Quantum field theory 2, lecture 03
Abstract index notation
The following is this convenient to use an abstract index notation where a greek index combines the continuous position index and the discrete field index, \[\alpha = (x, j)\] and sums over \(\alpha\) are abbreviations for integrals over \(x\) and sums over \(j\) with the appropriate factors, e. g. \[\sum_\alpha = \sum_j \int d^d x \sqrt{g},\] and we write \[\Phi^\alpha = \Phi_j(x), \quad\quad\quad J_\alpha = J_j(x).\] In abstract index notation one has also the unit element \[\delta^\alpha_{\;\;\beta} = \frac{1}{\sqrt{g}(x)} \delta^{(d)}(x-y) \delta_{j k},\] where \(\alpha=(x, j)\) and \(\beta = (y,k)\).
When working in Minkowski space one can expand fields in momentum modes, \[\Phi_j(x) = \int \frac{d^d p}{(2\pi)^d} e^{ipx} \Phi_j(p),\] and \(\alpha\) can then also represent the combination \((p,j)\) with \[\sum_\alpha = \sum_j \int \frac{d^d p}{(2\pi)^d}, \quad\quad\quad \delta^\alpha_{\;\;\beta} = (2\pi)^d \delta^{(d)}(p-q) \delta_{j k}.\] This works similarly for other expansion schemes. With a bit of experience one can easily translate expressions involving abstract indices to the concrete functional or integral expressions.
One should nevertheless keep in mind that infinite functional vector spaces have partly more involved mathematical properties than finite vector spaces, and the associated mathematical subtleties and complications should not be forgotten through the use of abstract indices.
Legendre transform
We have in abstract index notation \[\Gamma[\Phi] = \sup_J \left( J_\alpha \Phi^\alpha - W[J] \right).\] This is a Legendre transform and can be inverted where it is well defined, \[W[J] = \sup_\Phi \left( J_\alpha \Phi^\beta - \Gamma[\Phi] \right).\] We now take a few functional derivatives to see what the physical role of \(\Gamma[\Phi]\) is. We have the first derivatives \[\begin{split} W^\alpha = & \frac{\delta}{\delta J_\alpha} W[J] = \Phi^\alpha,\\ \Gamma_\alpha = & \frac{\delta}{\delta \Phi^\alpha} \Gamma[\phi] = J_\alpha, \end{split}\] and the second functional derivatives \[W^{\alpha\beta} = \frac{\delta^2}{\delta J_\alpha\delta J_\beta} W[J] = \frac{\delta}{\delta J_\alpha} \Phi^\beta,\] as well as its inverse \[\Gamma_{\alpha\beta} = \frac{\delta^2}{\delta\Phi^\alpha\delta\Phi^\beta} \Gamma[\Phi] = \frac{\delta}{\delta\Phi^\alpha} J_\beta.\] These are just relation we have derived previously, now written in terms of abstract indices.
Correlation functions and graphical representation
Taking further functional derivatives of \(W\) we obtain \[\begin{split} W^{\alpha\beta\gamma} = & \frac{\delta^3}{\delta J_\alpha \delta J_\beta \delta J_\gamma} W[J] = \frac{\delta}{\delta J_\alpha} \left( \Gamma^{(2)}[\Phi]^{-1} \right)^{\beta\gamma} \\ = & \frac{\delta\Phi^{\tilde\alpha}}{\delta J_\alpha} \frac{\delta}{\delta \Phi^{\alpha}} \left( \Gamma^{(2)}[\Phi]^{-1} \right)^{\beta\gamma} \\ = & - W^{\alpha\tilde\alpha} W^{\beta\tilde\beta} W^{\gamma\tilde\gamma} \Gamma^{(3)}_{\tilde\alpha\tilde\beta\tilde\gamma}. \end{split}\] We used here that \(\delta(M^{-1}) = M^{-1} \delta M M^{-1}\) for any matrix \(M\).
Similarly, the connected four-point correlation function is given by \[\begin{split} W^{\alpha\beta\gamma\delta} = & - W^{\alpha\tilde\alpha} W^{\beta\tilde\beta} W^{\gamma\tilde\gamma} W^{\delta\tilde\delta} \Gamma_{\tilde\alpha\tilde\beta\tilde\gamma\tilde\delta} \\ & + W^{\alpha\tilde\alpha} W^{\delta\tilde\delta} \Gamma_{\tilde\alpha\tilde\delta\lambda} W^{\lambda\kappa} \Gamma_{\kappa\tilde\beta\tilde\gamma} W^{\beta\tilde\beta} W^{\gamma\tilde\gamma}\\ & + \text{2 permutations}. \end{split}\] These are in fact tree-level diagrams where propagators are full propagators, \[\Delta = W^{(2)} = (\Gamma^{(2)})^{-1},\] and vertices are given by \(\Gamma^{(3)}\) and \(\Gamma^{(4)}\) !
One-particle irreducible
What is then actually the meaning of one-particle irreducible? This has a diagramatic meaning in terms of perturbation theory. For diagrams that are one-particle-irreducible it is not possible to cut them into pieces – to reduce them – by opening just a single internal line.
One observes that all diagrams can be composed out of sets of one=particle irreducible diagrams and it corresponds to the construction above, with full propagators and vertices from the quantum effective action \(\Gamma[\Phi]\). For this reason the latter is also known as one-particle irreducible or 1-P. I. effective action.
Quantum effective action and S-matrix
Recall that the S-matrix elements can be obtained from connceted correlation functions with external propagators removed, or “amputated”. But such a removing of external propagators also happens in the transition from connected correlation functions to one-particle-irreducible effective action!
In fact, one can write the transition matrix amplitude for \(n\to m\) particle scattering problems in the from \[\mathcal{T} (2\pi)^d \delta^{(d)}(p^\text{out}-p^\text{in}) = -\left\{\Gamma^{(n+m)}[\Phi_\text{eq}] + \text{tree terms} \right\}.\] The functional derivatives on the right hand side must be evaluated with \(\Phi_\text{eq}\) a solution of the field equation (usually a homogeneous or even vanishing field configuration), and the momenta of the incoing and outgoing particles must be on-shell.
The quantum effective action we use here is in the real time formalism and fundamentally based on the vacuum-to-vacuum transition amplitude. Functional derivatives of this effective action describe particle excitations. The additional tree terms involve lower order derivatives of the quantum effective action, \(\Gamma^{(p)}\) with \(p<n+m\).
An example is \(2\to 2\) scattering where the transition amplitude \(\mathcal{T}(p_1,p_2,p_3,p_4)\) has the contributions
The internal line corresponds to the full propagator. We observe that for known 1-P. I. -vertices only tree diagram appear. There are no more loops, since the fluctuating effects are already incorporated into the computation of the quantum effective action.
Summary
In summary, the quantum effects of fluctuations change the microscopic action \(S[\phi]\) to the quantum effective or macroscopic action \(\Gamma[\Phi]\). The latter includes all fluctuation effects. Once \(\Gamma[\Phi]\) is known, only tree diagrams have to be evaluated for the computation of the transition amplitude \(\mathcal{T}\)! The full propagator and the full vertices in tree diagrams are given by the propagator \((\Gamma^{(2)}[\Phi])^{-1}\) and the one-particle-irreducible vertices \(\Gamma^{(n)}[\Phi]\) with \(n\geq 3\). In order to compute the transition amplitude one can follow the receipe:
Compute \(\Gamma[\Phi]\).
Draw all tree diagrams.
Insert full propagator for lines and full vertices.
Funnctional integral representation for quantum effective action
We derive now an expresion for the quantum effective action in terms of functional integrals. We start with \[e^{-i\Gamma[\Phi]} = e^{iW[J]-i\int_x\{J^*\Phi+\Phi^* J\}},\] and insert there the definition of the Schwinger functional, leading to \[e^{-i\Gamma[\Phi]} = \int D\phi \, e^{iS[\phi]+i\int_x \{J^*(\phi-\Phi)+(\phi-\Phi)^* J\}}.\] One may use here the field equation \[\frac{\delta\Gamma[\Phi]}{\delta\Phi(x)} = \sqrt{g}(x) J^*(x),\] and make a change of variables, defining \(\phi^\prime = \phi-\Phi\). This leads to the identity \[\exp\left[-i\Gamma[\Phi]\right] = \int D\phi^\prime \, \exp\left[iS[\Phi+\phi^\prime]+i\int d^d x \left\{\frac{\delta \Gamma[\Phi]}{\delta \Phi(x)}\phi^\prime(x)+\phi^{\prime*}(x) \frac{\delta\Gamma[\Phi]}{\delta\Phi^*(x)}\right\}\right].\] Note that this is an implicit relation because the first derivative of the effective action appears also on the right hand side.