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QFT Propagators

Complex plane visualization of different propagator types in quantum field theory: retarded, advanced, and Feynman propagators. The diagram shows their analytic structure in relation to Matsubara frequencies (dots on imaginary axis). This representation helps understanding causality in quantum field theory and the connection between real-time and imaginary-time (thermal) formalisms.


QFT Propagators

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Code

qft-propagators.typ (78 lines)

#import "@preview/cetz:0.5.2": canvas, decorations, draw
#import draw: circle, content, line

#set page(width: auto, height: auto, margin: 5pt, fill: none)
#set text(size: 12pt)

#let re-range = 5
#let im-range = 4
#let arrow-style = (end: "stealth", fill: black, scale: 0.5)

#canvas({
  // cetz decorations can't carry a mark (cetz-package/cetz#446), so the zigzag stops
  // short of the end and a straight stub carries the arrowhead
  let tip = 0.4
  decorations.zigzag(
    line((-re-range, 0), (re-range - tip, 0)),
    amplitude: 0.1,
    segment-length: 0.2,
  )
  line((re-range - tip, 0), (re-range, 0), mark: arrow-style, name: "x-axis")
  content("x-axis.end", $"Re"(omega)$, anchor: "north-east", padding: (top: 5pt))

  // Imaginary axis
  line(
    (0, -im-range + 1),
    (0, im-range - 1),
    mark: arrow-style,
    name: "y-axis",
  )
  content("y-axis.end", $"Im"(omega)$, anchor: "north-west", padding: (
    left: 5pt,
  ))

  // Matsubara frequencies
  for n in range(-im-range, im-range + 1) {
    circle(
      (0, 2 / 3 * n),
      radius: 0.05,
      fill: black,
      name: "omega" + str(n),
    )
  }
  content((-1.1, 2), align(right)[Matsubara\ frequencies])

  for spec in (
    (
      name: "advanced",
      start: (-re-range, -1),
      end: (re-range, -1),
      paint: red,
      label: [advanced],
      label-pos: (rel: (0, -0.4), to: "advanced.start"),
      label-args: (anchor: "south-west", padding: (left: 5pt)),
    ),
    (
      name: "retarded",
      start: (-re-range, 1),
      end: (re-range, 1),
      paint: blue,
      label: [retarded],
      label-pos: "retarded.start",
      label-args: (anchor: "south-west", padding: 2pt),
    ),
    (
      name: "feynman",
      start: (-re-range, -1),
      end: (re-range, 1),
      paint: orange,
      label: [Feynman],
      label-pos: (rel: (-0.7, -0.55), to: "feynman.end"),
      label-args: (padding: 2pt),
    ),
  ) {
    line(spec.start, spec.end, stroke: (paint: spec.paint, dash: "dashed"), name: spec.name)
    content(spec.label-pos, text(spec.paint, spec.label), ..spec.label-args)
  }
})