LSZ formalism

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Here we review the LSZ or Lehmann-Symanzik-Zimmerman formalism[1] for a scalar field. The LSZ reduction formula provides an explicit way of calculating S-matrix elements (scattering amplitudes) in terms of correlation functions of an interacting field.

Scattering

At a fixed time t_0\,, then, we may construct the state a^{\dagger}_{\varphi_1}(t_0) a^{\dagger}_{\varphi_2}(t_0) ... \left| 0 \right\rangle\, which we consider to be a state of n\, particles with wave-packets \varphi_i(\mathbf{x}, t_0)\,, i = 1...n\, that are negligible mutual overlap. At a later time t_1\, we may consider a similar state a^{\dagger}_{\varphi_1'}(t_1) a^{\dagger}_{\varphi_2'}(t_1) ... \left| 0 \right\rangle\, containing n'\, particles. The state we constructed at t_0\, is still the state of the system (Heisenberg picture), but since the operators are time dependent and interacting, we need to interpret it in terms of states constructed at time t_1\,. Therefore we consider the overlap

S = \left\langle 0 | a_{\varphi_1'}(t_1) a_{\varphi_2'}(t_1) ... a^{\dagger}_{\varphi_1}(t_0) a^{\dagger}_{\varphi_2}(t_0) ... | 0 \right\rangle\,

and ask what the probability amplitude is that the particles created at time t_0\, have scattered into those prepared at time t_1\,. In order to be sure that all transient behaviour has died down, we then let t_0 \to -\infty\, and t_1 \to \infty\,. We may also insert a time-ordering symbol without changing anything:

S = \left\langle 0 | \mathcal{T}\left( a_{\varphi_1'}(\infty) a_{\varphi_2'}(\infty) ... a^{\dagger}_{\varphi_1}(-\infty) a^{\dagger}_{\varphi_2}(-\infty) ... \right)| 0 \right\rangle\,,

or

S = \left\langle \varphi'_1 \varphi'_2...(\infty) | \varphi_1 \varphi_2 ...(-\infty)\right\rangle\,.

LSZ reduction formula

To perform the above calculation, note the following trick

a^{\dagger}_{\varphi}(\infty) - a^{\dagger}_{\varphi}(-\infty)\, = \int_{-\infty}^{\infty}\!dt\, \partial_0 a^{\dagger}_{\varphi}(t)\,,
= \int_{-\infty}^{\infty}\!dt\, \int\!\frac{d^3k}{(2\pi)^3} \frac{\varphi^*(k)}{2\omega_k} \partial_0 \left( -i \int\!d^3x\ e^{-i k\cdot x} \stackrel{\leftrightarrow}{\partial}_0 \phi(x) \right)\,,
= -i\int\!d^4x\, \int\!\frac{d^3k}{(2\pi)^3} \frac{\varphi^*(k)}{2\omega_k} \partial_0 \left( e^{-i k\cdot x} \stackrel{\leftrightarrow}{\partial}_0 \phi(x) \right)\,,
= -i\int\!d^4x\, \int\!\frac{d^3k}{(2\pi)^3} \frac{\varphi^*(k)}{2\omega_k} e^{-i k\cdot x}\left( \omega_k^2 + \partial_0^2 \right) \phi(x)\,,
= -i\int\!d^4x\, \int\!\frac{d^3k}{(2\pi)^3} \frac{\varphi^*(k)}{2\omega_k} e^{-i k\cdot x}\left( \mathbf{k}^2 + m^2 + \partial_0^2 \right) \phi(x)\,,
= -i\int\!d^4x\, \int\!\frac{d^3k}{(2\pi)^3} \frac{\varphi^*(k)}{2\omega_k} e^{-i k\cdot x}\left( -\stackrel{\leftarrow}{\nabla}^2 + m^2 + \partial_0^2 \right) \phi(x)\,,
= -i\int\!d^4x\, \int\!\frac{d^3k}{(2\pi)^3} \frac{\varphi^*(k)}{2\omega_k} e^{-i k\cdot x}\left( - \vec{\nabla}^2 + m^2 + \partial_0^2 \right) \phi(x)\,,
= -i\int\!d^4x\, \int\!\frac{d^3k}{(2\pi)^3} \frac{\varphi^*(k)}{2\omega_k} e^{-i k\cdot x}\left( \partial^2 + m^2\right) \phi(x)\,.

So,

a^{\dagger}_{\varphi}(-\infty) = a^{\dagger}_{\varphi}(\infty) +i\int\!d^4x\, \int\!\frac{d^3k}{(2\pi)^3} \frac{\varphi^*(k)}{2\omega_k} e^{-i k\cdot x}\left( \partial^2 + m^2\right) \phi(x)\,,
a_{\varphi}(\infty) = a_{\varphi}(-\infty) +i\int\!d^4x\, \int\!\frac{d^3k}{(2\pi)^3} \frac{\varphi(k)}{2\omega_k} e^{i k\cdot x}\left( \partial^2 + m^2\right) \phi^{\dagger}(x)\,.

We may insert the expression

a^{\dagger}_{\varphi}(-\infty) = a^{\dagger}_{\varphi}(\infty) + \left( a^{\dagger}_{\varphi}(-\infty) - a^{\dagger}_{\varphi}(\infty)\right)\,

into S\,. For instance,

S\, = \left\langle 0 | \mathcal{T}\left( a_{\varphi_1'}(\infty) a_{\varphi_2'}(\infty) ... a^{\dagger}_{\varphi_1}(-\infty) a^{\dagger}_{\varphi_2}(-\infty) ... \right)| 0 \right\rangle\,,
= \left\langle 0 | \mathcal{T}\left( a_{\varphi_1'}(\infty) a_{\varphi_2'}(\infty) ... a^{\dagger}_{\varphi_1}(\infty) a^{\dagger}_{\varphi_2}(-\infty) ... \right)| 0 \right\rangle + \left\langle 0 | T\left( a_{\varphi_1'}(\infty) a_{\varphi_2'}(\infty) ... \left[a^{\dagger}_{\varphi}(-\infty) - a^{\dagger}_{\varphi}(\infty)\right] a^{\dagger}_{\varphi_2}(-\infty) ... \right)| 0 \right\rangle\,,
= \left\langle 0 | \mathcal{T}\left( a_{\varphi_1'}(\infty) a_{\varphi_2'}(\infty) a^{\dagger}_{\varphi_1}(\infty) ... a^{\dagger}_{\varphi_2}(-\infty) ... \right)| 0 \right\rangle + \left\langle 0 | \mathcal{T}\left( a_{\varphi_1'}(\infty) a_{\varphi_2'}(\infty) ... \left[a^{\dagger}_{\varphi}(-\infty) - a^{\dagger}_{\varphi}(\infty)\right] a^{\dagger}_{\varphi_2}(-\infty) ... \right)| 0 \right\rangle\,,


At this point we consider the limit in which each of the wave-packets are taken to have well-defined momentum \mathbf{k}_i\, and \mathbf{k}'_i\,. In this limit, each \varphi_i(\mathbf{k}) \to (2\pi)^3 2\omega_{k_i}\delta^{(3)}(\mathbf{k}-\mathbf{k_i})\,, which corresponds to plane waves. This limit is strictly taken after all other limits, so that the connection with real particles with narrow but finite widths in \mathbf{k}\,-space (i.e., nearly mononergetic wave-packets) can be made. We may then commute a^{\dagger}_{\mathbf{k}_1}(\infty)\, to the left in the first expression, where, unless any \mathbf{k}'_j = \mathbf{k}_1\, (giving a factor of \delta^{(3)}(\mathbf{k}'_j - \mathbf{k}_1)\,). These are referred to as disconnected processes, in which one or more of the particles do not scatter. After a^{\dagger}_{\varphi_1}(\infty)\, has been commuted to the left, it will encounter \left\langle 0\right|\,, but we have already argued that \lim_{t \to \pm \infty} \left\langle 0 | a^{\dagger}_\varphi(t) | \Psi _{mp} \right\rangle = 0\,, so that this term gives zero. Therefore


S\, = \sum_i \delta^{(3)}(\mathbf{k}^\prime_i - \mathbf{k}_1 ) \left\langle 0 | \mathcal{T}\left( a_{k_1^\prime}(\infty)a\!\!\!/_{k_i'}(\infty)...a_{k_n'}(\infty)... a^{\dagger}_{k_2}(-\infty) ... \right)| 0 \right\rangle\,
+ \left\langle 0 | \mathcal{T}\left( a^{\dagger}_{k_1}(\infty) a_{k_1'}(\infty) a_{k_2'}(\infty) ... a^{\dagger}_{k_2}(-\infty) ... \right)| 0 \right\rangle\,,
+ \left\langle 0 | \mathcal{T}\left( a_{k_1'}(\infty) a_{k_2'}(\infty) ... \left[a^{\dagger}_{k_1}(-\infty) - a^{\dagger}_{k_1}(\infty)\right] a^{\dagger}_{k_2}(-\infty) ... \right)| 0 \right\rangle\,.
= \sum_i \delta^{(3)}(\mathbf{k}^\prime_i - \mathbf{k}_1 ) \left\langle 0 | \mathcal{T}\left( a_{k_1^\prime}(\infty)a\!\!\!/_{k_i'}(\infty)...a_{k_n'}(\infty)... a^{\dagger}_{k_2}(-\infty) ... \right)| 0 \right\rangle\,,
+ \left\langle 0 | \mathcal{T}\left( a_{k_1'}(\infty) a_{k_2'}(\infty) ... \left[a^{\dagger}_{k_1}(-\infty) - a^{\dagger}_{k_1}(\infty)\right] a^{\dagger}_{k_2}(-\infty) ... \right)| 0 \right\rangle\,.

Also,

a^{\dagger}_{k_1}(-\infty) - a^{\dagger}_{k_1}(\infty)= i\int\!d^4x\, \int\!\frac{d^3k}{(2\pi)^3} \frac{(2\pi)^3 2\omega_{k_1}\delta^{(3)}(\mathbf{k}-\mathbf{k}_1)}{2\omega_k} e^{-i k\cdot x}\left( \partial^2 + m^2\right) \phi(x)\,,

or,

a^{\dagger}_{k_1}(-\infty) - a^{\dagger}_{k_1}(\infty)= i\int\!d^4x\, e^{-i k_1\cdot x}\left( \partial^2 + m^2\right) \phi(x)\,.

Repeated application of this process, commuting factors of a^{\dagger}_{\varphi_i}(\infty)\, to the left and factors of a_{\varphi_i'}(\infty)\, to the right gives


S\, = (i)^{n+m} \int\!d^4x_1\, e^{-i k_1\cdot x_1} \int\!d^4x_2\, e^{-i k_2\cdot x_2} ...\int\!d^4x_1\, e^{i k_1\cdot x_1} \int\!d^4x_2\, e^{i k_2\cdot x_2}... \times \,
\left( \partial^2_1 + m^2\right) \left( \partial_2^2 + m^2\right) ...\left( \partial^2_{1'} + m^2\right) \left( \partial^2_{2'} + m^2\right)...\left\langle 0 | \mathcal{T}\left( \phi(x_1) \phi(x_2) ... \phi^{\dagger}(x'_1) \phi^{\dagger}({x'}_2)... \right)| 0 \right\rangle\,

This is the famous LSZ reduction formula. To make connection with the original field, we must divide by an appropriate number of factors of Z^{\frac{1}{2}}\,, to account for field-strength renormalization. Here we have neglected the disconnected parts of the expression. To be precise,

S\, = (i)^{n+m} \frac{1}{Z^{\frac{n+m}{2}}} \int\!d^4x_1\, e^{-i k_1\cdot x_1} \int\!d^4x_2\, e^{-i k_2\cdot x_2} ...\int\!d^4x_1\, e^{i k_1\cdot x_1} \int\!d^4x_2\, e^{i k_2\cdot x_2}... \times \,
\left( \partial^2_1 + m^2\right) \left( \partial_2^2 + m^2\right) ...\left( \partial^2_{1'} + m^2\right) \left( \partial^2_{2'} + m^2\right)...\left\langle 0 | \mathcal{T}\left( \phi(x_1) \phi(x_2) ... \phi^{\dagger}(x'_1) \phi^{\dagger}({x'}_2)... \right)| 0 \right\rangle\,+\, \mathrm{disconnected\mbox{ }terms}\,.

References

Further reading: [2] [3]

  1. ↑ H. Lehmann, K. Symanzik, and W. Zimmerman, On the formulation of quantized field theories, Nuovo Cimento 1, 205 (1955)
  2. ↑ Mark Srednicki (2007). Quantum Field Theory. Cambridge University Press, Cambridge. ISBN 978-0521864497. 
  3. ↑ M.E. Peskin and D.V. Schroeder (1995). An Introduction to Quantum Field Theory. Addison-Wesley, Reading, Massachusetts. ISBN 978-0201503975. 
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