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Quantum Field Theory of Black-Swan Events

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Abstract

Free and weakly interacting particles are described by a second-quantized nonlinear Schrödinger equation, or relativistic versions of it. They describe Gaussian random walks with collisions. By contrast, the fields of strongly interacting particles are governed by effective actions, whose extremum yields fractional field equations. Their particle orbits perform universal Lévy walks with heavy tails, in which rare events are much more frequent than in Gaussian random walks. Such rare events are observed in exceptionally strong windgusts, monster or rogue waves, earthquakes, and financial crashes. While earthquakes may destroy entire cities, the latter have the potential of devastating entire economies.

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Notes

  1. See http://en.wikipedia.org/wiki/Black_swan_theory.

  2. See http://www.nytimes.com/2007/04/22/books/chapters/0422-1st-tale.html?pagewanted=all.

  3. Contrasting the 1994 book by R.J. Herrnstein and C. Murray entitled “The Bell Curve” http://de.wikipedia.org/wiki/The_Bell_Curve.

  4. See http://en.wikipedia.org/wiki/2010_Flash_Crash.

  5. A travelling pedestrian salesman is a Gaussian random walker, as a jetsetter he becomes a Lévy random walker.

  6. In two dimensions, a sequence of critical exponents have been tabulated in [23].

  7. For the so-called Riesz fractional derivative see [28, 29]. For the so-called Weyl derivative see [30]: \(\hat{p}_{4}^{1-\gamma}f(t)\equiv\varGamma^{-1}[1-\gamma]\int_{t}^{\infty}dt'(t-t'+i\epsilon)^{-2+\gamma} f(t')\).

  8. The relevant functional matrix is \(\langle\mathbf{x} |(-\boldsymbol{\nabla}^{2})^{\lambda/2}|\mathbf {x} '\rangle=\varGamma[-\lambda/2]^{-1} \int d\sigma\, \sigma^{-\lambda/2-1} {(4\pi\sigma)^{-D/2}}e^{R^{2}/4\sigma} = {}^{D} c_{\lambda}R^{-\lambda-D}\), where D c λ =2λ Γ((D+λ)/2)/π D/2 Γ(−λ/2), and R≡|x−x′|. If λ is close to an even integer, it needs a small positive shift λ→λ +≡λ+ϵ and we can replace ϵR ϵ−1/2 by \(\delta(R)=S_{D}R^{D-1}\delta^{(D)}({\bf R})\). For A>0 we have \(|\mathbf{x}'|^{-A}= {}^{D}c_{\lambda_{A}}^{-1}\langle\mathbf{x}' | (-\boldsymbol{\nabla}^{2})^{\lambda_{A}/2}|\mathbf{0} \rangle\) with λ A ≡A−D, so that we find \(\int d^{D}x'\langle\mathbf{x} |(- \boldsymbol{\nabla}^{2})^{\lambda/2}|\mathbf {x}' \rangle|\mathbf{x}'|^{-A}= {}^{D}c_{\lambda_{A}}^{-1}\langle\mathbf{x}|(- \boldsymbol{\nabla}^{2})^{(\lambda+A-D)/2} |\mathbf{0}\rangle= {}^{D}c_{\lambda+A-D}{}^{D}c^{-1}_{\lambda_{A}}|\mathbf{x}|^{-A-\lambda}\).

  9. This technique is explained in Chaps. 12 and 19 of Ref. [4]. The pseudotime s resembles the so-called Schwinger proper time used in relativistic physics.

  10. There should be no danger of confusing the fluctuating noise variable η in this equation with the constant critical exponent η in (9).

  11. See Eq. (29.165) in Ref. [4].

  12. For a pedagogical discussion see [44].

  13. We ignore the problem that near the resonance it is hard to confine the particles to the trap. See [45].

  14. For the universal use of this functional see [46].

  15. See Eq. (1.35) in Ref. [26].

  16. The decimal numbers are from seven-loop calculation in D=3 dimensions in Table 20.2 of Ref. [26].

  17. See Eq. (10.191) in Ref. [26] and expand \(f(t/M^{1/\beta})\sim\tilde{f} (\xi \varPhi^{2/(D-2+\eta)}) \) like \(\tilde{f}(x)=1+cx^{-\omega}+\cdots{}\).

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Acknowledgements

I am grateful to P. Jizba, N. Laskin, M. Lewenstein, A. Pelster, and M. Zwierlein for useful comments, and to Fabio Scardigli, Luca Di Fiore, and Matteo Nespoli for their hospitality in Taipeh during their Workshop Horizons of Quantum Physics (http://www.quantumhorizons.org).

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    H. Kleinert

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Appendix

Appendix

The lowest-order critical exponents can be extracted directly from the one-loop-corrected inverse Green function G −1(E,p) in D=2+ϵ dimensions after a minimal subtraction of the 1/ϵ -pole at [55]:

$$\begin{aligned} E-\mathbf{p}^2+ a \biggl( \frac{1}{3}\mathbf{p}^2-E \biggr)^{D-1}. \end{aligned}$$
(33)

For p=0, this has a power −(−E)1−aϵ, so that γ=aϵ. For E=0, on the other hand, we obtain (−p 2)1−aϵ/3, so that (1−γ)/z−1≈γ/3.

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Kleinert, H. Quantum Field Theory of Black-Swan Events. Found Phys 44, 546–556 (2014). https://doi.org/10.1007/s10701-013-9749-x

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