QFT Bridge: Optical Theorem and Unitarity

The optical theorem is a consequence of unitarity. In nonrelativistic scattering, it relates the imaginary part of the forward scattering amplitude to the total cross section. In QFT, the same principle relates the imaginary part of a forward invariant amplitude to a sum over all allowed on-shell intermediate states. Unitarity owns the nonrelativistic channel and partial-wave constraints; this bridge owns their relativistic translation.

The formulas look different because the normalization of states and the phase-space measure are different. The principle is the same:

$$S^\dagger S=I.$$

Quantum-Mechanical Form

With the potential-scattering amplitude convention

$$\psi(\mathbf r) \sim e^{ikz} + f(\theta) \frac{e^{ikr}}{r},$$

the optical theorem is

$$\sigma_{\mathrm{tot}} = \frac{4\pi}{k} \operatorname{Im}f(0).$$

This equation says that the total probability removed from the incident beam is encoded in the forward amplitude.

T-Matrix Unitarity

If

$$S=I+iT,$$

then unitarity gives

$$S^\dagger S=I \quad \Rightarrow \quad i(T^\dagger-T)=T^\dagger T.$$

Taking a forward matrix element produces a relation between the imaginary part of the forward amplitude and a sum over final states.

This is the algebraic heart of every optical-theorem formula. The rest is normalization.

QFT Form

In QFT, a schematic unitarity relation for an initial state $i$ is

$$2\,\operatorname{Im}\mathcal M_{i\to i} = \sum_n \int d\Phi_n\, |\mathcal M_{i\to n}|^2,$$

with convention-dependent flux and symmetry factors suppressed. The phase-space measure $d\Phi_n$ sums over allowed on-shell final states.

The right-hand side is the field-theory version of “all possible ways probability can leave the forward channel.”

Cuts and Imaginary Parts

In perturbative QFT, imaginary parts of amplitudes arise when intermediate states can go on shell. Diagrammatic cutting rules make this precise: cutting internal lines places them on shell and relates the discontinuity of a diagram to products of lower-order amplitudes.

This is not a new principle beyond the optical theorem. It is unitarity implemented in relativistic perturbation theory.

Resonance Connection

Resonances have complex poles and widths because probability can flow into open channels. In a simple one-channel model, the width appears in a denominator such as

$$E-E_R+i\Gamma/2.$$

In QFT, widths and imaginary self-energies similarly reflect allowed decay channels. The optical-theorem perspective explains why an imaginary part is not optional: it is required by unitarity once channels open.

Normalization Hazards

When translating the optical theorem, check:

• whether $S=I+iT$ or another $T$ convention is used;
• whether the amplitude is $f$, $T$, or $\mathcal M$;
• whether state normalization is nonrelativistic or relativistic;
• what phase-space measure is included;
• whether identical-particle factors are present;
• whether the “total” cross section includes all open channels.

Common Mistakes

• Treating the optical theorem as a special property of central potentials rather than of unitarity.
• Comparing $\operatorname{Im}f(0)$ with $\operatorname{Im}\mathcal M$ without converting normalization.
• Omitting inelastic channels from the unitarity sum.
• Thinking imaginary parts in amplitudes always signal dissipation rather than open quantum channels.
• Applying cutting intuition without putting intermediate states on shell.

Exercises

1. Derive the algebraic unitarity relation for $S=I+iT$.

Solution

Use

$$S^\dagger S = (I-iT^\dagger)(I+iT) = I+iT-iT^\dagger+T^\dagger T.$$

Setting this equal to $I$ gives

$$i(T-T^\dagger)+T^\dagger T=0,$$

or

$$i(T^\dagger-T)=T^\dagger T.$$

2. Why does the QFT optical theorem involve a phase-space integral?

Solution

QFT final states can contain several particles with continuous momenta. Summing over all final states therefore requires the Lorentz-invariant phase-space measure, including momentum conservation and the on-shell measures for each particle. This replaces the simpler angular integration of elementary two-body potential scattering.

References

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