Quantum Field Theory Course

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

Part I, Chapter 1 | Page 4 of 8

Symmetries and Conservation Laws

Noether's theorem for continuous symmetries

1.11 Continuous Symmetries

A symmetry of the action is a transformation of the fields that leaves the action invariant. Noether's theorem states that every continuous symmetry corresponds to a conserved current.

Infinitesimal Transformations

Consider an infinitesimal transformation:

$$\phi(x) \to \phi'(x) = \phi(x) + \epsilon \Delta \phi(x)$$

where Ξ΅ is infinitesimal and Δφ characterizes the transformation. For example:

  • Translation: $\Delta \phi = -a^\mu \partial_\mu \phi$ for constant aΞΌ
  • Lorentz transformation: $\Delta \phi = -\frac{1}{2}\omega^{\mu\nu}(x_\mu \partial_\nu - x_\nu \partial_\mu)\phi$
  • Internal symmetry: Δφ depends only on Ο†, not x (e.g., phase rotations)

1.12 Noether's Theorem

If the Lagrangian density changes by a total derivative under the transformation:

$$\delta \mathcal{L} = \epsilon \partial_\mu F^\mu$$

for some FΞΌ, then there exists a conserved current:

$$\boxed{j^\mu = \frac{\partial \mathcal{L}}{\partial(\partial_\mu \phi)} \Delta \phi - F^\mu}$$

satisfying the continuity equation:

$$\boxed{\partial_\mu j^\mu = 0}$$

Conserved Charge

The time component j0 is the charge density. The total charge:

$$Q = \int d^3x \, j^0(x)$$

is conserved (time-independent):

$$\frac{dQ}{dt} = \int d^3x \, \partial_0 j^0 = -\int d^3x \, \nabla \cdot \vec{j} = 0$$

where we assumed the current vanishes at spatial infinity.

1.13 Example: Phase Symmetry (U(1))

Consider a complex scalar field Ο† with Lagrangian:

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

Global U(1) Symmetry

The Lagrangian is invariant under the global phase transformation:

$$\phi \to e^{i\alpha} \phi, \quad \phi^* \to e^{-i\alpha} \phi^*$$

For infinitesimal Ξ±:

$$\Delta \phi = i\phi, \quad \Delta \phi^* = -i\phi^*$$

Noether Current

The conserved current is:

$$j^\mu = \frac{\partial \mathcal{L}}{\partial(\partial_\mu \phi)} \Delta \phi + \frac{\partial \mathcal{L}}{\partial(\partial_\mu \phi^*)} \Delta \phi^*$$

Computing the derivatives:

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

Therefore:

$$\boxed{j^\mu = i(\phi^* \partial^\mu \phi - \phi \partial^\mu \phi^*)}$$

This is exactly the probability current from quantum mechanics! The conserved charge Q is the total particle number.

1.14 Spacetime Symmetries

The most important symmetries are those of spacetime itself:

Time Translation Invariance

Symmetry: $t \to t + \epsilon$

Conserved quantity: Energy (Hamiltonian H)

Spatial Translation Invariance

Symmetry: $\vec{x} \to \vec{x} + \vec{\epsilon}$

Conserved quantity: Momentum $\vec{P}$

Rotation Invariance

Symmetry: Rotations about axes

Conserved quantity: Angular Momentum $\vec{L}$

Lorentz Boost Invariance

Symmetry: Boosts to moving frames

Conserved quantity: Motion of center of energy

All these symmetries are unified in the energy-momentum tensor TΞΌΞ½, which we'll construct in detail on the next pages. This tensor encodes all conserved quantities associated with spacetime translations.

Key Concepts (Page 4)

  • β€’ Continuous symmetry β†’ conserved current (Noether's theorem)
  • β€’ Conserved current satisfies: $\partial_\mu j^\mu = 0$
  • β€’ U(1) phase symmetry β†’ conserved particle number
  • β€’ Spacetime translations β†’ energy and momentum conservation
  • β€’ Energy-momentum tensor unifies all spacetime symmetries
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