Quantum Field Theory Course

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

Part I, Chapter 7

Energy-Momentum Tensor

Conserved currents from spacetime translation symmetry

7.1 Introduction

The energy-momentum tensor Tμν is one of the most important objects in field theory. It encodes the energy, momentum, and stress carried by fields, and serves as the source term in Einstein's equations of general relativity.

By Noether's theorem, invariance under spacetime translations leads to conservation of energy and momentum, unified in a single tensorial quantity.

Physical Meaning of Components

  • • T00: Energy density
  • • T0i: Momentum density (i-component)
  • • Ti0: Energy flux (flow of energy in i-direction)
  • • Tij: Stress tensor (momentum flux)

7.2 Canonical Energy-Momentum Tensor

Definition from Noether's Theorem

Under an infinitesimal spacetime translation xμ → xμ + εμ, the field changes as:

$$\delta \phi = -\epsilon^\mu \partial_\mu \phi$$

Applying Noether's theorem, the conserved current for translation in the ν-direction is:

$$\boxed{T^\mu_{\phantom{\mu}\nu} = \frac{\partial \mathcal{L}}{\partial(\partial_\mu \phi)} \partial_\nu \phi - \delta^\mu_\nu \mathcal{L}}$$

This is the canonical energy-momentum tensor. With both indices up:

$$\boxed{T^{\mu\nu} = \frac{\partial \mathcal{L}}{\partial(\partial_\mu \phi)} \partial^\nu \phi - g^{\mu\nu} \mathcal{L}}$$

Multiple Fields

For a theory with multiple fields φa:

$$T^{\mu\nu} = \sum_a \frac{\partial \mathcal{L}}{\partial(\partial_\mu \phi_a)} \partial^\nu \phi_a - g^{\mu\nu} \mathcal{L}$$

Conservation Law

If the Lagrangian has no explicit spacetime dependence:

$$\boxed{\partial_\mu T^{\mu\nu} = 0}$$

This is four equations (one for each ν), expressing conservation of energy (ν=0) and the three components of momentum (ν=1,2,3).

7.3 Example: Real Scalar Field

For the Klein-Gordon Lagrangian:

$$\mathcal{L} = \frac{1}{2}\partial_\mu \phi \partial^\mu \phi - \frac{1}{2}m^2 \phi^2$$

The canonical momentum is:

$$\frac{\partial \mathcal{L}}{\partial(\partial_\mu \phi)} = \partial^\mu \phi$$

Therefore:

$$\boxed{T^{\mu\nu} = \partial^\mu \phi \partial^\nu \phi - g^{\mu\nu}\left[\frac{1}{2}\partial_\alpha \phi \partial^\alpha \phi - \frac{1}{2}m^2 \phi^2\right]}$$

Explicit Components

Energy Density (T00):

$$T^{00} = \frac{1}{2}\dot{\phi}^2 + \frac{1}{2}(\nabla \phi)^2 + \frac{1}{2}m^2 \phi^2 = \mathcal{H}$$

Kinetic + Gradient + Potential energy density

Momentum Density (T0i):

$$T^{0i} = \dot{\phi} \partial^i \phi = \pi \nabla^i \phi$$

Field momentum density

Energy Flux (Ti0):

$$T^{i0} = \partial^i \phi \dot{\phi} = T^{0i}$$

Rate of energy flow (equals momentum density for scalar)

Stress Tensor (Tij):

$$T^{ij} = \partial^i \phi \partial^j \phi - \delta^{ij}\left[\frac{1}{2}\dot{\phi}^2 - \frac{1}{2}(\nabla \phi)^2 - \frac{1}{2}m^2 \phi^2\right]$$

Momentum flux / stress

7.4 Electromagnetic Field

Canonical Tensor

For the Maxwell Lagrangian ℒ = -¼FμνFμν, the canonical tensor is:

$$T^{\mu\nu}_{\text{can}} = -F^{\mu\lambda}\partial^\nu A_\lambda + \frac{1}{4}g^{\mu\nu}F_{\alpha\beta}F^{\alpha\beta}$$

However, this is not gauge invariant and not symmetric!

Symmetric Tensor

The physically correct tensor is:

$$\boxed{T^{\mu\nu} = -F^{\mu\lambda}F^\nu_{\phantom{\nu}\lambda} + \frac{1}{4}g^{\mu\nu}F_{\alpha\beta}F^{\alpha\beta}}$$

This is gauge invariant and symmetric: Tμν = Tνμ.

In Terms of E and B

The energy density:

$$T^{00} = \frac{1}{2}(\vec{E}^2 + \vec{B}^2)$$

The momentum density / energy flux (Poynting vector):

$$T^{0i} = T^{i0} = (\vec{E} \times \vec{B})^i = \vec{S}^i$$

The Maxwell stress tensor:

$$T^{ij} = -E^i E^j - B^i B^j + \frac{1}{2}\delta^{ij}(\vec{E}^2 + \vec{B}^2)$$

7.5 Dirac Field

For the Dirac Lagrangian ℒ = $\bar{\psi}(i\gamma^\mu \partial_\mu - m)\psi$:

$$\frac{\partial \mathcal{L}}{\partial(\partial_\mu \psi)} = i\bar{\psi}\gamma^\mu$$

The canonical energy-momentum tensor:

$$T^{\mu\nu}_{\text{can}} = i\bar{\psi}\gamma^\mu \partial^\nu \psi$$

This is not symmetric. The symmetric (Belinfante) tensor is:

$$\boxed{T^{\mu\nu} = \frac{i}{4}\bar{\psi}\gamma^\mu \overleftrightarrow{\partial}^\nu \psi}$$

where $\overleftrightarrow{\partial}^\nu = \overrightarrow{\partial}^\nu - \overleftarrow{\partial}^\nu$.

Energy and Momentum

The energy density is:

$$T^{00} = i\psi^\dagger \dot{\psi} = \bar{\psi}(-i\vec{\gamma} \cdot \nabla + m)\psi$$

Total energy:

$$E = \int d^3x \, T^{00}$$

7.6 Symmetry of the Energy-Momentum Tensor

Why Symmetry Matters

A symmetric energy-momentum tensor Tμν = Tνμ is crucial for:

  • Coupling to gravity in General Relativity
  • Proper definition of angular momentum
  • Physical interpretation as stress tensor
  • Conservation of angular momentum

Belinfante Improvement

The canonical tensor can be symmetrized by adding an improvement term:

$$\Theta^{\mu\nu} = T^{\mu\nu} + \partial_\lambda K^{\lambda\mu\nu}$$

where Kλμν is antisymmetric in its first two indices: Kλμν = -Kμλν.

Since ∂λ∂μKλμν = 0 (antisymmetry), the improved tensor is still conserved:

$$\partial_\mu \Theta^{\mu\nu} = \partial_\mu T^{\mu\nu} = 0$$

When is Tμν Already Symmetric?

  • ✓ Scalar fields (real or complex)
  • ✗ Vector fields (Maxwell, needs improvement)
  • ✗ Spinor fields (Dirac, needs improvement)

7.7 Conserved Charges

Total 4-Momentum

Integrating the energy-momentum tensor over all space:

$$\boxed{P^\mu = \int d^3x \, T^{0\mu}}$$

This gives the total 4-momentum of the field:

$$P^0 = E = \int d^3x \, \mathcal{H}$$
$$\vec{P} = \int d^3x \, T^{0i} = \int d^3x \, \pi \nabla \phi$$

Time Evolution

From conservation ∂μTμν = 0:

$$\frac{dP^\nu}{dt} = \int d^3x \, \partial_0 T^{0\nu} = -\int d^3x \, \nabla_i T^{i\nu} = 0$$

(assuming fields vanish at spatial infinity). Therefore energy and momentum are conserved.

7.8 Coupling to Gravity

Einstein's Equations

In General Relativity, the energy-momentum tensor is the source of spacetime curvature:

$$R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R + \Lambda g_{\mu\nu} = 8\pi G T_{\mu\nu}$$

The right-hand side is the energy-momentum tensor of all matter and fields. This couples QFT to gravity!

Conservation in Curved Spacetime

In curved spacetime, ordinary derivatives are replaced by covariant derivatives:

$$\nabla_\mu T^{\mu\nu} = 0$$

This is the covariant statement of energy-momentum conservation.

Importance for Cosmology

The energy-momentum tensor determines:

  • Expansion rate of the universe (Friedmann equations)
  • Evolution of perturbations (structure formation)
  • Equation of state: w = p/ρ
  • Dark energy behavior

Chapter 7 Summary

  • • Definition: $T^{\mu\nu} = \frac{\partial \mathcal{L}}{\partial(\partial_\mu \phi)} \partial^\nu \phi - g^{\mu\nu} \mathcal{L}$
  • • Conservation: ∂μTμν = 0 (4 equations for energy-momentum)
  • • Components: T00 (energy), T0i (momentum), Tij (stress)
  • • Symmetry: Tμν = Tνμ required for coupling to gravity
  • • Examples: Scalar, electromagnetic, and Dirac fields
  • • Conserved charges: Pμ = ∫d³x T0μ
  • • Gravity coupling: Source term in Einstein's equations
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