Quantum Field Theory Course

Total: 440+ pages covering classical field theory through the Standard Model

Part III, Chapter 4 — Page 3 of 3

Spectral Representation

The Kallen-Lehmann representation: exact structure of the interacting propagator

4.12 Motivation: The Exact Propagator

In the free theory, the 2-point function is the Feynman propagator with a single pole at $p^2 = m^2$. In the interacting theory, the exact 2-point function receives corrections from all loop diagrams. What is the most general structure this exact propagator can have?

The Kallen-Lehmann spectral representation answers this question using only Lorentz invariance, unitarity (completeness of states), and positivity of the Hilbert space norm. It is an exact, non-perturbative result.

💡Why Is This Important?

The spectral representation tells us the exact propagator is a "weighted sum" of free propagators with different masses. The weight function (spectral density) encodes all the information about the particle spectrum of the theory — bound states, resonances, and multi-particle continua. It is one of the few exact results in interacting QFT.

4.13 Derivation of the Kallen-Lehmann Representation

Consider the exact 2-point function in the interacting theory:

$$G^{(2)}(x-y) = \langle\Omega|T\{\phi(x)\phi(y)\}|\Omega\rangle$$

where $|\Omega\rangle$ is the exact vacuum of the interacting theory. Insert a complete set of intermediate states. By Lorentz invariance, these states can be labelled by total 4-momentum $p^\mu$ and invariant mass squared $\mu^2 = p^2$:

$$\mathbf{1} = |\Omega\rangle\langle\Omega| + \sum_{\lambda}\int \frac{d^3p}{(2\pi)^3}\frac{1}{2E_{\mathbf{p}}(\lambda)}|\lambda,\mathbf{p}\rangle\langle\lambda,\mathbf{p}|$$

where $|\lambda,\mathbf{p}\rangle$ represents a state with quantum numbers $\lambda$ boosted to momentum $\mathbf{p}$. Consider $t_x > t_y$ (the other case is analogous). Insert the completeness relation:

$$\langle\Omega|\phi(x)\phi(y)|\Omega\rangle = \sum_\lambda \int \frac{d^3p}{(2\pi)^3}\frac{1}{2E_\mathbf{p}(\lambda)}\langle\Omega|\phi(x)|\lambda,\mathbf{p}\rangle\langle\lambda,\mathbf{p}|\phi(y)|\Omega\rangle$$

(The vacuum term vanishes since $\langle\Omega|\phi|\Omega\rangle = 0$ by the $\mathbb{Z}_2$ symmetry $\phi\to-\phi$.) Using translation invariance $\phi(x) = e^{iP\cdot x}\phi(0)e^{-iP\cdot x}$:

$$\langle\Omega|\phi(x)|\lambda,\mathbf{p}\rangle = \langle\Omega|\phi(0)|\lambda,\mathbf{0}\rangle\, e^{-ip\cdot x}\Big|_{p^0 = E_\mathbf{p}(\lambda)}$$

Define the spectral density (or spectral function):

$$\boxed{\rho(\mu^2) = \sum_\lambda (2\pi)\delta(\mu^2 - m_\lambda^2)\,|\langle\Omega|\phi(0)|\lambda,\mathbf{0}\rangle|^2}$$

where $m_\lambda^2$ is the invariant mass squared of state $|\lambda\rangle$. After combining both time orderings and performing the $d^3p$ integral:

$$\boxed{G^{(2)}(x-y) = \int_0^\infty \frac{d\mu^2}{2\pi}\,\rho(\mu^2)\,D_F(x-y;\mu^2)}$$

where $D_F(x-y;\mu^2)$ is the free Feynman propagator with mass $\mu$. This is the Kallen-Lehmann spectral representation in position space.

4.14 Momentum-Space Form and the Exact Propagator

Fourier transforming to momentum space:

$$\boxed{\tilde{G}^{(2)}(p^2) = \int_0^\infty \frac{d\mu^2}{2\pi}\frac{i\,\rho(\mu^2)}{p^2 - \mu^2 + i\epsilon}}$$

The spectral density $\rho(\mu^2)$ has a specific structure dictated by the particle content:

$$\rho(\mu^2) = 2\pi Z\,\delta(\mu^2 - m^2) + \text{(multi-particle continuum for } \mu^2 \geq (2m)^2\text{)}$$

This gives the exact propagator the structure:

$$\boxed{\tilde{G}^{(2)}(p^2) = \frac{iZ}{p^2 - m^2 + i\epsilon} + \int_{4m^2}^\infty \frac{d\mu^2}{2\pi}\frac{i\,\rho(\mu^2)}{p^2 - \mu^2 + i\epsilon}}$$

The first term is an isolated pole at $p^2 = m^2$ corresponding to the single-particle state with residue $Z$. The second term is a branch cut starting at $p^2 = 4m^2$ (the two-particle threshold) from the continuum of multi-particle states.

💡Reading the Spectral Density

The spectral density is like a "mass spectrum" of the theory:

  • Delta function at $m^2$: The stable single-particle state (pole in propagator)
  • Continuum above $4m^2$: Two-particle states and beyond (branch cut)
  • Additional delta functions: Would indicate bound states below the continuum threshold
  • Peaks in the continuum: Resonances (unstable particles)

4.15 Connection to Unitarity

The spectral density satisfies a crucial sum rule from the equal-time commutation relation $[\phi(x),\pi(y)]_{t_x=t_y} = i\delta^3(\mathbf{x}-\mathbf{y})$:

$$\boxed{\int_0^\infty \frac{d\mu^2}{2\pi}\,\rho(\mu^2) = 1}$$

Since $\rho(\mu^2) \geq 0$ (it is a sum of squared matrix elements), the single-particle contribution gives:

$$Z + \int_{4m^2}^\infty \frac{d\mu^2}{2\pi}\,\rho_{\text{cont}}(\mu^2) = 1$$

Since the continuum contribution is strictly positive in an interacting theory:

$$\boxed{0 < Z < 1 \quad \text{(in interacting theory)}}$$

This is a non-perturbative constraint on the wavefunction renormalization. In the free theory, $\rho_{\text{cont}} = 0$ and $Z = 1$. Interactions always reduce $Z$ below 1 because the field operator creates multi-particle states as well as the single particle.

The optical theorem relates the imaginary part of the propagator to the spectral density:

$$\text{Im}\,\tilde{G}^{(2)}(p^2 + i\epsilon) = -\frac{1}{2}\rho(p^2) \quad \text{for } p^2 > 0$$

This connects the spectral representation directly to unitarity: the imaginary part of scattering amplitudes is related to the total rate of producing intermediate states.

4.16 Worked Example: One-Loop Self-Energy

Let us verify the spectral structure using the one-loop self-energy in $\phi^4$ theory. The exact propagator is related to the self-energy $\Sigma(p^2)$ by Dyson resummation:

$$\tilde{G}^{(2)}(p^2) = \frac{i}{p^2 - m_0^2 - \Sigma(p^2) + i\epsilon}$$

The physical mass $m$ is defined by the pole condition:

$$m^2 = m_0^2 + \text{Re}\,\Sigma(m^2)$$

Near the pole, expand $\Sigma(p^2) \approx \Sigma(m^2) + (p^2-m^2)\Sigma'(m^2) + \cdots$:

$$\tilde{G}^{(2)}(p^2) \approx \frac{i}{(p^2-m^2)(1-\Sigma'(m^2))} = \frac{iZ}{p^2-m^2+i\epsilon} + \text{regular}$$

where the wavefunction renormalization is:

$$\boxed{Z = \frac{1}{1 - \Sigma'(m^2)}}$$

At one loop in $\phi^4$, $\Sigma'(m^2) \sim \lambda^2/(4\pi)^2$ from the bubble diagram (the tadpole is momentum-independent so $\Sigma'_{\text{tadpole}} = 0$). Thus $Z = 1 - O(\lambda^2) < 1$, consistent with the non-perturbative bound. The imaginary part of $\Sigma(p^2)$ is nonzero for $p^2 > 4m^2$, confirming the onset of the two-particle continuum at the expected threshold.

💡The Physical Mass vs the Bare Mass

The pole of the exact propagator defines the physical mass $m$, which differs from the bare mass $m_0$ in the Lagrangian. The residue $Z$ at this pole is the probability that the field creates a single-particle state rather than a multi-particle cloud. Renormalization conditions fix $m$ and $Z$ to their physical values.

Key Concepts — Spectral Representation

  • Kallen-Lehmann: exact propagator = integral of free propagators weighted by $\rho(\mu^2)$
  • Spectral density $\rho(\mu^2) \geq 0$ encodes the full particle spectrum
  • Isolated pole at $p^2 = m^2$ with residue $Z$ = stable single particle
  • Branch cut at $p^2 \geq 4m^2$ = multi-particle continuum
  • Sum rule: $Z + \int\rho_{\text{cont}} = 1$ implies $0 < Z < 1$ in interacting theory
  • Optical theorem: $\text{Im}\,G^{(2)} \propto \rho$, connecting to unitarity
  • Physical mass defined by pole of exact propagator, $Z$ by its residue
  • Non-perturbative result: valid to all orders and beyond perturbation theory
← Back: LSZ Formula
Page 3 of 3
Next Chapter: Feynman Diagrams →