[Paper Review] Tachyonic Dirac sea

Ernst Trojan|arXiv (Cornell University)|Jan 27, 2012

Black Holes and Theoretical Physics11 references3 citations

TL;DR

This paper investigates the thermodynamic properties of a many-tachyon Fermi system, showing that tachyons with imaginary energy (k < m) form a tachyonic Dirac sea that contributes constant terms to energy density and pressure, while leaving sound speed unaffected. The key result is that excluding these unstable states leads to inconsistent thermodynamic functions incompatible with the ordinary Fermi gas upon analytic continuation, validating their inclusion for consistency and causality at high densities.

ABSTRACT

We consider a system of many fermions with tachyonic energy spectrum \varepsilon_k=\sqrt{k^2-m^2} and clarify that tachyons with imaginary energy and low momentum (k

Motivation & Objective

• To resolve the physical interpretation of tachyons with imaginary energy (k < m) in a many-fermion system governed by the tachyonic energy spectrum εk = √(k² − m²).
• To determine whether unstable states with k < m should be excluded from thermodynamic calculations in a cold tachyon Fermi gas.
• To establish the consistency of thermodynamic functions under analytic continuation m → im, ensuring agreement with the standard relativistic Fermi gas.
• To identify the critical density nT ≈ 1.84n⋆ above which the system remains hydrodynamically stable.

Proposed method

• Derive thermodynamic functions (energy density, pressure, particle density) from the grand canonical partition function using the tachyonic energy spectrum εk = √(k² − m²), split into real and imaginary parts via the Heaviside step function.
• Apply the Fermi-Dirac distribution fε = 1 / (exp[(εk − μ)/T] + 1) at zero temperature, leading to a step function in momentum space (fε = Θ(kF − k)).
• Compute the energy density and pressure by integrating over momentum space, separating contributions from the stable (k > m) and unstable (k < m) sectors.
• Use analytic continuation m → im to map the tachyonic system to the standard relativistic Fermi gas, verifying consistency of thermodynamic functions.
• Analyze the sound speed cs² = dP/dE and show it remains finite only when the unstable sector (k < m) is included.
• Assess hydrodynamic stability by comparing the critical Fermi momentum kF < √(3/2)m to the onset of instability, identifying nT as the minimum stable density.

Experimental results

Research questions

• RQ1 Do tachyons with imaginary energy (k < m) contribute meaningfully to the thermodynamic functions of a many-fermion system with a tachyonic energy spectrum?
• RQ2 Is the inclusion of the unstable sector (k < m) necessary to ensure consistency of thermodynamic functions under the transformation m → im, which maps tachyons to massive fermions?
• RQ3 What is the critical particle density nT above which a cold tachyon Fermi gas remains hydrodynamically stable?
• RQ4 How does the sound speed behave when the Fermi momentum kF approaches m, and why does it diverge only when the unstable sector is excluded?
• RQ5 Why does excluding the unstable states (k < m) lead to thermodynamic functions incompatible with the standard Fermi gas upon analytic continuation?

Key findings

• The energy density and pressure of a cold tachyon Fermi gas acquire additional constant terms from the unstable sector (k < m), given by E = E_real + iγ/2π² ∫₀^m k² √(m² − k²) dk and P = P_real + iγ/6π² ∫₀^m k³ (−k/(2√(m² − k²))) dk.
• The sound speed remains unaffected by the imaginary-energy contributions and is given by cs² = (1/3) kF² / (kF² − m²), which diverges as kF → m.
• The system becomes hydrodynamically unstable when kF < √(3/2)m, corresponding to a critical particle density nT ≈ 1.84 n⋆, where n⋆ = γm³/(6π²).
• Excluding the unstable sector (k < m) leads to thermodynamic functions that do not reduce to those of the ordinary Fermi gas under m → im, violating consistency with standard field theory.
• The inclusion of the tachyonic Dirac sea (k < m) ensures that the sound speed remains consistent with causality and analytic continuation, validating its physical role in the many-body system.
• The critical density nT ≈ 1.84 n⋆ is significantly higher than the density at kF = m, indicating that only high-density tachyon Fermi gases are physically viable for stable, causal behavior.