Effective field theory in time-dependent settings

The observables of interest to us are the time-dependent expectation values of operators ${\cal O}(t)$,

$${\rm tr}\bigl{(}\rho(t){\cal O}(t)\bigr{)},$$

(1)

where $\rho(t)$ is the density matrix describing the system. For most situations we can take $\rho(t)$ to describe a pure state such as the vacuum state of a free field theory, in which case $\rho(t)=|0(t)\rangle\langle 0(t)|$ and where

$${\rm tr}\bigl{(}\rho(t){\cal O}(t)\bigr{)}=\langle 0(t)|{\cal O}(t)|0(t)\rangle,$$

(2)

but we leave open the possibility that the state may actually be mixed, as for a thermal state. For practical calculations, it is often convenient to evaluate the time-evolution in the interaction picture. If we separate the Hamiltonian into its free and interacting parts, $H=H_{0}+H_{I}$, then in the interaction picture, the time-evolution of the operators is given by the free part, while the evolution of the density matrix is given by the interacting part. The density matrix satisfies the Liouville equation,

$$i{\partial\rho(t)\over\partial t}=[H_{I}(t),\rho(t)],$$

(3)

where $H_{I}(t)$ is the interaction Hamiltonian in the interaction picture.

Given an initial state $\rho(t_{0})$, the Liouville equation can be solved in terms of the time evolution operator $U_{I}(t,t_{0})$ satisfying the usual Dyson equation,

$$i{\partial U_{I}(t,t_{0})\over\partial t}=H_{I}U_{I}(t,t_{0})$$

(4)

where $U_{I}(t_{0},t_{0})=\mathbb{I}$ and

$$\rho(t)=U_{I}(t,t_{0})\rho(t_{0})U_{I}^{\dagger}(t,t_{0}).$$

(5)

The solution that satisfies this initial condition is

$$U_{I}(t,t_{0})=Te^{-i\int_{t_{0}}^{t}dt^{\prime}\,H_{I}(t^{\prime})}.$$

(6)

In the interaction picture then, the expectation value of an operator is given by

	$\displaystyle{\rm tr}\bigl{(}\rho(t){\cal O}(t)\bigr{)}$	$\displaystyle=$	$\displaystyle{\rm tr}\bigl{[}\rho(t_{0})U_{I}^{\dagger}(t,t_{0}){\cal O}(t)U_{I}(t,t_{0})\bigr{]}$		(7)
		$\displaystyle=$	$\displaystyle{\rm tr}\Bigl{[}\rho(t_{0})\Bigl{(}Te^{-i\int_{t_{0}}^{t}dt^{\prime}\,H_{I}(t^{\prime})}\Bigr{)}^{\dagger}{\cal O}(t)\Bigl{(}Te^{-i\int_{t_{0}}^{t}dt^{\prime}\,H_{I}(t^{\prime})}\Bigr{)}\Bigr{]}.$

Reading the terms inside the trace from right to left, the first operator evolves from $t_{0}$ to $t$, where the operator ${\cal O}(t)$ is inserted, and then we turn around and evolve back to $t_{0}$ again. Because of the cyclicity of the trace, we are starting and ending in the same state, which is defined by $\rho(t_{0})$.

We can use this way of viewing the time evolution to define a contour time-ordered product. Let us introduce $\pm$ labels for the fields, where a $+$ field occurs on the on the forward part of this contour, running from $t_{0}$ to $t$, and a $-$ field occurs on the backward part that runs from $t$ back to $t_{0}$ again. The $+$ fields are then associated with the operator $U_{I}(t,t_{0})$ while the $-$ fields are associated with the $U_{I}^{\dagger}(t,t_{0})$ operator. We can extend the time integral into the infinite future as well by judiciously inserting a factor of the identity, ${\mathbb{I}}=U_{I}^{\dagger}(\infty,t)U_{I}(\infty,t)$ between the $U^{\dagger}_{I}(t,t_{0})$ and ${\cal O}(t)$ in the previous equation, writing¹¹1We typically assume that the operator being evaluated is placed on the $+$ part of the contour. However, evaluating it on the $-$ contour would give exactly the same result.

$${\rm tr}\bigl{(}\rho(t){\cal O}(t)\bigr{)}={\rm tr}\Bigl{[}\rho(t_{0})T_{c}\Bigl{(}{\cal O}^{+}(t)e^{-i\int_{t_{0}}^{\infty}dt^{\prime}\,[H_{I}^{+}(t^{\prime})-H_{I}^{-}(t^{\prime})]}\Bigr{)}\Bigr{]}.$$

(8)

The $H^{\pm}_{I}$ are of exactly the same form as the original interacting part of the Hamiltonian, except that in $H^{+}_{I}$ we label all the fields within it as $+$ fields (e.g. $H_{I}^{+}(t)=H_{I}[\Phi^{+}(t,\vec{x})]$) and in $H_{I}^{-}$ they are all labeled as $-$ fields. The subscript on the time-ordering symbol $T_{c}$ indicates that the time-ordering is that inherited from the closed time contour: times that occur in a $-$ field always occur after and in the opposite order of those in a $+$ field. This follows from the original Hermitian conjugation $U_{I}^{\dagger}(t,t_{0})$ and the fact that it occurs furthest to the left—the ‘latest’ in the contour sense—in the matrix element.

When $H_{I}$ is small in comparison with the free Hamiltonian, we can evaluate the expectation value perturbatively, expanding in terms of propagators of the free theory. In the interaction picture, the state does not evolve for a free theory, so the propagator is evaluated using the initial state, $\rho(t_{0})$. There are four possible ways of contracting the field depending on whether each of the two fields is on the forward ($+$) or backward ($-$) part of the contour. This means that in any process, we have four possible time-ordered Green’s functions,

$$G^{\pm\pm}(t,\vec{x};t^{\prime},\vec{y})={\rm tr}\left(\rho(t_{0})T_{c}\bigl{(}\Phi^{\pm}(t,\vec{x})\Phi^{\pm}(t^{\prime},\vec{y})\bigr{)}\right).$$

(9)

Using the definition of the time-ordering along the contour, these Green’s functions are

	$\displaystyle G^{++}(t,\vec{x};t^{\prime},\vec{y})$	$\displaystyle=$	$\displaystyle\Theta(t-t^{\prime})\,G^{>}(t,\vec{x};t^{\prime},\vec{y})+\Theta(t^{\prime}-t)\,G^{<}(t,\vec{x};t^{\prime},\vec{y})$
	$\displaystyle G^{+-}(t,\vec{x};t^{\prime},\vec{y})$	$\displaystyle=$	$\displaystyle G^{<}(t,\vec{x};t^{\prime},\vec{y})$
	$\displaystyle G^{-+}(t,\vec{x};t^{\prime},\vec{y})$	$\displaystyle=$	$\displaystyle G^{>}(t,\vec{x};t^{\prime},\vec{y})$
	$\displaystyle G^{--}(t,\vec{x};t^{\prime},\vec{y})$	$\displaystyle=$	$\displaystyle\Theta(t^{\prime}-t)\,G^{>}(t,\vec{x};t^{\prime},\vec{y})+\Theta(t-t^{\prime})\,G^{<}(t,\vec{x};t^{\prime},\vec{y})$		(10)

where the two Wightman functions are given by the two possible orderings of the fields depending on whether $t>t^{\prime}$ or $t<t^{\prime}$,

$$G^{>}(t,\vec{x};t^{\prime},\vec{y})={\rm tr}\left(\rho(t_{0})\Phi(t,\vec{x})\Phi(t^{\prime},\vec{y})\right)\qquad\hbox{and}\qquad G^{<}(t,\vec{x};t^{\prime},\vec{y})={\rm tr}\left(\rho(t_{0})\Phi(t^{\prime},\vec{y})\Phi(t,\vec{x})\right).$$

We shall mostly be examining the case where $\rho(t_{0})$ is the free field vacuum state, which we shall write as $|0(t_{0})\rangle\equiv|0\rangle$. Since this state is invariant under spatial translations, we can conveniently express these Wightman functions in their Fourier transformed form,

	$\displaystyle G^{>}_{k}(t,t^{\prime})$	$\displaystyle=$	$\displaystyle{1\over 2\omega_{k}}e^{-i\omega_{k}(t-t^{\prime})}$
	$\displaystyle G^{<}_{k}(t,t^{\prime})$	$\displaystyle=$	$\displaystyle{1\over 2\omega_{k}}e^{i\omega_{k}(t-t^{\prime})},$		(11)

where $k\equiv|\!|\vec{k}|\!|$ is the magnitude of the spatial momentum and $\omega_{k}\equiv\sqrt{k^{2}+m^{2}}$.

For many purposes it is useful to represent the expectation value as a path integral,

	$\displaystyle{\rm tr}\bigl{(}\rho(t){\cal O}(t)\bigr{)}$	$\displaystyle=$	$\displaystyle\int d\phi\,\langle\phi|\rho(t_{0})U_{I}^{\dagger}(t,t_{0}){\cal O}(t)U_{I}(t,t_{0})|\phi\rangle$
		$\displaystyle=$	$\displaystyle\int d\phi^{+}d\phi^{-}\,\rho(\phi^{+},\phi^{-};t_{0})\int_{\rm bc}{\cal D}\Phi^{+}{\cal D}\Phi^{-}\,e^{i(S[\Phi^{+}]-S[\Phi^{-}])}{\cal O}(t),$

where we have first used cyclicity of the trace and then evaluated it in the field basis $\left\{|\phi\rangle\right\}$. Just as before, we can interpret this trace as follows: start at $t_{0}$ with the initial state, evolve to $t$ where the operator ${\cal O}(t)$ is inserted and then evolve back to $t_{0}$. The boundary conditions on the path integrals over $\Phi^{\pm}$ are $\Phi^{+}(t_{0},\vec{x})=\phi^{+}(\vec{x})$, $\Phi^{-}(t_{0},\vec{x})=\phi^{-}(\vec{x})$, and $\Phi^{+}(t,\vec{x})=\Phi^{-}(t,\vec{x})$. The last condition can be interpreted as a continuity condition for a field $\Phi_{c}$ defined along a time contour starting at $t_{0}$, going to $t$, turning around and then returning to $t_{0}$. This is the origin of the closed time contour and the difference in actions in the exponential is just the integral of the Lagrangian density along this closed time contour, taking the orientation of the contour into account.

We can define a generating functional from this path integral by introducing sources $J^{\pm}$ which couple linearly to the $\Phi^{\pm}(t,\vec{x})$ fields,

$$Z[J^{+},J^{-}]=\int d\phi^{+}d\phi^{-}\,\rho(\phi^{+},\phi^{-};t_{0})\int_{\rm bc}{\cal D}\Phi^{+}{\cal D}\Phi^{-}\,e^{i(S[\Phi^{+}\!\!,\,J^{+}]-S[\Phi^{-}\!\!,\,J^{-}])}.$$

(13)

As before, we can always extend the time integrals, which are implicit in the actions,

$$S[\Phi^{\pm}\!\!,\,J^{\pm}]=\int_{t_{0}}^{t}dt^{\prime}\int d^{3}\vec{x}\,\Bigl{\{}{\cal L}[\Phi^{\pm}(t^{\prime},\vec{x})]+J^{\pm}(t^{\prime},\vec{x})\Phi^{\pm}(t^{\prime},\vec{x})\Bigr{\}},$$

(14)

into the infinite future since the forward and the backward parts of the contour beyond $t$ (the time at which we are evaluating a particular operator) exactly cancel—as they must if the theory is to remain causal.

Notice that there are two marked differences between this closed time path formalism and the more commonly used $S$-matrix treatment. In a scattering process, we start in the infinite past with an eigenstate of the free theory, evolve it into the infinite future, and find its overlap with the eigenstates of the free theory again. In that case, there is only one occurrence of the time-evolution operator, $U_{I}(\infty,-\infty)$. In the Schwinger-Keldysh approach, in contrast, we are evolving the entire matrix element forward, evolving both states, and accordingly both $U_{I}^{\dagger}$ and $U_{I}$ appear.

The second difference is that we are evolving over a finite interval and from a general initial state defined at an arbitrary initial time. We are free to consider a situation where $t_{0}\to-\infty$ or to choose the system to be the vacuum state of the free theory, but whatever state we have chosen, once we have specified its value at $t_{0}$, the preceding history is irrelevant for its subsequent evolution. The lower limit of the integrals in the exponents of the time-evolution operators is always $t_{0}$, so any intermediate vertices or any loop corrections are never evaluated prior to the initial time.

In some instances, it can be useful to extend the integral formally into the infinite past. Whether this should be done depends on the detailed physical situation and initial state that we are studying. For example, if we arrange the interactions prior to $t_{0}$ so that our system will be in precisely the desired state by that time, then we can extend the time integral in these actions to cover the entire real line. To arrange the state thus will almost always require an explicit time-dependence in the action—in the couplings for instance—if we do not happen to start with an eigenstate of the interacting theory.

For most of this article we shall be examining a system that begins in its free vacuum state at $t_{0}$ and then evolves under the influence of some interactions. This could be arranged by multiplying the interactions in $H_{I}$ by a step-function, $\Theta(t-t_{0})$, so that if the theory has been evolving freely before $t_{0}$ it will have the behavior that we wish afterwards. This perspective is heuristically useful since it shows that by starting in the vacuum state of the free theory rather than the interacting one we can expect some interesting evolution, since the $\Theta$-function represents a very sudden change to the system. But ultimately the details of how the system arrived in a particular state at $t_{0}$ are irrelevant for its subsequent evolution.

One practical difference between calculating a matrix element in the Schwinger-Keldysh and the $S$-matrix frameworks is that it is usually not possible to amputate the external legs of a process in the former approach. While this fact is not in itself anything profound, it does mean that the calculations of processes in the Schwinger-Keldysh picture will typically be lengthier than those for the analogous graphs in a scattering process. It is therefore helpful to have tools or tricks that can simplify the analysis whenever possible. One such tool is the tadpole method Weinberg:1973ua ; Boyanovsky:1994me , which uses the equation of motion for a classical background field to follow the quantum loop corrections of a theory.

Consider a quantum field $\Phi(t,\vec{x})$ that has a time-dependent expectation value when it is in a state specified by the density matrix $\rho(t)$,

$$\phi(t)={\rm tr}\bigl{(}\rho(t)\Phi(t,\vec{x})\bigr{)}.$$

(15)

In most models of inflation, it is just such a classical expectation value of a field which drives the accelerated cosmological expansion, while the quantum fluctuations about this background provide a source for all of the subsequent inhomogeneity of the universe. Condensed matter systems that undergo a phase transition can similarly have order parameters that evolve in time.

Setting a constant one-point function, or tadpole diagram, to zero in a field theory ensures that the quantum field is an excitation above its correct vacuum state; this condition is equivalent to calculating the amount by which we must shift the field such that we expand about the correct vacuum expectation value. The analogue of this statement is also true for non-constant backgrounds, where setting the one-point function to zero ensures that the background satisfies the correct equation of motion.

When we evaluate an expectation value in the Schwinger-Keldysh picture, we expand about the same classical background $\phi(t)$ on both branches of the time-contour,

$$\Phi^{\pm}(t,\vec{x})=\phi(t)+\varphi^{\pm}(t,\vec{x}),$$

where $\varphi^{\pm}(t,\vec{x})$ is the part of the quantum field representing the fluctuations about the classical part. By doing so, we arrive at the same equation of motion for $\phi(t)$ regardless of whether we place $\varphi(t,\vec{x})$ on either part of the contour when evaluating its one-point function,

$${\rm tr}\bigl{(}\rho(t)\varphi^{\pm}(t,\vec{x})\bigr{)}=0.$$

Notice that although the one-point function for $\varphi^{\pm}(t,\vec{x})$ has only one external leg for the fluctuations, it can have arbitrarily many external (nonpropagating) legs for the classical field $\phi(t)$.²²2We shall refer to the appearances of the classical field $\phi(t)$ as legs of a diagram—and we shall draw them as external dotted lines in graphs—though of course $\phi(t)$ is not a propagating field. We adopt this language and use this notation nonetheless since through it the graphs more closely resemble the corresponding $N$-point processes for which they are the analogues.

The requirement that the sum of all the tadpole graphs must vanish produces an equation for the background field $\phi(t)$. This equation contains the loop corrections from the quantum part of the field so it reproduces most of the familiar renormalization properties of the theory. It therefore provides a simpler route for analyzing the structures of the effective theory than is possible by evaluating the higher-order $N$-point functions of $\Phi^{\pm}(t,\vec{x})$ directly.

At lowest order, the tadpole condition is just the statement that the field $\phi(t)$ satisfies its classical equation of motion. To show this, write the action for the field on each of the two parts of the contour as $S^{\pm}[\Phi^{\pm}]$ and expand the field about its background value, $\Phi^{\pm}(t^{\prime},\vec{y})=\phi(t^{\prime})$. Keeping only the linear terms, we find

$$S^{\pm}[\Phi^{\pm}]=S^{\pm}[\phi]+\int_{t_{0}}^{\infty}dt^{\prime}\int d^{3}\vec{y}\,\biggl{\{}{\delta S^{\pm}(\Phi^{\pm})\over\delta\Phi^{\pm}}\biggr{|}_{\Phi^{\pm}(t^{\prime},\vec{y})=\phi(t^{\prime})}\varphi^{\pm}(t^{\prime},\vec{y})\biggr{\}}+{\cal O}({\varphi^{\pm}}^{2}).$$

(16)

Since $S^{+}[\phi]=S^{-}[\phi]$, the zeroth order terms cancel in the difference that appears in the generating functional. At linear order in both the expansion of the action and the exponentials, the one-point function of $\varphi(t,\vec{x})$ is then

			$\displaystyle\!\!\!\!\!\!\!\!\!{\rm tr}\bigl{(}\rho(t)\varphi^{+}(t,\vec{x})\bigr{)}$
		$\displaystyle=$	$\displaystyle\int{\cal D}\varphi^{+}{\cal D}\varphi^{-}\,\biggl{\{}\varphi^{+}(t,\vec{x})e^{i\int_{t_{0}}^{\infty}\int dt^{\prime}d^{3}\vec{y}\,\left[{\delta S^{+}(\Phi^{+})\over\delta\Phi^{+}(t^{\prime},\vec{y})}\bigr{|}_{\phi(t^{\prime})}\varphi^{+}(t^{\prime},\vec{y})-{\delta S^{-}(\Phi^{-})\over\delta\Phi^{-}(t^{\prime},\vec{y})}\bigr{|}_{\phi(t^{\prime})}\varphi^{-}(t^{\prime},\vec{y})\right]\ +\ {\cal O}(\varphi^{\pm 2})}\biggr{\}}$
		$\displaystyle=$	$\displaystyle i\int_{t_{0}}^{\infty}dt^{\prime}\int d^{3}\vec{y}\,\biggl{\{}{\delta S^{+}\over\delta\Phi^{+}(t^{\prime},\vec{y})}\biggr{|}_{\phi(t^{\prime})}G^{++}(t,\vec{x};t^{\prime},\vec{y})-{\delta S^{-}\over\delta\Phi^{-}(t^{\prime},\vec{y})}\biggr{|}_{\phi(t^{\prime})}G^{+-}(t,\vec{x};t^{\prime},\vec{y})\biggr{\}}+\cdots.$

The Green’s functions for the fluctuations that appear in these two terms are independent so that for the one-point function to vanish, this integrand must also vanish, which in turn requires that

$${\delta S^{\pm}\over\delta\Phi^{\pm}}\biggr{|}_{\Phi^{\pm}=\phi(t)}=0.$$

(18)

This condition is nothing more than the statement that the action is stationary at $\phi(t)$ to leading order—precisely the condition that defines the classical equation of motion for $\phi(t)$.

What makes the tadpole condition such a useful tool is that it also incorporates all of the higher order operators of $\varphi^{\pm}(t,\vec{x})$ once we have proceeded beyond linear order. Any product of operators with an odd number of $\varphi^{\pm}$ fields, when contracted with the external $\varphi^{+}(t,\vec{x})$ being evaluated, will produce loop corrections to the equation of motion for the background field $\phi(t)$. This procedure can be best illustrated through a familiar example.

Consider a scalar field with a quartic self-interaction,

$$S_{\Phi}=\int dtd^{3}\vec{x}\,\Bigl{\{}{\textstyle{1\over 2}}\partial_{\mu}\Phi\partial^{\mu}\Phi-{\textstyle{1\over 2}}m^{2}\Phi^{2}-{\textstyle{1\over 24}}\lambda\Phi^{4}\Bigr{\}}.$$

(19)

If we expand the field about its expectation value, $\Phi(t,\vec{x})=\phi(t)+\varphi^{\pm}(t,\vec{x})$, and gather the operators according to their powers of the quantum fluctuations, we find

	$\displaystyle S^{+}_{\Phi}-S^{-}_{\Phi}$	$\displaystyle=$	$\displaystyle\int dtd^{3}\vec{x}\,\Bigl{\{}{\textstyle{1\over 2}}\partial_{\mu}\varphi^{+}\partial^{\mu}\varphi^{+}-{\textstyle{1\over 2}}m^{2}\varphi^{+2}-\varphi^{+}\Bigl{[}\ddot{\phi}+m^{2}\phi+{\textstyle{1\over 6}}\lambda\phi^{3}\Bigr{]}$		(20)
			$\displaystyle\qquad\quad\ \,-\,{\textstyle{1\over 4}}\lambda\,\varphi^{+2}\phi^{2}-{\textstyle{1\over 6}}\lambda\,\varphi^{+3}\phi-{\textstyle{1\over 24}}\lambda\,\varphi^{+4}-(\varphi^{+}\leftrightarrow\varphi^{-})\Bigr{\}}.$

The first two terms correspond to the free Lagrangian for the field $\varphi^{+}(t,\vec{x})$. From the perspective of these fluctuations, the rest of the terms represent a set of interactions with the background $\phi(t)$. Notice that we have chosen to treat the ${1\over 2}\lambda\phi^{2}\varphi^{+2}$ as an interaction, even though it is quadratic in $\varphi^{+}(t,\vec{x})$. As long as ${1\over 2}\lambda\phi^{2}\ll m^{2}$, we can do so consistently; when this condition is not satisfied, we must instead define an effective, time dependent mass, $m^{2}_{\rm eff}(t)=m^{2}+{1\over 2}\lambda\phi^{2}(t)$. The behavior of the field in this regime can then be found numerically Boyanovsky:1994me .

For simplicity, let us choose the field to be in its free vacuum state initially, at $t=t_{0}$, $\rho(t_{0})=|0(t_{0})\rangle\langle 0(t_{0})|$, with its subsequent evolution given by the interaction terms in the action, $|0(t)\rangle=U_{I}(t,t_{0})|0(t_{0})\rangle$. If we evaluate the tadpole condition for this state to second order in $\lambda$, then we generate the set of graphs shown in figure 2.

The structure of the one-point function to this order is

$$\langle 0(t_{1})|\varphi(t_{1},\vec{x})|0(t_{1})\rangle=-\int_{t_{0}}^{t_{1}}dt\,{\sin[m(t_{1}-t)]\over m}\,\Bigl{\{}\cdots\Bigr{\}}=0;$$

(21)

the first factor in the integrand corresponds to the $\varphi$-propagator, with no spatial momentum flowing through it, and the ellipses corresponds to an equation of motion for the background field $\phi(t)$,

	$\displaystyle\ddot{\phi}(t)+m^{2}\phi(t)+{\lambda\over 6}\phi^{3}(t)+{\lambda\over 2}\phi(t)\int{d^{3}\vec{k}\over(2\pi)^{3}}\,{1\over 2\omega_{k}}$
	$\displaystyle\qquad-{\lambda^{2}\over 4}\phi(t)\int^{t}_{t_{0}}dt^{\prime}\,\phi^{2}(t^{\prime})\int{d^{3}\vec{k}\over(2\pi)^{3}}\,{\sin\left[2\omega_{k}(t-t^{\prime})\right]\over 2\omega^{2}_{k}}+{\cal O}(\lambda^{3})=0.\qquad$		(22)

The first three terms are just an instance of the general result that we found before, that the tree-level terms correspond to the classical equation of motion for $\phi(t)$. The first of the integral terms—and the second of the graphs in the figure—is a quadratically divergent loop integral. It is linear in $\phi(t)$, and it can be recognized as the familiar divergent correction to the mass, which can be absorbed by defining a renormalized mass for the field.

The last of the terms in the equation of motion, represented by the last of the graphs, is cubic in the background field, although two of the fields are evaluated at the intermediate time $t^{\prime}$ over which we are integrating. The integral over the spatial loop momentum is divergent, though the sine function obscures the precise form of this divergence somewhat since it vanishes at $t=t^{\prime}$. To isolate the divergent part of this graph, let us define the kernel,

$$K(t-t^{\prime})=\int{d^{3}\vec{k}\over(2\pi)^{3}}\,{\cos[2\omega_{k}(t-t^{\prime})]\over\omega_{k}^{3}}.$$

(23)

This function clearly has a logarithmic divergence for large values of $|\!|\vec{k}|\!|$ when $t=t^{\prime}$. Its $t^{\prime}$ derivative is essentially the loop integral that appears in the cubic correction to the equation of motion for the background field $\phi(t)$. We use this property to integrate it by parts to produce three terms

	$\displaystyle\int_{t_{0}}^{t}dt^{\prime}\,\phi^{2}(t^{\prime})\int{d^{3}\vec{k}\over(2\pi)^{3}}\,{\sin[2\omega_{k}(t-t^{\prime})]\over 2\omega^{2}_{k}}$	$\displaystyle=$	$\displaystyle{1\over 4}\int_{t_{0}}^{t}dt^{\prime}\,\phi^{2}(t^{\prime}){d\over dt^{\prime}}K(t-t^{\prime})$		(24)
		$\displaystyle=$	$\displaystyle{1\over 4}\phi^{2}(t)K(0)-{1\over 4}\phi^{2}(t_{0})K(t-t_{0})$
			$\displaystyle-{1\over 2}\int_{t_{0}}^{t}dt^{\prime}\,\phi(t^{\prime})\dot{\phi}(t^{\prime})K(t-t^{\prime}).$

Multiplying by the ${1\over 4}\lambda^{2}\phi(t)$ factor that appeared in the equation of motion, the first of these terms becomes ${1\over 8}\lambda^{2}K(0)\,\phi^{3}(t)$. It is the standard logarithmically divergent one-loop correction to the quartic coupling, which can be absorbed by a renormalization of $\lambda$. The final term in the equation includes dissipative effects, as is shown by the $\dot{\phi}(t^{\prime})$ within the integral. It occurs because the nonlinear coupling allows the background to transfer its energy into excitations of the quantum fluctuations.

The second term, once we have included the necessary prefactor, has the form

$$\textstyle-{1\over 16}\lambda^{2}\phi^{2}(t_{0})K(t-t_{0})\,\phi(t).$$

(25)

It is proportional to $\phi(t)$, which might seem to suggest that it is a mass term, albeit one with a time-dependent mass since the kernel depends on time. However, the field itself is also evaluated on the boundary, as shown by the factor $\phi^{2}(t_{0})$, which indicates that this term is actually associated with a quartic operator at the level of the full action for the original quantum field $\Phi(t,\vec{x})$. As we shall show later, the amplitude of the kernel decays as its argument grows large, $K(t-t_{0})\to 0$ as $t-t_{0}\to\infty$.

If we had prepared the system in the infinite past, $t_{0}\to-\infty$, then this term would vanish. Alternatively, if we had started with a free theory in the infinite past and turned on the interactions adiabatically, then this term would also never have appeared. The reason that the time-dependent terms have appeared is that we began evolving our system at $t_{0}$ from an initial state, the free vacuum, that is not an eigenstate of the interacting theory. This can alternatively be viewed as evolving the free theory starting in the infinite past with the free vacuum as the initial state, and then turning on the interactions at $t_{0}$. From either perspective, this term cannot be neglected, at least during a transient stage after $t_{0}$. We are interested here in studying theories where there is an explicit time-dependence in a setting where there also is a hierarchy of scales, so we shall next investigate how to construct effective field theories in such a setting.

Let us now examine a concrete example. Consider a theory where the action for the light $\Phi(t,\vec{x})$ and heavy $\chi(t,\vec{x})$ fields is given by

$$S=\int dtd^{3}\vec{x}\,\Bigl{\{}{\textstyle{1\over 2}}\partial_{\mu}\Phi\partial^{\mu}\Phi-{\textstyle{1\over 2}}m^{2}\Phi^{2}+{\textstyle{1\over 2}}\partial_{\mu}\chi\partial^{\mu}\chi-{\textstyle{1\over 2}}M^{2}\chi^{2}-{\textstyle{1\over 24}}\lambda\Phi^{4}-{\textstyle{1\over 24}}\tilde{\lambda}\chi^{4}-{\textstyle{1\over 2}}g\chi^{2}\Phi^{2}\Bigr{\}}.$$

(31)

and where there is a clear hierarchy of scales, $M\gg m$. Aside from their individual actions, the two fields interact through a term which is quadratic in both of the fields,

$$H_{I}(t)=\int d^{3}\vec{x}\,\Bigl{\{}{\textstyle{1\over 2}}g\Phi^{2}\chi^{2}\Bigr{\}}.$$

(32)

The time-dependence in this system is fairly subtle. As before we shall let the light field have a classical expectation value,

$$\langle 0(t)|\Phi(t,\vec{x})|0(t)\rangle=\phi(t),$$

while letting that of the heavy field vanish, $\langle 0(t)|\chi(t,\vec{x})|0(t)\rangle=0$. The background field $\phi(t)$ is a solution to its own equation of motion and while it has an explicit time-dependence, it does not itself break the symmetry of the original action. The true origin of the breaking of the time-translation symmetry resides in the fact that the evolution starts at a finite initial time, $t_{0}$, and in our choice of the states. Both fields are chosen initially to be in their free vacuum states, which we write as $|0(t_{0})\rangle=|0\rangle$, but the subsequent evolution is determined by the full action, which includes the interaction term. Therefore we do not start the system in the vacuum state of the interacting theory nor in any energy eigenstate of the interacting theory. We choose the free vacuum as a simple example of a nonequilibrium initial state and expect it to entail some interesting time evolution. One issue here is that it possibly contains excitations of heavy degrees of freedom or of light degrees of freedom at high momenta. Their presence would invalidate the usual description as a local effective field theory in terms of the light field and should cause some relaxation phenomenon as the heavy or high momentum degrees of freedom diminish with time.

We could make this time-dependence, which is present in the action through the limits of the integration, into one that appears explicitly in the Lagrangian by writing the interaction term as ${\textstyle{1\over 2}}\Theta(t-t_{0})g\chi^{2}\Phi^{2}$. We could then study the equivalent problem where the system is placed in the free vacua for both fields at $t\to-\infty$ and then allowed to evolve freely until $t=t_{0}$ when they begin to interact. From this perspective it is clear that the time-dependence is a sudden one, although it may not have initially appeared so. It should not be surprising therefore when later we find that the effective theory contains operators beyond the usual Poincaré invariant set.

To explore the structure of the effective theory, we evaluate the one-point function for the fluctuations of the light field, $\varphi(t,\vec{x})$, in the presence of a classical background, $\phi(t)$. Expanding the interaction term about this background, $\Phi^{\pm}(t,\vec{x})=\phi(t)+\varphi^{\pm}(t,\vec{x})$, we generate the following contributions to the interaction Hamiltonian,³³3We are again working in the limit where we can treat the ${1\over 2}g\phi^{2}\chi^{2}$ term as a part of the interactions rather than as a part of the mass term of the free part of the theory. This treatment should be consistent as long as $g\phi^{2}\ll M^{2}$, which is a natural requirement since $\phi$ is associated with the light and $M$ is associated with the heavy part of the theory.

$$H_{I}^{\pm}(t)=\int d^{3}\vec{x}\,\Bigl{\{}{\textstyle{1\over 2}}g\phi^{2}{\chi^{\pm}}^{2}+g\phi{\varphi^{\pm}}{\chi^{\pm}}^{2}+{\textstyle{1\over 2}}g{\varphi^{\pm}}^{2}{\chi^{\pm}}^{2}\Bigr{\}}.$$

(33)

This interaction produces a series of tadpole graphs which can be arranged according to the number of loops and external $\phi$ legs. At any given order in the coupling $g$ these graphs lead to a finite number of corrections to the $\phi(t)$ equation of motion. We evaluate the leading nontrivial one-loop correction completely, showing how it generates a series of terms with time-derivative of $\phi(t)$ which are suppressed by an appropriate power of $1/M$.

The truly leading one-loop contribution to the one-point function of the fluctuation is given by the contraction of the external $\varphi^{+}(t,\vec{x})$ field with the operator $g\phi\varphi^{\pm}(\chi^{\pm})^{2}$ in the interaction. This leads to the first graph shown in figure 3. It is another quadratically divergent correction to the mass of the light field, which can be removed by defining an appropriate physical mass for it. We assume that we have done so and consider the order $g^{2}$, one-loop correction to the equation of motion for the background. This graph has three external $\phi$-legs as shown in the figure.

Assuming that we have appropriately renormalized the mass of the light field, the equation of motion for $\phi(t)$ assumes the following form when we include the contribution from the second diagram in the figure,

$$\ddot{\phi}(t)+m^{2}\phi(t)+{1\over 6}\lambda\phi^{3}(t)-{1\over 2}g^{2}\phi(t)\int_{t_{0}}^{t}dt^{\prime}\,\phi^{2}(t^{\prime})\int{d^{3}\vec{k}\over(2\pi)^{3}}\,{\sin\bigl{[}2\omega_{k}(t-t^{\prime})\bigr{]}\over\omega_{k}^{2}}+\cdots=0.$$

(34)

Here $\omega_{k}$ is the once again the energy of the field running around the loop,

$$\omega_{k}=\sqrt{k^{2}+M^{2}},$$

though this time it is the heavy $\chi$ field that is doing so.

We recognize the structure of the last term from our earlier analysis of the $\lambda\Phi^{4}$ self-coupling. A crucial change from before is that now the integral over the loop momentum depends on the mass of the heavy field rather than the light one. It is this property that will allow us to expand this loop integral in a series of operators with derivatives of the background field $\phi(t)$ accompanied by appropriate powers of $1/M$. In the low energy limit, where $\phi(t)$ changes little during an interval $1/M$, operators with more derivatives will correspondingly be more suppressed.

As a first step in this expansion, we integrate by parts with respect to the $t^{\prime}$ dependence inside the loop-momentum integral. This step simultaneously increases the convergence of the integral, by adding another power of $\omega_{k}$ in the denominator, while moving a derivative to the $\phi^{2}(t^{\prime})$ factor. The formal structure of this loop correction is exactly what we met before in analyzing the loop corrections from the $\lambda\Phi^{4}$ self-coupling. Once again we define a kernel function,

$$K(t-t^{\prime})\equiv\int{d^{3}\vec{k}\over(2\pi)^{3}}\,{\cos\bigl{[}2\omega_{k}(t-t^{\prime})\bigr{]}\over\omega_{k}^{3}},$$

(35)

and integrate the loop correction to the $\phi$ equation by parts to generate three terms,

			$\displaystyle\!\!\!\!\!\!\!\!\!\!\!\!\!{1\over 2}g^{2}\phi(t)\int_{t_{0}}^{t}dt^{\prime}\,\phi^{2}(t^{\prime})\int{d^{3}\vec{k}\over(2\pi)^{3}}\,{\sin\bigl{[}2\omega_{k}(t-t^{\prime})\bigr{]}\over\omega_{k}^{2}}={1\over 4}g^{2}\phi(t)\int_{t_{0}}^{t}dt^{\prime}\,\phi^{2}(t^{\prime}){d\over dt^{\prime}}K(t-t^{\prime})$		(36)
		$\displaystyle=$	$\displaystyle{1\over 4}g^{2}\phi^{3}(t)K(0)-{1\over 4}g^{2}\phi(t)\phi^{2}(t_{0})K(t-t_{0})-{1\over 4}g^{2}\phi(t)\int_{t_{0}}^{t}dt^{\prime}\,{d\phi^{2}(t^{\prime})\over dt^{\prime}}K(t-t^{\prime}).$

The first term is another logarithmically divergent correction to the quartic self-interaction of the light field which can be renormalized by defining a physical coupling,

$$\lambda_{\rm phys}=\lambda+{\textstyle{3\over 2}}g^{2}K(0)+\cdots.$$

The second term is a mass term – $\phi^{2}(t_{0})$ is just a constant—but it is one with a time-dependent mass since the kernel $K(t-t_{0})$ depends explicitly on the time. We can evaluate this time-dependence exactly, once we have integrated over the loop-momentum, and express the result as a Meijer $G$-function,

$$K(t-t_{0})={M(t-t_{0})\over 8\pi}G^{20}_{13}\bigl{(}M^{2}(t-t_{0})^{2}\big{|}^{\ \ 1}_{-{1\over 2},-{1\over 2},0}\bigr{)}.$$

(37)

What is relevant for the effective theory of the light field, however, is only what happens over intervals that are long compared with the natural scale of the heavy theory, which in this case means $M(t-t_{0})\gg 1$. In this limit, the leading time-dependence of the kernel can be extracted through a saddle-point approximation of the momentum integral,

$$K(t-t_{0})={1\over 2\sqrt{e\pi^{3}}}{\cos\bigl{[}2M(t-t_{0})+{3\pi\over 4}\bigr{]}\over[2M(t-t_{0})]^{3/2}}+\cdots.$$

(38)

Although we have found that the two boundary terms from the integration by parts produce two corrections to the equation of motion—one local and familiar and the other nonlocal and a little less so—there still remains another unevaluated time integral. It might seem that we have merely postponed our difficulties by pushing them into this term. Does it also contain any further nonlocal dependence on $t-t_{0}$ and if so, what is it? Does it decay and is its decay rate different from what we have already seen?

For the one-loop correction of the previous section, where the loop contained a light field, there was little to be gained by further integrating the kernel by parts, once we extracted its logarithmically divergent piece. But when the loop field is heavy and the external legs are light, then each further integration by parts moves a time derivative from the part of the graph depending on the heavy field, which scales as ${d\over dt^{\prime}}\sim M$, to the light part, which scales as ${d\over dt^{\prime}}\sim m$. Iterating this process infinitely produces a series of contributions to the equation of motion which are further and further suppressed in the low energy effective theory.

Let us define the kernels that occur in each of the successive integrations by parts to be

	$\displaystyle K^{+}_{3+2n}(t-t^{\prime})$	$\displaystyle=$	$\displaystyle\int{d^{3}\vec{k}\over(2\pi)^{3}}\,{\cos[2\omega_{k}(t-t^{\prime})]\over\omega_{k}^{3+2n}}$
	$\displaystyle K^{-}_{2+2n}(t-t^{\prime})$	$\displaystyle=$	$\displaystyle\int{d^{3}\vec{k}\over(2\pi)^{3}}\,{\sin[2\omega_{k}(t-t^{\prime})]\over\omega_{k}^{2+2n}}.$		(39)

These function are even ($+$) or odd ($-$) in their arguments under $t-t^{\prime}\to t^{\prime}-t$. The original kernel that we introduced before is thus $K(t-t^{\prime})=K^{+}_{3}(t-t^{\prime})$. Each kernel is related to the one just above or just below it by a very simple recursion relation,

	$\displaystyle K^{-}_{2n}(t-t^{\prime})={1\over 2}{d\over dt^{\prime}}K^{+}_{2n+1}(t-t^{\prime})$
	$\displaystyle K^{+}_{2n+1}(t-t^{\prime})=-{1\over 2}{d\over dt^{\prime}}K^{-}_{2n+2}(t-t^{\prime}).$		(40)