Quantum Field Theory Course

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

Part I, Chapter 1 | Page 5 of 8

Energy-Momentum Tensor

Conserved currents from spacetime translation symmetry

1.15 Canonical Energy-Momentum Tensor

Under an infinitesimal spacetime translation:

$$x^\mu \to x^\mu + \epsilon^\mu$$

the field transforms as:

$$\phi(x) \to \phi(x - \epsilon) = \phi(x) - \epsilon^\mu \partial_\mu \phi(x)$$

Applying Noether's theorem, the conserved current for translation in the Ξ½ direction is:

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

We define the canonical energy-momentum tensor:

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

or with both indices up:

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

Conservation Law

From Noether's theorem, if the Lagrangian has no explicit spacetime dependence:

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

This is four conservation equations (Ξ½ = 0, 1, 2, 3), one for each component of momentum.

1.16 Physical Interpretation

The components of TΞΌΞ½ have clear physical meanings:

T00: Energy Density

The energy density (Hamiltonian density):

$$T^{00} = \pi \dot{\phi} - \mathcal{L} = \mathcal{H}$$

T0i: Momentum Density

The i-th component of momentum density:

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

Ti0: Energy Flux

Energy flux in the i-th direction (energy current):

$$T^{i0} = \frac{\partial \mathcal{L}}{\partial(\partial_i \phi)} \dot{\phi}$$

Tij: Stress Tensor

Momentum flux = stress (force per unit area):

$$T^{ij} = \frac{\partial \mathcal{L}}{\partial(\partial_i \phi)} \partial^j \phi - \delta^{ij}\mathcal{L}$$

Conserved Quantities

Integrating over space gives the total 4-momentum:

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

Explicitly:

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

Conservation $\partial_\mu T^{\mu\nu} = 0$ implies $dP^\nu/dt = 0$, i.e., energy and momentum are conserved.

1.17 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 conjugate momentum is:

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

Therefore the energy-momentum tensor is:

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

Explicit Components

The energy density (T00 component):

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

The momentum density:

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

The stress tensor:

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

1.18 Symmetry of the Energy-Momentum Tensor

The canonical energy-momentum tensor is not always symmetric: TΞΌΞ½ β‰  TΞ½ΞΌ in general.

However, one can always add an improvement term to make it symmetric. The symmetric (Belinfante) energy-momentum tensor is:

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

where K is antisymmetric in its first two indices. The symmetric tensor is crucial for:

  • Coupling to gravity via Einstein's equations
  • Defining angular momentum properly
  • Physical interpretation as stress tensor in continuum mechanics

For scalar fields, the canonical TΞΌΞ½ is already symmetric. For fields with spin (like the electromagnetic field or Dirac field), symmetrization is necessary.

Key Concepts (Page 5)

  • β€’ Energy-momentum tensor: $T^{\mu\nu} = \frac{\partial \mathcal{L}}{\partial(\partial_\mu \phi)} \partial^\nu \phi - g^{\mu\nu} \mathcal{L}$
  • β€’ Conservation: $\partial_\mu T^{\mu\nu} = 0$ (4 equations)
  • β€’ T00 = energy density, T0i = momentum density
  • β€’ Total 4-momentum: $P^\nu = \int d^3x \, T^{0\nu}$
  • β€’ Symmetric tensor couples to gravity
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