Spontaneous collapse effects on relativistic fermionic matter

TL;DR

By embedding CSL in a relativistic Dirac field within the Keldysh formalism, the paper derives an effective action and Dyson-Schwinger equations to study spontaneous collapse effects on fermions. It shows that the collapse induces an imaginary self-energy, producing a complex dynamical mass and Lorentz-violating behavior in the infrared that is suppressed in the ultraviolet. The hydrodynamic analysis reveals a conserved current but a non-conserved energy-momentum tensor, reflecting heating from the noise. Non-perturbative results impose bounds on CSL parameters via proton stability, offering a framework to test collapse models against high-energy phenomena and explore the quantum-to-classical transition in relativistic systems.

Abstract

This study expands the spontaneous collapse assumptions into the relativistic quantum field theory framework for Dirac fields. By solving Lindblad's master equation using the Keldysh formalism, the effective action is derived, which captures the dynamics of fermions with spontaneous collapse represented as an imaginary self-interaction term. Utilizing the corresponding Dyson-Schwinger equations at 1-loop approximation, the effective mass induced by the nonlinearity is computed. The findings indicate the presence of a new mechanism that introduces a qualitative change in the mass spectrum, where the particle's mass becomes complex. This mechanism, which generates a Lorentz invariance violation in the infrared regime, recovers the Lorentz invariance in the ultraviolet regime. The corresponding hydrodynamics of the system is analyzed through the Keldysh component of the propagator, and a conserved charge is found. In contrast, the energy-momentum tensor is shown to be non-conserving. This phenomenon represents a new contribution to the understanding of the spontaneous collapse and the transition from quantum to the classical realm.

Figures (4)

• Figure 1: Plot of the solutions of the non-perturbative gap equations for $Im[\mathcal{Z}(u)]$ , normalized with respect to $\mathcal{Z}_0=\lim_{{\bf p \to 0}}\mathcal{Z}({\bf p})$, with $u= \log_{10}|{\bf p}|/\sqrt{4\sigma}$.
• Figure 2: Plot of the solutions of the non-perturbative gap equations for $\Gamma(u)=Im[M(u)]$, normalized with respect to $\Gamma_0=-\lim_{{\bf p \to 0}}Im[M({\bf p})]$, with $u= \log_{10}|{\bf p}|/\sqrt{4\sigma}$.
• Figure 3: Non-perturbative solutions of the mean lifetime fraction $\tau/\tau_U$, with $\tau=1/\Gamma_0$ and $\tau_U =13.8 \times 10^9$years, as a function of $\log_{10}[m/eV]$. The solid green, dashed green, and dotted green lines represent the non-perturbative solutions for $\lambda= 10^{-17} s^{-1}$ and $r_C =10^{-7}m$, $r_C =10^{-6} m$, $r_C =10^{-5} m$, respectively. The solid red, dashed red, and dotted red lines represent the non-perturbative solutions for $\lambda= 10^{-18} s^{-1}$ and $r_C =10^{-7}m$, $r_C =10^{-6} m$, $r_C =10^{-5} m$, respectively. The black dashed line represents the proton ($m_p=928$ MeV) and the red region represents the instability region.
• Figure 4: Parameter space plot for the SC model. The light green exclusion region comes from the proton stability in the non-perturbative approach. For reference, one includes the GRW GRWref values ($\lambda = 10^{-16} s^{-1}$, $r_C = 10^{-7} m$ ) and the values proposed by Adler Adlerref ($\lambda = 10^{-8\pm 2} s^{-1}$, $r_C = 10^{-7} m$ ) and ($\lambda = 10^{-6\pm 2} s^{-1}$, $r_c = 10^{-6} m$).