===== Extended quantum action functional . finite =====
==== Partial function ====
| @#55CCEE: context     | @#55CCEE: $ \mathbb K = \mathbb C \lor \mathbb R $ |
| @#55CCEE: context     | @#55CCEE: $ m\in\mathbb N $ |
| @#55CCEE: context     | @#55CCEE: $ D $ .... self-adjoint operator in $\mathbb K^m$ with well behaved inverse at least for $D+i\,\varepsilon\,\mathrm 1$ |
| @#FFBB00: definiendum | @#FFBB00: $Z:(\mathbb K^2\to\mathbb R)\to \mathbb K^4\to \mathbb K $ |
| @#FFBB00: definiendum | @#FFBB00: $Z_{\mathcal L_\mathrm{int}}(J,K,\phi,\psi):=\mathrm{e}^{i\hbar^{-1}\sum_{i=1}^m\mathcal L_\mathrm{int}\left(-i\,\hbar\frac{\partial}{\partial J_i},-i\,\hbar\frac{\partial}{\partial K_i}\right)} \left( \mathrm{e}^{i\,\hbar^{-1} \left\langle J\left|\,\mathcal{R}_\varepsilon\,\right|K\right\rangle}\cdot\mathrm{e}^{Z_\text{source}(J,K,\phi,\psi)}\right)$ |

| @#BBDDEE: $\mathcal{R}_\varepsilon\equiv-\left(D+i\,\varepsilon\,\mathrm{1}\right)^{-1}$ |
| @#BBDDEE: $Z_\text{source}(J,K,\phi,\psi):=i\,\hbar^{-1}\left(\left\langle J\left|\right. \phi\right\rangle+\left\langle K\left|\right. \psi\right\rangle\right)$ |

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=== Discussion ===
The operator $\mathcal{R}_\varepsilon$ is denoted the response function because it relates the classical free field to the source, $(D+i\,\varepsilon)\phi=-J$. The brackets are the standard inner product on $\mathbb C^n$ or $\mathbb R^n$. The finite quantum action functional presented here can be generalized by 

  * considering an infinite set (spacetime manifold) instead of $\{1,2,\dots,m\}$, in which case the sums $\langle a|b \rangle$ become integrals
  * consider non-scalar fields, which introduces even more indices and the possibility for much more complicated operators $D$.

The factor involving $Z_\text{source}$ is the only one involving arguments $\phi$ and $\psi$ and it's essentially only there to formulate some identies (LSZ reduction formula). In the discussion below we only consider the case $\phi=\psi=0$.

The expression 

$Z_{\mathcal L_\mathrm{int}}(J,K,0,0)=\mathrm{e}^{i\hbar^{-1}\sum_{i=1}^m\mathcal L_\mathrm{int}\left(-i\hbar\frac{\partial}{\partial J_i},-i\hbar\frac{\partial}{\partial K_i}\right)}\mathrm{e}^{i\hbar^{-1} \left\langle J\left|\,\mathcal{R}_\varepsilon\,\right|K\right\rangle}$ 

should be interpreted as the solution of an integral as presented in [[magic Gaussian integral]]. Reminder: In one dimension it reads 

$\int_{-\infty}^\infty\,f(\phi)\,\mathrm{e}^{-\tfrac{1}{2}a\,\phi^2+i\,\phi\,j}\mathrm d\phi \ \propto \ f\left(-i\frac{\partial}{\partial j}\right)\,\mathrm e^{-\tfrac{1}{2}j^2\,a^{-1}}$

Indeed, for $\mathbb K:=\mathbb R$ it equals a finite dimensional (path) integral

$Z_{\mathcal L_\mathrm{int}}(J,K,0,0) = \mathcal N\cdot\lim_{\varepsilon,\eta\to 0}\int_{-\infty}^\infty \mathrm e^{i\,\hbar^{-1}\,S(J,K,\phi,\psi)}\prod_{i=1}^m \mathrm d\phi_i\,\mathrm d\psi_i$

Here $\mathcal N$ is a normalization constant and $S(J,K,\phi,\psi)$ is the sum of the following terms

$S_\text{free}:=\left\langle\psi\left|\,D\,\right|\phi\right\rangle$

$S_\text{int}:=\sum_{i=1}^m {\mathcal L_\mathrm{int}}(\phi_i,\psi_i) $

$S_\text{source}:=\left\langle J\left|\right. \phi\right\rangle+\left\langle K\left|\right. \psi\right\rangle$

$\mathfrak{reg}:=-\varepsilon\,\left\langle\phi\left|\right.\phi\right\rangle-\eta\,\left\langle\psi\left|\right.\psi\right\rangle$

To elaborate on that consider now, for simplicity, the "only one kind of particle"-case 

$\mathcal N\cdot\lim_{\varepsilon\to 0}\int_{-\infty}^\infty \mathrm e^{i\,\hbar^{-1}\,S(J,\phi)}\prod_{i=1}^m \mathrm d\phi_i$ 

with one source $J$ and field $\phi$. Consider 

$"\mathcal N'\cdot\int_{-\infty}^\infty\,(1+(i\,\hbar^{-1})\,\phi\,J+(i\,\hbar^{-1})^2\,\phi^2\,J^2+\dots)\,\mathrm{e}^{i\,\hbar^{-1}\,\left(S_\text{free}(\phi)+S_\text{int}(\phi)\right)}\mathcal D\phi"$

which is analog to the probability theoretic characteristic function $\mathrm{{\bf E}}[\mathrm{e}^{i\,\hbar^{-1}\left\langle J\left|\right. \phi\right\rangle}]$. The point is that this generating functional of the path integrals equals the quantum action functional which, by the interpretation given in [[Retarded propagator . time-independent Hamiltonian|retarded propagator]], encodes the correlation functions/transition amplitudes $C_k(i_1,i_2,\dots,i_m)$ of the theory:

$ Z(J) = \sum_{j=0}^\infty \frac{i^j}{j!\hbar^j}\sum_{i_0,\dots,i_j=1}^m C_j(i_1,\dots,i_j)\prod_{l=0}^jJ_j = $

$ =C_0^\phi+\frac{i}{\hbar}\sum_{k=1}^m C_2(k)\,J_k+\frac{i^2}{2!\hbar^2}\sum_{k,j=1}^m C_1(k,j)\,J_k\,J_j+\dots $

=== Example ===
Consider the so called $\phi^4$-theory given via $D=\Box+m^2$ and $\mathcal L_\text{int}(\phi)=-\lambda\,\phi^4$. Then the quantum action functional reads

$\mathrm{e}^{i\,\hbar^{-1}\cdot\lambda\sum_{j=1}^m\left(-i\,\hbar\frac{\partial}{\partial J_j}\right)^4} \mathrm{e}^{i\,\hbar^{-1} \left\langle J\left|\,-\left(\Box+m^2+i\,\varepsilon\,\mathrm{1}\right)^{-1}\,\right|J\right\rangle}$

One is interested in the functions $C$ which encode all observables of the theory. For a given process of interest (e.g. scattering of particles from and to some spacetime points which are given by certain $j$-values) one must expand $Z$ in $J$'s and powers of the coupling, $\lambda^k$, and read off the $C$-coefficient. However, the expression is very labourious to compute. Note that the $J$-derivatives produce complicated sums, e.g.

$\sum_j \frac{\partial}{\partial J_j}\sum_i \sum_l\,f(J_i,J_l)=\sum_j \sum_i \left(f^{(1,0)}(J_j,J_i)+f^{(0,1)}(J_i,J_j)\right)$,

and so a summand

$\lambda^k\sum_{j=1}^m\left(-i\,\hbar\frac{\partial}{\partial J_j}\right)^{2k}\,\left\langle J\left|\,-\left(\Box+m^2+i\,\varepsilon\,\mathrm{1}\right)^{-1}\,\right|J\right\rangle^n$

is //extremely// messy. The nice thing, though, is that each term in the expansion can be represented by a //Feynman diagram//. The external lines translate to the choosen spacetime points $j$, the internal lines to the inverse operator of the theory (propagator) and the internal vertices to the coupling constants $\lambda$ plus sums like above. Below are the $\lambda^0,\lambda^1$ and $\lambda^2$ terms for the process with four points (expressed in momentum space).

{{phi4_one-zero_order_neufeld.png}}

{{phi4_one-loop_neufeld.png}}

In the discrete case, the "Differential operators" are just value differences of the field on neighboring points. The operator is a matrix e.g. with some entries above the diagonal. The action involves sums over term involving that matrix. In the continuum limit the derivatives of the inner products $\langle J|(\dots)^{-1}|J\rangle$ become very tricky integrals. By passing to momentum space, the different or differential operators can be multiplicatively invertex, because e.g $\Box$ just becomes $k^2$. The Feynman rules of a theory are the ad hoc rules of how to compute the correlation functions from the diagrams

{{phi4-feynman-rules-peskin-schroeder.png?X550}}

The order of the internal vertices is four because the expression $\left(-i\hbar\frac{\partial}{\partial J_i}\right)^4$ always brings down four $\phi$'s at a time.

Conversely, in the quantum electrodynamics Lagrangian

$\mathcal{L}^\text{QED} = - \tfrac{1}{4} (\partial_\mu A_\nu - \partial_\nu A_\mu)(\partial^\mu A^\nu - \partial^\nu A^\mu) + \sum_n^\text{#fermions} \bar\psi_n (i \gamma^\mu (\partial_\mu+ieA_\mu) -m_n) \psi_n$

the interaction term 

$\mathcal{L}_\text{int}^\text{QED}=-e\,\left({\bar\psi}\,\gamma^\mu\,\psi\right)\,A_\mu$ 

always brings together two fermion lines and one wiggly photon line

{{qed_two-three-four_vertex-diagrams.png}}

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=== Context ===
[[Square matrix]], [[Self-adjoint operator]]
=== Requirements ===
[[Function integral on ℝⁿ]], [[Exponential function]]
=== Related ===
[[Magic Gaussian integral]], [[Retarded propagator . time-independent Hamiltonian]]