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Representations of the fractional d'Alembertian and initial conditions in fractional dynamics

Gianluca Calcagni, Giuseppe Nardelli

Abstract

We construct representations of complex powers of the d'Alembertian operator $\Box$ in Lorentzian signature and pinpoint one which is self-adjoint and suitable for classical and quantum fractional field theory. This self-adjoint fractional d'Alembertian is associated with complex-conjugate poles, which are removed from the physical spectrum via the Anselmi--Piva prescription. As an example of empty spectrum, we consider a purely fractional propagator and its Källén--Lehmann representation. Using a cleaned-up version of the diffusion method, we formulate and solve the problem of initial conditions of the classical dynamics with a standard plus a fractional d'Alembertian, showing that the number of initial conditions is two. We generalize this result to a much wider class of nonlocal theories and discuss its applications to quantum gravity.

Representations of the fractional d'Alembertian and initial conditions in fractional dynamics

Abstract

We construct representations of complex powers of the d'Alembertian operator in Lorentzian signature and pinpoint one which is self-adjoint and suitable for classical and quantum fractional field theory. This self-adjoint fractional d'Alembertian is associated with complex-conjugate poles, which are removed from the physical spectrum via the Anselmi--Piva prescription. As an example of empty spectrum, we consider a purely fractional propagator and its Källén--Lehmann representation. Using a cleaned-up version of the diffusion method, we formulate and solve the problem of initial conditions of the classical dynamics with a standard plus a fractional d'Alembertian, showing that the number of initial conditions is two. We generalize this result to a much wider class of nonlocal theories and discuss its applications to quantum gravity.

Paper Structure

This paper contains 32 sections, 1 theorem, 106 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Complex powers of positive operators
  3. Complex powers of the d'Alembertian
  4. Properties of the d'Alembertian
  5. Balakrishnan--Komatsu representation of Boxg
  6. Balakrishnan representation of Boxg
  7. Logarithmic representation of Boxg
  8. Self-adjoint Fresnel representation of |Box|g
  9. Self-adjoint Balakrishnan--Komatsu representation of |Box|g
  10. Spectrum of the purely fractional operator
  11. Propagator of (-Box)g
  12. Spectrum of (-Box)g
  13. Propagator of (Box2)g/2
  14. Spectrum of (Box2)g/2
  15. Comparison with the ultra-distributions approach
  16. ...and 17 more sections

Key Result

Theorem 1

The Balakrishnan--Komatsu representation (Bdot) of $|\Box|^\gamma$ holds on the space of ultra-distributions and one can restrict its domain on ${\rm D}(\boxdot^\gamma)=\zeta'$.

Figures (9)

  • Figure 1: Maximal contour $\Gamma$ of the Cauchy representation of the propagator (\ref{['Gz1']}) of the operator $(-\Box)^\gamma$. The contour encompasses the pole at $z=-k^2$. The half-lines ${\rm cut}^\pm$ run along the branch cut at $\text{Im}\,z=0$, $\text{Re}\,z>0$, which starts at the branch point $z=0$.
  • Figure 2: Contour $\Gamma=\Gamma_+\cup\Gamma_-$ of the Cauchy representation of the propagator (\ref{['Gtota']}). $\Gamma_+$ and $\Gamma_-$ are mutually disjoint pieces covering, respectively, the $\text{Re}\,z>0$ and the $\text{Re}\,z<0$ half-plane making up definition (\ref{['Gtota']}). The vertical lines $L_\pm$ run along the discontinuity at $\text{Re}\,z=0$.
  • Figure 3: Path $\Gamma_{\rm u}$ (dashed lines) in the $(\text{Re}\, k^0,\text{Im}\, k^0)$ plane and its deformation $\Gamma_+^{\rm u}\cup\Gamma_-^{\rm u}$ (solid thick curves) around the branch cuts $k^0\leqslant -\omega$ and $k^0\geqslant \omega$ (gray thick lines) for a massive field. Credit: adaptation of Calcagni:2021ljs.
  • Figure 4: Non-maximal contour $\Gamma_{\rm F}$ of the Cauchy representation (\ref{['fey']}) of the standard Feynman propagator $1/(k^2-i\epsilon)$. The radius $r$ of the circle is arbitrary and small but does not have to be infinitesimal. The gray dot marks the pole of $\tilde{G}(z)$.
  • Figure 5: Transformed Feynman paths $\Gamma'_+$ (curves in the $\text{Re}\,z'>0$ half-plane) and $\Gamma'_-$ (curves in the $\text{Re}\,z'<0$ half-plane) for decreasing parameter $r$ (increasing thickness) in the cases (a) $|k^0|>\omega$ (time-like $k^2<0$) and (b) $|k^0|<\omega$ (space-like $k^2>0$), for the same value of $k^0$ and opposite value of $k^2$. Actually the paths do not cross the imaginary axis $\text{Re}\,z'=0$ and points thereon are not part of $\Gamma_+'\cup\Gamma_-'$. All paths are counter-clockwise.
  • ...and 4 more figures

Theorems & Definitions (2)

  • Theorem
  • proof