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Physics · Document 14

Quantum Field Theory

Fields, particles, and the quantum vacuum

1. From Particles to Fields

Quantum Field Theory (QFT) unifies quantum mechanics with special relativity. Instead of particles, the fundamental objects are quantum fields that permeate all of spacetime.

Particles are excitations of fields! An electron is a quantum of the electron field, just as a photon is a quantum of the electromagnetic field.
Quantum Mechanics
  • * Fixed number of particles
  • * Particles are fundamental
  • * Non-relativistic
Quantum Field Theory
  • * Particle creation/annihilation
  • * Fields are fundamental
  • * Fully relativistic

2. The Lagrangian Density

QFT is formulated using the Lagrangian density L. The action is:

S = integral of L d^4x
Action in Field Theory

Example: Free Scalar Field

L = (1/2)(partial_mu phi)(partial^mu phi) - (1/2)m^2 phi^2
Klein-Gordon Lagrangian

This describes a spin-0 particle of mass m (like the Higgs boson before symmetry breaking).

Example: Dirac Field

L = psi-bar (i*gamma^mu partial_mu - m) psi
Dirac Lagrangian

This describes spin-1/2 fermions (electrons, quarks).

3. Feynman Diagrams

Feynman diagrams are pictorial representations of terms in perturbation theory. Each diagram corresponds to a mathematical expression for the scattering amplitude.

🎮 Interactive: Feynman Diagram Examples
e-e-gammavertex
Electron-Photon Vertex
Basic QED interaction: e- emits or absorbs a photon
Amplitude: -i * e * gamma^mu
Solid line
Fermion (e-, e+)
Wavy line
Photon
Dot
Vertex (-ie*gamma)

Feynman Rules

* External lines: Incoming/outgoing particles

* Internal lines: Virtual particles (propagators)

* Vertices: Interaction points (coupling constants)

* Loops: Integrate over all internal momenta

4. Propagators

The propagator describes how a particle moves from one point to another. In momentum space, it encodes the particle's mass and spin.

🎮 Interactive: Propagators for Different Particles
p
i / (p^2 - m^2 + i*epsilon)
Klein-Gordon propagator for spin-0 particles (Higgs)
p^2
4.00
m^2
1.00
On-shell?
No (virtual)
The propagator has a pole at p^2 = m^2. This is where the particle is "on-shell" (physical). Virtual particles can be off-shell (p^2 != m^2).

5. The Quantum Vacuum

In QFT, the vacuum is not empty. The uncertainty principle allows virtual particles to briefly exist, creating vacuum fluctuations.

🎮 Interactive: Vacuum Fluctuations
Virtual particle-antiparticle pairs
Uncertainty Principle
Delta E * Delta t >= h-bar/2
Zero-Point Energy
E_0 = (1/2) h-bar * omega
The vacuum is not empty! Virtual particles constantly appear and disappear. This is observable through the Casimir effect and the Lamb shift.

Observable Effects

Casimir Effect
Two metal plates attract due to vacuum fluctuations
Lamb Shift
Energy level shift in hydrogen from vacuum polarization
Z²
<0|phi^2|0> != 0
The vacuum expectation value of field fluctuations is non-zero

6. Renormalization

Loop diagrams give infinite results! Renormalization is the procedure of absorbing these infinities into redefined (physical) parameters.

alpha(mu) = alpha(mu_0) + beta * ln(mu/mu_0) + ...
Running Coupling
Physical predictions are finite and testable. The fine structure constant alpha ~ 1/137 "runs" with energy scale.
Z²
alpha^(-1) = 4Z^2 + 3
The Z^2 framework predicts alpha at a specific scale from geometry

7. Connection to Z^2 Framework

Quantum field theory provides the language for understanding the Standard Model:

Z²
Z = integral exp(-S[phi]) D[phi]
The partition function sums over all field configurations
Z²
Feynman diagrams
Perturbation theory organizes calculations by number of vertices
Z²
Renormalization group
Physical quantities depend on the energy scale of observation

Why QFT Matters for Z^2

  • * Field content: The Standard Model is a specific QFT with particular fields
  • * Interactions: Determined by gauge symmetry (next document!)
  • * Coupling constants: The values alpha, alpha_s, etc. require explanation
  • * Vacuum energy: Connected to the cosmological constant problem

Exercises

  1. Derive the Klein-Gordon equation from the scalar field Lagrangian using the Euler-Lagrange equations.
  2. Count the number of vertices in a 2-to-2 electron scattering diagram at tree level.
  3. Why does the photon propagator have no mass term (m^2 = 0)?
  4. The electron self-energy diagram is divergent. What physical quantity does it renormalize?
  5. At what energy scale does the electromagnetic coupling become strong (alpha ~ 1)?