Weyl Fields and Chirality

Chirality is a representation-theoretic grading, not a choice of gamma-matrix basis and not a synonym for helicity. In even dimension a normalized Clifford volume element can split a complex spinor into two chiral modules. In four-dimensional Minkowski spacetime this split is implemented by $P_L$ and $P_R$; the massless Dirac action becomes two independent two-component Weyl actions, while a Dirac mass couples the two sectors. On the nonzero massless particle shell, left and right chirality select helicities $-\tfrac12$ and $+\tfrac12$, respectively, but the antiparticle created by the same complex Weyl field has the opposite helicity.

This page develops that free four-dimensional field and then states exactly which parts survive a change of dimension, signature, orientation, or matrix representation. Chiral gauge assignments, Yukawa interactions, and anomalies remain outside its scope.

Required background. The Dirac Field supplies the free four-component action and equation, Lorentz covariance, the inherited Clifford and adjoint structure, and the fact that a Dirac mass couples chiral sectors.

Helpful background. Spinors, Conjugations, Bilinears, Chirality, and Fierz Identities supplies the reusable even-dimensional grading, half-spin modules, conjugation intertwiners, and dimension- and signature-dependent bilinear rules.

Chirality is a Lorentz-representation grading

Begin in four-dimensional Minkowski spacetime and restrict spacetime transformations to the proper orthochronous Lorentz group. Parity will be added later as a diagnostic. We inherit the metric, orientation, Clifford, and $\gamma_5$ choices from the chapter’s shared declarations. The central operators used below are

$$\gamma_5 = i\gamma^0\gamma^1\gamma^2\gamma^3, \qquad P_L=\frac{1-\gamma_5}{2}, \qquad P_R=\frac{1+\gamma_5}{2}.$$

The Clifford relation gives

$$\begin{gathered} \gamma_5^\dagger=\gamma_5, \qquad \gamma_5^2=1, \qquad \{\gamma_5,\gamma^\mu\}=0, \\ P_L^2=P_L, \qquad P_R^2=P_R, \qquad P_LP_R=0, \qquad P_L+P_R=1. \end{gathered}$$

Thus every Dirac spinor has the unique decomposition

$$\psi=\psi_L+\psi_R, \qquad \psi_L=P_L\psi, \qquad \psi_R=P_R\psi.$$

The Lorentz generators contain even products of gamma matrices and therefore commute with $\gamma_5$. Proper orthochronous Lorentz transformations preserve the two eigenspaces. In four dimensions they are the inequivalent complex representations customarily denoted $(\tfrac12,0)$ and $(0,\tfrac12)$. Their construction and transformation laws are given in Schwartz 2014, § 10.2, pp. 163–166.

This statement is basis independent. If

$$\gamma'^\mu=U\gamma^\mu U^{-1}, \qquad \psi'=U\psi,$$

then

$$\gamma'_5=U\gamma_5U^{-1}, \qquad P'_{L,R}=UP_{L,R}U^{-1}.$$

The equation $P_L\psi=\psi$ is therefore equivalent to $P'_L\psi'=\psi'$. Choosing a basis in which $\gamma_5$ is diagonal makes the split visible; it does not create the split.

Parity behaves differently. With the standard parity action proportional to $\gamma^0$,

$$\gamma^0\gamma_5\gamma^0=-\gamma_5, \qquad \gamma^0P_L=P_R\gamma^0.$$

Parity exchanges the two chiral representations. A theory containing only one Weyl sector is consequently not closed under parity by itself.

The massless Dirac field separates into two Weyl fields

The anticommutation of $\gamma_5$ with $\gamma^\mu$ gives the two identities

$$\gamma^\mu P_L=P_R\gamma^\mu, \qquad \gamma^\mu P_R=P_L\gamma^\mu.$$

It also reverses the projector in the Dirac adjoint:

$$\overline{\psi_L} = \overline\psi P_R, \qquad \overline{\psi_R} = \overline\psi P_L.$$

This reversal is consequential. It makes the same-chirality kinetic terms nonzero and the cross kinetic terms vanish. Up to the boundary term already controlled on the Dirac page,

$$\begin{aligned} \overline\psi\,i\gamma^\mu\partial_\mu\psi &= \overline{\psi_L}\, i\gamma^\mu\partial_\mu\psi_L + \overline{\psi_R}\, i\gamma^\mu\partial_\mu\psi_R, \\ \overline\psi\psi &= \overline{\psi_L}\psi_R + \overline{\psi_R}\psi_L. \end{aligned}$$

Projecting the massive Dirac equation gives

$$\begin{aligned} i\gamma^\mu\partial_\mu\psi_L &=m\psi_R, \\ i\gamma^\mu\partial_\mu\psi_R &=m\psi_L. \end{aligned}$$

At $m=0$ the two equations and the two kinetic actions are independent. Each one defines a free Weyl field. At $m\ne0$ the projectors still exist, but the Dirac dynamics transfers amplitude between them. The projector algebra, mass mixing, and massless decoupling are derived in Schwartz 2014, § 11.1, pp. 185–188.

The massless free action also admits independent constant phases for the two sectors,

$$\psi_L\longmapsto e^{-i\alpha_L}\psi_L, \qquad \psi_R\longmapsto e^{-i\alpha_R}\psi_R,$$

with separately conserved free currents

$$j_L^\mu = \overline\psi\gamma^\mu P_L\psi, \qquad j_R^\mu = \overline\psi\gamma^\mu P_R\psi.$$

These are free-field statements. Gauging a chiral phase, coupling it through a Yukawa interaction, and deciding whether the quantum current is anomalous require additional representation and regulator data and belong to later volumes.

A Dirac mass is not the only algebraically possible mass construction. A two-component Majorana-type mass may be allowed when charge assignments, conjugation, and reality data permit it. That distinct question is treated on the Majorana page; the calculation above establishes only that a Dirac mass pairs opposite chiralities.

A chiral basis exposes the two-component equations

To realize the first application explicitly, now choose a chiral gamma-matrix basis. This is a local computational choice, not the invariant definition of a Weyl field:

$$\gamma^\mu = \begin{pmatrix} 0&\sigma^\mu\\ \overline\sigma^\mu&0 \end{pmatrix}, \qquad \gamma_5 = \begin{pmatrix} -\mathbf1_2&0\\ 0&+\mathbf1_2 \end{pmatrix},$$

where

$$\sigma^\mu=(\mathbf1_2,\boldsymbol\sigma), \qquad \overline\sigma^\mu=(\mathbf1_2,-\boldsymbol\sigma).$$

The Pauli-matrix algebra is equivalent to

$$\begin{aligned} \sigma^\mu\overline\sigma^\nu + \sigma^\nu\overline\sigma^\mu &= 2\eta^{\mu\nu}\mathbf1_2, \\ \overline\sigma^\mu\sigma^\nu + \overline\sigma^\nu\sigma^\mu &= 2\eta^{\mu\nu}\mathbf1_2. \end{aligned}$$

Write the projected fields as

$$\psi_L = \begin{pmatrix} \chi_\alpha\\ 0 \end{pmatrix}, \qquad \psi_R = \begin{pmatrix} 0\\ \rho^{\dot\alpha} \end{pmatrix}.$$

The two free actions and equations are

$$\begin{aligned} S_L &= \int\mathrm d^4x\, i\chi^\dagger_{\dot\alpha} \overline\sigma^{\mu\dot\alpha\alpha} \partial_\mu\chi_\alpha, & i\overline\sigma^\mu\partial_\mu\chi &=0, \\ S_R &= \int\mathrm d^4x\, i\rho^\dagger_\alpha \sigma^{\mu\alpha\dot\alpha} \partial_\mu\rho_{\dot\alpha}, & i\sigma^\mu\partial_\mu\rho &=0. \end{aligned}$$

Each action is real after integration by parts with the same boundary conditions used for the Dirac action. The field dimension remains $[\chi]=[\rho]=3/2$. The two-component kinetic terms and dotted/undotted index placement are developed in Schwartz 2014, § 10.6.2, pp. 179–180.

For a future-directed momentum $p^\mu=(E_{\mathbf p},\mathbf p)$, the site contraction gives

$$\begin{aligned} p_\mu\sigma^\mu &= E_{\mathbf p}\mathbf1_2 - \boldsymbol\sigma\mathbin{\cdot}\mathbf p, \\ p_\mu\overline\sigma^\mu &= E_{\mathbf p}\mathbf1_2 + \boldsymbol\sigma\mathbin{\cdot}\mathbf p. \end{aligned}$$

Consequently,

$$(p_\mu\sigma^\mu) (p_\nu\overline\sigma^\nu) = p^2\mathbf1_2.$$

At nonzero null momentum each factor has rank one. One Weyl equation therefore leaves one complex plane-wave amplitude in each frequency sector. Applying the opposite sigma matrix also gives $\Box\chi=0$ or $\Box\rho=0$, an independent check that the Weyl equation has the massless relativistic dispersion relation.

For an ordinary complex left-handed solution,

$$j_L^\mu = \chi^\dagger\overline\sigma^\mu\chi, \qquad j_L^0=\chi^\dagger\chi\ge0, \qquad \partial_\mu j_L^\mu=0.$$

The last two statements check both the sign in $\overline\sigma^\mu=(1,-\boldsymbol\sigma)$ and the adjoint placement. Positivity here concerns ordinary complex solutions, not an ordering relation among Grassmann-valued classical fields.

The massless specialization of the two-component action and its canonical equal-time structure can also be read from Srednicki 2007, § 37, p. 236. Srednicki uses the paired conventions $\eta_{\mathrm S}=(-,+,+,+)$ and $\{\gamma^\mu,\gamma^\nu\}=-2\eta_{\mathrm S}^{\mu\nu}$, so the complex Clifford relation agrees after $\eta_{\text{site}}=-\eta_{\mathrm S}$. The momentum and Fourier signs above were derived directly in the site convention rather than copied in isolation.

Helicity matches chirality only on the massless shell

Helicity is the spin projection along a nonzero momentum. For a two-component spin-$\tfrac12$ wavefunction,

$$h = \frac12 \boldsymbol\sigma\mathbin{\cdot}\widehat{\mathbf p}, \qquad \widehat{\mathbf p} = \frac{\mathbf p}{|\mathbf p|}.$$

Take positive-frequency plane waves

$$\chi(x)=\xi(\mathbf p)e^{-ip\cdot x}, \qquad \rho(x)=\zeta(\mathbf p)e^{-ip\cdot x}, \qquad E_{\mathbf p}=|\mathbf p|>0.$$

The Weyl equations reduce to

$$\begin{aligned} \left( E_{\mathbf p} + \boldsymbol\sigma\mathbin{\cdot}\mathbf p \right)\xi &=0, \\ \left( E_{\mathbf p} - \boldsymbol\sigma\mathbin{\cdot}\mathbf p \right)\zeta &=0. \end{aligned}$$

Therefore

$$h\xi=-\frac12\xi, \qquad h\zeta=+\frac12\zeta.$$

For positive-energy particle modes in four-dimensional Lorentzian spacetime, left chirality selects helicity $-\tfrac12$ and right chirality selects $+\tfrac12$. This agreement requires $p^2=0$, $p^0>0$, and $\mathbf p\ne0$. It is not an operator identity on an arbitrary off-shell spinor.

A complex quantum field also creates antiparticles. The antiparticle belongs to the conjugate one-particle representation, so its physical helicity is opposite to that of the particle annihilated by the same chiral field.

Four-dimensional massless helicities in the declared γ₅ convention

Field sector	γ₅ eigenvalue	Particle annihilated	Antiparticle created
Left Weyl	−1	h = −1/2	h = +1/2
Right Weyl	+1	h = +1/2	h = −1/2

The particle/antiparticle distinction is visible in the massless projector limits in Srednicki 2007, § 38, p. 244: the right projector selects the $u_+$ particle coefficient but the $v_-$ antiparticle coefficient, with the complementary pair in the left sector. The identification of the $b^\dagger$ and $d^\dagger$ states and their opposite charges is made explicit in Srednicki 2007, § 39, pp. 249–250.

At $\mathbf p=0$ helicity has no direction and is undefined; the rank-one argument above also degenerates. For a massive particle the chiral projectors remain meaningful, but the mass mixes them and a boost can reverse the three-momentum relative to the spin. Massive helicity is therefore not a Lorentz-invariant replacement for chirality. Parity supplies another check: it reverses momentum relative to spin and simultaneously exchanges the two chiral sectors.

A complex Weyl field contains particles and antiparticles

The physical field content can be checked without repeating the full Dirac Fock-space construction. Define

$$d\Pi_p = \frac{\mathrm d^3\mathbf p} {(2\pi)^3\,2E_{\mathbf p}}, \qquad E_{\mathbf p}=|\mathbf p|,$$

and, for this equal-time calculation, label the two fixed-basis components by $a,b=1,2$.

Use the same Minkowski Fock representation as on the canonical page. Hatted fields and momentum-labelled operators are operator-valued distributions, so the equations below are interpreted after smearing with wave packets. Let $|\Omega\rangle$ be the vacuum characterized, in that distributional sense, by

$$a_-(\mathbf p)|\Omega\rangle=0, \qquad b_+(\mathbf p)|\Omega\rangle=0.$$

The Hamiltonian and charge below are normal-ordered quadratic forms on the dense finite-particle wave-packet domain. Equivalently, one may first impose the finite regulator used in the full canonical construction and then remove it on that domain.

With this setup, expand one complex left Weyl field as

$$\widehat\chi_a(x) = \int d\Pi_p \left[ a_-(\mathbf p)x_a(p)e^{-ip\cdot x} + b_+^\dagger(\mathbf p)y_a(p)e^{+ip\cdot x} \right].$$

The subscripts record the helicities of the physical states annihilated or created. Both coefficient spinors satisfy

$$\begin{gathered} p_\mu\overline\sigma^\mu x(p)=0, \qquad p_\mu\overline\sigma^\mu y(p)=0, \\ x^\dagger x=y^\dagger y=2E_{\mathbf p}, \qquad xx^\dagger=yy^\dagger=p_\mu\sigma^\mu. \end{gathered}$$

Although $x$ and $y$ obey the same two-component kernel equation, the creation operator $b_+^\dagger$ transforms in the conjugate one-particle representation. Its subscript is not obtained by naively reading the Pauli eigenvalue of $y$ as a particle-state label.

Use the nonzero anticommutators

$$\begin{aligned} \{a_-(\mathbf p),a_-^\dagger(\mathbf q)\} &= (2\pi)^3\,2E_{\mathbf p}\, \delta^{(3)}(\mathbf p-\mathbf q), \\ \{b_+(\mathbf p),b_+^\dagger(\mathbf q)\} &= (2\pi)^3\,2E_{\mathbf p}\, \delta^{(3)}(\mathbf p-\mathbf q). \end{aligned}$$

Changing $\mathbf p\mapsto-\mathbf p$ in the antiparticle contribution gives the completeness check

$$x(\mathbf p)x^\dagger(\mathbf p) + y(-\mathbf p)y^\dagger(-\mathbf p) = 2E_{\mathbf p}\mathbf1_2.$$

It follows that

$$\{ \widehat\chi_a(t,\mathbf x), \widehat\chi_b^\dagger(t,\mathbf y) \} = \delta_{ab} \delta^{(3)}(\mathbf x-\mathbf y).$$

This is the fixed-frame canonical anticommutator. Its component notation is not being promoted to a Lorentz-covariant tensor equation.

On that domain, after the same vacuum-relative reordering used for the Dirac field, the free Hamiltonian and the charge for $\chi\mapsto e^{-i\alpha}\chi$ are

$$\begin{aligned} \widehat H_{\mathrm{vac\text{-}rel}} &= \int d\Pi_p\,E_{\mathbf p} \left( a_-^\dagger a_- + b_+^\dagger b_+ \right), \\ \widehat Q &= \int d\Pi_p \left( a_-^\dagger a_- - b_+^\dagger b_+ \right). \end{aligned}$$

Thus a complex left Weyl field has a negative-helicity particle and a positive-helicity antiparticle, both with nonnegative excitation energy and opposite phase charge. A right Weyl field reverses the helicities. The full constraint reduction, vacuum prescription, and positivity proof remain with Canonical Quantization of the Free Dirac Field; identifying particle and antiparticle modes does not impose a Majorana relation between them.

Dimension, signature, orientation, and basis change the caveats

The four-dimensional formulas are one instance of an even-dimensional construction. In an oriented even dimension $d$, choose the signature-dependent phase $\kappa$ so that the normalized Clifford volume element

$$\Gamma_* = \kappa\, \gamma^0\gamma^1\cdots\gamma^{d-1}$$

satisfies $\Gamma_*^2=1$. Moving any gamma matrix through the other $d-1$ factors gives

$$\Gamma_*\gamma^\mu = (-1)^{d-1}\gamma^\mu\Gamma_*.$$

For even $d$ this is an anticommutation relation. The projectors

$$P_\pm=\frac{1\pm\Gamma_*}{2}$$

split the complex spin representation into two chiral modules, and Clifford multiplication maps one into the other. For odd $d$, the volume element commutes with every gamma matrix. On an irreducible complex module it is a scalar, so it supplies no nontrivial internal Weyl split.

The even-dimensional construction and the odd-dimensional obstruction are proved in Weinberg 2000, Ch. 32 appendix, pp. 401–405, especially equations (32.A.17)–(32.A.22) on p. 403 and (32.A.36) on p. 405. Weinberg uses $(-,+,\ldots,+)$ with $\{\gamma_{\mathrm W},\gamma_{\mathrm W}\}=2\eta_{\mathrm W}$. The map $\gamma_{\text{site}}=i\gamma_{\mathrm W}$ gives the site complex Clifford sign, while the phase and name of $\Gamma_*$ are fixed anew. The invariant conclusion—two half-spin modules in even dimension and no such split in odd dimension—survives that translation.

Several qualifications now become precise:

• Reversing orientation or reversing the conventional sign of $\Gamma_*$ exchanges the labels on the two chiral subspaces.
• A gamma-matrix similarity transformation conjugates $\Gamma_*$ and its projectors. It changes components, not whether a field is chiral.
• Euclidean even-dimensional spinors still have a chirality grading, but Euclidean space has no positive-energy null particle orbit, so the Lorentzian helicity table does not apply.
• In Lorentzian dimension greater than four, the finite-spin massless rotational little group is $\operatorname{Spin}(d-2)$ rather than the one-dimensional rotation group of four-dimensional massless kinematics. Chirality therefore does not reduce to a universal scalar equation “helicity equals $\pm\tfrac12$.”
• Whether a chiral module also admits a Majorana or symplectic reality condition depends on dimension and signature. Chirality alone does not answer that question.

Common pitfalls

Using the wrong projector on the adjoint. If $\psi_L=P_L\psi$, then $\overline{\psi_L}=\overline\psi P_R$, not $\overline\psi P_L$. Missing this reversal makes the Weyl kinetic term appear to vanish.

Saying chirality equals helicity without hypotheses. The comparison on this page assumes a four-dimensional, Lorentzian, nonzero, massless, positive-energy on-shell momentum and distinguishes particles from antiparticles. Chirality remains a field-representation grading away from that regime; helicity does not.

Removing the antiparticle. A complex Weyl field has one particle helicity and the opposite antiparticle helicity. Keeping only the annihilation term would not produce a local complex quantum field.

Claiming that chirality forbids every mass. A Dirac mass couples opposite chiralities. A Majorana-type mass is a different construction whose availability depends on charge, conjugation, dimension, and signature.

Defining a Weyl field by upper or lower components. That description is valid only after choosing a chiral gamma basis. The invariant definition is an eigenfield of the normalized chirality operator.

Exporting the four-dimensional helicity table. Odd dimensions have no Weyl split, higher-dimensional massless little groups are not labeled by one helicity number, and Euclidean chirality has no Lorentzian particle-helicity interpretation.

Check your understanding

Check 1: recover the two chiral kinetic terms

Starting from $\psi=\psi_L+\psi_R$, show which of the four terms in $\overline\psi\,i\gamma^\mu\partial_\mu\psi$ survive.

Solution

Use $\overline{\psi_L}=\overline\psi P_R$ and $P_R\gamma^\mu=\gamma^\mu P_L$. Then

$$\overline{\psi_L}\gamma^\mu\psi_R = \overline\psi\gamma^\mu P_LP_R\psi =0,$$

and similarly $\overline{\psi_R}\gamma^\mu\psi_L=0$. The two surviving terms are

$$\overline{\psi_L}\, i\gamma^\mu\partial_\mu\psi_L + \overline{\psi_R}\, i\gamma^\mu\partial_\mu\psi_R.$$

Their sum reconstructs the massless Dirac kinetic term.

Check 2: derive the left-particle helicity

Let $\chi(x)=\xi e^{-ip\cdot x}$ with $p^0=|\mathbf p|>0$. Derive the helicity of $\xi$ from the left Weyl equation.

Solution

The equation $i\overline\sigma^\mu\partial_\mu\chi=0$ gives

$$\left( |\mathbf p|\mathbf1_2 + \boldsymbol\sigma\mathbin{\cdot}\mathbf p \right)\xi=0.$$

Division by $|\mathbf p|$ yields

$$\boldsymbol\sigma\mathbin{\cdot} \widehat{\mathbf p}\,\xi=-\xi.$$

Therefore $h\xi=(\boldsymbol\sigma\cdot\widehat{\mathbf p}/2)\xi =-\tfrac12\xi$. This is the particle coefficient. The antiparticle created by the same field has the opposite physical helicity.

Check 3: recover the equal-time field anticommutator

Explain why the two rank-one coefficient matrices in the left-Weyl expansion sum to the identity after the antiparticle momentum is reversed.

Solution

The particle coefficient gives

$$x(\mathbf p)x^\dagger(\mathbf p) = E_{\mathbf p}\mathbf1_2 - \boldsymbol\sigma\mathbin{\cdot}\mathbf p.$$

After $\mathbf p\mapsto-\mathbf p$, the antiparticle coefficient gives

$$y(-\mathbf p)y^\dagger(-\mathbf p) = E_{\mathbf p}\mathbf1_2 + \boldsymbol\sigma\mathbin{\cdot}\mathbf p.$$

Their sum is $2E_{\mathbf p}\mathbf1_2$, which cancels the same factor in $d\Pi_p$ and leaves the spatial Fourier representation of $\delta^{(3)}(\mathbf x-\mathbf y)\mathbf1_2$.

Where the distinction is used next

• Majorana Fields and Reality Conditions compares a chiral projection with an antilinear reality condition and determines which free mass terms and neutral identifications are allowed.
• Plane Waves, Spin Sums, and Bilinears supplies the full four-component normalization and the physical antiparticle spin-label convention used in chiral projector limits.
• The Fermion Propagator derives the massless first-order inverse and its chiral off-diagonal numerator with the Feynman prescription and contact term intact.
• Electroweak Gauge and Matter Structure develops interacting chiral gauge assignments.
• Yukawa Couplings and Fermion Masses develops gauge-compatible mass generation and flavor matrices.
• Perturbative Chiral and Gauge Anomalies develops anomalous currents, coefficients, and cancellation conditions.

References

• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge University Press, 2014. DOI.
• Srednicki, Mark. Quantum Field Theory. Cambridge University Press, 2007. DOI.
• Weinberg, Steven. The Quantum Theory of Fields. Volume III: Supersymmetry. Cambridge University Press, 2000. DOI.