phenomena [6]. Al- P (s, x)= H1,1 , α π(D+1)/2|x|D+1 1,1 |x|2 ternatively, it can be formulated as to strong-coupling (0,1) ! limit of the quantum field theory [7, 8]. This theory has exlpained critical phenomena with high accuracy. In par- which is simply the Cauchy distribution function ticular it has predicted the value of the critical exponent [Γ(D/2+1/2)/π(D+1)/2]s/(s2 + |x|2)D/2+1/2. −α α in the singularity of the specific heat C ∝ |T −Tc| to From what we explained above it is clear that these be α ≈−0.0127 in excellent agreement with the satellite nongaussian L´evy walks are contained in the theory based measurement αexp ≈−0.0129 [9]. on the action (4) in the strong-coupling limit. Using the 2 results of Ref. [7], and the textbook [8] the effective action (McGraw-Hill, New York, 1965). of this strong-coupling limit reads [4] H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, − g World Scientific, Singapore, 2006. Aeff = d3x φ(x)(pˆ2)1 η/2φ(x)+ c |φ|δ+1(x) , (7) 4! [5] H. Kleinert, Gauge Fields in Condensed Matter, Vol. I, Z h i World Scientific, Singapore, 1989. where 1 − η/2 ≈ 0.985, δ = (D +2 − η)/(D − 2+ η) ≈ [6] J. Zinn-Justin, Quantum Field Theory and Critical Phe- 5(1 − 6η/5) ≈ 4.83, and g ≈ 1.4. nomena, Clarendon, Oxford, 1996. c [7] H. Kleinert, Phys. Rev. D 57, 2264 (1998). The original field theory based on the action (4) is [8] H. Kleinert and V. Schulte-Frohlinde, Critical Proper- 4 extremal for fields that satisfy the typical Schr¨odinger ties of φ -Theories, World Scientific, Singapore 2001 boundary conditions that φ(x) must be single valued. (klnrt.de/b8). In the quantum theory this is the origin of all quantum [9] J.A. Lipa, J.A. Nissen, D. A. Stricker, D.R. Swanson, numbers. In the strong-coupling limit, on the other hand, and T.C.P. Chui, Phys. Rev. B 68, 174518 (2003); M. these conditions are no longer satisfied. Instead, the field Barmatz, I. Hahn J.A. Lipa, R.V. Duncan, Rev. Mod. 79 is multivalued [16], and by proposing such wave funtions Phys. , 1 (2007). Also see picture on the titlepage of the textbook [8]. for strongly interacting electron systems in two dimen- [10] F. Black and M. Scholes, J. Pol. Economy 81, 637 (1973). sions has earned Laughlin [17] the Nobel prize in 1998. [11] W. Feller An Introduction to Probability Theory and The strong-coupling limit may also be needed for an un- Its Applications vol. 2, Wiley, New York, 1991; J.-P. derstanding of high-Tc conductivity, which is often asso- Bouchaud and M. Potters, Theory of Financial Risks, ciated with a non-Fermi liquid behavior of the electrons From Statistical Physics to Risk Management, Cam- [18]. bridge U. Press, 2000. See also Ch. 20 in [4]. [12] P. Bhattacharyya and B.K. Chakrabarti (Eds.) Modelling Critical and Catastrophic Phenomena in Geoscience: A Acknowledgment: I am grateful to P. Jizba, N. Laskin, Statistical Physics Approach, Lecture Notes in Physics and A. Pelster for useful comments. 705 (Springer, Berlin, 2006); P. Bhattacharyya, A. Chat- terjee, B.K. Chakrabarti, Physica A 381, 377 (2007) (physics/0510038) . [13] See Subsec. 20.1.2 of Ref. [4]. 46 ∗ [14] Jun-Sheng Duan, J. 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