Classical limit of a scalar quantum field theory

TL;DR

The paper addresses how a scalar quantum field theory manifests classical behavior when observed at macroscopic spatial scales, using a gliding UV cutoff $k$ and the Closed Time Path formalism to treat the system as open. Its central methodology combines a one-loop renormalization group treatment with an open-dynamics action that generally acquires complex parameters, reflecting decoherence and dissipation; the subtraction point is chosen at the spectral peak to extract running couplings. The key findings include a second-order phase transition between weakly and strongly open theories, the emergence of a non-relativistic correlation length in the weakly open phase, and decoherence-driven suppression of quantum fluctuations in the IR for closed bare theories; the RG flow reveals rich crossovers between relativistic and non-relativistic regimes and highlights the nontrivial role of the UV environment in the quantum-to-classical transition. This framework clarifies how classical laws arise from quantum fields and provides a structured route to study open-system effects, non-relativistic scaling, and potential universality in scalar field theories.

Abstract

It is well known that a minimal distance emerges in quantum field theories owing to the need to regularize the UV divergences. The macroscopical limit at large minimal distance, weak spatial resolution, is investigated for a self interacting scalar quantum field theory by the help of the renormalization group. The lowering of the cutoff always opens the dynamics hence the renormalization group has to be implemented for open quantum field theories. A strongly coupled non-relativistic scaling regime is found supporting a second order phase transition between weakly and strongly open theories. The weakly (strongly) open bare theories develop into strongly (weakly) open dynamics during the renormalization group flow. The two known conditions of classical limit, the strong decoherence and the suppression of the quantum fluctuations are confirmed for closed bare theories at distances beyond a non-relativistic correlation length.

Figures (7)

• Figure 1: Lorentz boost to the right tilts the space-time axes, $x\to x'$, $t\to t'$, and the initial conditions on the $x'$ axis are found by solving the equation of motion along the dotted lines.
• Figure 2: The Feynman diagramms: (a) ${\cal O}(u_4^*u_4)$ contribution to the beta function of $v_{2,2}$, (b) ${\cal O}(v_{2,2}^2)$ contribution to the beta function of $u_4$. The vertical dotted line separates the bra and the ket factors, the $\phi_+$ and the $\phi_-$ modes, respectively and the horizontal dotted lines connect the $\phi_+^2$ and the $\phi_-^2$ factors in the vertex $v_{2,2}$.
• Figure 3: Renormalized trajectories close to the decoherence phase transition. The initial conditions are $m^2_B=0.1$, $\tilde{\nu}_B=0.1g^2$, $d_{0B}=0.1g^2$, $\tilde{d}_{2B}=0.1g^2$, $u_{4rB}=0.1$. Different trajectories belong to the values $g^2\in[0.016478,0.0164788]$.
• Figure 4: Running of $u_{4r}$ in the non-relativistic regime (solid line) and the STP evolution (\ref{['stpevol']}) (dashed line).
• Figure 5: The crossing of the trajectories. (a): The trajectory of $u_{4r}$ for $m^2_B=0.1$, $g=1$, $\tilde{\nu}_B=d_{0B}=\tilde{d}_{2B}=0.01$, $u_{4rB}\in[10^{-3},7\times10^{-3}]$. The three largest $u_{4r}$ trajectories are in the strongly open phase.(b): The finite difference $\Delta\tilde{\nu}/\Delta\tilde{\nu}_B$ for $m^2_B=10^{-4}$, $d_{0B}=\tilde{d}_{2B}=0.1$, $g^2\in[0.166,0.167]$. The trajectories are in the weakly open phase, the finite difference is increasing with $g$ and changes sign at the lowest two $g$ values.