Quantum field theory 2, lecture 02

Partition function

To summarize, the quantum partition function for a transition amplitude, in the presence of a source field \(J\), an external U\((1)\) gauge field \(A_\mu(x)\) and a metric \(g_{\mu\nu}(x)\), is given by a functional integral, \[Z[J,A,g] = \int D\phi \, e^{iS[\phi,A,g]+i\int_x \{ J^* \phi+\phi^* J \}},\] with the action \[S[\phi, A, g] = \int_x \left\{ -g^{\mu\nu} [\partial_\mu + i q A_\mu] \phi^* [\partial_\nu-iqA_\nu]\phi - V(\phi^*\phi) + i \epsilon \phi^* \phi \right\}.\] We use here the abbreviation \(\int_x = \int d^d x \sqrt{g(x)}\) for the integral over space and time.

Field expectation value and correlation function

From the partition function one can take one functional derivative with respect to the source to obtain \[\langle \Omega_\text{f} | \phi(x) | \Omega_\text{in} \rangle = \frac{1}{Z[J,A,g]} \left[-i \frac{1}{\sqrt{g(x)}}\frac{\delta}{\delta J^*(x)} \right] Z[J,A,g].\] For the special case where initial and final quantum states agree, \(|\Omega_\text{in} \rangle = | \Omega_\text{f} \rangle = | \Omega \rangle\), this is an expectation value of the field in that particular quantum state and we use the notation \[\langle \phi(x) \rangle = \langle \Omega | \phi(x) | \Omega \rangle.\] We assume this situation in the following, and \(|\Omega \rangle\) is the vacuum state if we use the \(i\epsilon\)-prescription and send \(t_\text{in}\to -\infty\) and \(t_\text{f}\to \infty\).

Similarly, second functional derivatives yield correlation functions, \[\begin{split} \langle \phi(x) \phi(y) \rangle = & \langle \Omega | \phi(x) \phi(y) | \Omega \rangle \\ = & \frac{1}{Z[J,A,g]} \left[-i \frac{1}{\sqrt{g(x)}}\frac{\delta}{\delta J^*(x)} \right] \left[-i \frac{1}{\sqrt{g(y)}}\frac{\delta}{\delta J^*(y)} \right] Z[J,A,g]. \end{split}\] Typically this will be evaluated at vanishing source, i. e. by setting \(J=0\) after the functional derivatives have been taken. The gauge field \(A_\mu\) and metric \(g_{\mu\nu}\) might be non-trivial, however.

In a similar way one can obtain expectation values of the electromagnetic current \(J^\mu(x)\) or the energy-momentum tensor \(T^{\mu\nu}(x)\) by taking functional derivatives of \(Z[J,A,g]\) with respect to the gauge field \(A_\mu(x)\) or the metric \(g_{\mu\nu}(x)\), respectively.

Schwinger functional

The generating functional for connected correlation functions can be introduced by setting \[W[J,A,g] = -i \ln Z[J,A,g],\] or equivalently, \[e^{iW[J,A,g]} = Z[J,A,g].\] This is the generating functional for conneced correlation functions, e. g. for \(g_{\mu\nu}=\eta_{\mu\nu}\) and \(A_\mu=0\), \[\begin{split} \langle \phi(x) \phi(x) \rangle_c = & \left[ -i\frac{\delta}{\delta J^*(x)} \right] \left[ -i\frac{\delta}{\delta J^*(y)} \right] i W[J] \\ = & \langle \phi(x) \phi(x) \rangle - \langle \phi(x) \rangle \langle \phi(x) \rangle. \end{split}\]

Field equation for the Schwinger functional

The Schwinger functional satisfies in particular \[\frac{1}{\sqrt{g(x)}} \frac{\delta}{\delta J^*(x)} W[J,A,g] = \langle \phi(x) \rangle,\] which can be seen as a field equation. Because we need the expectation value more often in the following we introduce a separate notation, \[\Phi(x) = \langle \phi(x) \rangle.\]

Propagator

The full connected two-point function, or Feynman propagator, follows from the Schwinger functional within the \(i\epsilon\) formalism as \[\Delta(x,y) = i \langle \phi(x) \phi^*(y) \rangle_c = \frac{1}{\sqrt{g(x)}\sqrt{g(y)}}\left[ \frac{\delta}{\delta J^*(x)} \right] \left[ \frac{\delta}{\delta J(y)} \right] W[J,A,g].\]

The connected correlation function has the property that it decays for large space-like separation, \[\lim_{|\mathbf{x}-\mathbf{y}|\to \infty} \Delta(x,y) =0.\] For the full correlation function \(\langle \phi(x) \phi(y) \rangle = \Phi(x) \Phi(y) + \langle \phi(x) \phi(y) \rangle_c\) this may not hold, but it factorizes then into a product of expectation values in the large separation limit.

Quantum effective action

We introduce another generating functional, the quantum effective action or one-particle irreducible effective action. It is defined as a Legendre transform of the Schwinger functional, \[\Gamma[\Phi] = \sup_J \left( \int_x \left\{ J^*(x) \Phi(x) + \Phi^*(x) J(x) \right\} - W[J] \right).\] The field \(\Phi(x)\) is here the expectation value of \(\phi(x)\), as follows from the condition that the variation with respect to \(J^*(x)\) must vanish as the supremum, \[\Phi(x) = \frac{1}{\sqrt{g(x)}} \frac{\delta}{\delta J^*(x)} W[J] = \langle \phi(x)\rangle.\] Because \(J\) is always evaluated at the supremum, one should understand it to be a functional of the expectation value \(J[\Phi]\) on the right hand side of the definition for \(\Gamma[\Phi]\). Because \(W\) depends also on \(A_\mu\) and \(g_{\mu\nu}\), this is also the case for the effective action, \(\Gamma[\Phi, A, g]\).

Quantum field equation

The quantum effective action satisfies a particularly interesting field equation, \[\frac{\delta}{\delta \Phi(x)} \Gamma[\Phi] = \sqrt{g(x)} J^*(x).\] In particular, for \(J=0\) it is stationary, \(\delta\Gamma/\delta\Phi(x)=0\). This is very similar to a classical action, which is stationary on the equations of motion (principle of stationary action). In fact, one should see \(\Gamma[\Phi]\) as playing the role of the classical action.

Let us prove the field equation. We first use the chain rule, \[\begin{split} \frac{\delta\Gamma}{\delta \Phi(x)} = & - \frac{\delta}{\delta \Phi(x)} W[J] + \int d^d y \sqrt{g} \left\{ \frac{\delta J^*(y)}{\delta\Phi(x)} \Phi(y) + \Phi^*(y) \frac{\delta J(y)}{\delta \Phi(x)} \right\} + \sqrt{g} J^*(x), \end{split}\] and can use then \[\begin{split} \frac{\delta}{\delta \Phi(x)} W[J] = & \int d^d y \left\{ \frac{\delta W}{\delta J(y)} \frac{\delta J(y)}{\delta \Phi(x)} + \frac{\delta W}{\delta J^*(y)} \frac{\delta J^*(y)}{\delta \Phi(x)} \right\} \\ = & \int d^d y \sqrt{g} \left\{ \Phi^*(y) \frac{\delta J(y)}{\delta \Phi(x)} + \Phi(y) \frac{\delta J^*(y)}{\delta \Phi(x)} \right\}. \end{split}\] This implies a cancelation of terms and directly yields the field equation as claimed.

Inverse propagator

From the second functional derivative of the quantum effective action one obtains the inverse propagator, \[\Gamma^{(2)}_{jk}(x,y)[\Phi] = \left[ \frac{1}{\sqrt{g(x)}} \frac{\delta}{\delta \Phi_j(x)} \right] \left[ \frac{1}{\sqrt{g(x)}} \frac{\delta}{\delta \Phi_k(y)} \right] \Gamma[\Phi] = P_{jk}(x,y).\] We work here for convenience with a basis of real fields \(\Phi_j(x)\). As an (infinite dimensional) matrix, or operator, this is in fact inverse to the second functional derivative of the Schwinger functional, \[W^{(2)}_{jk}(x,y)[J] = \left[ \frac{1}{\sqrt{g(x)}} \frac{\delta}{\delta J_j(x)} \right] \left[ \frac{1}{\sqrt{g(x)}} \frac{\delta}{\delta J_k(y)} \right] W[J] = \Delta_{jk}(x,y),\] in the sense that \(P^{-1} = \Delta\) or, more explicitely, \[\int d^d \sqrt{g} \left\{ P_{jk}(x,y) \Delta_{kl}(y,z) \right\} = \frac{1}{\sqrt{g(x)}} \delta^{(d)}(x-z) \delta_{jl}.\] Typically \(P=\Gamma^{(2)}\) is actually a generalized derivative operator, and \(\Delta(x,y)\) is its Greens function. To prove the above we note that \[\begin{split} P_{jk}(x,y) = & \frac{1}{\sqrt{g(x)}} \frac{\delta}{\delta \Phi_j(x)} J_k(y),\\ \Delta_{kl}(y,z) = & \frac{1}{\sqrt{g(y)}} \frac{\delta}{\delta J_k(y)} \Phi_l(z), \end{split}\] and therefore \[\begin{split} & \int d^dy \sqrt{g(y)} P_{jk}(x,y) \Delta_{kl}(y,z) = \frac{1}{\sqrt{g(x)}} \int d^d y \frac{\delta J_k(y)}{\delta \Phi_j(x)} \frac{\delta \Phi_l(z)}{\delta J_k(y)} \\ & = \frac{1}{\sqrt{g(x)}} \frac{\delta\Phi_l(z)}{\delta \Phi_j(x)} = \frac{1}{\sqrt{g(x)}} \delta^{(d)}(x-z) \delta_{jl}, \end{split}\] as claimed.

From the inverse of the second functional derivative of the microscopic action \(S[\phi]\) one would obtain a microscopic (or bare or un-renormalized) propagator, while the inverse of \(\Gamma^{(2)}\) gives the full propagator, including all quantum corrections!