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Is a covariant virtual tachyon viable?

Krzysztof Jodłowski

Abstract

Sidney Coleman has noted that superluminal particles or observers would be able to go back in time and have no definite trajectory according to subluminal observers, while not violating Lorentz invariance [1]. Recently, Dragan and Ekert have significantly developed similar ideas even further, which lead to formulation of ``quantum principle of relativity'' that intimately links the two theories [2]. However, field theory descriptions of an on-shell tachyon, described by scalar field $φ$ with negative mass squared parameter, lead to violation of basic principles of relativity or quantum mechanics. In this work, we investigate whether purely virtual tachyons can be consistent within the fakeon framework-the only known viable formulation of purely virtual particles. We identify two fatal obstructions. First, Lorentz boosts mix creation and annihilation operators, rendering the canonical commutation relations non-invariant despite formal invariance of the vacuum. Second, the real part of the tachyon Feynman propagator and Wheeler propagator have disjoint support, preventing application of both the fakeon prescription and Wheeler-Feynman absorber mechanism. Interactions with stable Standard Model fields further violate Lorentz invariance and the equivalence principle, and we provide quantitative limit on coupling strength of such scenario. Our analysis excludes possibility of formulating covariant quantum field theory of interacting virtual tachyons.

Is a covariant virtual tachyon viable?

Abstract

Sidney Coleman has noted that superluminal particles or observers would be able to go back in time and have no definite trajectory according to subluminal observers, while not violating Lorentz invariance [1]. Recently, Dragan and Ekert have significantly developed similar ideas even further, which lead to formulation of ``quantum principle of relativity'' that intimately links the two theories [2]. However, field theory descriptions of an on-shell tachyon, described by scalar field with negative mass squared parameter, lead to violation of basic principles of relativity or quantum mechanics. In this work, we investigate whether purely virtual tachyons can be consistent within the fakeon framework-the only known viable formulation of purely virtual particles. We identify two fatal obstructions. First, Lorentz boosts mix creation and annihilation operators, rendering the canonical commutation relations non-invariant despite formal invariance of the vacuum. Second, the real part of the tachyon Feynman propagator and Wheeler propagator have disjoint support, preventing application of both the fakeon prescription and Wheeler-Feynman absorber mechanism. Interactions with stable Standard Model fields further violate Lorentz invariance and the equivalence principle, and we provide quantitative limit on coupling strength of such scenario. Our analysis excludes possibility of formulating covariant quantum field theory of interacting virtual tachyons.
Paper Structure (9 sections, 33 equations, 3 figures)

This paper contains 9 sections, 33 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Tachyonic Green's functions
  3. Commutator function
  4. Advanced/retarded Green's functions
  5. Feynman propagator
  6. Wheeler propagator
  7. Canonical quantization of a free tachyon fakeon
  8. Tachyon-bradyon interactions: equivalence principle and Lorentz invariance violation
  9. Conclusions

Figures (3)

  • Figure 1: Integration contours in the complex $k^0$ plane for: (a) the commutator function; (b) and (c) the retarded Green's function for $t>0$ and $t<0$, respectively; and (d) the Feynman propagator for $t>0$.
  • Figure 2: Interactions of the virtual tachyon $\phi$: (a) exchange between fermions inducing a long-range oscillating potential that violates Lorentz invariance; (b) mixing with the Higgs boson; (c) mixed quartic interaction $\phi^2 h^2$.
  • Figure 3: (a) Elastic electron scattering mediated by virtual tachyon exchange. (b) The integration contour used to evaluate the non-relativistic potential.