Lorentz Field Representations and Poincaré Particle Representations

A Lorentz field representation and a Poincaré particle representation act on different spaces and answer different questions. A field representation tells how a finite list of components at a spacetime point mixes under Lorentz transformations. In four dimensions it is ordinarily a finite-dimensional representation of $\operatorname{Spin}^+(1,3)\cong SL(2,\mathbb C)$, and it is generally nonunitary because the Lorentz group is noncompact.

A particle representation acts instead on a one-particle Hilbert space. It must be unitary, includes translations as well as Lorentz transformations, and is normally infinite-dimensional because momentum ranges over a mass shell. Under the usual irreducibility and positive-energy hypotheses, its labels come from a momentum orbit and a unitary representation of that orbit’s little group.

The two are connected by covariant coefficient functions—intertwiners—and by field equations, constraints, or gauge equivalences. They are not the same representation. In particular, decomposing a field’s components under spatial rotations does not by itself determine the particle spins in the theory.

Required background. Lie Groups, Lie Algebras, and Exponential and Adjoint Maps supplies Lie algebras, exponentials, semidirect products, and covers; Representations, Intertwiners, Invariants, and Tensor Decomposition supplies restrictions, direct sums, tensor products, and intertwiners.

Field modules and particle Hilbert spaces

Unless stated otherwise, “Lorentz group” means the proper orthochronous component

$$SO^+(1,3) = \left\{ \Lambda\in GL(4,\mathbb R): \Lambda^{\mathsf T}\eta\Lambda=\eta,\ \det\Lambda=1,\ \Lambda^0{}_0\geq1 \right\}.$$

Spinorial representations live on its double and universal cover $SL(2,\mathbb C)$. Parity and time reversal are not in this connected component. All finite-dimensional field modules below are complex; real structures are additional data.

Hermitian quantum generators are used. Commutators involving unbounded Hilbert-space generators are understood on a common invariant domain. The Fourier convention is the site’s forward phase $e^{+ip\cdot x}$ and inverse phase $e^{-ip\cdot x}$, so a positive-energy plane wave below is $e^{-ip\cdot x}$.

This page defines the two representation problems, classifies the finite-dimensional Lorentz labels needed to compare them, and gives only the induced-representation structure needed for that comparison. The full Wigner classification, relativistic normalization, discrete symmetries, continuous-spin sectors, multiparticle states, and physical interpretation remain at the Foundations continuation.

Two representation problems

The distinction is easiest to see before introducing any classification. The full action on the space of field configurations is of course infinite-dimensional; “finite-dimensional field representation” refers to the component matrix $D(A)$.

Question	Lorentz field representation	Poincaré particle representation
Acting group	$SL(2,\mathbb C)$ on component indices; spacetime arguments also transform	$\widetilde{\mathcal P}^{\,\uparrow}_+=\mathbb R^{1,3}\rtimes SL(2,\mathbb C)$
Carrier	A finite-dimensional component space $F$	A one-particle Hilbert space $\mathcal H_1$
Typical dimension	Finite	Infinite, because momentum is continuous
Inner product	No positive-definite Lorentz-invariant inner product is required	A positive-definite Hilbert inner product preserved by every $U(g)$
Main data	A label $(j_L,j_R)$ and any direct sums, reality conditions, or parity pairing	A translation-spectrum orbit and a unitary little-group representation
What it answers	How do field components mix under a change of inertial frame?	How do physical one-particle states transform?

For a component field $\Phi:\mathbb R^{1,3}\to F$, let $A\in SL(2,\mathbb C)$ project to $\Lambda(A)\in SO^+(1,3)$, and let $D:SL(2,\mathbb C)\to GL(F)$. The induced active action on field configurations is

$$\bigl((a,A)\cdot\Phi\bigr)(x) = D(A)\, \Phi\!\left(\Lambda(A)^{-1}(x-a)\right).$$

The inverse spacetime map is the function-action convention established on the Lie-group page. With

$$(a,A)(b,B) = \bigl(a+\Lambda(A)b,AB\bigr),$$

direct substitution verifies $g_1\cdot(g_2\cdot\Phi)=(g_1g_2)\cdot\Phi$. This action needs no positive-definite inner product on $F$.

By contrast, a Poincaré representation on states is a strongly continuous homomorphism

$$U:\widetilde{\mathcal P}^{\,\uparrow}_+ \longrightarrow\mathcal U(\mathcal H_1), \qquad U(g_1g_2)=U(g_1)U(g_2),$$

where every $U(g)$ preserves the Hilbert norm. In the one-particle problem one also asks that $U$ be irreducible and that the translation spectrum satisfy a positive-energy or spectrum condition. Omitting translations, the Hilbert space, unitarity, or the spectral hypothesis changes the problem.

Fix the translation-generator convention by

$$U(a,\mathbf1)=\exp(-ia_\mu P^\mu).$$

The connected Poincaré group and its cover

The connected Poincaré group is the semidirect product

$$\mathcal P^{\uparrow}_+ = \mathbb R^{1,3}\rtimes SO^+(1,3).$$

Its universal cover is

$$\widetilde{\mathcal P}^{\,\uparrow}_+ = \mathbb R^{1,3}\rtimes SL(2,\mathbb C).$$

The Lorentz factor acts on translations through the covering projection $SL(2,\mathbb C)\to SO^+(1,3)$. Ordinary unitary representations of this cover encode the usual projective unitary representations of the connected Poincaré group in the four-dimensional Wigner problem. The cover is essential: its central element $-\mathbf1$ projects to the identity Lorentz transformation but distinguishes integer from half-integer spin.

Let $P^\mu$ generate translations and let $M^{\mu\nu}=-M^{\nu\mu}$ generate Lorentz transformations. With the Hermitian convention used here, the Poincaré algebra is

$$\begin{aligned} [P^\mu,P^\nu]&=0,\\ [M^{\mu\nu},P^\rho] &= i\left( \eta^{\nu\rho}P^\mu-\eta^{\mu\rho}P^\nu \right),\\ [M^{\mu\nu},M^{\rho\sigma}] &= i\bigl( \eta^{\nu\rho}M^{\mu\sigma} -\eta^{\mu\rho}M^{\nu\sigma}\\ &\quad -\eta^{\nu\sigma}M^{\mu\rho} +\eta^{\mu\sigma}M^{\nu\rho} \bigr). \end{aligned}$$

These relations are the local data. They do not by themselves decide whether a representation descends through a cover, is unitary, has positive energy, or is irreducible.

Two commuting invariants organize the ordinary one-particle cases. Define the Pauli–Lubanski vector

$$W^\mu = -\frac12\epsilon^{\mu\nu\rho\sigma} P_\nu M_{\rho\sigma}.$$

Then $P^2$ and $W^2$ commute with the connected Poincaré action. For a massive spin-$s$ irreducible representation,

$$P^2=m^2, \qquad W^2=-m^2s(s+1).$$

These eigenvalues belong to the state representation, not to the finite matrix $D(A)$ carried by a field index. The algebra, cover, projective formulation, and induced-representation construction are developed in Weinberg 1995, §§2.3–2.5 and 2.7 and originate in the unitary-representation analysis of Wigner 1939, §§4–7.

Finite-dimensional Lorentz field labels

Introduce rotations and boosts by

$$J_i=\frac12\epsilon_{ijk}M^{jk}, \qquad K_i=M^{0i}.$$

Their finite component matrices are parametrized by

$$D(R(\boldsymbol\theta)) = \exp(-i\boldsymbol\theta\cdot\mathbf J), \qquad D(B(\boldsymbol\zeta)) = \exp(-i\boldsymbol\zeta\cdot\mathbf K).$$

In a field module these are finite matrices; in a unitary state representation they are generally unbounded Hilbert-space generators. The same symbols express two representations of the same abstract algebra, not an identification of their carriers.

The Lorentz commutators become

$$\begin{aligned} [J_i,J_j]&=i\epsilon_{ijk}J_k,\\ [J_i,K_j]&=i\epsilon_{ijk}K_k,\\ [K_i,K_j]&=-i\epsilon_{ijk}J_k. \end{aligned}$$

After complexification, set

$$A_i=\frac12(J_i+iK_i), \qquad B_i=\frac12(J_i-iK_i).$$

A short calculation gives

$$[A_i,A_j]=i\epsilon_{ijk}A_k, \qquad [B_i,B_j]=i\epsilon_{ijk}B_k, \qquad [A_i,B_j]=0.$$

Thus the complexified Lorentz algebra is a direct sum of two copies of $\mathfrak{sl}_2(\mathbb C)$. The symbols $A_i$ and $B_i$ do not exhibit two compact $SU(2)$ subgroups inside the real Lorentz group; they classify finite-dimensional modules after complexification.

Every finite-dimensional irreducible complex representation is labeled by two nonnegative half-integers,

$$(j_L,j_R), \qquad \dim(j_L,j_R) =(2j_L+1)(2j_R+1).$$

Restricting to spatial rotations identifies the diagonal $SU(2)\subset SL(2,\mathbb C)$. The usual tensor-product rule gives

$$(j_L,j_R)\big|_{SU(2)} \cong \bigoplus_{j=|j_L-j_R|}^{j_L+j_R}V_j,$$

where $j$ increases in integer steps. Representative field labels are:

Field component module	Dimension	Content under spatial rotations
Scalar $(0,0)$	$1$	$0$
One Weyl module $(\tfrac12,0)$	$2$	$\tfrac12$
The conjugate Weyl module $(0,\tfrac12)$	$2$	$\tfrac12$
Four-vector $(\tfrac12,\tfrac12)$	$4$	$0\oplus1$
Dirac module $(\tfrac12,0)\oplus(0,\tfrac12)$	$4$	$\tfrac12\oplus\tfrac12$

Which Weyl module is called left- or right-handed depends on the declared boost and chirality conventions; their conjugate pairing is invariant. Parity exchanges $(j_L,j_R)$ and $(j_R,j_L)$, so a module with $j_L\ne j_R$ does not extend by itself to a parity representation. A paired direct sum can.

Global descent supplies a second check. The central element $-\mathbf1\in SL(2,\mathbb C)$ acts on $(j_L,j_R)$ by

$$(-1)^{2j_L}(-1)^{2j_R} = (-1)^{2(j_L+j_R)}.$$

Consequently, $(j_L,j_R)$ descends to an ordinary representation of $SO^+(1,3)$ exactly when $j_L+j_R\in\mathbb Z$. Weyl modules do not descend; the vector module does. This is the Lorentz analogue of the $SU(2)\to SO(3)$ descent test, but it is a test on field labels, not yet a particle classification.

This construction and the table are supported by Tong 2006, §4.1, “The Lorentz Group” and Weinberg 1995, §5.6. Both sources use convention choices that must be translated before comparing boost or chirality signs.

Why finite field representations are generally nonunitary

The connected Lorentz group is noncompact. A nontrivial finite-dimensional representation of its simple real Lie algebra cannot preserve a positive-definite Hermitian form. Equivalently, there are no nontrivial finite-dimensional unitary representations of $SL(2,\mathbb C)$; see Weinberg 1995, §5.6.

The vector representation makes the point without a theorem. A boost of rapidity $\chi$ in one spatial direction may be written

$$\Lambda(\chi) = \begin{pmatrix} \cosh\chi&\sinh\chi&0&0\\ \sinh\chi&\cosh\chi&0&0\\ 0&0&1&0\\ 0&0&0&1 \end{pmatrix}.$$

It satisfies

$$\Lambda(\chi)^{\mathsf T}\eta\Lambda(\chi)=\eta,$$

but it does not preserve the positive Euclidean norm on $\mathbb C^4$: the vector $(1,0,0,0)$ is sent to a vector of Euclidean squared norm $\cosh^2\chi+\sinh^2\chi$. Lorentz covariance preserves the indefinite form $\eta$, not a positive component-space probability norm.

There is no conflict with quantum unitarity. Field components are not probability amplitudes in the one-particle Hilbert space. The Poincaré operators $U(g)$ acting on physical states remain unitary even when the finite component matrices $D(A)$ appearing in a covariance law are not. Unitarity under the compact rotation subgroup is compatible with nonunitary boosts.

Positive-energy particle representations

Strong continuity of translations supplies commuting self-adjoint generators $P^\mu$. Their joint spectrum transforms under the Lorentz group. For an irreducible representation, the relevant spectral support is one Lorentz orbit. With the usual spectrum condition it lies in the closed future cone.

Choose a standard momentum $k$ on an orbit $\mathcal O$ and a standard lift $L(p)\in SL(2,\mathbb C)$ satisfying $\Lambda(L(p))k=p$. The little group

$$G_k = \{A\in SL(2,\mathbb C):\Lambda(A)k=k\}$$

acts inside the fiber over $k$. For a general Lorentz transformation $A$,

$$\mathcal W(A,p) = L\!\left(\Lambda(A)p\right)^{-1} A\,L(p) \in G_k$$

is its Wigner little-group element. Changing the choice of $L(p)$ makes a momentum-dependent little-group change of basis in each fiber, not a change of the induced representation’s equivalence class.

For the positive massive orbit

$$\mathcal H_m^+ = \{p:p^2=m^2,\ p^0>0\}, \qquad k=(m,\mathbf0),$$

the little group on the cover is $SU(2)$. A spin-$s$ particle space has the form

$$\mathcal H_{m,s} \cong L^2(\mathcal H_m^+,\mathrm d\mu_m) \otimes\mathbb C^{\,2s+1}, \qquad \mathrm d\mu_m(p) = \frac{\mathrm d^3\mathbf p}{(2\pi)^3\,2p^0},$$

up to normalization convention. The fiber is finite, but the momentum factor makes the representation infinite-dimensional.

For a positive-energy massless orbit, the little group is the double cover of $ISO(2)$. If its translation subgroup acts trivially, the remaining rotation character gives helicity and $W^\mu=hP^\mu$. Nontrivial action of that translation subgroup leads to continuous-spin representations. This sentence marks the boundary; the cases, normalization, and physical assumptions belong to One-Particle States. The orbit, little-group, and induced-representation statements above are developed in Weinberg 1995, §2.5 and Wigner 1939, §§6–7.

The logical structure is therefore

$$\text{translation spectrum} \longrightarrow \text{Lorentz orbit} \longrightarrow \text{little group} \longrightarrow \text{unitary induced representation}.$$

No entry in this chain is supplied merely by writing a finite $(j_L,j_R)$ label.

Fields and particles meet through intertwiners

A free operator-valued field has a schematic mode expansion

$$\Phi_r(x) = \sum_\sigma \int\mathrm d\mu_m(p)\, \left[ u_r(p,\sigma)a(p,\sigma)e^{-ip\cdot x} + v_r(p,\sigma)b^\dagger(p,\sigma)e^{+ip\cdot x} \right].$$

Here $r$ is a component index in the finite Lorentz module $F$, while $(p,\sigma)$ labels vectors in particle and antiparticle representations. The coefficient functions $u$ and $v$ connect these spaces. At a standard massive momentum, covariance requires a relation of the form

$$D(R)\,u(k,\sigma) = \sum_{\sigma'} u(k,\sigma')\,d^{(s)}_{\sigma'\sigma}(R), \qquad R\in SU(2),$$

so $u(k,\cdot)$ is an intertwiner from the spin-$s$ little-group module into the restriction of the field module. A convention may move inverses between $D$, $d^{(s)}$, and the state action, but the intertwining statement is invariant.

Existence of such an intertwiner is only a kinematic compatibility condition. Field equations can remove component combinations, gauge equivalence can identify them, and a parity-covariant description can require a paired direct sum. An interacting local field may have overlap with several one-particle species and with multiparticle continuum states. Consequently:

• one particle type can be described by several covariant field choices;
• one field can interpolate more than one particle or antiparticle type;
• a rotation component of a Lorentz field need not survive as a physical one-particle polarization; and
• a Lorentz label alone does not establish that the theory contains any stable particle.

The field expansion and its covariance intertwiners are developed in Weinberg 1995, §§5.1 and 5.6. Dynamics, statistics, and normalization have deliberately not been inferred from the schematic formula.

Controlled QFT example: a massive vector

Let $A^\mu(x)$ transform in the four-vector module $(\tfrac12,\tfrac12)$. Its four component labels restrict under spatial rotations as

$$\left(\frac12,\frac12\right)\Big|_{SU(2)} \cong V_0\oplus V_1.$$

This says that $A^0$ is a rotational scalar and $\mathbf A$ a rotational vector. It does not say that the field creates both a spin-$0$ particle and a spin-$1$ particle.

For the free massive Proca system, the equations are

$$\partial_\mu F^{\mu\nu}+m^2A^\nu=0, \qquad F^{\mu\nu} = \partial^\mu A^\nu-\partial^\nu A^\mu.$$

Taking a divergence gives $\partial_\mu A^\mu=0$ when $m\ne0$. For a positive-energy plane wave

$$A^\mu(x)=\varepsilon^\mu(p)e^{-ip\cdot x}, \qquad p^2=m^2,$$

the subsidiary condition becomes

$$p_\mu\varepsilon^\mu(p)=0.$$

At the standard momentum $k=(m,\mathbf0)$, this is $\varepsilon^0(k)=0$. The three remaining spatial polarization vectors form the spin-$1$ representation of the massive little group $SU(2)$. This bounded Proca calculation follows Weinberg 1995, §5.3, translated to the $(+---)$ convention.

The result compares three different numbers:

$$\begin{array}{c|c|c} \text{object}&\text{finite fiber dimension}&\text{meaning}\\ \hline (\tfrac12,\tfrac12)&4&\text{Lorentz field components}\\ V_0\oplus V_1&1+3&\text{rotation content before equations}\\ V_1&3&\text{massive physical polarizations} \end{array}$$

The full one-particle Hilbert space is still infinite-dimensional because each polarization occurs for every $p\in\mathcal H_m^+$. Thus the field label, rotational restriction, subsidiary equation, and particle representation each do a distinct job. The developed action, constraint, Hamiltonian, and massless-limit analysis belongs to The Proca Field.

Distinguishing fields from particle states

When a representation label appears in a relativistic calculation, ask:

1. What is the carrier? Finite component space, classical configurations, operator-valued distributions, or a Hilbert space?
2. Which global group acts? $SO^+(1,3)$, its $SL(2,\mathbb C)$ cover, the connected Poincaré group, or an extension including parity?
3. Is unitarity required? It is required for physical state transformations, not for a finite covariant component matrix.
4. Do translations act? If yes, identify their joint spectrum and the momentum orbit.
5. What connects components to states? Name the intertwiner and any equations, constraints, gauge equivalences, or locality conditions.

Only after all five answers are known is it safe to translate “spin” from one context to another.

Common pitfalls

Reading particle spin directly from $(j_L,j_R)$. Restriction to rotations lists how unconstrained field components transform. The particle spin is a little-group label in a unitary Poincaré representation and may be selected by equations or equivalences.

Demanding unitary finite boost matrices. A finite Lorentz field module normally preserves no positive-definite component norm. Quantum unitarity belongs to $U$ on the Hilbert space; the two requirements are compatible.

Dropping translations from the particle problem. A unitary Lorentz representation alone has no momentum orbit, mass shell, or translation spectrum. It is not yet a Poincaré particle representation.

Ignoring the cover. Half-integer labels are honest representations of $SL(2,\mathbb C)$ but only projective representations downstairs. A Lie algebra calculation cannot replace the central-element descent test.

Calling the two complex factors physical rotation groups. The $A_i,B_i$ split occurs after complexifying the Lorentz algebra. Physical spatial rotations form the diagonal compact $SU(2)$ subgroup.

Exercises

1. Type three representation statements

Classify each object as field data, particle data, or insufficient data:

1. a four-component $A^\mu(x)$ with $A'(x)=\Lambda A(\Lambda^{-1}x)$;
2. square-integrable wavefunctions $\psi_\sigma(p)$ on $\mathcal H_m^+$ with a unitary Poincaré action;
3. one $(2s+1)\times(2s+1)$ unitary matrix for every spatial rotation.

Solution

The first is finite-dimensional Lorentz field data: the spacetime argument and a four-vector component index transform. The second is particle data: translations label momentum, the Lorentz group moves momentum, and the action is unitary on a Hilbert space. The third is only a rotation-group representation. It could become the massive little-group fiber, but without translations, a momentum orbit, and induction it is not a Poincaré representation.

2. Verify the complex Lorentz split

Starting from the $J_i,K_i$ commutators, compute $[A_i,A_j]$, $[B_i,B_j]$, and $[A_i,B_j]$.

Solution

For example,

$$\begin{aligned} [A_i,A_j] &= \frac14\left( [J_i,J_j] +i[J_i,K_j] +i[K_i,J_j] -[K_i,K_j] \right)\\ &= \frac12\epsilon_{ijk}(iJ_k-K_k) = i\epsilon_{ijk}A_k. \end{aligned}$$

Changing $i$ to $-i$ gives $[B_i,B_j]=i\epsilon_{ijk}B_k$, and the mixed terms cancel, so $[A_i,B_j]=0$.

3. Test descent and rotation content

For $(j_L,j_R)=(\tfrac12,\tfrac12)$ and $(1,\tfrac12)$, determine the dimension, the spatial-rotation content, and whether the representation descends to $SO^+(1,3)$.

Solution

The vector module has dimension $2\cdot2=4$, restricts as $0\oplus1$, and descends because $j_L+j_R=1$ is integral. The $(1,\tfrac12)$ module has dimension $3\cdot2=6$, restricts as $\tfrac12\oplus\tfrac32$, and does not descend because $j_L+j_R=\tfrac32$. The central element $-\mathbf1$ acts as $-1$ on the second module.

4. Identify the massive little group

Let $k=(m,\mathbf0)$ with $m>0$. Show that a proper orthochronous Lorentz transformation fixing $k$ is a spatial rotation. What is the little group on the universal cover?

Solution

Fixing $k$ fixes its timelike unit direction. Preservation of $\eta$ then preserves the orthogonal spatial subspace and its negative Euclidean inner product. Properness and time orientation leave an element of $SO(3)$ on that subspace. Its inverse image in $SL(2,\mathbb C)$ is $SU(2)$, whose irreducible unitary representations are labeled by $s=0,\tfrac12,1,\ldots$.

5. Remove the false scalar polarization

For a Proca plane wave at $k=(m,\mathbf0)$, use $k_\mu\varepsilon^\mu=0$ to count the physical polarization vectors. Explain why the $V_0$ in the rotation restriction of the four-vector is not a spin-$0$ particle.

Solution

The constraint is $m\varepsilon^0=0$, so $\varepsilon^0=0$. Three independent spatial polarizations remain and rotate as $V_1$. The rotational scalar was a component of the unconstrained Lorentz vector; the field equation removes it from the on-shell one-particle fiber. A spin-$0$ particle would require its own invariant one-particle subspace, which this calculation does not produce.

Where to continue

• For the first physical application to field labels and one-particle states, continue to One-Particle States: Mass, Spin, and Relativistic Normalization. That Foundations page recommends this mathematical preparation but does not hard-require it.
• Clifford Algebras and Pin and Spin Groups and Spinors, Conjugations, Bilinears, Chirality, and Fierz Identities develop independent algebraic branches. Their later position in the chapter does not make them prerequisites for this page.

References

• David Tong (2006), Quantum Field Theory, §4, “The Dirac Equation”, especially §4.1 on the Lorentz group and its finite-dimensional spinor representations, Cambridge Part III lecture notes, supports the Lorentz-algebra split, finite field labels, and convention checks.
• Steven Weinberg, The Quantum Theory of Fields, Volume I: Foundations, Cambridge University Press, 1995, §§2.3–2.5, 2.7, 5.1, 5.3, and 5.6. This is the QFT-application and structural source for the connected Poincaré group, one-particle induced representations, projective representations, field covariance intertwiners, finite Lorentz modules, and vector-field comparison. Its metric and transformation conventions have been translated to those declared above.
• Eugene P. Wigner, “On Unitary Representations of the Inhomogeneous Lorentz Group”, Annals of Mathematics 40 (1939), 149–204, §§4–7; see also the CERN bibliographic record. This paper develops the classification by unitary representations using momentum orbits, little groups, and induction. This page uses only the framework needed for the field–particle distinction.