Currents, Charges, and Quantum Ward Identities

A continuous symmetry constrains correlation functions through currents and Ward identities, but a conserved local current does not automatically define a finite conserved charge. This chapter connects those constructions while keeping contact terms, domains, and boundary flux visible.

Enter this chapter

Begin with Continuous Symmetries, Generators, and Charges to distinguish the local and integrated questions.

Helpful background. Classical Symmetries, Currents, and Stress Tensors supply the classical current and stress-tensor examples.

Topics

1. Continuous Symmetries, Generators, and Charges — Tests the existence and conservation of symmetry generators.
2. Quantum Currents, Improvements, and Conservation — Explains quantum currents and the freedom to improve their local representatives.
3. Spacetime Currents, Stress Tensors, and Charge Algebras — Relates spacetime transformations to energy–momentum and charge algebras.
4. Localized Transformations and Ward–Takahashi Identities — Derives identities by allowing a symmetry parameter to vary in spacetime.
5. Contact Terms, Equal-Time Commutators, and Schwinger Terms — Tracks insertion terms produced by time ordering and possible central contributions.

How the ideas fit together

Localizing a transformation produces a current divergence, but it also varies every operator insertion. Inside a time-ordered correlator these variations occur as contact terms, so a conservation law away from insertions is not the complete Ward identity. See Schwartz 2014, § 14.8.

Integrating that identity turns local conservation into charge balance only when boundary flux and operator-domain conditions are controlled. The interior and wall Ward-identity diagram separates the origins of these terms. Improvements may change boundary charges even when they leave the bulk divergence unchanged.

Review the chapter

If $\partial_\mu j^\mu=0$, why can the charge inside a fixed finite region change?

Solution

Integrating over the region gives the time derivative of its charge as minus the outward spatial flux. Conservation says charge is transported consistently; it does not require the flux through an artificial or physical boundary to vanish.

Where to go next

Continue to Background Fields and Gauging to organize current insertions and global symmetry probes through external sources.

References

• Matthew D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press, 2014. DOI.