Decoherence in an interacting quantum field theory: The vacuum case

We apply the decoherence formalism to an interacting scalar field theory. In the spirit of the decoherence literature, we consider a “system field” and an “environment field” that interact via a cubic coupling. We solve for the propagator of the system field, where we include the self-energy corrections due to the interaction with the environment field. In this paper, we consider an environment in the vacuum state ($T=0$). We show that neglecting inaccessible non-Gaussian correlators increases the entropy of the system as perceived by the observer. Moreover, we consider the effect of a changing mass of the system field in the adiabatic regime, and we find that at late times no additional entropy has been generated.

Figure 1

Schwinger-Keldysh contour with finite initial time $t_0$ and final time $t_f$.

Figure 2

Schwinger-Keldysh contour where the initial and final times in Fig. 1 have been extended to negative and positive infinity, respectively.

Figure 3

Contributions to the 2PI effective action up to two-loop order. The double solid lines denote $\varphi$ propagators, whereas the double dashed lines correspond to $\chi$ propagators.

Figure 4

Contributions to the self-masses up to one-loop order. Again, the double solid lines denote $\varphi$ propagators, whereas the double dashed lines correspond to $\chi$ propagators. Hence, the first two Feynman diagrams contribute to the self-mass of $\chi (x)$, and only the third diagram contributes to the self-mass of $\varphi (x)$.

Figure 5

Statistical propagator in Fourier space for a small coupling $h/\rho =3/2$ (solid line) and a larger one $h/\rho =4$ (dashed line). Because of a nonzero coupling, we observe that the $\delta$ function, present in the original dispersion relation, has broadened to a “quasiparticle peak,” roughly of a Breit-Wigner form. If the coupling increases, the quasiparticle peak broadens further. Clearly, when $h\gg {\omega }_{\varphi }$ in the strongly coupled regime, we have a “collection of quasiparticles.” We used $k/\rho =1$, $m_{\varphi }/\rho =1$, and $\mu /\rho =1$.

Figure 6

Statistical particle number density $n_k$ as a function of $k/\rho$. We used $h/\rho =1$ and $m_{\varphi }/\rho =1$ (black solid line), $h/\rho =2$ and $m_{\varphi }/\rho =1$ (black dashed line), $h/\rho =1$ and $m_{\varphi }/\rho =0.5$ [gray (blue) line] and $h/\rho =2$ and $m_{\varphi }/\rho =0.5$ [gray (blue) dashed line]. In the UV $n_k$ vanishes irrespective of the value of $h/\rho$ or $m_{\varphi }/\rho$. Particles are only produced by the interaction in the statistical sense in the IR. The mass only influences the IR behavior. Moreover, one can show by appropriate rescalings of the functions above that $n_k$ is given by Eq. (94) in the UV.

Figure 7

Phase space evolution for constant $m_{\varphi }$ for pure state initial conditions ${\Delta }_k(t_0)=1$ (black line) and mixed state initial conditions [gray (red) line] ${\Delta }_k(t_0)={\Delta }_{\mathrm{ms}}$. In both cases the phase space area settles to the constant value ${\Delta }_{\mathrm{ms}}$, indicated by the dashed line and calculated in Eq. (92). We use $k/\rho =1$, $m_{\varphi }/\rho =1$, $h/\rho =4$, and $N=2000$.

Figure 8

Entropy generation for the system field $\varphi$ through interaction with the environment $\chi$ in the vacuum state. As both fields are in a vacuum state, the entropy generation is relatively small. We used pure state initial conditions $S_k(t_0)=0$ and $k/\rho =1$, $m_{\varphi }/\rho =1$, $h/\rho =4$ and $N=2000$. The entropy settles to a constant value $S_{\mathrm{ms}}$ calculated from the value of ${\Delta }_{\mathrm{ms}}$ of Fig. 7 and Eq. (10).

Figure 9

Convergence for the phase space evolution presented in Fig. 7. The black and red (the darker ones) lines are identical to the ones in Fig. 7 and most accurate ($N=2000$). The gray and yellow lines (the lighter ones) are calculated with $N=1000$. The other parameters are kept fixed. Clearly, the difference between ${\Delta }_{\mathrm{ms}}$ and the numerical asymptotes decreases as the accuracy increases.

Figure 10

Test of our numerical code. This plot shows the difference of the phase space area calculated from Eqs. (85, 87) for different values of $N$ but the same values for the other parameters $k/\rho =1$, $m_{\varphi }/\rho =1$, and $h/\rho =4$ for pure state initial conditions. We used $N=1000$ at $t=100$ (solid line), $N=1000$ at $t=50$ (dotted line), and $N=1000$ at $t=25$ (dashed line). The difference between the two methods disappears as the accuracy of the numerical evolution increases.

Figure 11

Exponential approach to ${\Delta }_{\mathrm{ms}}$. We study, for different initial conditions, differences of ${\Delta }_k(t)$ on a logarithmic scale defined by Eq. (101). Clearly, the decoherence rate does not depend on the initial conditions. We used $k/\rho =1$, $m_{\varphi }/\rho =1$, $h/\rho =4$, and $N=2000$.

Figure 12

Decoherence time scale. We confirm the scaling relation for the decoherence time scale anticipated in Eq. (100). For all plots we took $k/\rho =1$ and $N=1000$, and furthermore we used $m_{\varphi }/\rho =1$, $h/\rho =4$ (solid red line), $m_{\varphi }/\rho =1$, $h/\rho =1$ (dashed red line), $m_{\varphi }/\rho =4$, $h/\rho =4$ (solid blue line, higher frequency), and $m_{\varphi }/\rho =4$, $h/\rho =1$ (dashed blue line, higher frequency).

Figure 13

Phase space area decrease due to a mass increase from $m_{\varphi }/\rho =0.75$ to $m_{\varphi }/\rho =2$ (solid black line). We used $h/\rho =4$, $k/\rho =1$, and $N=2000$. We thus observe a slight entropy decrease. The solid gray line denotes the constant mass phase space evolution for $m_{\varphi }/\rho =2$. As the two asymptotes coincide, we conclude no entropy has been generated at late times by the mass change. ${\Delta }_{\mathrm{ms}}^{(1)}$ and ${\Delta }_{\mathrm{ms}}^{(2)}$ are the constant mixed phase space areas calculated for $m_{\varphi }/\rho =0.75$ and $m_{\varphi }/\rho =2$, respectively.

Figure 14

Entropy decrease for the case presented in Fig. 13. Because of the mass increase, the phase space area decreases which results consequently in a drop in the entropy.

Figure 15

Phase space area increase due to a mass decrease from $m_{\varphi }/\rho =2$ to $m_{\varphi }/\rho =0.75$ (solid black line). The other parameters are the same as in Fig. 13. The solid gray line denotes the constant mass phase space evolution for $m_{\varphi }/\rho =0.75$. Again the two asymptotes coincide and no entropy has been generated at late times by the mass change.

Figure 16

Entropy increase for the case presented in Fig. 15. Clearly, a mass decrease gives rise to a slight entropy increase.

Figure 17

Causal propagator with a constant mass. Parameters: $k/\rho =1$, $m_{\varphi }/\rho =1$.

Figure 18

Causal propagator for a small change in the mass. Parameters: $k/\rho =1$, $A/{\rho }^2=B/{\rho }^2=1/2$.

Figure 19

Causal propagator for a large change in the mass. Parameters: $k/\rho =1$, $A/{\rho }^2=B/{\rho }^2=2$.

Figure 20

Statistical propagator with a constant mass. Parameters: $k/\rho =1$, $m_{\varphi }/\rho =1$.

Figure 21

Statistical propagator for a small change in the mass. Parameters: $k/\rho =1$, $A/{\rho }^2=B/{\rho }^2=1/2$.

Figure 22

Statistical propagator for a large change in the mass. Parameters: $k/\rho =1$, $A/{\rho }^2=B/{\rho }^2=2$.

Figure 23

This plot shows $|{\beta }_k{|}^2$ as a function of the final mass $m_{\mathrm{out}}/\rho$, for fixed $k/\rho =0.01$. The dashed line shows the adiabatic regime ($m_{\mathrm{in}}/\rho =0.2$), whereas the solid line shows the nonadiabatic regime ($m_{\mathrm{in}}/\rho =0.02$).