Quantum Field Theory Course

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

Part I, Chapter 6

Internal & Spacetime Symmetries

Classification and physical consequences of field symmetries

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Video Lecture

Lecture 2: Symmetries and Conservation Laws - MIT 8.323

Noether's theorem and conserved currents in field theory (MIT QFT Course)

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6.1 Introduction: Symmetries in Physics

Symmetries are the backbone of modern physics. By Noether's theorem, every continuous symmetry corresponds to a conserved quantity. In QFT, symmetries are classified into two fundamental types:

Spacetime Symmetries

Transformations of spacetime coordinates xΞΌ β†’ x'ΞΌ

  • Translations
  • Rotations
  • Lorentz boosts
  • PoincarΓ© group

Internal Symmetries

Transformations in field space Ο† β†’ Ο†', independent of x

  • Global phase (U(1))
  • Isospin (SU(2))
  • Color (SU(3))
  • Gauge symmetries

6.2 Spacetime Translations

Time Translation

Under time translation t β†’ t + Ξ΅:

$$\phi(t, \vec{x}) \to \phi(t + \epsilon, \vec{x}) = \phi(t, \vec{x}) + \epsilon \partial_t \phi$$

If the Lagrangian has no explicit time dependence (βˆ‚β„’/βˆ‚t = 0), then by Noether's theorem, the energy (Hamiltonian) is conserved:

$$H = \int d^3x \, \mathcal{H} = \int d^3x \, [\pi \dot{\phi} - \mathcal{L}] = \text{constant}$$

Spatial Translation

Under spatial translation x β†’ x + Ξ΅:

$$\phi(t, \vec{x}) \to \phi(t, \vec{x} + \vec{\epsilon}) = \phi(t, \vec{x}) + \epsilon^i \partial_i \phi$$

If the Lagrangian is spatially homogeneous (βˆ‚β„’/βˆ‚xi = 0), then momentum is conserved:

$$\vec{P} = \int d^3x \, \pi \nabla \phi = \text{constant}$$

4-Momentum Conservation

Combined, these give conservation of 4-momentum:

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

where TΞΌΞ½ is the energy-momentum tensor.

6.3 Rotations and Angular Momentum

Rotation Transformation

An infinitesimal rotation by angle ΞΈ about the z-axis:

$$x \to x - \theta y, \quad y \to y + \theta x, \quad z \to z$$

For a scalar field:

$$\delta \phi = -\theta(x \partial_y - y \partial_x)\phi = -\theta \vec{L}_z \cdot \nabla \phi$$

Angular Momentum Tensor

The conserved quantity is angular momentum:

$$\vec{L} = \int d^3x \, \vec{x} \times (\pi \nabla \phi)$$

More generally, the angular momentum tensor is:

$$M^{\mu\nu\rho} = x^\nu T^{\mu\rho} - x^\rho T^{\mu\nu}$$

For fields with spin, there's an additional intrinsic angular momentum term.

Spin

For the Dirac field, the total angular momentum includes orbital and spin parts:

$$\vec{J} = \vec{L} + \vec{S}$$

where the spin operator for Dirac fermions is:

$$\vec{S} = \int d^3x \, \psi^\dagger \frac{1}{2}\vec{\Sigma} \psi, \quad \vec{\Sigma} = \begin{pmatrix} \vec{\sigma} & 0 \\ 0 & \vec{\sigma} \end{pmatrix}$$

6.4 Lorentz Transformations

Lorentz Group SO(1,3)

The proper orthochronous Lorentz group consists of transformations preserving:

$$x'^{\mu} = \Lambda^\mu_{\phantom{\mu}\nu} x^\nu, \quad \Lambda^\mu_{\phantom{\mu}\rho} g_{\mu\nu} \Lambda^\nu_{\phantom{\nu}\sigma} = g_{\rho\sigma}$$

Infinitesimal Generators

An infinitesimal Lorentz transformation:

$$\Lambda^\mu_{\phantom{\mu}\nu} = \delta^\mu_\nu + \omega^\mu_{\phantom{\mu}\nu}$$

where ωμν is antisymmetric: ωμν = -ωνμ. This gives 6 independent generators:

  • 3 rotations: Ο‰ij (i, j = 1,2,3)
  • 3 boosts: Ο‰0i

Field Representations

Different fields transform differently under Lorentz transformations:

Scalar Field (spin 0):

$$\phi'(x') = \phi(x)$$

Vector Field (spin 1):

$$A'^\mu(x') = \Lambda^\mu_{\phantom{\mu}\nu} A^\nu(x)$$

Spinor Field (spin 1/2):

$$\psi'(x') = S(\Lambda)\psi(x)$$

where S is a 4Γ—4 representation of the Lorentz group

6.5 Internal Symmetries

Global U(1) Symmetry

The simplest internal symmetry is a global phase transformation:

$$\phi(x) \to e^{i\alpha}\phi(x)$$

where Ξ± is a constant (independent of x). For the complex scalar field Lagrangian:

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

this symmetry leads to conserved electric charge (or particle number):

$$Q = i\int d^3x \, (\phi^* \dot{\phi} - \dot{\phi}^* \phi) = \text{constant}$$

Isospin SU(2)

Consider a doublet of fields:

$$\Phi = \begin{pmatrix} \phi_1 \\ \phi_2 \end{pmatrix}$$

SU(2) transformations:

$$\Phi \to U\Phi = e^{i\vec{\alpha} \cdot \vec{\tau}/2}\Phi$$

where Ο„i are the Pauli matrices. This gives 3 conserved charges(isospin components).

Color SU(3)

In QCD, quarks come in a triplet:

$$q = \begin{pmatrix} q_r \\ q_g \\ q_b \end{pmatrix}$$

SU(3) color symmetry:

$$q \to e^{i\alpha^a T^a}q$$

where Ta (a = 1,...,8) are the Gell-Mann matrices. This gives 8 conserved color charges.

6.6 Discrete Symmetries: C, P, T

Three fundamental discrete symmetries in QFT:

Parity (P)

Spatial inversion: x β†’ -x, t β†’ t

$$P: (t, \vec{x}) \to (t, -\vec{x})$$

β€’ Scalars: Ο†(t, x) β†’ Β±Ο†(t, -x)
β€’ Vectors: A0 β†’ A0, A β†’ -A
β€’ Violated in weak interactions!

Charge Conjugation (C)

Particle ↔ Antiparticle

$$C: \phi \to \phi^*, \quad \psi \to C\bar{\psi}^T$$

β€’ Swaps particles and antiparticles
β€’ Also violated in weak interactions
β€’ Changes sign of all charges

Time Reversal (T)

Time inversion: t β†’ -t

$$T: (t, \vec{x}) \to (-t, \vec{x})$$

β€’ Antiunitary operator
β€’ Reverses momentum and spin
β€’ Small violations observed (CP violation implies T violation via CPT)

CPT Theorem

One of the most fundamental theorems in QFT:

Any local, Lorentz-invariant quantum field theory must be invariant under the combined CPT transformation.

Consequences: particle and antiparticle have same mass and lifetime; CP violation implies T violation

6.7 Global vs. Local (Gauge) Symmetries

Global Symmetry

Transformation parameter is constant everywhere:

$$\phi(x) \to e^{i\alpha}\phi(x), \quad \alpha = \text{constant}$$

Local (Gauge) Symmetry

Transformation parameter depends on spacetime:

$$\phi(x) \to e^{i\alpha(x)}\phi(x)$$

This requires introducing a gauge field AΞΌ and replacing derivatives:

$$\partial_\mu \to D_\mu = \partial_\mu + ieA_\mu \quad \text{(covariant derivative)}$$

The gauge field transforms as:

$$A_\mu \to A_\mu + \frac{1}{e}\partial_\mu \alpha$$

This mechanism requires the existence of force carriers (photons, gluons, W/Z bosons)!

Chapter 6 Summary

  • β€’ Spacetime symmetries: translations β†’ energy/momentum, rotations β†’ angular momentum
  • β€’ Lorentz group: 3 rotations + 3 boosts, fields transform by representation
  • β€’ Internal symmetries: U(1) β†’ charge, SU(2) β†’ isospin, SU(3) β†’ color
  • β€’ Discrete symmetries: P (parity), C (charge conjugation), T (time reversal)
  • β€’ CPT theorem: combined CPT is exact symmetry of all local QFTs
  • β€’ Gauge symmetries: local symmetries require gauge fields (force carriers)
  • β€’ Noether's theorem: every continuous symmetry β†’ conserved current
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Internal & Spacetime Symmetries
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