LSZ reduction formula
The LSZ reduction formula lets us calculate S-matrix elements $\braket{f|S|i}$ from time ordered correlation functions.
LSZ reduction formula
For some initial and final states $\ket{i}$ and $\ket{f}$, which are created by the field operators $\phi(x_i)$ we can calculate the S-matrix elements $\braket{f|S|i}$ with $$ \begin{align*} \braket{f|S|i} = &\left[i\int d^4x_1 e^{-ip_1x_1}(\square_1 + m^2)\right]\dotsc \left[i\int d^4x_n e^{-ip_nx_n}(\square_n + m^2)\right] \\ &\times \braket{\Omega|T{\phi(x_1)\dotsc\phi(x_n)}|\Omega} \end{align*} $$ The integrals with the $-i$ exponent applies for initial states and the $+i$ for final states The state $\ket{\Omega}$ is the ground state of the interacting theory
For free fields $(\Box + m^2)\phi(x)=0$, however these zeros can cancel with divergences from propagators. Since othUnier terms are killed by these zeros, we basically project onto initial and final states.
Derivation of LSZ formula
For a $2 \to n$ process the initial and final states at $t = \pm \infty$ can be written as $$ \begin{align} \ket{i}=\sqrt{2w_1}\sqrt{2w_2}a^\dagger_{p_1}(-\infty)a^\dagger_{p_2}(-\infty)\ket{\Omega}\\ \ket{f}=\sqrt{2w_3}\dotsc \sqrt{2w_n}a^\dagger_{p_3}(\infty)\dotsc a^\dagger_{p_n}(-\infty)\ket{\Omega} \end{align} $$ with $\omega^2_p = \vec{p}^2 + m^2$ If $\ket{i}\neq\ket{f}$ the S-matrix is just the probability of $\ket{i}$ evolving to $\ket{f}$ $$\braket{f|S|i}=2^{n/2}\sqrt{w_1w_2w_3\dotsc w_n} \braket{\Omega|a_{p_3}(\infty)\dotsc a_{p_n}(\infty)a^\dagger_{p_1}(-\infty)a^\dagger_{p_2}(-\infty)|\Omega}.$$
To relate the expression to field operators we use $$\boxed{i\int d^4x e^{ipx}(\Box + m^2)\phi(x) = \sqrt{2w_p}(a_p(\infty) - a_p(-\infty))}$$ This can be proven by assuming:
- The theory is free at $\pm\infty$
- the fields vanish at $\vec{x}\to\pm\infty$ Using 2. we can integrate by parts over $\vec{x}$ to resolve the $\partial^2_x$ leading to $$i\int d^4x e^{ipx}(\Box + m^2)\phi(x)=i\int d^4x e^{ipx}(d^2_t + w^2_p)\phi(x)$$ From this expression we can factor out a time derivative to cancel the integral over $t$ $$i\int d^4x e^{ipx}(d^2_t + w^2_p)\phi(x) = \int dt \partial_t \left[e^{i\w_p t}\int d^3x e^{-i\vec{p}\vec{x}}(i\partial_t + w_p)\phi(x)\right]$$ The $d^3x$ integral can be executed by inserting the field operator which leads to a $\delta$ function. We can then execute the $dt$ integral which evaluates the expression at $t=\pm\infty$ $$ \begin{align} i\int d^4x e^{ipx}(d^2_t + w^2_p)\phi(x) &= \int dt \partial_t [(e^{iw_p t})(\sqrt{2w_p}a_p(t)e^{iw_pt})]\\ &= \sqrt{2w_p}[a_p(\infty) - a_p(-\infty)] \end{align} $$ Hermitian conjugation gives a similar expression for $a^\dagger_p(\pm\infty)$ To obtain the wanted expresion of the S-matrix element we use the Time-ordering operator $T$ with $$ \begin{align} \braket{f|S|i}&=2^{n/2}\sqrt{w_1w_2w_3\dotsc w_n} \braket{\Omega|a_{p_3}(\infty)\dotsc a_{p_n}(\infty)a^\dagger_{p_1}(-\infty)a^\dagger_{p_2}(-\infty)|\Omega}\\ &=\sqrt{2^nw_1\dotsc w_n}\braket{\Omega|T{[a_{p_3}(\infty)-a_{p_3}(-\infty)]\dotsc[a_{p_n}(\infty)-a_{p_n}(-\infty)]\\ \times[a^\dagger_{p_1}(\infty)-a^\dagger_{p_1}(-\infty)] [a^\dagger_{p_2}(\infty)-a^\dagger_{p_2g}(-\infty)]}|\Omega} \end{align} $$ Using the boxed expression above this can be translated to the LSZ formula There are details to be considered when pulling the $\Box$ out of the time ordering
Also there are subtleties when some momenta are identical
The LSZ formula works not only for the field operators $\phi(x)$ but for any operator $\mathcal{O}$ which creates one particle states from the ground state $$\braket{p|\mathcal{O}(x)|\Omega}=We^{ipx}.$$ It can therefore be used to calculate S-matrix elements for states that are not associated to fundamental fields, like bound states in Effective Field Theories