QFT Bridge: S-Matrix

The scattering matrix maps asymptotic incoming states to asymptotic outgoing states. S-Matrix constructs the nonrelativistic operator from Møller wave operators. This bridge asks what survives when the asymptotic states become multiparticle states of quantum fields.

The shared abstraction is

$$S_{fi} = \langle f,\mathrm{out}|i,\mathrm{in}\rangle.$$

The difference is what the states mean and how amplitudes are normalized.

In and Out States

In nonrelativistic scattering, one often starts from a free incoming state and constructs an outgoing-boundary scattering state:

$$|\psi^{(+)}\rangle = |\phi\rangle + G_0^{(+)}(E)V|\psi^{(+)}\rangle.$$

The potential acts within a fixed-particle Hilbert space.

In QFT, the in and out states are Fock-space states. Particle number can change, so an $S$-matrix element can describe processes such as

$$2\to2,\qquad 2\to3,\qquad 2\to n.$$

Exact asymptotic particles must be stable. An unstable excitation has no normalizable one-particle in-state persisting to infinite time; decay amplitudes and resonance poles describe it through a controlled preparation or pole approximation. The free Fock-space description also needs modification for long-range infrared problems such as charged particles coupled to massless photons.

Normalization Changes

Nonrelativistic scattering often uses amplitudes $f(\theta,\phi)$ defined by

$$\psi(\mathbf r) \sim e^{i\mathbf k\cdot\mathbf r} + f(\theta,\phi) \frac{e^{ikr}}{r}.$$

Use natural units $\hbar=c=1$. The common relativistic normalization is

$$\langle\mathbf p',b|\mathbf p,a\rangle =2E_{\mathbf p}(2\pi)^3\delta^{(3)}(\mathbf p'-\mathbf p)\delta_{ab}.$$

With these states, define the connected invariant amplitude $\mathcal M$ after factoring out momentum conservation:

$$\langle f|S|i\rangle = \langle f|i\rangle + i(2\pi)^4\delta^{(4)}(P_f-P_i) \mathcal M_{fi}$$

for the connected transition under consideration. In multiparticle sectors, disconnected spectator contributions must also be separated. The invariant state normalization supplies the final-state measure $d^3p/[(2\pi)^3 2E]$.

For a two-particle initial state, these conventions give

$$d\sigma=\frac{\overline{|\mathcal M|^2}}{4\sqrt{(p_1\cdot p_2)^2-m_1^2m_2^2}}\,d\Phi_n.$$

Initial spin populations determine the average; unobserved final spins are summed. If $d\Phi_n$ integrates labeled identical particles over the whole phase space, include the appropriate factorials. Relativistic scattering defines $d\Phi_n$ and obtains the two-body angular formula.

LSZ Preview

In potential scattering, the Lippmann–Schwinger equation constructs scattering states from a Hamiltonian and boundary conditions. In QFT, the LSZ reduction formula extracts scattering amplitudes from time-ordered correlation functions by isolating external stable one-particle poles, removing their propagators, and including the field-to-particle residue factors. A pole $G(p)\sim iZ/(p^2-m^2+i0)$ supplies the factor needed to convert a field insertion into a normalized asymptotic particle; changing the field normalization cannot change the final cross section.

The analogy is:

$$\text{asymptotic free states} \quad \leftrightarrow \quad \text{external particle poles}.$$

The implementation differs because QFT uses fields, relativistic normalization, and multiparticle phase space.

From Correlation Functions to QFT Observables explains why LSZ starts from a connected time-ordered correlator and what remains before it becomes a cross section.

Exercises

1. What is the shared meaning of $S_{fi}$ in QM and QFT?

Solution

In both settings, $S_{fi}$ is the amplitude for an incoming asymptotic state $i$ to be detected as an outgoing asymptotic state $f$:

$$S_{fi} = \langle f,\mathrm{out}|i,\mathrm{in}\rangle.$$

The meaning and normalization of the states differ between nonrelativistic quantum mechanics and QFT.

2. Why does particle production force a conceptual change from potential scattering to QFT scattering?

Solution

Potential scattering usually acts in a fixed-particle Hilbert space. QFT scattering acts on Fock space, where states with different particle numbers coexist. Processes such as $2\to3$ or decays cannot be represented as ordinary one-particle scattering from a fixed potential without adding extra structure by hand.

References

• M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory, Addison-Wesley, 1995.
• M. D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press, 2014.
• J. R. Taylor, Scattering Theory: The Quantum Theory of Nonrelativistic Collisions, Dover, 2006.
• S. Weinberg, The Quantum Theory of Fields, Vol. I: Foundations, Cambridge University Press, 1995.