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 other 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 $$i\int d^4x e^{ipx}(\Box + m^2)\phi(x)=i\int d^4x e^{ipx}(\partial^2_t + w^2_p)\phi(x),$$ since $\omega^2_p=\vec{p}^2+m^2$. From this expression we can factor out a time derivative to cancel the integral over $t$ $$i\int d^4x e^{ipx}(\partial^2_t + w^2_p)\phi(x) = \int dt \partial_t \left[e^{iw_p t}\int d^3x e^{-i\vec{p}\vec{x}}(i\partial_t + w_p)\phi(x)\right]$$ 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 insert the above identity into the form of the S-matrix form the begin of the derivation. To relate the two form we use the Time-ordering operator $T$ so that $$ \begin{align} \braket{f|S|i}&=2^{n/2}\sqrt{w_1w_2w_3\dotsc w_n} \braket{\Omega|a_{p_3}(\infty)\dotsc a_{p_n}(\infty)a^\dagger_{p_1}(-\infty)a^\dagger_{p_2}(-\infty)|\Omega}\\ &=\sqrt{2^nw_1\dotsc w_n}\braket{\Omega|T{[a_{p_3}(\infty)-a_{p_3}(-\infty)]\dotsc[a_{p_n}(\infty)-a_{p_n}(-\infty)]\\ \times[a^\dagger_{p_1}(\infty)-a^\dagger_{p_1}(-\infty)] [a^\dagger_{p_2}(\infty)-a^\dagger_{p_2g}(-\infty)]}|\Omega} \end{align} $$ 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, which technically is not allowed. However it can be shown that this only introduced Contact terms, which do not contribute.
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