Real-time quantum evolution in the classical approximation and beyond

With the goal in mind of deriving a method to compute quantum corrections for the real-time evolution in quantum field theory, we analyze the problem from the perspective of the Wigner function. We argue that this provides the most natural way to justify and extend the classical approximation. A simple proposal is presented that can allow us to give systematic quantum corrections to the evolution of expectation values and/or an estimate of the errors committed when using the classical approximation. The method is applied to the case of a few degrees of freedom and compared with other methods and with the exact quantum results. An analysis of the dependence of the numerical effort involved as a function of the number of variables is given, which allows us to be optimistic about its applicability in a quantum field theoretical context.

Figure 1

The Wigner function $W_{\mathrm{uq}}$ in the ultraquantum limit, for fixed values of $x$ and $t$.

Figure 2

(a): Time evolution of $⟨Q^2⟩$ compared to the classical approximation, the 2PI truncation to next-to-leading order in $1/N$ expansion, and the method proposed in this paper. (b): Relative error committed in each of the approximations as a function of time.

Figure 3

The value of $⟨Q^2⟩$ at the third maximum in time computed using several values of $\kappa$ (see text). The lines are linear and quadratic fits.

Figure 4

(a): The same as Fig. 2a but for the potential in Eq. (52). (b): Relative differences as in Fig. 2b.

Figure 5

Time evolution of $⟨\sum _k|{\varphi }_k{|}^2(mt)⟩$ for $N=2$, for the fully quantum, classical approximation and LQC method for $\kappa =1/3$.

Figure 6

The same as Fig. 5 but for $N=8$ (a) and $N=32$ (b).