Lie Groups, Representations, and Spinors

Symmetry calculations depend on which group acts, on which space, and through which representation. This chapter connects group actions, Lie algebras, tensor decompositions, and spinors while keeping local infinitesimal data distinct from global group structure.

Enter this chapter

Begin with Groups, Actions, Quotients, and Covers to identify kernels and global identifications. For relativistic fields, use Lorentz Field Representations and Poincaré Particle Representations after the basic representation language.

Topics

1. Groups, Actions, Quotients, and Covers — Distinguishes an action, its invisible kernel, a quotient, and a covering group.
2. Lie Groups, Lie Algebras, and Exponential and Adjoint Maps — Connects infinitesimal generators with local finite transformations.
3. Representations, Intertwiners, Invariants, and Tensor Decomposition — Uses compatible maps and invariant tensors to decompose representations.
4. Compact Lie Groups, Roots, Weights, and Weyl Structure — Organizes compact-group representations by roots and weights.
5. Lorentz Field Representations and Poincaré Particle Representations — Separates finite field components from unitary one-particle state spaces.
6. Clifford Algebras and Pin and Spin Groups — Builds Clifford multiplication and the groups covering orthogonal transformations.
7. Spinors, Conjugations, Bilinears, Chirality, and Fierz Identities — Determines the dimension-dependent meaning of conjugation, chirality, and spinor identities.

How the ideas fit together

A Lie algebra determines the local commutator structure, but it does not determine all global representations. The groups $SU(2)$ and $SO(3)$ share a Lie algebra, yet a representation of the former descends to the latter only when the central element $-I$ acts trivially. Half-integer-spin representations fail that test. See Hall 2015, Chapter 4.

The same attention to the acted-on space prevents a second confusion: a Lorentz transformation of a finite field multiplet is not the unitary action on the full particle Hilbert space. Momentum labels, physical inner products, and the little group enter the particle problem.

Review the chapter

Why can two fields with the same Lie-algebra transformation law differ in their allowed global symmetry group?

Solution

The infinitesimal law cannot detect a discrete kernel of a covering map. Exponentiating around a noncontractible loop may act nontrivially on one field. Descent to the quotient requires every element of the covering kernel to act as the identity.

Where to go next

Continue to Symmetry, Actions, and Redundancy to distinguish transformations of physical states from gauge redundancy.

References

• Brian C. Hall, Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, 2nd ed., Graduate Texts in Mathematics 222, Springer, 2015. DOI.