Contour integration is a powerful technique in complex analysis, used extensively in physics for evaluating integrals that arise in quantum field theory, especially when calculating Feynman propagators. Here’s an explanation of how contour integration applies to Feynman propagator calculations:

Feynman Propagator Overview

The Feynman propagator is a key object in quantum field theory, representing the probability amplitude for a particle to travel from one point to another. In the context of a scalar field theory, the Feynman propagator GF(x−y)G_F(x – y) in position space is related to the propagator G~F(p)\tilde{G}_F(p) in momentum space by a Fourier transform:

GF(x−y)=∫d4p(2π)4G~F(p)e−ip⋅(x−y)G_F(x – y) = \int \frac{d^4 p}{(2\pi)^4} \tilde{G}_F(p) e^{-ip \cdot (x – y)}

In momentum space, the Feynman propagator for a scalar field is:

G~F(p)=1p2−m2+iϵ\tilde{G}_F(p) = \frac{1}{p^2 – m^2 + i\epsilon}

where:

  • p2=p02−p2p^2 = p_0^2 – \mathbf{p}^2 (the Minkowski space four-momentum squared),
  • mm is the mass of the particle,
  • iϵi\epsilon is a small imaginary term ensuring the correct causal structure and convergence properties.

Contour Integration

To evaluate integrals involving the propagator, such as when transforming back to position space, contour integration is often employed. This technique leverages the residues of complex functions and the Cauchy integral theorem.

Step-by-Step Process

  1. Identify Poles: The integrand 1p2−m2+iϵ\frac{1}{p^2 – m^2 + i\epsilon} has poles at p2=m2−iϵp^2 = m^2 – i\epsilon. For the energy component p0p_0 (the time component of momentum), this corresponds to poles at: p0=±p2+m2−iϵ≈±p2+m2∓iϵp_0 = \pm \sqrt{\mathbf{p}^2 + m^2 – i\epsilon} \approx \pm \sqrt{\mathbf{p}^2 + m^2} \mp i\epsilon
  2. Choose the Contour: Depending on whether you’re evaluating an integral over p0p_0 or the spatial components, you’ll choose an appropriate contour in the complex plane to perform the integration. For the time-ordered propagator, the contour typically runs along the real axis but must be deformed to avoid the poles by going above or below them.
  3. Deform the Contour:
    • For the positive energy pole p0=p2+m2−iϵp_0 = \sqrt{\mathbf{p}^2 + m^2} – i\epsilon, the contour is deformed to pass below the pole.
    • For the negative energy pole p0=−p2+m2+iϵp_0 = -\sqrt{\mathbf{p}^2 + m^2} + i\epsilon, the contour is deformed to pass above the pole.
  4. Evaluate the Residues: Using the residue theorem, the integral around a closed contour enclosing the poles is 2πi2\pi i times the sum of the residues at the poles. The residue of 1p2−m2+iϵ\frac{1}{p^2 – m^2 + i\epsilon} at a pole p0=p0polep_0 = p_0^{\text{pole}} is given by:

    Res(1p2−m2+iϵ,p0pole)=lim⁡p0→p0pole(p0−p0pole)1p02−(p2+m2−iϵ)=12p2+m2\text{Res}\left(\frac{1}{p^2 – m^2 + i\epsilon}, p_0^{\text{pole}}\right) = \lim_{p_0 \to p_0^{\text{pole}}} (p_0 – p_0^{\text{pole}}) \frac{1}{p_0^2 – (\mathbf{p}^2 + m^2 – i\epsilon)} = \frac{1}{2\sqrt{\mathbf{p}^2 + m^2}}

  5. Integrate: The integral over p0p_0 can now be performed using the contour that picks up the contributions from the residues. This usually results in an expression involving an exponential decay or oscillation term, indicating the propagation in time and space.

Example Integral

Consider the integral:

I=∫−∞∞dp02πe−ip0(t−t′)p02−ωp2+iϵI = \int_{-\infty}^{\infty} \frac{dp_0}{2\pi} \frac{e^{-ip_0 (t – t’)}}{p_0^2 – \omega_p^2 + i\epsilon}

where ωp=p2+m2\omega_p = \sqrt{\mathbf{p}^2 + m^2}. The poles are at p0=±ωp∓iϵp_0 = \pm \omega_p \mp i\epsilon.

  • For t>t′t > t’, close the contour in the lower half-plane (picking up the residue at p0=ωp−iϵp_0 = \omega_p – i\epsilon).
  • For t<t′t < t’, close the contour in the upper half-plane (picking up the residue at p0=−ωp+iϵp_0 = -\omega_p + i\epsilon).

Evaluating the residues and summing the contributions gives the time-ordered Feynman propagator.

Summary

Contour integration allows us to handle the complex poles of the Feynman propagator, ensuring causality and correct boundary conditions in quantum field calculations. By deforming the integration path around these poles, we can systematically evaluate the integrals and derive expressions for the propagators in various scenarios.