Lie Groups, Representations, and Spinors

Use this chapter when a QFT question asks what transformations act, how continuous transformations are generated, how a vector space carries an action, whether global form or a cover matters, or how spinors arise. Begin with groups and actions if the vocabulary is unfamiliar. For continuous symmetries, follow the group–Lie-algebra–representation core and then choose one branch: roots and weights for compact internal groups, Lorentz and Poincaré representations for fields and particles, or Clifford algebras and spinors for fermions.

The chapter’s structural distinction is that a group, its Lie algebra, a representation, a field label, and a particle representation are different objects. In particular, finite-dimensional Lorentz representations label fields. Positive-energy irreducible unitary representations of the universal cover of the proper orthochronous Poincaré group—equivalently, irreducible projective unitary representations of that connected group—classify elementary one-particle transformation types under the Wigner hypotheses. Roots and weights are optional, not preparation that every Lorentz or spinor route must cross.

The chapter supplies reusable symmetry grammar. Developed physical symmetry tests, charges, multiplet classifications, gauge dynamics, Wigner classification, and fermion field theory remain in their physical volumes. This bounded chapter does not attempt theorem-first global representation theory; geometric spin structures continue in Differential Geometry and Bundles.

Parent volume: Mathematical Methods

Enter this chapter

Use these readiness checks to choose the smallest route that repairs the actual gap.

Readiness check	Ready	If unsure	Repair and return
Can you distinguish a group from an action of that group and identify transformations that act trivially?	Begin at groups, actions, quotients, and covers.	Test whether the acting group, acted-on set, and action kernel have all been named.	Use Groups, Actions, Quotients, and Covers.
Can you distinguish a vector, covector, tensor product, and basis change?	Enter the representation or Clifford branch as required.	Check whether a dual action or tensor-product action has been confused with component notation.	Review Vector Spaces, Duals, Linear Maps, and Bases and Direct Sums, Tensor Products, and Index Structure.
Can you tell a homomorphism $\rho:G\to GL(V)$ from an arbitrary family of matrices?	Enter representations and intertwiners after its hard inputs.	Verify $\rho(g_1g_2)=\rho(g_1)\rho(g_2)$ and identify the vector space $V$ .	Read Groups, Actions, Quotients, and Covers before Representations, Intertwiners, Invariants, and Tensor Decomposition.
Can you check $\Lambda^{\mathsf T}\eta\Lambda=\eta$ and explain why a Lorentz field label is not yet a particle spin label?	Take the Lorentz/Poincaré branch.	State which group acts, on which space, and whether the representation is finite-dimensional or unitary.	Use Relativity, Lorentz symmetry, and spin repair, then return to Lorentz Field Representations and Poincaré Particle Representations.
Can you track tensor-algebra relations and signature-dependent signs without choosing a preferred gamma-matrix basis?	Take the Clifford route, then the spinor route.	Check the sign in $\{\gamma^\mu,\gamma^\nu\}=2\eta^{\mu\nu}$ under the site’s metric.	Review Direct Sums, Tensor Products, and Index Structure, then use Clifford Algebras and Pin and Spin Groups.
Can you work with commuting generators and their simultaneous eigenvalues?	Add roots and weights when compact-group classification is the goal.	If not, skip this advanced branch initially.	Return after Lie Groups, Lie Algebras, and Exponential and Adjoint Maps and Representations, Intertwiners, Invariants, and Tensor Decomposition, then use Compact Lie Groups, Roots, Weights, and Weyl Structure; this is not a universal prerequisite.

For connected remediation of the linear-algebra inputs, use Linear and tensor methods repair. For spacetime, causal, and spin-label preparation, use the Lorentz and spin repair.

Choose a route

In the table, an arrow marks a required step, and a plus sign means that both inputs are required. Later branch rows begin from their named inputs and assume any prerequisites stated on those pages; material explicitly called recommended remains optional.

Reader goal	Immediate route from established inputs	Capability at the end
First graduate encounter	Groups and Actions $\to$ Lie Groups and Lie Algebras; separately, Groups and Actions + Vector Spaces and Duals $\to$ Representations	Separate the symmetry object, its infinitesimal algebra, and its realization on a vector space, then choose one branch
Internal symmetry actions	Groups and Actions	Identify orbits, stabilizers, kernels, quotients, and cover data before asking whether an action is a physical QFT symmetry
Continuous transformations and generators	Groups and Actions $\to$ Lie Groups and Lie Algebras	Pass between a finite transformation and its infinitesimal generator while retaining global-form limitations
Multiplets, invariant tensors, and selection rules	Groups and Actions + Vector Spaces and Duals $\to$ Representations	Type representations, intertwiners, invariant subspaces, and tensor decompositions
Compact non-Abelian representation data	Lie Groups and Lie Algebras + Representations $\to$ Roots and Weights	Use Cartan, root, weight, and Weyl data under the compact/semisimple hypotheses stated on the roots-and-weights page
Field labels versus particle states	Lie Groups and Lie Algebras + Representations $\to$ Lorentz and Poincaré Representations	Keep finite-dimensional field covariance separate from unitary one-particle classification
Dirac and spinor preparation	Tensor Products $\to$ Clifford, Pin, and Spin; Clifford + Representations $\to$ Spinors and Bilinears	Construct the algebraic spinor language and track conjugation, chirality, and Fierz hypotheses; add the Lorentz/Poincaré page before the physical Dirac-field exit

The sidebar is a reference order, not a compulsory course. In particular, the compact roots-and-weights topic is an independent branch, while the Clifford branch hard-requires tensor products but only recommends the groups-and-actions page.

The symmetry grammar

Groups act; Lie algebras encode infinitesimal commutator structure; representations realize groups or algebras as linear transformations; roots and weights organize selected compact representations; Clifford algebras encode a quadratic form; Pin and Spin groups project to orthogonal groups; and spinors are modules carrying the resulting action. Each arrow changes the type of object, so none can be replaced by a change of notation.

The broad progression from elementary actions through Lie groups, representations, $SU(2)$ , $SO(3)$ , and spinors is treated in Kosmann-Schwarzbach 2022, chapters “General Facts About Groups” through “Spin Groups and Spinors”. The Lie-theoretic and compact representation statements are developed in Hall 2015, Chapters 2–12 and Fulton and Harris 1991, Chapters 1–24.

Object	Relation used in the chapter	What it does not determine
Group $G$ and action on $X$	A homomorphism $G\to\operatorname{Sym}(X)$ specifies how transformations act; its kernel records invisible elements	The abstract group alone does not select an action, a field, or a physical symmetry
Quotients and covers	A quotient group $G/N$ requires $N\triangleleft G$ and identifies $g\sim g'$ exactly when $g^{-1}g'\in N$ ; a coset space $G/H$ with nonnormal $H$ and an orbit space are quotient sets or spaces and generally have no induced group law; a cover retains global information while mapping to another group	The common Lie algebra does not recover global form or which representations descend
Lie group $G$ and Lie algebra $\mathfrak g$	The tangent space at the identity carries the bracket; the exponential map supplies a local bridge	The exponential map need not be globally injective or surjective, and $\mathfrak g$ does not fix $G$
Representation $\rho$	A homomorphism into $GL(V)$ turns symmetry into linear maps; intertwiners commute with the declared action	A representation is not the group, the vector being transformed, or an arbitrary matrix assignment
Roots, weights, and Weyl action	Under the stated compact/semisimple hypotheses they organize generators and irreducible representation data	They are not a universal language for arbitrary noncompact representations
Lorentz field representation	A finite-dimensional representation specifies how field components mix under spacetime transformations	It does not by itself classify particle states or prove the existence of particles
Physical Poincaré representation	Under the Wigner hypotheses, a positive-energy irreducible unitary representation of the universal cover of the proper orthochronous Poincaré group organizes one-particle mass and spin or helicity data	It is not the finite-dimensional transformation law of a local field and does not prove that a particle exists in an interacting theory
Clifford algebra and spinor module	The quadratic form fixes the Clifford relation; a module carries the spinor action	A Clifford algebra, a Spin group, a spinor module, and a geometric spin structure are distinct

The field/particle distinction and its physical hypotheses are developed in Weinberg 1995, Chapters 2 and 5. The Clifford, Pin, Spin, and module distinctions are treated structurally in Lawson and Michelsohn 1989, Chapter I. Exact definitions and proofs are given on the leaf pages.

Shared conventions and invariant checks

The chapter inherits the site’s active mostly-minus convention,

\eta_{\mu\nu} = \operatorname{diag}(+1,-1,-1,-1).

Group actions are written as active left actions unless a leaf declares otherwise. An induced action on functions therefore normally contains an inverse, $(g\cdot f)(x)=f(g^{-1}\cdot x)$ . Passive component changes and right actions must be translated rather than mixed into the same formula.

In a unitary matrix representation of a compact or internal symmetry, the site’s gauge convention uses Hermitian generators:

[T^a,T^b]=if^{abc}T^c.

Mathematical sources often use anti-Hermitian matrices $t^a=-iT^a$ , for which the translated bracket is

[t^a,t^b]=f^{abc}t^c.

Mapping back recovers the Hermitian convention, which is the required round-trip check. Trace normalization, root length, Dynkin index, and coupling placement remain local choices when they affect a result. Finite-dimensional nonunitary Lorentz field representations must state their generator and exponential conventions.

In four-dimensional Lorentzian spinor sections,

\{\gamma^\mu,\gamma^\nu\}=2\eta^{\mu\nu}, \qquad \overline\psi=\psi^\dagger\gamma^0, \qquad \gamma_5=i\gamma^0\gamma^1\gamma^2\gamma^3.

No gamma-matrix representation is preferred. With $p\!\!\!/\equiv\gamma^\mu p_\mu$ , the invariant checkpoint is

\begin{aligned} (p\!\!\!/)^{\,2} &= \frac12 \{\gamma^\mu,\gamma^\nu\}p_\mu p_\nu\\ &= p^2. \end{aligned}

Dimensions or signatures outside this baseline must restate chirality, reality, charge-conjugation, and adjoint choices. A Majorana or Weyl condition and every Fierz sign carries dimension, signature, base field ( $\mathbb R$ or $\mathbb C$ ), Grassmann parity, and conjugation hypotheses.

Global form is also convention-relevant data. Name the group rather than only its Lie algebra, specify the connected component when material, and identify the cover kernel. A double cover is not automatically the universal cover in every signature or component.

One thread and one recurring comparison

The Dirac thread. The Lorentz page separates finite-dimensional field labels from one-particle Poincaré data. The Clifford page then turns the metric into an algebraic relation and constructs the relevant double cover. The spinor page adds conjugations, bilinears, chirality, and Fierz rearrangements. At every stage the dimension, signature, and selected group or cover remain explicit while the route adds a representation, Clifford algebra, module and bilinears, and then geometric bundle and operator data. Any Lorentzian-to-Riemannian transition before the elliptic index bridge must be stated and its Clifford and sign conventions rechecked. This Dirac route does not classify all spin systems or guarantee that every manifold admits the required spin structure. For physical dynamics and observables, it first exits to The Dirac Field and then to Plane Waves, Spin Sums, and Bilinears. It later resumes at Spin Structures and Dirac Operators when the question is geometric, after its independent manifold/curvature and bundle prerequisites. A later optional continuation is Fredholm and Dirac Index Theorems and Zero-Mode Counting, once its spectra/resolvents and characteristic-classes inputs are also in place.

Recurring $SU(2)$ – $SO(3)$ comparison. The groups page introduces the double cover $SU(2)\to SO(3)$ with kernel $\{\pm\mathbf1\}$ . The Lie page shows that the groups have isomorphic Lie algebras, while the representation page asks whether the kernel acts trivially. In the spin- $j$ irreducible representation, $-\mathbf1$ acts as $(-1)^{2j}$ , so integer-spin representations descend to $SO(3)$ and half-integer-spin representations do not. This is the recurring check that group, algebra, cover, and representation have not been conflated; see Kosmann-Schwarzbach 2022, pp. 89–118.

Topics in this chapter

The seven topics are:

Groups, Actions, Quotients, and Covers asks how groups act, what data survive a quotient, and when a cover is essential. It is the foundation entry with no hard prerequisite. Its endpoint is the ability to separate actions, kernels, orbits, stabilizers, faithful quotients, coset spaces, and cover descent. Continue to the Lie page for continuous structure or to What Is a Symmetry of a QFT? for the physical symmetry test.
Lie Groups, Lie Algebras, and Exponential and Adjoint Maps hard-requires the groups-and-actions page. It asks how the local Lie algebra encodes infinitesimal transformations and their finite action. The page develops the group/algebra distinction, bracket, exponential map, and adjoint maps. Continue to Continuous Symmetries, Generators, and Charges for the physical charge interpretation.
Representations, Intertwiners, Invariants, and Tensor Decomposition hard-requires groups/actions and the external vector-spaces page. It asks how symmetry becomes linear maps and how invariant tensors and irreducible pieces are recognized. Its output is representation and intertwiner type discipline, not an unrestricted complete-reducibility theorem. Continue to Multiplets, Invariants, and Selection Rules for the developed physical use.
Compact Lie Groups, Roots, Weights, and Weyl Structure requires the Lie and representation pages. This branch organizes compact-group representations through Cartan subalgebras, roots, weights, and Weyl symmetry. Its compact/semisimple hypotheses stay visible. Continue to Electroweak Gauge and Matter Structure for a physical multiplet application.
Lorentz Field Representations and Poincaré Particle Representations requires the Lie and representation pages. It establishes the chapter’s central category distinction: finite-dimensional Lorentz field representations and unitary Poincaré particle representations answer different questions. The physical Wigner classification is developed in Foundations; continue to One-Particle States: Mass, Spin, and Relativistic Normalization.
Clifford Algebras and Pin and Spin Groups requires tensor products; groups/actions is recommended. It asks how a quadratic form produces a Clifford algebra and the double covers of orthogonal groups. It stops before geometric spin structures and physical fermion dynamics. Continue to The Dirac Field for physical fermion treatment, to the spinor leaf for bilinear algebra, or, after its independent manifold/curvature and bundle prerequisites, to Spin Structures and Dirac Operators.
Spinors, Conjugations, Bilinears, Chirality, and Fierz Identities requires Clifford theory and representations. This branch asks how spinor representations and invariant bilinears depend on dimension and signature. It requires every reality, chirality, conjugation, and Fierz claim to retain its hypotheses. Continue to the two Foundations fermion pages in the Dirac thread for physical fields, states, and amplitudes.

Chapter synthesis and stopping boundary

The chapter’s minimum durable conclusions are:

a group does not determine how it acts, and an action can have a nontrivial kernel;
a Lie algebra captures local infinitesimal structure but not the global group, quotient, cover, or allowed representation spectrum;
the exponential map is a local bridge, not generally a global bijection;
complete reducibility requires hypotheses and cannot be assumed for every noncompact or nonunitary representation;
roots and weights are a specialized compact/semisimple tool, not a mandatory preface to Lorentz or spinor work;
finite-dimensional Lorentz representations label fields, whereas physical Poincaré representations organize elementary one-particle transformation types under the Wigner hypotheses stated above;
a Clifford algebra depends on its scalar field, quadratic form, and signature; Pin and Spin groups, spinor modules, and spin structures are separate constructions; and
chirality, reality conditions, bilinear symmetry, and Fierz signs are dimension- and signature-dependent.

The chapter stops before deciding whether an action is an exact, approximate, anomalous, spontaneously broken, or gauge symmetry of a QFT. It does not derive Noether currents or Ward identities, classify interacting multiplets, choose a gauge-group global form, prove particle existence, build a fermion theory, or construct a spin structure on a manifold.

Review the chapter

Retrieval and explanation — type the objects. Given a matrix group acting on a field multiplet, identify the group, action, kernel, representation space, and representation map. A successful response does not call the matrix list “the group” without checking multiplication and does not confuse the field vector with the representation. Repair with Groups, Actions, Quotients, and Covers and Representations, Intertwiners, Invariants, and Tensor Decomposition.

Derivation reconstruction — differentiate a representation. Starting from a smooth group representation, explain why its derivative must satisfy

\mathrm d\rho([X,Y]) = [\mathrm d\rho(X),\mathrm d\rho(Y)].

Success identifies smoothness, differentiation at the identity, and the commutator bracket; it does not infer the global representation from the Lie algebra without checking topology. Repair with Lie Groups, Lie Algebras, and Exponential and Adjoint Maps and Representations, Intertwiners, Invariants, and Tensor Decomposition.

Convention translation — change generator basis. Translate $[T^a,T^b]=if^{abc}T^c$ to $t^a=-iT^a$ and back. Success obtains $[t^a,t^b]=f^{abc}t^c$ and returns to the original formula. A sign guessed without the round trip is incomplete. Repair with Lie Groups, Lie Algebras, and Exponential and Adjoint Maps.

Comparison and failure diagnosis — global form and particles. Explain why $SU(2)$ and $SO(3)$ can share a Lie algebra but have different representation spectra, then explain why a Lorentz field label is not a one-particle spin label. Success checks the cover kernel in the first part and names the Poincaré representation and its Hilbert-space unitarity in the second. Repair with Groups, Actions, Quotients, and Covers, Representations, Intertwiners, Invariants, and Tensor Decomposition, and Lorentz Field Representations and Poincaré Particle Representations.

Transfer — test descent on a new cover. Let $q:\mathbb R\to U(1)$ be $q(\theta)=e^{i\theta}$ . For $s\in\mathbb R$ , let $\chi_s(\theta)=e^{is\theta}$ be a unitary character of $\mathbb R$ . Determine when $\chi_s$ descends through $q$ to a representation of $U(1)$ . Success checks that the kernel $2\pi\mathbb Z$ acts trivially and concludes $s\in\mathbb Z$ ; merely observing that $\mathbb R$ and $U(1)$ have the same Lie algebra is incomplete. Repair with Groups, Actions, Quotients, and Covers and Representations, Intertwiners, Invariants, and Tensor Decomposition.

Synthesis — follow the Dirac thread. Starting from a Lorentz transformation, name the successive mathematical objects needed to reach a covariant spinor bilinear, then identify where physical field dynamics and geometric spin structures leave this chapter. Success names the Lorentz representation, Clifford algebra, Spin action, spinor module, conjugation, and bilinear before taking the two explicit exits above. Repair with Lorentz Field Representations and Poincaré Particle Representations, Clifford Algebras and Pin and Spin Groups, and Spinors, Conjugations, Bilinears, Chirality, and Fierz Identities.

Optional exercise — test a dominant integral weight. For a compact semisimple Lie algebra with simple roots $\alpha_i$ and a proposed highest weight $\lambda$ , evaluate $2(\lambda,\alpha_i)/(\alpha_i,\alpha_i)$ for every simple root. Success identifies the nonnegative-integer condition, states the compact/semisimple and invariant-inner-product hypotheses, and does not impose this test on an arbitrary noncompact representation problem. Repair with Compact Lie Groups, Roots, Weights, and Weyl Structure.

Exit routes

These links lead to later topics; this chapter alone is not sufficient for them. Each physical and geometric continuation states its prerequisites.

Direct prerequisite map

The arrows below indicate required preparation. Each named input may have additional prerequisites; recommended preparation is omitted.

Destination	Direct hard-input route
What Is a Symmetry of a QFT?	Groups, Actions, Quotients, and Covers $\to$ What Is a Symmetry of a QFT?
Continuous Symmetries, Generators, and Charges	What Is a Symmetry of a QFT? + Classical Symmetries, Currents, and Stress Tensors $\to$ Continuous Symmetries, Generators, and Charges
Multiplets, Invariants, and Selection Rules	What Is a Symmetry of a QFT? + Representations, Intertwiners, Invariants, and Tensor Decomposition $\to$ Multiplets, Invariants, and Selection Rules
Electroweak Gauge and Matter Structure	The Yang–Mills Action and Gauge Self-Interaction + Compact Lie Groups, Roots, Weights, and Weyl Structure $\to$ Electroweak Gauge and Matter Structure
One-Particle States	No direct hard prerequisite
The Dirac Field $\to$ Plane Waves, Spin Sums, and Bilinears	The Action Principle and Field Equations + One-Particle States $\to$ The Dirac Field $\to$ Plane Waves, Spin Sums, and Bilinears
Spin Structures and Dirac Operators	Levi–Civita Connections, Geodesics, and Riemann Curvature + Vector, Principal, and Associated Bundles + Clifford Algebras and Pin and Spin Groups $\to$ Spin Structures and Dirac Operators
Fredholm and Dirac Index Theorems and Zero-Mode Counting	Spectra, Resolvents, Spectral Measures, and Functional Calculus + Characteristic Classes and Chern–Weil Theory + Spin Structures and Dirac Operators $\to$ Fredholm and Dirac Index Theorems and Zero-Mode Counting

For exact QFT symmetry actions, continue to What Is a Symmetry of a QFT?. For generators and charges, use Continuous Symmetries, Generators, and Charges.
For physical multiplets and invariant couplings, continue to Multiplets, Invariants, and Selection Rules. For compact electroweak data, use Electroweak Gauge and Matter Structure.
For particle interpretation, use One-Particle States. For fermion dynamics and observables, continue first to The Dirac Field, then to Plane Waves, Spin Sums, and Bilinears.
For spinors on manifolds and geometric Dirac operators, continue—after the independent manifold/curvature and bundle prerequisites—to Spin Structures and Dirac Operators. With its additional spectra/resolvents and characteristic-classes inputs, the later optional exit is Fredholm and Dirac Index Theorems and Zero-Mode Counting.
For a connected learning route, use Fermions, spin, and anticommutation. For capability repair, use Relativity, Lorentz symmetry, and spin.
To choose another mathematical branch, return to Mathematical Methods.

References

William Fulton and Joe Harris, Representation Theory: A First Course, Graduate Texts in Mathematics 129, Springer, 1991, Parts I–III. These parts develop finite-dimensional representations, tensor constructions, classical Lie algebras, and root-and-weight organization. Its hypotheses are not extended to arbitrary noncompact representation categories.
Brian C. Hall, Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, second edition, Graduate Texts in Mathematics 222, Springer, 2015, Chapters 1–5 and 7–13. These chapters establish matrix Lie groups, exponentials, Lie algebras, basic representations, roots, semisimple structure, and compact groups.
Yvette Kosmann-Schwarzbach, Groups and Symmetries: From Finite Groups to Lie Groups, second edition, Springer, 2022, chapters “General Facts About Groups” through “Spin Groups and Spinors.” These chapters organize the chapter map, especially the $SU(2)$ – $SO(3)$ comparison. The book’s physics examples are not used to replace this site’s physical continuations.
H. Blaine Lawson Jr. and Marie-Louise Michelsohn, Spin Geometry, Princeton Mathematical Series 38, Princeton University Press, 1989, Chapter I, §§1–6. These sections establish Clifford algebras, Pin and Spin groups, their representations, and the relevant Lie structures. Later geometric and index-theorem material belongs to subsequent chapters and pages.
Steven Weinberg, The Quantum Theory of Fields, Volume I: Foundations, Cambridge University Press, 1995, Chapters 2 and 5. This is the structural source for the bounded field-representation versus particle-representation distinction and the Foundations continuation; no physical classification theorem is reproduced on this overview.