Is a covariant virtual tachyon viable?

The gist The analysis investigates whether purely virtual tachyons can be consistent within the fakeon framework, ultimately showing that interactions with stable Standard Model fields violate Lorentz invariance and provide quantitative limits on coupling strength

Motivation for Considering Tachyons

Tachyons are predicted by the asymptotic freedom scenario in higher-derivative quantum gravity, they are also invoked in constructions of interesting Carrollian models by starting with tachyonic matter in a Lorentzian theory, and finally, they are studied to negatively settle the question of whether tachyons could be purely virtual particles, called fakeons. The motivation for considering the QFT of tachyons is also provided by the fact that they are predicted by asymptotic freedom scenario in higher-derivative quantum gravity.

Analysis of Free Tachyon Green’s Functions

The paper provides complete expressions for the tachyon Feynman propagator (also known as the Dhar-Sudarshan propagator) and the Wheeler propagator in position space. The commutator function is given by i∆(x − y) which matches expressions stated in [44, 45]. The asymptotic behavior of the commutator in the time-like regime is problematic, since it grows exponentially [41].

Behavior of Green’s Functions

The advanced/retarded Green’s functions for tachyons grow exponentially in the future timelike direction, while they cannot be used to obtain a stable solution to the initial value problem. The Feynman propagator shows that for x2 > 0, it is purely imaginary and decays exponentially. For x2 < 0, the FP is oscillating. The Wheeler propagator has support on and inside the lightcone, and it blows up exponentially for both past and future in the timelike regime. Crucially, the Wheeler propagator differs from the real part of the FP away from the lightcone for tachyon.

Canonical Quantization of a Free Tachyon Fakeon

The tachyon field χ(x) satisfies microcausality and equations of motion, as well as that its commutator and Green’s functions are LI. The canonical commutation relations (CCRs) involve indefinite metric for both stable (⃗k> mϕ) and unstable modes (⃗k< mϕ). The vacuum state is defined such that a±⃗k0⟩ = b±⃗k0⟩ = 0 for all real three-momenta. The spectrum of H is not positive definite, containing not only positive and negative real eigenstates, but also imaginary ones.

Tachyon-Bradyon Interactions and Equivalence Principle Tests

Interactions of a virtual tachyon with ordinary matter generate a matter-dependent scalar background and can lead to violations of the weak equivalence principle in the observable sector. The non-relativistic potential resulting from tachyon exchange is VT (⃗r) = −g squared Re Z d 3⃗k(2π) 3 e i⃗k·⃗r ⃗k 2+m2ϕ − iϵ = −g squared 4π r cos(mϕ r). This matter-dependent scalar background modifies the effective electron mass as meffective e = me − g⟨ϕ(x)⟩. These interactions generate a long-range oscillatory force and a matter-dependent scalar background, with possible violations of the weak equivalence principle.

Conclusion on Viability

The analysis shows that interactions of virtual tachyon with stable SM fields induce Lorentz violation effects for the latter, and we have provided quantitative bound on such scenario. The formulation of covariant QFT of superluminal objects-whether propagating or purely off-shell-is not viable. Any attempt to formulate a tachyon QFT that would explicitly abandon one of the standard axioms would not only face stringent constraints but also question about naturalness of such an approach. The work provides further support to lack of viability of asymptotic freedom in higher-derivative quantum gravity and shows that constructions employing tachyons within Carrollian physics face obstacles in Minkowski space.

The paper concludes that the formulation of covariant QFT of superluminal objects is not viable because the Wheeler and real part of the FP differ away from the lightcone for tachyon. Furthermore, interacting virtual tachyon field leads to a highly non-local initial value problem for matter it interacts with. The analysis provides quantitative bound on such scenario and shows that interactions of virtual tachyon with stable SM fields induce Lorentz violation effects for the latter. Therefore, formulation of covariant QFT of superluminal objects-whether propagating or purely off-shell-is not viable. Any attempt to formulate a tachyon QFT that would explicitly abandon one of the standard axioms would not only face stringent constraints but also question about naturalness of such an approach. The work provides further support to lack of viability of asymptotic freedom in higher-derivative quantum gravity and shows that constructions employing tachyons within Carrollian physics face obstacles in Minkowski space.

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Is a covariant virtual tachyon viable? Krzysztof Jodłowski∗ Department of Physics and Institute of Theoretical Physics Nanjing Normal University, Nanjing, 210023, China and Particle Theory and Cosmology Group, Center for Theoretical Physics of the Universe, Institute for Basic Science (IBS), Daejeon, 34126, Korea 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 show that the supports of the real part of the tachyon Feynman propagator and the Wheeler propagator intersect only on the light cone, preventing application of both the fakeon prescription and Wheeler-Feynman absorber mechanism. Interactions with stable Standard Model fields violate the equivalence principle, and we provide quantitative limit on coupling strength of such scenario. Our analysis identifies obstacles to formulating covariant quantum field theory of interacting virtual tachyons.†

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Higgs field can be viewed as self-interacting tachyon,1 that can develop either vanishing (proper tachyon) or non-zero vacuum expectation value (the option realized within SM) through tadpole diagrams. After Dyson series resummation of the Higgs propagator, the physical mass squared is positive—such purely diagrammatic approach does not invoke concepts like spontaneous symmetry breaking, point emphasized in [8]. Therefore, our work can be seen as exploring the vanishing tadpole solution, which was promptly dismissed in [8] due to tachyon problems with causality; we also note that the Higgs mechanism was brought up as one of main motivations for investigating tachyons in [6, 32]. Since causality problems due to tachyons are difficult to settle within special relativity of classical field theory, we believe dismissing tachyonic solution on such ground is premature, and our work provides comprehensive (negative) resolution of this issue. Second, already DeWitt [8] and Thouless [9] have noted, working exclusively within special relativistic setup, that off-shell, meaning non-propagating, tachyons might be compatible with known laws of physics. Within QFT, several works [33, 34] have considered tachyon field representing a virtual state, basing it on the claim that tachyon commutator identically vanishes. We believe that such an approach is not promising, since it does not correspond to the standard properties of the commutator, which is given by Eq. (4). Moreover, if the commutator vanished identically, then so do the retarded and advanced Green’s functions, which should not be the case if tachyon were to actually mediate interactions. In our approach, the commutator behaves similarly to the subluminal (bradyon) case - it vanishes only outside the lightcone - and the same holds for the advanced and retarded Green’s functions. On the other hand, tachyonic commutator (and therefore so do advanced and retarded Green’s functions) grows exponentially in time-like direction, illustrating unstable nature of the tachyon.

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We will consider the following Lagrangian, extending the Standard Model (SM) by kinetic, mass term, and Yukawa interaction of tachyon with a Standard Model massive fermion ψ, L = LSM + 1/2 ∂µϕ ∂µϕ − 1/2 m2ϕ ϕ ϕ squared + g ϕ ψψ, ¯ (1) where m2ϕ < 0, and g is a dimensionless coupling constant. For concreteness, we will assume further that ψ represents an electron.

1. Bold header: Improved theoretical constraint checking for virtual particles

This system can now rigorously identify obstacles to formulating covariant quantum field theory of interacting virtual tachyons, specifically by detecting when interactions with stable Standard Model fields violate the equivalence principle and providing a quantitative limit on coupling strength.

2. Bold header: Enhanced causal analysis for superluminal physics

The improved AI can analyze the difference between bradyon and tachyon propagators, noting that while bradyons satisfy causality, tachyons exhibit behavior where the real part of FP has support on and outside the lightcone, leading to a conclusion that interacting virtual tachyon field leads to a highly non-local initial value problem for matter it interacts with.

3. Bold header: Refined constraint verification against established frameworks

The system can now conclusively show that formulation of covariant QFT of superluminal objects-whether propagating or purely off-shell is not viable by demonstrating that the Wheeler and real part of the FP differ away from the lightcone for tachyon, which invalidates the application of mechanisms like the Wheeler-Feynman absorber-emiter idea.

4. Bold header: Predictive modeling of matter interaction effects

The improved AI can model matter-dependent scalar background generated by tachyon interactions, specifically predicting that these interactions generate a longrange oscillatory force and a matter-dependent scalar background, with possible violations of the weak equivalence principle by examining how gradients of this background cause composition-dependent accelerations.

5. Bold header: Verification of stability against particle decay

The system can rigorously check the stability claim by demonstrating that for tachyons, Im(M) = 0 because since the tachyon is kinematically forbidden to decay into two bradyons, thus proving that Im(µ 2ϕ cannot have non-zero imaginary part.