Is the effective potential effective for dynamics?

We critically examine the applicability of the effective potential within dynamical situations and find, in short, that the answer is negative. An important caveat of the use of an effective potential in dynamical equations of motion is an explicit violation of energy conservation. An adiabatic effective potential is introduced in a consistent quasistatic approximation, and its narrow regime of validity is discussed. Two ubiquitous instances in which even the adiabatic effective potential is not valid in dynamics are studied in detail: parametric amplification in the case of oscillating mean fields, and spinodal instabilities associated with spontaneous symmetry breaking. In both cases profuse particle production is directly linked to the failure of the effective potential to describe the dynamics. We introduce a consistent, renormalized, energy conserving dynamical framework that is amenable to numerical implementation. Energy conservation leads to the emergence of asymptotic highly excited, entangled stationary states from the dynamical evolution. As a corollary, decoherence via dephasing of the density matrix in the adiabatic basis is argued to lead to an emergent entropy, formally equivalent to the entanglement entropy. The results suggest novel characterization of asymptotic equilibrium states in terms of order parameter vs energy density.

Figure 1

Unstable bands for ${\kappa }_{n,-}^2\le {\kappa }^2=\frac{k^2}{m^2}\le {\kappa }_{n,+}^2$ for $n=2,3,4$. The range is constrained by ${\kappa }^2>0$.

Figure 2

Two linearly independent solutions of Mathieu’s equation (4.4), $h0(\tau ),h1(\tau )$ with initial conditions $h0(0)=1,h{0}^{\prime }(0)=0;h1(0)=0,h{1}^{\prime }(0)=1$, for the unstable band for $n=2$, with $\eta =4$ and $\alpha =1$, corresponding to ${\kappa }^2=1$, approximately in the middle of the first physical unstable band for $\kappa$. A general solution for a mode function $g_k(\tau )$ is a complex linear combination of $h0(\tau )$ and $h1(\tau )$ satisfying the condition (3.15).

Figure 3

Two stable solutions of Mathieu’s equation (4.4), $F(\tau )$ with initial conditions $F(0)=1,F^{\prime }(0)=0$, for $\eta =3,5$ and $\alpha =1$, respectively, on either side of the first physical unstable band at $\eta =4$.