Quantum Field Theory – Proper vs Effective Action and Their Quantum Corrections

1pi-effective-actionactionlagrangian-formalismpath-integralquantum-field-theory

There is a difference between the classical field $\phi(x)$ (which appears in the classical action $S[\phi]$) and the quantity $\phi_c$ defined as $$\phi_c(x)\equiv\langle 0|\hat{\phi}(x)|0\rangle_J$$ which appears in the effective action. Even though $\phi_c(x)$ is referred to as the "classical field", I don't see why $\phi(x)$ and $\phi_c$ should be the same.

In what sense, therefore, is the effective action $\Gamma[\phi_c]$ a quantum-corrected classical action $S[\phi]$? How can we compare the functionals of two different objects (namely, $\phi(x)$ and $\phi_c(x)$) and claim that $\Gamma[\phi_c]$ is a correction over $S[\phi]$?

I apologize for any lack of clarity in the question and the confusion I'm hoping to clear up.

Best Answer

There is already a good answer by Solenodon Paradoxus. Here we provide a formal proof (via the stationary phase/WKB approximation).

  1. To fix notation, we define the 1PI effective/proper action $$ \Gamma[\phi_{\rm cl}]~=~W_c[J]-J_k \phi_{\rm cl}^k, \tag{1}$$ as the Legendre transformation of the generating functional $W_c[J]$ for connected diagrams. We assume that the Legendre transformation is regular, i.e. the formula $$\begin{align} \phi_{\rm cl}^k~=~&\frac{\delta W_c[J]}{\delta J_k} \cr \Updownarrow~& \cr J_k~=~&-\frac{\delta \Gamma[\phi_{\rm cl}]}{\delta \phi_{\rm cl}^k}\end{align} \tag{2}$$ is invertible. Here $J_k$ are the sources and $\phi_{\rm cl}^k$ are the so-called classical fields. (The latter terminology is a bit of a misnormer as $\phi_{\rm cl}^k[J]$ as a function of the sources $J_{\ell}$ could depend explicitly on $\hbar$. See also section 8 below.)

  2. The partition function/path integral is $$\begin{align} \exp&\left\{ \frac{i}{\hbar} W_c[J]\right\}\cr ~=~&Z[J]\cr ~:=~&\int \! {\cal D}\frac{\phi}{\sqrt{\hbar}}~\exp\left\{ \frac{i}{\hbar} \left(S[\phi]+J_k \phi^k\right)\right\} . \end{align}\tag{3}$$ The first equality in eq. (3) is the linked cluster theorem, cf. e.g. this Phys.SE post.

  3. At this place it is customary to mention some elementary facts. The 1-pt function/quantum averaged field is by definition $$\begin{align} \langle \phi^k \rangle_J ~:=~&\frac{1}{Z[J]} \int \! {\cal D}\frac{\phi}{\sqrt{\hbar}}~\phi^k\exp\left\{ \frac{i}{\hbar} \left(S[\phi]+J_{\ell} \phi^{\ell}\right)\right\}\cr ~=~&\frac{1}{Z[J]} \frac{\hbar}{i} \frac{\delta }{\delta J_k}\int \! {\cal D}\frac{\phi}{\sqrt{\hbar}}~\exp\left\{ \frac{i}{\hbar} \left(S[\phi]+J_{\ell} \phi^{\ell}\right)\right\}\cr ~\stackrel{(3)}{=}~&\frac{1}{Z[J]} \frac{\hbar}{i}\frac{\delta Z[J]}{\delta J_k}\cr ~\stackrel{(3)}{=}~&\frac{\delta W_c[J]}{\delta J_k} ~\stackrel{(2)}{=}~\phi_{\rm cl}^k. \end{align} \tag{4}$$

  4. The 2-pt function is by definition $$\begin{align} \langle \phi^k \phi^{\ell}\rangle_J ~:=~&\frac{1}{Z[J]} \int \! {\cal D}\frac{\phi}{\sqrt{\hbar}}~\phi^k\phi^{\ell}\exp\left\{ \frac{i}{\hbar} \left(S[\phi]+J_m \phi^m\right)\right\}\cr ~\stackrel{(3)}{=}~&\frac{1}{Z[J]} \left(\frac{\hbar}{i}\right)^2\frac{\delta^2 Z[J]}{\delta J_k\delta J_{\ell}}~\cr \stackrel{(3)}{=}~&\frac{1}{Z[J]} \frac{\hbar}{i} \frac{\delta}{\delta J_k} \left(Z[J]\frac{\delta W_c[J]}{\delta J_{\ell}}\right)\cr ~\stackrel{(4)}{=}~&\frac{\hbar}{i} \frac{\delta^2 W_c[J]}{\delta J_k\delta J_{\ell}} + \langle \phi^k \rangle_J \langle \phi^{\ell} \rangle_J,\end{align} \tag{5}$$ i.e. the connected 2-pt function plus a disconnected piece.

  5. Now let us return to OP's question. By formal inverse Fourier transformation of the path integral (3), we get $$\begin{align} \exp&\left\{ \frac{i}{\hbar}S[\phi_{\rm cl}]\right\}\cr ~\stackrel{(3)}{=}~&\int \! {\cal D}\frac{J}{\sqrt{\hbar}}~\exp\left\{ \frac{i}{\hbar} \left(W_c[J]-J_k \phi_{\rm cl}^k\right)\right\} \cr ~\stackrel{\text{WKB}}{\sim}& {\rm Det}\left(\frac{1}{i}\frac{\delta^2 W_c[J[\phi_{\rm cl}]]}{\delta J_k \delta J_{\ell}}\right)^{-1/2} \exp\left\{ \frac{i}{\hbar}\Gamma[\phi_{\rm cl}]\right\}\left(1+ {\cal O}(\hbar)\right) \cr ~\stackrel{(8)}{=}~& {\rm Det}\left(\frac{1}{i}\frac{\delta^2 \Gamma[\phi_{\rm cl}]}{\delta \phi_{\rm cl}^k \delta \phi_{\rm cl}^{\ell}}\right)^{1/2} \exp\left\{ \frac{i}{\hbar}\Gamma[\phi_{\rm cl}]\right\}\left(1+ {\cal O}(\hbar)\right)\cr &\quad\text{for}\quad\hbar~\to~0 \end{align} \tag{6}$$ in the stationary phase/WKB approximation $J_k=J_k[\phi_{\rm cl}]+\sqrt{\hbar}\eta_k$. In the last equality of eq. (6), we used that $$\begin{align}\delta^k_{\ell} ~=~&\frac{\delta \phi_{\rm cl}^k[J[\phi_{\rm cl}]]}{\delta\phi_{\rm cl}^{\ell}}\cr ~=~&\frac{\delta \phi_{\rm cl}^k[J[\phi_{\rm cl}]]}{\delta J_m} \frac{\delta J^m[\phi_{\rm cl}]}{\delta\phi_{\rm cl}^{\ell}} \cr ~\stackrel{(2)}{=}~& -\frac{\delta^2 W_c[J[\phi_{\rm cl}]]}{\delta J_k\delta J_m} \frac{\delta^2 \Gamma[\phi_{\rm cl}]}{\delta\phi_{\rm cl}^m\delta\phi_{\rm cl}^{\ell}},\end{align} \tag{7}$$ i.e.

    $$\begin{align}&\text{The 2-pt functions }\cr & \frac{1}{i}\frac{\delta^2 W_c[J]}{\delta J_k\delta J_m} \text{ and } \frac{1}{i}\frac{\delta^2 \Gamma[\phi_{\rm cl}]}{\delta\phi_{\rm cl}^m\delta\phi_{\rm cl}^{\ell}}\cr& \text{ are inverses of each other.} \end{align}\tag{8}$$

  6. We will assume that the action $S$ has no explicit $\hbar$-dependence. The effective action $\Gamma[\phi_{\rm cl}]=\sum_{n=0}^{\infty}\Gamma_n[\phi_{\rm cl}]$ becomes a $\hbar$/loop-expansion. Eq. (6) shows that the effective action $$\begin{align} \Gamma[\phi_{\rm cl}] ~\stackrel{(6)}{=}~& S[\phi_{\rm cl}] +\frac{i\hbar}{2}\ln {\rm Det}\left(\frac{1}{i}\frac{\delta^2 \Gamma[\phi_{\rm cl}]}{\delta \phi_{\rm cl}^k \delta \phi_{\rm cl}^{\ell}}\right) +{\cal O}(\hbar^2) \tag{9} \cr ~\stackrel{(9)}{=}~& S[\phi_{\rm cl}] +\frac{i\hbar}{2}\ln {\rm Det}\left(\frac{1}{i} H_{k\ell}[\phi_{\rm cl}]\right) +{\cal O}(\hbar^2) \tag{10}\end{align}$$ agrees with the action $S$ up to quantum corrections. In eq. (10) we have defined the Hessian $$ H_{k\ell}[\phi]~:=~ \frac{\delta^2 S[\phi]}{\delta\phi^k\delta\phi^{\ell}}. \tag{11} $$ (The square root factor in eq. (6) only contributes at one-loop and beyond.)

    In other words, we deduce that to zeroth-order in $\hbar$/tree diagrams in the effective action

    $$\text{Tree-level}:~~ \Gamma_0[\phi_{\rm cl}] ~\stackrel{(9)}{=}~S[\phi_{\rm cl}] \tag{12}$$

    is equal to the action $S$ itself. Similarly, we deduce that to first-order in $\hbar$/one-loop diagrams in the effective action

    $$\text{1-loop}:~~ \Gamma_1[\phi_{\rm cl}] ~\stackrel{(10)}{=}~\frac{i\hbar}{2}\ln {\rm Det}\left(\frac{1}{i} H_{k\ell}[\phi_{\rm cl}] \right) \tag{13}$$

    is equal to a functional determinant of the Hessian of the action $S$. Eqs. (10), (12) & (13) answer OP's question. See also this related Phys.SE post.

  7. At this place it is customary to mention some elementary facts. Let there be given fixed sources $J_k$. From$^1$ $$\begin{align} \frac{\delta \Gamma[\phi_{\rm cl}]}{\delta \phi_{\rm cl}^k} ~\stackrel{(2)}{=}~~&-J_k\cr ~\stackrel{\text{EL eqs.}}{\approx}& \frac{\delta S[\phi_0]}{\delta \phi^k} \cr ~=:~~&E_k[\phi_0], \end{align} \tag{14} $$ we deduce that the so-called classical solution $\phi_{\rm cl}^k$ and the Euler-Lagrange (EL) solution $\phi_0^k$ agree$^1$ $$ \phi_{\rm cl}^k[J]~\stackrel{(9)+(14)}{\approx}~\phi_0^k[J] +{\cal O}(\hbar) \tag{15} $$ up to quantum corrections. Eq. (15) justifies the practice to call $\phi_{\rm cl}^k$ the classical field. (We assume that each solution to eq. (14) is unique, due to pertinent boundary conditions. We have excluded instantons for simplicity.)

    Conversely, if we are given a $\phi_{\rm cl}$, we can consider the corresponding shifted source $$\begin{align} J_k^{>0}[\phi_{\rm cl}]~:=~&E_k[\phi_{\rm cl}]+J_k[\phi_{\rm cl}]\cr ~\stackrel{(2)}{=}~&\frac{\delta S[\phi_{\rm cl}]}{\delta \phi^k_{\rm cl}} -\frac{\delta \Gamma[\phi_{\rm cl}]}{\delta \phi^k_{\rm cl}}\cr ~\stackrel{(12)}{=}~&-\frac{\delta \Gamma_{>0}[\phi_{\rm cl}]}{\delta \phi^k_{\rm cl}} ~=~{\cal O}(\hbar). \end{align}\tag{16} $$

  8. Alternatively, from the background field method $$ \underbrace{\phi^k}_{\text{quan. field}} ~=~\overbrace{\underbrace{\phi^k_{\rm cl}}_{\text{clas. field}}}^{\text{backgr. field}}+\underbrace{\eta^k}_{\text{fluctuation}}, \tag{17}$$ the effective action (1) becomes $$\begin{align}\exp&\left\{\frac{i}{\hbar}\Gamma[\phi_{\rm cl}]\right\}\cr ~\stackrel{(1)+(3)}{=}& \int\!{\cal D}\frac{\phi}{\sqrt{\hbar}} ~\exp\left\{\frac{i}{\hbar} \left(S[\phi] +J_k[\phi_{\rm cl}](\phi^k-\phi^k_{\rm cl}) \right) \right\} \cr ~\stackrel{(17)}{=}~& \int\!{\cal D}\frac{\eta}{\sqrt{\hbar}} ~\exp\left\{\frac{i}{\hbar} \left(S[\phi_{\rm cl}+\eta] +J_k[\phi_{\rm cl}] \eta^k \right)\right\} \cr ~=~& \int\!{\cal D}\frac{\eta}{\sqrt{\hbar}} ~\exp\left\{\frac{i}{\hbar} \left( S[\phi_{\rm cl}] +\underbrace{\left(E_k[\phi_{\rm cl}] +J_k[\phi_{\rm cl}]\right)}_{={\cal O}(\hbar)} \eta^k +\frac{1}{2}\eta^k H_{k\ell}[\phi_{\rm cl}] \eta^{\ell} +{\cal O}(\eta^3) \right)\right\} \cr ~\stackrel{\text{WKB}}{\sim}& {\rm Det}\left(\frac{1}{i}H_{mn}[\phi_{\rm cl}] \right)^{-1/2}\left(1+ {\cal O}(\hbar)\right) \exp\left\{ \frac{i}{\hbar}\left(S[\phi_{\rm cl}] -\frac{1}{2}J_k^{>0}[\phi_{\rm cl}] (H^{-1})^{k\ell}[\phi_{\rm cl}] J_{\ell}^{>0}[\phi_{\rm cl}] \right)\right\} \cr ~\stackrel{(2)+(15)}{=}& {\rm Det}\left(\frac{1}{i}H_{mn}[\phi_{\rm cl}]\right)^{-1/2}\exp\left\{ \frac{i}{\hbar}S[\phi_{\rm cl}]\right\}\left(1+ {\cal O}(\hbar)\right)\cr &\quad\text{for}\quad\hbar~\to~0 \end{align} \tag{18}$$ in the stationary phase/WKB approximation $$\eta^k~=~ -(H^{-1})^{k\ell}[\phi_{\rm cl}]J_{\ell}^{>0}[\phi_{\rm cl}] + \underbrace{{\cal O}(\sqrt{\hbar})}_{\text{fluctuation}}.\tag{19}$$ Eq. (18) again leads to the sought-for eq. (10).

  9. More generally, if we separate the action $$ S[\phi]~=~ \underbrace{E_k[\phi_{\rm cl}]\eta^k}_{\text{linear part}} + \underbrace{\frac{1}{2}\eta^k H_{k\ell}[\phi_{\rm cl}]\eta^{\ell}}_{\text{quadratic part}} +\underbrace{S_{\neq 12}[\phi_{\rm cl},\eta]}_{\text{the rest}}, \tag{20}$$ then the effective action reads to all orders $$\begin{align}\exp&\left\{\frac{i}{\hbar}\Gamma[\phi_{\rm cl}]\right\}\cr ~\stackrel{\begin{array}{c}\text{Gauss.}\cr\text{int.}\end{array}}{\sim}& {\rm Det}\left(\frac{1}{i}H_{mn}[\phi_{\rm cl}]\right)^{-1/2} \cr &\exp\left\{ \frac{i}{\hbar} S_{\neq 12}\left[\phi_{\rm cl},\frac{\hbar}{i}\frac{\delta}{\delta J_k[\phi_{\rm cl}]} \right]\right\} \cr &\exp\left\{ -\frac{i}{2\hbar}J_k^{>0}[\phi_{\rm cl}] (H^{-1})^{k\ell}[\phi_{\rm cl}] J_{\ell}^{>0}[\phi_{\rm cl}] \right\}\end{align}\tag{21}$$ after a Gaussian integration. It follows that $$\begin{align}\frac{i}{\hbar}&\Gamma_{>1}[\phi_{\rm cl}]\cr ~\stackrel{(12)+(13)+(21)}{=}& \ln\left(\exp\left\{ \frac{i}{\hbar} S_{\neq 012}\left[\phi_{\rm cl},\frac{\hbar}{i}\frac{\delta}{\delta J_k[\phi_{\rm cl}]} \right]\right\}\right. \cr &\left. \exp\left\{ -\frac{i}{2\hbar}J_k^{>0}[\phi_{\rm cl}] (H^{-1})^{k\ell}[\phi_{\rm cl}] J_{\ell}^{>0}[\phi_{\rm cl}] \right\}\right)\end{align}\tag{22}$$ is the sum of all connected diagrams made out of propagators $-(H^{-1})^{k\ell}[\phi_{\rm cl}]$; shifted external sources $J_k^{>0}[\phi_{\rm cl}]$; and $\eta$-vertices with $\geq 3$ $\eta$-legs.

    After substituting $J^{>0}_k[\phi_{\rm cl}]=-\delta \Gamma_{>0}[\phi_{\rm cl}]/\delta \phi_{\rm cl}^k$ on the RHS of eq. (22) via the relation (16), then one may show that eq. (22) becomes an all-order recursion relation for the effective action $\Gamma[\phi_{\rm cl}]$.

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$^1$ The $\approx$ symbol means here equality modulo the Euler-Lagrange (EL) equations.

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