Quantum Field Theory Course

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

Part VII, Chapter 4

Finite Temperature QFT

Quantum fields in thermal equilibrium: the early universe and quark-gluon plasma

Why Thermal QFT?

At finite temperature, quantum fields are in thermal equilibrium with a heat bath. This describes systems like the early universe (hot Big Bang),quark-gluon plasma (RHIC/LHC collisions), and phase transitionsin particle physics.

Temperature introduces new physics: thermal fluctuations, phase transitions, and modifications to particle propagation. The formalism combines quantum field theory with statistical mechanics!

4.1 Thermal Density Matrix

A system in thermal equilibrium at temperature T is described by the thermal density matrix:

$$\rho = \frac{1}{Z} e^{-\beta H}$$

where Ξ² = 1/kBT (we set kB = 1) and Z is the partition function:

$$Z = \text{Tr}(e^{-\beta H})$$

Expectation values are computed using the thermal average:

$$\langle \mathcal{O} \rangle_\beta = \text{Tr}(\rho \mathcal{O}) = \frac{1}{Z}\text{Tr}(e^{-\beta H} \mathcal{O})$$

4.2 Imaginary Time Formalism

The key insight: the thermal trace looks like time evolution in imaginary time!

$$e^{-\beta H} = e^{-iH(i\beta)} = e^{-iHt}\Big|_{t \to -i\beta}$$

This means we can use the Euclidean path integral with periodic (bosons) or antiperiodic (fermions) boundary conditions in imaginary time:

\begin{align*} \phi(\mathbf{x}, 0) &= \phi(\mathbf{x}, \beta) \quad \text{(bosons)} \\ \psi(\mathbf{x}, 0) &= -\psi(\mathbf{x}, \beta) \quad \text{(fermions)} \end{align*}

Euclidean Time = Temperature

The imaginary time direction is compactified to a circle of circumference Ξ². Temperature is literally the inverse size of the Euclidean time dimension!

Ο„ ∈ [0, Ξ²] ⟷ Temperature T = 1/Ξ²

4.3 Matsubara Frequencies

Due to periodic/antiperiodic boundary conditions, the energy (Ο‰β‚€) is quantizedin the Euclidean time direction:

\begin{align*} \omega_n &= \frac{2\pi n}{\beta} = 2\pi nT \quad \text{(bosons)} \\ \omega_n &= \frac{2\pi (n + 1/2)}{\beta} = 2\pi (n + \tfrac{1}{2})T \quad \text{(fermions)} \end{align*}

where n ∈ β„€. These are the Matsubara frequencies.

Frequency vs Temperature

The Matsubara frequency spacing is:

$$\Delta\omega = \frac{2\pi}{\beta} = 2\pi T$$

At high temperature (T β†’ ∞), the spacing becomes large and we approach classical thermal physics. At low temperature (T β†’ 0), the spacing is tiny and we recover zero-temperature QFT.

4.4 Thermal Propagators

The thermal propagator for a scalar field is:

$$\Delta(\mathbf{p}, \omega_n) = \frac{1}{\omega_n^2 + \mathbf{p}^2 + m^2}$$

where ωn is a bosonic Matsubara frequency. Note: this is the Euclidean propagator with p₀ → iωn.

To get real-time correlators, we perform analytic continuation:

$$i\omega_n \to \omega + i\epsilon$$

This gives the retarded thermal propagator:

$$\Delta^R(\omega, \mathbf{p}) = \frac{1}{\omega^2 - \mathbf{p}^2 - m^2 + i\epsilon}$$

4.5 Thermal Distributions

At finite temperature, particle states are populated according to thermal statistics:

\begin{align*} n_B(\omega) &= \frac{1}{e^{\beta\omega} - 1} \quad \text{(Bose-Einstein)} \\ n_F(\omega) &= \frac{1}{e^{\beta\omega} + 1} \quad \text{(Fermi-Dirac)} \end{align*}

These modify propagators and loop integrals. For example, the thermal part of the propagator is:

$$\Delta_T(\omega, \mathbf{p}) = 2\pi\delta(\omega^2 - \omega_p^2) \, n_B(\omega_p)$$

where Ο‰p = √(pΒ² + mΒ²).

High Temperature Limit

For Ο‰ β‰ͺ T (high temperature):

$$n_B(\omega) \approx \frac{T}{\omega}, \quad n_F(\omega) \approx \frac{1}{2} - \frac{\omega}{4T}$$

Bosons have large thermal occupation ~ T/Ο‰. Fermions are limited by Pauli exclusion to n ≀ 1.

4.6 Finite Temperature Effective Potential

At finite temperature, the effective potential Veff(Ο†, T) receives thermal corrections. For a scalar field with tree-level potential Vβ‚€(Ο†):

$$V_{\text{eff}}(\phi, T) = V_0(\phi) + V_{\text{1-loop}}(\phi, T=0) + V_T(\phi, T)$$

The thermal contribution (high-T expansion) is:

$$V_T(\phi, T) \approx \frac{\pi^2}{90}g_*T^4 + \frac{1}{24}m^2(\phi)T^2 + \ldots$$

where g* counts relativistic degrees of freedom and mΒ²(Ο†) is the field-dependent mass.

4.7 Thermal Phase Transitions

The shape of Veff(Ο†, T) changes with temperature, potentially leading tophase transitions!

Example: Electroweak Phase Transition

The Higgs potential at finite temperature:

$$V_{\text{eff}}(\phi, T) = -\frac{m^2}{2}\phi^2 + \frac{\lambda}{4}\phi^4 + \frac{T^2}{24}\left(3g^2 + g'^2 + 4\lambda\right)\phi^2$$
  • High T: Thermal term dominates β†’ minimum at Ο† = 0 (symmetric phase)
  • Low T: Tree-level dominates β†’ minimum at Ο† = v β‰  0 (broken phase)
  • Critical temperature: Tc ~ v ~ 246 GeV (electroweak scale)

4.8 First vs Second Order Transitions

Second Order (Continuous)

  • VEV changes continuously: Ο†(T) β†’ 0 as T β†’ Tc
  • No latent heat
  • Correlation length diverges: ΞΎ ~ |T - Tc|-Ξ½
  • Critical exponents describe universality classes
  • Example: Ising model critical point

First Order (Discontinuous)

  • VEV jumps discontinuously at Tc
  • Latent heat released
  • Barrier between phases β†’ supercooling/superheating
  • Nucleation of bubbles of true vacuum
  • Example: Water β†’ ice transition

The order of the transition depends on the shape of Veff(Ο†, T) near Tc.

4.9 Phase Transitions in the Early Universe

As the early universe cooled from the Big Bang, it underwent several phase transitions:

1. QCD Phase Transition (T ~ 150 MeV)

Quark-gluon plasma β†’ hadrons. Likely a crossover (smooth, not sharp). Studied at RHIC and LHC by colliding heavy ions.

2. Electroweak Phase Transition (T ~ 100 GeV)

Higgs field acquires VEV, giving masses to W, Z, and fermions. In the Standard Model, this is a crossover. Extensions (e.g., MSSM) could make it first-order β†’ gravitational waves!

3. GUT Phase Transition (T ~ 10¹⁡ GeV)

Grand unified symmetry breaking. Could produce magnetic monopoles (problem: none observed!). Inflation may dilute them.

4. Inflation (T ~ 10¹⁴ GeV?)

Not a thermal phase transition, but related to scalar field dynamics. Ends with reheating β†’ thermal universe.

4.10 Baryogenesis and Sakharov Conditions

A first-order electroweak phase transition is crucial for electroweak baryogenesisβ€” explaining why the universe has more matter than antimatter.

Sakharov Conditions (1967)

To generate a baryon asymmetry, you need:

  1. Baryon number violation: Otherwise B stays conserved at zero
  2. C and CP violation: Otherwise processes and their conjugates cancel
  3. Departure from thermal equilibrium: Needed for rates to differ

A first-order phase transition naturally provides condition (3) via bubble nucleation and expansion!

Unfortunately, the Standard Model electroweak transition is not first-order and doesn't have enough CP violation. New physics is needed!

4.11 Quark-Gluon Plasma

At temperatures T > Tc ~ 150 MeV, QCD undergoes deconfinement: hadrons melt into a quark-gluon plasma (QGP).

Properties of QGP

  • Free quarks and gluons (approximately)
  • Energy density: Ξ΅ ~ (π²/30)g*T⁴ where g* ~ 50 for QCD
  • Behaves like a nearly perfect fluid (low viscosity!)
  • Studied at RHIC (Brookhaven) and LHC (CERN) via heavy-ion collisions
  • Existed in early universe for t ~ 10-5 seconds

Key Takeaways

  • Finite-T QFT: combine quantum fields with statistical mechanics
  • Imaginary time formalism: Ο„ ∈ [0, Ξ²] with Ξ² = 1/T
  • Matsubara frequencies: Ο‰n = 2Ο€nT (bosons), 2Ο€(n+Β½)T (fermions)
  • Thermal distributions: nB, nF modify propagators
  • Effective potential Veff(Ο†,T) changes with temperature
  • Phase transitions: QCD, electroweak, GUT transitions in early universe
  • First-order transitions: crucial for baryogenesis
  • Quark-gluon plasma: deconfined QCD at T > 150 MeV
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