07.07.09 · representation-theory / compact-lie

Representations of the Lorentz group: $SL (2, C)$ , the $(j_{1}, j_{2})$ reps, and Wigner's theorem

shipped3 tiersLean: none

Anchor (Master): Sternberg *Group Theory and Physics* Chs. 8–9; Weinberg *The Quantum Theory of Fields, Vol. I* (CUP 1995) §2.7; Wigner *On unitary representations of the inhomogeneous Lorentz group* (Ann. Math. 40, 1939); Naimark *Linear Representations of the Lorentz Group* (Pergamon 1964); Bargmann *Irreducible unitary representations of the Lorentz group* (Ann. Math. 48, 1947)

Intuition Beginner

The rotations you can do to an object in three-dimensional space form one symmetry group. Special relativity adds a second kind of move: a boost, which mixes space and time when you change to a frame moving at constant velocity. Rotations and boosts together make up the Lorentz group, the symmetries of flat spacetime that fix the origin. A boost is like a rotation, except the geometry it preserves is the spacetime interval, where time enters with the opposite sign from space.

The deep fact is that the Lorentz group splits into two independent copies of the rotation story. There is a left-handed copy and a right-handed copy, and each one behaves like the spin theory you already met for ordinary rotation. A relativistic object is labelled by two spins at once: one number $j_{1}$ for the left copy and one number $j_{2}$ for the right copy. A left-handed electron carries spin in the left copy only; its mirror image carries spin in the right copy only.

Why bother? Because every relativistic particle and every relativistic field is built by choosing those two labels. The pair $(j_{1}, j_{2})$ is the blueprint. Pick $(\frac{1}{2}, 0)$ and you get a left-handed Weyl spinor; pick $(\frac{1}{2}, \frac{1}{2})$ and you get an ordinary four-vector like the electromagnetic potential. The whole zoo of relativistic objects comes from this one two-label scheme.

Visual Beginner

A diagram showing the Lorentz group as two side-by-side dials, one labelled "left" and one labelled "right". Each dial is the rotation-spin dial you saw for $S U (2)$ , marked with the allowed spin values $0, \frac{1}{2}, 1, \frac{3}{2}, \dots$ . A relativistic object is a pin placed on each dial: the left pin sits at $j_{1}$ and the right pin sits at $j_{2}$ . Familiar objects are marked: the four-vector at $(\frac{1}{2}, \frac{1}{2})$ , the left Weyl spinor at $(\frac{1}{2}, 0)$ , the right Weyl spinor at $(0, \frac{1}{2})$ , and the plain scalar at $(0, 0)$ .

The picture captures the central idea: the Lorentz group is two rotation stories running in parallel, and naming a relativistic object means choosing a spin on each.

Worked example Beginner

Count the components of three relativistic objects by reading off their two labels.

Step 1. The rule for the size of a $(j_{1}, j_{2})$ object: the left dial contributes $2 j_{1} + 1$ slots and the right dial contributes $2 j_{2} + 1$ slots, and you multiply, because the two choices are independent. So the number of components is $(2 j_{1} + 1) \times (2 j_{2} + 1)$ .

Step 2. The scalar $(0, 0)$ . Left contributes $2 \times 0 + 1 = 1$ . Right contributes $2 \times 0 + 1 = 1$ . Product: $1 \times 1 = 1$ . A scalar has one component, like temperature at a point. Correct.

Step 3. The left-handed Weyl spinor $(\frac{1}{2}, 0)$ . Left contributes $2 \times \frac{1}{2} + 1 = 2$ . Right contributes $2 \times 0 + 1 = 1$ . Product: $2 \times 1 = 2$ . A Weyl spinor has two components. This is the smallest object that feels the spin doubling.

Step 4. The four-vector $(\frac{1}{2}, \frac{1}{2})$ . Left contributes $2$ . Right contributes $2$ . Product: $2 \times 2 = 4$ . Four components, exactly matching the four coordinates of spacetime — time and the three space directions. The electromagnetic potential is one of these.

What this tells us: the two spin labels are not decoration. They predict the exact number of components an object carries, and the answers line up with the objects physics already uses — one number for a scalar, two for a Weyl spinor, four for a vector.

Check your understanding Beginner

Formal definition Intermediate+

The Lorentz group $O (3, 1)$ is the group of real $4 \times 4$ matrices $Λ$ preserving the Minkowski form $η = diag (+ 1, - 1, - 1, - 1)$ , meaning $Λ^{T} η Λ = η$ . Two homomorphisms $det : O (3, 1) \to {\pm 1}$ and the sign of $Λ^{0}_{0}$ split the group into four connected components. The proper orthochronous component, $SO^{+} (3, 1)$ , is the identity component: those $Λ$ with $det Λ = + 1$ and $Λ^{0}_{0} \geq 1$ . The other three components are reached from this one by space inversion $P$ , time reversal $T$ , and their product $P T$ . The signature convention $(+, -, -, -)$ is fixed throughout this unit.

Identify Minkowski space with the real vector space of Hermitian $2 \times 2$ matrices through $$ x = (x^0, x^1, x^2, x^3) \longmapsto X = x^\mu \sigma_\mu = \begin{pmatrix} x^0 + x^3 & x^1 - i x^2 \ x^1 + i x^2 & x^0 - x^3 \end{pmatrix}, $$ where $σ_{0} = I$ and $σ_{1}, σ_{2}, σ_{3}$ are the Pauli matrices. Then $det X = (x^{0})^{2} - (x^{1})^{2} - (x^{2})^{2} - (x^{3})^{2}$ is the Minkowski norm. For $A \in SL (2, C)$ the map $X \mapsto A X A^{†}$ sends Hermitian matrices to Hermitian matrices and preserves the determinant, so it acts on Minkowski space by a Lorentz transformation $Λ (A) \in SO^{+} (3, 1)$ . The assignment $A \mapsto Λ (A)$ is a surjective Lie-group homomorphism with kernel ${\pm I}$ ; it is the spin covering $SL (2, C) \to SO^{+} (3, 1)$ , a two-to-one universal covering map.

At the Lie-algebra level write $J$ for the rotation generators and $K$ for the boost generators, satisfying $[J_{a}, J_{b}] = i ϵ_{ab c} J_{c}$ , $[J_{a}, K_{b}] = i ϵ_{ab c} K_{c}$ , $[K_{a}, K_{b}] = - i ϵ_{ab c} J_{c}$ . Pass to the complex combinations $$ \vec J_+ = \tfrac12 (\vec J + i \vec K), \qquad \vec J_- = \tfrac12 (\vec J - i \vec K). $$ These satisfy $[J_{+ a}, J_{+ b}] = i ϵ_{ab c} J_{+ c}$ , $[J_{- a}, J_{- b}] = i ϵ_{ab c} J_{- c}$ , and $[J_{+ a}, J_{- b}] = 0$ . So over $C$ the Lorentz Lie algebra factors: $$ \mathfrak{so}(3,1){\mathbb{C}} \cong \mathfrak{sl}(2,\mathbb{C}) \oplus \mathfrak{sl}(2,\mathbb{C}), $$ two commuting copies of the complexified angular-momentum algebra. A finite-dimensional irreducible representation of $SL (2, C)$ is therefore a pair: a highest weight $j_1 \in \tfrac12 \mathbb{Z}{\geq 0} $f or t h e$ \vec J_+ $a l g e b r aan d ahi g h es tw e i g h t$ j_2 \in \tfrac12 \mathbb{Z}{\geq 0} $f or t h e$ \vec J- $a l g e b r a . T h er e p r ese n t a t i o ni s d e n o t e d$ (j_1, j_2) $, ha s d im e n s i o n$ (2 j_1 + 1)(2 j_2 + 1) $, an d i sr e a l i se d a s$ \mathrm{Sym}^{2 j_1}(\mathbb{C}^2) \boxtimes \overline{\mathrm{Sym}^{2 j_2}(\mathbb{C}^2)} $, t h e h o l o m or p hi cs p in -$ j_1 $r e p r ese n t a t i o n t e n sor e d w i t h t h e an t ih o l o m or p hi cs p in -$ j_2$ representation. The formal definition follows Sternberg ^{[Sternberg Chs. 8–9]} and Weinberg ^{[Weinberg §2.7]}.

Counterexamples to common slips

The factorisation $so (3, 1)_{C} ≅ sl (2, C) \oplus sl (2, C)$ holds only after complexifying. The real algebra $so (3, 1)$ is the real form $sl (2, C)_{R}$ , a simple real Lie algebra; it does not split as a real direct sum. The combinations $J_{\pm}$ require the factor $i$ multiplying $K$ , so $J_{+}$ and $J_{-}$ are not generators of two real subalgebras. The labels $j_{1}, j_{2}$ are honest, but $(j_{2}, j_{1})$ is the complex conjugate of $(j_{1}, j_{2})$ , and complex conjugation swaps the two factors rather than acting inside one.
A finite-dimensional $(j_{1}, j_{2})$ representation is not unitary unless $j_{1} = j_{2} = 0$ . The rotation subgroup acts unitarily, but a boost is represented by a non-compact one-parameter group whose generators are not anti-Hermitian, so no inner product is preserved. This is forced by non-compactness, not by a poor choice of basis.
The four-vector representation is $(\frac{1}{2}, \frac{1}{2})$ , dimension $4$ , not $(1, 0) \oplus (0, 1)$ . The latter, dimension $6$ , is the antisymmetric two-tensor (the field-strength representation). The decomposition $(\frac{1}{2}, 0) \otimes (0, \frac{1}{2}) = (\frac{1}{2}, \frac{1}{2})$ is what builds the vector from two Weyl spinors of opposite handedness.

Key theorem with proof Intermediate+

Theorem (classification of finite-dimensional irreducibles of $SL (2, C)$ ). The finite-dimensional irreducible complex representations of $SL (2, C)$ , viewed as a real Lie group, are exactly the representations $(j_{1}, j_{2})$ with $j_{1}, j_{2} \in \frac{1}{2} Z_{\geq 0}$ , of dimension $(2 j_{1} + 1) (2 j_{2} + 1)$ . Two such are isomorphic if and only if their labels agree. None is unitary except $(0, 0)$ .

Proof. A finite-dimensional complex representation of the real Lie group $SL (2, C)$ differentiates to a representation of the real Lie algebra $sl (2, C)_{R}$ , and on a complex vector space this extends $C$ -linearly to the complexification $sl (2, C)_{R} \otimes_{R} C$ . The complexification of a complex simple Lie algebra $g$ regarded as real is $g \oplus \overset{ˉ}{g} ≅ sl (2, C) \oplus sl (2, C)$ , the two summands being the holomorphic and antiholomorphic parts spanned by $J_{+}$ and $J_{-}$ .

A finite-dimensional irreducible representation of a direct sum $a \oplus b$ of semisimple Lie algebras is an external tensor product $V_{a} ⊠ V_{b}$ of irreducibles of the factors; this is the standard consequence of Schur's lemma together with complete reducibility, applied to the commuting actions of the two summands. The finite-dimensional irreducibles of each $sl (2, C)$ factor are the highest-weight modules $Sym^{2 j} (C^{2})$ , $j \in \frac{1}{2} Z_{\geq 0}$ , of dimension $2 j + 1$ , from the ladder-operator theory of $sl_{2}$ . Pairing a highest weight $j_{1}$ for $J_{+}$ with a highest weight $j_{2}$ for $J_{-}$ produces the irreducible $(j_{1}, j_{2})$ of dimension $(2 j_{1} + 1) (2 j_{2} + 1)$ . The highest-weight pair is a complete invariant, so distinct labels give non-isomorphic representations and these exhaust the finite-dimensional irreducibles.

Each such representation integrates to the group: because $SL (2, C)$ is simply connected, every finite-dimensional representation of its Lie algebra exponentiates to a representation of the group. Concretely, the $J_{+}$ factor sees a group element $A$ through the holomorphic action $A \mapsto Sym^{2 j_{1}} (A)$ , and the $J_{-}$ factor sees it through the antiholomorphic action $A \mapsto Sym^{2 j_{2}} (\overset{ˉ}{A})$ .

Non-unitarity: in any representation the rotation generators $J$ and the boost generators $K$ are related by $J_{\pm} = \frac{1}{2} (J \pm i K)$ . In the $(j_{1}, j_{2})$ representation $J_{+}$ acts as spin $j_{1}$ and $J_{-}$ as spin $j_{2}$ , so $J = J_{+} + J_{-}$ and $K = - i (J_{+} - J_{-}) = i (J_{-} - J_{+})$ . For the representation to be unitary one needs $J$ and $K$ Hermitian. The operators $J_{\pm}$ are Hermitian on the spin modules, so $J$ is Hermitian, but $K = i (J_{-} - J_{+})$ is then anti-Hermitian rather than Hermitian whenever $J_{-} - J_{+} \neq = 0$ , that is whenever $(j_{1}, j_{2}) \neq = (0, 0)$ . The boost generators exponentiate to operators of unbounded norm, so no positive-definite inner product is invariant. This reflects a general fact: a non-compact simple Lie group has no finite-dimensional unitary representation other than the one sending every element to the identity action. $□$

Bridge. This classification builds toward the construction of every relativistic field, and it appears again in the Dirac-equation unit 12.11.01 in the guise of the four-component spinor. The foundational reason the Lorentz group carries two spin labels is exactly the complexified splitting $so (3, 1)_{C} ≅ sl (2, C) \oplus sl (2, C)$ : each factor is a copy of the angular-momentum algebra whose representation theory was settled in 07.06.11, and putting these together identifies a Lorentz irrep with a pair of $sl_{2}$ irreps. This is exactly the same highest-weight mechanism that classifies $S U (2)$ spin in 07.07.05, now run twice in parallel. The central insight is that the handedness of a Weyl spinor is the statement that one of the two labels vanishes: $(\frac{1}{2}, 0)$ is left-handed, $(0, \frac{1}{2})$ is right-handed, and parity is the operation that swaps the factors. The bridge is that the finite-dimensional theory, non-unitary because the group is non-compact, governs how field components transform, while the genuinely unitary representations needed for quantum states are infinite-dimensional and generalises the compact-group story to the full Poincaré group.

Exercises Intermediate+

Exercise 5 (medium, short-answer).

Show that complex conjugation of a representation sends $(j_{1}, j_{2})$ to $(j_{2}, j_{1})$ , and deduce that the Weyl spinors $(\frac{1}{2}, 0)$ and $(0, \frac{1}{2})$ are complex-conjugate to one another.

Hint

Conjugating the matrices swaps the holomorphic action $A$ with the antiholomorphic action $\overset{ˉ}{A}$ .

Answer

The $(j_{1}, j_{2})$ representation acts by $A \mapsto Sym^{2 j_{1}} (A) ⊠ Sym^{2 j_{2}} (\overset{ˉ}{A})$ . Complex-conjugating every matrix entry replaces $A$ by $\overset{ˉ}{A}$ and $\overset{ˉ}{A}$ by $A$ , which exchanges the holomorphic and antiholomorphic factors and so sends $(j_{1}, j_{2})$ to $(j_{2}, j_{1})$ . Applied to $(\frac{1}{2}, 0)$ this gives $(0, \frac{1}{2})$ : the conjugate of a left-handed Weyl spinor is a right-handed one. A representation is self-conjugate exactly when $j_{1} = j_{2}$ , which is why the four-vector $(\frac{1}{2}, \frac{1}{2})$ admits a real form while a single Weyl spinor does not. Rubric: full credit for the factor-swap argument and the Weyl-spinor conclusion.

Exercise 6 (hard, short-answer).

Prove that the antisymmetric part of $(\frac{1}{2}, \frac{1}{2})^{\otimes 2}$ is $(1, 0) \oplus (0, 1)$ , and identify these with the self-dual and anti-self-dual parts of the electromagnetic field strength $F_{μν}$ .

Hint

The antisymmetric square of a four-vector is a rank-two antisymmetric tensor, six-dimensional. Use the eigenspaces of the Hodge dual operator $⋆$ on two-forms in Minkowski signature.

Answer

From Exercise 4, $(\frac{1}{2}, \frac{1}{2})^{\otimes 2} = (1, 1) \oplus (1, 0) \oplus (0, 1) \oplus (0, 0)$ . The symmetric square accounts for $(1, 1) \oplus (0, 0)$ , dimension $10$ , so the antisymmetric square is the remaining $(1, 0) \oplus (0, 1)$ , dimension $6 = 3 + 3$ , matching the six independent components of an antisymmetric $F_{μν}$ . The Hodge dual $⋆$ acts on real two-forms with $⋆^{2} = - 1$ in Lorentzian signature, so its eigenvalues are $\pm i$ on the complexified space; the $+ i$ eigenspace is the self-dual part $F^{+}$ and the $- i$ eigenspace is the anti-self-dual part $F^{-}$ . Under $J_{\pm}$ the self-dual part carries $(1, 0)$ and the anti-self-dual part carries $(0, 1)$ , because $J_{+}$ generates self-dual rotations and $J_{-}$ anti-self-dual ones. The Maxwell field is the real combination $F = F^{+} + F^{-}$ with $F^{-} = \overline{F^{+}}$ , consistent with $(1, 0)$ and $(0, 1)$ being complex conjugates. Rubric: full credit for the dimension bookkeeping plus the $⋆$ -eigenspace identification.

Exercise 7 (hard, short-answer).

Explain why a single massless particle of helicity $\pm 1$ (the photon) is described in field theory by the non-unitary $(1, 0) \oplus (0, 1)$ field-strength representation, yet its quantum states form a unitary representation of the Poincaré group. Where does the unitarity come from?

Hint

Distinguish the finite-dimensional representation acting on field components from the infinite-dimensional unitary representation acting on one-particle states, classified by Wigner.

Answer

The field strength $F_{μν} (x)$ at each spacetime point transforms in the finite-dimensional, non-unitary $(1, 0) \oplus (0, 1)$ representation of the homogeneous Lorentz group: this governs how the six components mix under a Lorentz transformation. Unitarity is not expected here, because the homogeneous Lorentz group is non-compact and its finite-dimensional representations are non-unitary. The quantum states of the photon, by contrast, live in an infinite-dimensional Hilbert space carrying a unitary representation of the full Poincaré group; Wigner's classification labels it by zero mass and helicity $\pm 1$ , built by inducing from the little group $ISO (2)$ . The inner product that makes this unitary is the relativistically invariant one-particle norm integrated over the mass-zero orbit with its invariant measure, not any norm on the six field components. The field and the states are two different representations: the finite-dimensional one organises the local field operator, the infinite-dimensional one carries the physical inner product. Rubric: full credit for separating the two representations and locating unitarity in the infinite-dimensional state space.

Lean formalization Intermediate+

Mathlib carries Matrix.SpecialLinearGroup (so $SL (2, C)$ exists as a group) and the Pauli matrices can be written down, but no Lorentz group, spin covering, or $(j_{1}, j_{2})$ classification ships as named objects. The intended formalisation reads schematically:

import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
import Mathlib.LinearAlgebra.Matrix.Hermitian
import Mathlib.Analysis.InnerProductSpace.Basic

/-- The spin covering SL(2,ℂ) → SO⁺(3,1) via the action on
    Hermitian 2×2 matrices X ↦ A X Aᴴ, which preserves det X. -/
noncomputable def spinCover (A : Matrix.SpecialLinearGroup (Fin 2) ℂ) :
    LorentzGroup :=
  sorry  -- X ↦ A * X * Aᴴ on Hermitian matrices ≅ Minkowski space

/-- Kernel of the spin covering is {±I}. -/
theorem spinCover_ker :
    (spinCover.ker : Set (Matrix.SpecialLinearGroup (Fin 2) ℂ)) = {1, -1} :=
  sorry

/-- The finite-dimensional irreducible (j₁,j₂) of SL(2,ℂ). -/
noncomputable def lorentzIrrep (j₁ j₂ : ℕ) :
    Representation ℂ (Matrix.SpecialLinearGroup (Fin 2) ℂ)
      (Sym j₁ (Fin 2 → ℂ) ⊗ (Sym j₂ (Fin 2 → ℂ))ᶜ) :=
  sorry  -- holomorphic Sym^{2j₁} ⊠ antiholomorphic Sym^{2j₂}

/-- No finite-dimensional unitary irrep except (0,0). -/
theorem lorentzIrrep_not_unitary (j₁ j₂ : ℕ) (h : (j₁, j₂) ≠ (0, 0)) :
    ¬ IsUnitary (lorentzIrrep j₁ j₂) :=
  sorry  -- boost generators are non-Hermitian; non-compactness obstruction

The proof gap is substantial. Mathlib needs the Minkowski form and $SO^{+} (3, 1)$ as a connected Lie group, the spin-covering homomorphism with its kernel computation, the complexified splitting into two $sl_{2}$ factors, the symmetric-power construction realising $Sym^{2 j}$ , the external tensor product $⊠$ of representations of a product algebra, and the non-compactness obstruction to finite-dimensional unitarity. Wigner's theorem on projective symmetries is a separate target requiring the geometry of rays in a complex Hilbert space.

Advanced results Master

Theorem (Wigner's theorem on symmetries). Let $H$ be a complex Hilbert space and let $T$ be a bijection of the set of rays of $H$ preserving transition probabilities, $∣ ⟨ T ψ, T ϕ ⟩ ∣^{2} = ∣ ⟨ ψ, ϕ ⟩ ∣^{2}$ for all rays. Then $T$ is implemented by an operator on $H$ that is either unitary or antiunitary, and this operator is unique up to a phase.

A continuous symmetry group connected to the identity is implemented by unitaries, since antiunitarity cannot vary continuously to the identity; the discrete operations $P$ and $C$ are unitary, while $T$ is antiunitary. The consequence for relativistic physics is structural: a relativistic quantum theory carries a projective unitary representation of $SO^{+} (3, 1)$ (or of the Poincaré group), and lifting the projective representation to a genuine one passes to the simply connected cover $SL (2, C)$ . The states of a single particle therefore furnish an honest unitary representation of $SL (2, C) ⋉ R^{1, 3}$ , and by the classification of 07.07.06 this representation is infinite-dimensional, labelled by mass and spin.

Theorem (infinite-dimensional unitary irreducibles of $SL (2, C)$ ; Naimark, Gelfand–Naimark). The irreducible unitary representations of $SL (2, C)$ beyond the identity action form two families. The principal series is labelled by a pair $(k, ν)$ with $k \in \frac{1}{2} Z$ and $ν \in R$ , acting on $L^{2}$ sections over the sphere; the complementary series is labelled by $k = 0$ and a real parameter $s \in (0, 2)$ . Every irreducible unitary representation other than the identity action is infinite-dimensional.

These are the homogeneous-Lorentz analogues of Bargmann's $SL (2, R)$ series ^{[Bargmann 1947]}. They are not the representations carried by physical one-particle states — those are representations of the full Poincaré group, where the translations select a mass shell — but they appear in the harmonic analysis of fields on the light cone, in conformal field theory, and in the Regge-pole expansion. The contrast with the finite-dimensional $(j_{1}, j_{2})$ family is sharp: the same group has a non-unitary finite-dimensional theory and a unitary infinite-dimensional theory, and physics uses both, for fields and for states respectively.

Proposition (parity exchanges the two factors). Space inversion $P$ conjugates the spin covering by an element implementing $Λ (P) = diag (1, - 1, - 1, - 1)$ , and its action on the Lie algebra sends $J \mapsto J$ , $K \mapsto - K$ , hence $J_{+} \leftrightarrow J_{-}$ . Consequently $P$ maps the representation $(j_{1}, j_{2})$ to $(j_{2}, j_{1})$ .

This is why a parity-invariant theory of spin- $\frac{1}{2}$ matter cannot use a single Weyl spinor: $(\frac{1}{2}, 0)$ and $(0, \frac{1}{2})$ are exchanged by $P$ , so parity invariance forces the reducible Dirac sum $(\frac{1}{2}, 0) \oplus (0, \frac{1}{2})$ , which is irreducible under the group generated by $SO^{+} (3, 1)$ together with $P$ . The weak interaction, which violates parity, couples to a single Weyl component.

Theorem (covariant field constraints; Weinberg). A free field $ψ (x)$ transforming in $(j_{1}, j_{2})$ creates and annihilates one-particle states of spin $j$ for each $∣ j_{1} - j_{2} ∣ \leq j \leq j_{1} + j_{2}$ . Matching the field representation to a definite-spin particle requires subsidiary conditions (the Dirac equation for $(\frac{1}{2}, 0) \oplus (0, \frac{1}{2})$ , the transversality and mass-shell conditions for $(\frac{1}{2}, \frac{1}{2})$ ), which project the field's $(2 j_{1} + 1) (2 j_{2} + 1)$ components onto the $2 j + 1$ physical polarisations.

The field representation is finite-dimensional and non-unitary; the wave equation is exactly the device that reconciles it with the unitary one-particle representation of definite spin. For the Dirac field, $(\frac{1}{2}, 0) \oplus (0, \frac{1}{2})$ has four components, the Dirac equation halves the dynamical content, and the result describes a spin- $\frac{1}{2}$ particle and its antiparticle. See 12.11.01.

Synthesis. The two-label scheme is the foundational reason relativistic objects come in the variety they do, and it is exactly the complexified splitting $so (3, 1)_{C} ≅ sl (2, C) \oplus sl (2, C)$ run through the $sl_{2}$ representation theory of 07.06.11. The central insight is that the finite-dimensional $(j_{1}, j_{2})$ representations, non-unitary because $SL (2, C)$ is non-compact, govern field components, while the unitary representations demanded by Wigner's theorem are infinite-dimensional and generalise the compact-group classification to the Poincaré group of 07.07.06. Putting these together, a relativistic field theory uses two representations of the same symmetry at once: a finite-dimensional one to write down covariant fields and an infinite-dimensional one to carry the physical inner product, with wave equations as the bridge that projects one onto the other. The whole structure is dual to the compact case in a precise sense — the angular-momentum theory of $S U (2)$ supplies the building block, and the boost direction, encoded by the factor $i$ in $J_{\pm} = \frac{1}{2} (J \pm i K)$ , is what turns one copy into two and unitary into non-unitary.

Full proof set Master

Proposition (the spin covering is two-to-one with kernel ${\pm I}$ ). The homomorphism $ϕ : SL (2, C) \to SO^{+} (3, 1)$ defined by $ϕ (A) X = A X A^{†}$ on Hermitian matrices $X = x^{μ} σ_{μ}$ is surjective with kernel ${\pm I}$ .

Proof. First, $ϕ (A)$ is a Lorentz transformation: for Hermitian $X$ , the matrix $A X A^{†}$ is Hermitian, and $det (A X A^{†}) = ∣ det A ∣^{2} det X = det X$ because $det A = 1$ , so $ϕ (A)$ preserves the Minkowski norm $det X$ and is real-linear on the four-dimensional real space of Hermitian matrices. Continuity in $A$ and connectedness of $SL (2, C)$ place $ϕ (A)$ in the identity component $SO^{+} (3, 1)$ .

Kernel: $ϕ (A) = id$ means $A X A^{†} = X$ for every Hermitian $X$ . Taking $X = I$ gives $A A^{†} = I$ , so $A$ is unitary. Then $A X A^{- 1} = X$ for all Hermitian $X$ , hence for all $X$ by complex-linear span, so $A$ is central in $GL (2, C)$ , meaning $A = λ I$ . The condition $det A = 1$ forces $λ^{2} = 1$ , so $A = \pm I$ . Both lie in the kernel, and $ker ϕ = {\pm I}$ .

Surjectivity: the image is a subgroup of $SO^{+} (3, 1)$ containing the images of the rotations and boosts. For a rotation about the $3$ -axis take $A = diag (e^{i θ /2}, e^{- i θ /2})$ , giving the spatial rotation by $θ$ . For a boost along the $3$ -axis take $A = diag (e^{χ /2}, e^{- χ /2})$ , giving the boost of rapidity $χ$ . Rotations and boosts generate $SO^{+} (3, 1)$ , so $ϕ$ is onto. The map is therefore a two-to-one surjective homomorphism, and since $SL (2, C)$ is simply connected it is the universal cover. $□$

Proposition (non-unitarity of $(j_{1}, j_{2})$ for $(j_{1}, j_{2}) \neq = (0, 0)$ ). No finite-dimensional representation $(j_{1}, j_{2})$ with $(j_{1}, j_{2}) \neq = (0, 0)$ admits a Lorentz-invariant positive-definite Hermitian form.

Proof. Suppose a positive-definite invariant form exists. Then every group element acts by an operator preserving it, so the one-parameter boost subgroups act by operators with bounded inverse and bounded norm uniformly, which forces the boost generators $K$ to be represented by Hermitian operators with bounded exponential — that is, anti-Hermitian generators times $i$ , hence $K$ Hermitian and $e^{i χK}$ unitary. In the $(j_{1}, j_{2})$ representation, $J_{+}$ acts as spin $j_{1}$ and $J_{-}$ as spin $j_{2}$ , both by Hermitian operators on the respective spin modules. Then $K = i (J_{-} - J_{+})$ is $i$ times a Hermitian operator, so $K$ is anti-Hermitian unless $J_{-} - J_{+} = 0$ . The operator $J_{-} - J_{+}$ vanishes only when both spins are zero, since on a spin- $j$ module the generators are nonzero for $j > 0$ . Hence for $(j_{1}, j_{2}) \neq = (0, 0)$ the boost generators are anti-Hermitian, $e^{i χK}$ has eigenvalues $e^{\pm χ \cdot (real)}$ growing without bound as $χ \to \infty$ , and no positive-definite form is preserved. $□$

Proposition (parity sends $(j_{1}, j_{2})$ to $(j_{2}, j_{1})$ ). Conjugation by a parity-implementing element exchanges the two $sl_{2}$ factors.

Proof. Parity acts on the Lie algebra by $J \mapsto J$ , $K \mapsto - K$ , since rotations are even and boosts are odd under space inversion. Under $J_{\pm} = \frac{1}{2} (J \pm i K)$ this sends $J_{+} \mapsto \frac{1}{2} (J - i K) = J_{-}$ and likewise $J_{-} \mapsto J_{+}$ . An automorphism exchanging the two commuting $sl_{2}$ summands carries the highest-weight pair $(j_{1}, j_{2})$ to $(j_{2}, j_{1})$ , since it relabels which factor carries which weight. Thus the parity-conjugate of $(j_{1}, j_{2})$ is $(j_{2}, j_{1})$ . $□$

Theorem (Wigner's theorem), stated with proof outline — full proof in Weinberg §2.A ^{[Wigner 1939]}. Choose unit representatives for the rays. A probability-preserving ray map $T$ determines, after fixing phases on an orthonormal basis, a map on vectors that is additive and either complex-linear or complex-antilinear; the dichotomy comes from the two ways to preserve $∣ ⟨ \cdot, \cdot ⟩ ∣$ while fixing the modulus structure of the field $C$ . Norm preservation then makes the linear case unitary and the antilinear case antiunitary. Uniqueness up to phase follows because two implementing operators differ by an operator acting as the identity on every ray, which is a scalar of modulus one. The continuity argument rules out antiunitary implementation for elements connected to the identity.

Connections Master

Representations of $sl (2, C)$ 07.06.11. Each of the two commuting factors in the complexified Lorentz algebra is a copy of $sl_{2}$ , and the highest-weight modules $Sym^{2 j} (C^{2})$ classified there are exactly the building blocks of the $(j_{1}, j_{2})$ representations. The Lorentz classification is the $sl_{2}$ theory applied twice in parallel, which is why the labels are pairs.
$S U (2)$ and $S O (3)$ double cover 07.07.05. The compact rotation story — the double cover, the spin- $j$ irreducibles, the Clebsch-Gordan series — is the rotation subgroup of the Lorentz group. The Lorentz case adds boosts, which complexify the algebra and split it into two copies of the compact story; the descent criterion for half-integer spin and the projective-representation language carry over directly, now governing whether a Lorentz representation lifts to the cover $SL (2, C)$ .
Special relativity and Lorentz transformations 10.05.01. The four connected components of $O (3, 1)$ , the proper orthochronous subgroup, and the boost-rapidity parametrisation come from the kinematics established there. This unit supplies the representation theory that sits on top of those transformations: how spinors, vectors, and tensors transform under the very boosts and rotations defined in the relativity unit.
Dirac equation and relativistic spin 12.11.01. The four-component Dirac spinor is the reducible sum $(\frac{1}{2}, 0) \oplus (0, \frac{1}{2})$ , irreducible once parity is adjoined, and the Dirac equation is the subsidiary condition that projects its four components onto the two physical spin states of a massive spin- $\frac{1}{2}$ particle. The non-unitarity of the field representation and the unitarity of the one-particle states are reconciled there explicitly.
Wigner's classification of the Poincaré group 07.07.06. Wigner's theorem on projective symmetries, developed in this unit, is what forces physical states into unitary representations of the cover, and those unitary representations are the infinite-dimensional ones classified by mass and spin in the Poincaré-group unit. The homogeneous Lorentz reps here are the field-component half of the story; the Poincaré reps there are the physical-state half.

Historical & philosophical context Master

The two-component spinor calculus underlying the $(j_{1}, j_{2})$ scheme was introduced by Élie Cartan in his 1913 work on the representations of simple Lie groups, where spinors first appeared as the objects on which the covering groups act. Hermann Weyl gave the two-component spinor its place in relativistic physics in his 1929 paper Elektron und Gravitation I (Z. Phys. 56, 330–352), proposing the massless two-component (Weyl) equation. In the same year B. L. van der Waerden introduced the dotted-and-undotted index calculus (Nachr. Ges. Wiss. Göttingen 1929, 100–109) that encodes the two factors as transformation under $A$ and under $\overset{ˉ}{A}$ , the notation still standard in the subject.

Eugene Wigner's 1939 memoir On unitary representations of the inhomogeneous Lorentz group (Ann. Math. 40, 149–204) ^{[Wigner 1939]} established the framework that separates the finite-dimensional field representations from the infinite-dimensional unitary state representations, and proved that physical states must carry the latter. The symmetry theorem that quantum symmetries are unitary or antiunitary up to phase had appeared in Wigner's 1931 book Gruppentheorie und ihre Anwendung auf die Quantenmechanik der Atomspektren; its role in forcing projective-unitary, hence cover-unitary, representations of the Lorentz group is the conceptual hinge of the relativistic theory. Valentine Bargmann's 1947 paper Irreducible unitary representations of the Lorentz group (Ann. Math. 48, 568–640) ^{[Bargmann 1947]} worked out the $SL (2, R)$ case in full and supplied the methods that Naimark and Gelfand carried to $SL (2, C)$ , completed in Naimark's 1964 monograph Linear Representations of the Lorentz Group ^{[Naimark 1964]}. Steven Weinberg's systematic field-theoretic treatment in The Quantum Theory of Fields, Vol. I (1995, §2.7, §5.6) organised the $(A, B)$ field representations and their wave-equation constraints into the form used in modern particle physics.

Bibliography Master

@book{SternbergGroupTheoryPhysics,
  author    = {Sternberg, Shlomo},
  title     = {Group Theory and Physics},
  publisher = {Cambridge University Press},
  year      = {1994}
}

@book{WeinbergQFT1,
  author    = {Weinberg, Steven},
  title     = {The Quantum Theory of Fields, Vol. I: Foundations},
  publisher = {Cambridge University Press},
  year      = {1995}
}

@article{Wigner1939,
  author  = {Wigner, Eugene P.},
  title   = {On unitary representations of the inhomogeneous Lorentz group},
  journal = {Annals of Mathematics},
  volume  = {40},
  number  = {1},
  year    = {1939},
  pages   = {149--204}
}

@book{Wigner1931Gruppentheorie,
  author    = {Wigner, Eugene P.},
  title     = {Gruppentheorie und ihre Anwendung auf die Quantenmechanik der Atomspektren},
  publisher = {Vieweg},
  address   = {Braunschweig},
  year      = {1931}
}

@article{Weyl1929,
  author  = {Weyl, Hermann},
  title   = {Elektron und Gravitation. I},
  journal = {Zeitschrift f{\"u}r Physik},
  volume  = {56},
  year    = {1929},
  pages   = {330--352}
}

@article{vanderWaerden1929,
  author  = {van der Waerden, B. L.},
  title   = {Spinoranalyse},
  journal = {Nachrichten von der Gesellschaft der Wissenschaften zu G{\"o}ttingen, Mathematisch-Physikalische Klasse},
  year    = {1929},
  pages   = {100--109}
}

@article{Bargmann1947,
  author  = {Bargmann, Valentine},
  title   = {Irreducible unitary representations of the Lorentz group},
  journal = {Annals of Mathematics},
  volume  = {48},
  number  = {3},
  year    = {1947},
  pages   = {568--640}
}

@book{Naimark1964,
  author    = {Naimark, M. A.},
  title     = {Linear Representations of the Lorentz Group},
  publisher = {Pergamon Press},
  address   = {Oxford},
  year      = {1964}
}

@incollection{Cartan1913,
  author    = {Cartan, {\'E}lie},
  title     = {Les groupes projectifs qui ne laissent invariante aucune multiplicit{\'e} plane},
  booktitle = {Bulletin de la Soci{\'e}t{\'e} Math{\'e}matique de France},
  volume    = {41},
  year      = {1913},
  pages     = {53--96}
}

Prerequisites

07.07.05
07.06.11
10.05.01

Tier anchors

beginner: Sternberg *Group Theory and Physics* (CUP 1994), Ch. 8 informal opening on rotations versus boosts; Woit *Quantum Theory, Groups and Representations* Ch. 40 §1
intermediate: Sternberg *Group Theory and Physics* Chs. 8–9 (the Lorentz group, $\mathrm{SL}(2,\mathbb{C})$, and the $(j_1,j_2)$ classification); Woit *Quantum Theory, Groups and Representations* Ch. 40; Hall *Quantum Theory for Mathematicians* §17
master: Sternberg *Group Theory and Physics* Chs. 8–9; Weinberg *The Quantum Theory of Fields, Vol. I* (CUP 1995) §2.7; Wigner *On unitary representations of the inhomogeneous Lorentz group* (Ann. Math. 40, 1939); Naimark *Linear Representations of the Lorentz Group* (Pergamon 1964); Bargmann *Irreducible unitary representations of the Lorentz group* (Ann. Math. 48, 1947)

References

Sternberg — Group Theory and Physics · CUP 1994, Chs. 8–9; the Lorentz group, the spin-covering $\mathrm{SL}(2,\mathbb{C}) \to \mathrm{SO}^+(3,1)$, the finite-dimensional $(j_1,j_2)$ representations, and Wigner's theorem
Weinberg — The Quantum Theory of Fields, Vol. I · CUP 1995, §2.7 and §5.6; the $(A,B)$ (= $(j_1,j_2)$) finite-dimensional representations of the homogeneous Lorentz group and their field realisations
Wigner — On unitary representations of the inhomogeneous Lorentz group · Ann. Math. 40 (1939) 149–204; the unitary/antiunitary dichotomy for quantum symmetries and the requirement of infinite-dimensional unitary representations for physical states
Naimark — Linear Representations of the Lorentz Group · Pergamon 1964; the complete theory of finite- and infinite-dimensional representations of $\mathrm{SL}(2,\mathbb{C})$ and the proof that the finite-dimensional ones are non-unitary except the identity action
Bargmann — Irreducible unitary representations of the Lorentz group · Ann. Math. 48 (1947) 568–640; the unitary irreducible representations of $\mathrm{SL}(2,\mathbb{R})$ and the methods that extend to $\mathrm{SL}(2,\mathbb{C})$

Estimated time

beginner: 18m
intermediate: 42m
master: 80m