Symmetries

Quantum-mechanical symmetries organize states, observables, Hamiltonians, degeneracies, and selection rules. In field theory, the same organizing role expands into spacetime symmetries, internal symmetries, Noether currents, Ward identities, gauge redundancy, spontaneous symmetry breaking, and anomalies.

Quantum Mechanics Starting Point

A unitary symmetry of a time-independent Hamiltonian satisfies

$$UHU^\dagger=H.$$

For a continuous symmetry generated by $G$,

$$U(\alpha)=e^{-i\alpha G/\hbar}, \qquad [H,G]=0$$

when the symmetry is exact and time independent.

Field-Theory Continuation

In Lagrangian field theory, continuous global symmetries lead to conserved currents under the hypotheses of Noether’s theorem:

$$\partial_\mu j^\mu=0.$$

Quantum field theory then studies how these currents act on fields and correlation functions. Ward identities express the consequences of symmetry at the level of Green functions and scattering amplitudes.

What Carries Over

• Symmetry transformations act on states and operators.
• Generators produce continuous transformations.
• Conserved quantities label states and constrain dynamics.
• Selection rules follow from representation theory.
• Broken symmetries split degeneracies or allow new transitions.

What Changes

Field theory distinguishes global symmetries from gauge redundancies. A global symmetry acts on physical states. A gauge transformation is a redundancy in description, and physical states or observables must be gauge-invariant in the appropriate sense.

Field theory also adds:

• spacetime symmetry and Lorentz covariance,
• local currents,
• spontaneous symmetry breaking,
• Goldstone modes,
• anomalies,
• Ward and Slavnov–Taylor identities.

Common Mistakes

• Treating gauge symmetry as an ordinary physical degeneracy.
• Assuming every classical symmetry survives quantization.
• Confusing a conserved charge with a superselection rule.
• Importing nonrelativistic rotation intuition into Lorentz boosts without checking representation theory.
• Forgetting antiunitary symmetries such as time reversal.

Canonical Links

• Why Symmetry Matters
• Quantum Symmetries
• Unitary Symmetries
• Antiunitary Symmetries
• Charge Conjugation Preview
• CPT Preview
• Superselection Sectors Preview
• Galilean Boosts
• Generators
• Commutators and Conservation Laws
• From Quantum Generators to Noether Currents
• From Selection Rules to Ward Identities
• From Discrete Symmetries to CPT
• From Symmetry Breaking to Goldstone Theorem
• From Projective Representations to Anomalies Preview
• From Phase Symmetry to Gauge Theory
• Noether Theorem in QM
• Wigner–Eckart Theorem

References

• E. Noether, “Invariante Variationsprobleme,” Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, 235-257 (1918).
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• S. Weinberg, The Quantum Theory of Fields, Volume I, Cambridge University Press, 1995.
• M. D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press, 2014.

Exercises

1. Why is a gauge symmetry not the same kind of statement as a degeneracy from rotational symmetry?

Solution

A rotational symmetry can relate distinct physical states or classify them into representations. A gauge transformation is a redundancy in the description: gauge-related configurations represent the same physical configuration. Treating gauge redundancy as an ordinary degeneracy overcounts physical states.