Plane Waves, Spin Sums, and Bilinears

An on-shell Dirac equation has a two-dimensional solution space in each frequency sector, so an individual spinor depends on a spin basis and phase choice. The basis-independent statements are the complete sums $\sum_r u_r\bar u_r=p\!\!\!/+m$ and $\sum_r v_r\bar v_r=p\!\!\!/-m$. They encode the normalization, turn spin labels into covariant projectors, and explain the numerator of the free fermion propagator. This page derives those results for commuting free wavefunctions, develops their principal bilinears, and identifies what changes in the massless limit. Operator anticommutators, pole prescriptions, amplitude traces, and interacting form factors are left to their dedicated pages.

Required background. The Dirac Field supplies the Dirac equation, adjoint equation, Clifford factorization, and positive Cauchy-surface inner product used below.

Helpful background. Spinors, Conjugations, Bilinears, and Fierz Identities explains why the five gamma-matrix bilinears have their stated Lorentz types and why Grassmann parity matters when spinors are exchanged.

On-shell spinors and the normalization choice

For the massive derivation, take $m>0$ and label both sectors by the same future-directed on-shell momentum

$$p^\mu=(E_{\mathbf p},\mathbf p), \qquad E_{\mathbf p}=\sqrt{\mathbf p^2+m^2}.$$

The objects $u_r(p)$ and $v_r(p)$ are commuting spinor-valued wavefunctions; the index $r=1,2$ labels a chosen orthonormal spin basis. With the chapter’s Fourier and slash conventions, substitution of

$$\psi^{(+)}_r(x)=u_r(p)e^{-ip\cdot x}, \qquad \psi^{(-)}_r(x)=v_r(p)e^{+ip\cdot x}$$

into the Dirac equation gives

$$\begin{aligned} (p\!\!\!/-m)u_r(p)&=0,\\ (p\!\!\!/+m)v_r(p)&=0. \end{aligned}$$

The label on $v_r(p)$ is still the positive-energy vector $p$. Its exponential has four-wavevector $-p$, which is why its algebraic equation contains $p\!\!\!/+m$. Later, after quantization, the corresponding creation operator will create a positive-energy antiparticle; that interpretation is not needed for the present linear algebra.

Choose the covariant massive normalization

$$\begin{aligned} \bar u_r(p)u_s(p)&=2m\,\delta_{rs},\\ \bar v_r(p)v_s(p)&=-2m\,\delta_{rs}. \end{aligned}$$

The minus sign is a property of the Dirac adjoint, not a negative norm. Indeed, using $\{\gamma^0,p\!\!\!/\}=2E_{\mathbf p}$ together with the on-shell equations and their adjoints gives

$$u_r^\dagger(p)u_s(p) = v_r^\dagger(p)v_s(p) = 2E_{\mathbf p}\,\delta_{rs}.$$

Thus both sectors have positive Cauchy-surface norm. At equal $p$ one also has $\bar u_r(p)v_s(p)=0$. By contrast, $u_r^\dagger(p)v_s(p)$ need not vanish. If the spinors are written as functions of their spatial label while $E_{\mathbf p}$ remains positive, the Hamiltonian orthogonality statement is $u_r^\dagger(\mathbf p)v_s(-\mathbf p)=0$. Keeping the barred and daggered pairings distinct prevents a common false identity.

This normalization naturally accompanies the invariant positive-mass-shell measure

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

Rescaling every $u$ and $v$ is allowed, but it rescales the spin sums and must be compensated in any later field expansion. The formulas on this page all use the choice above.

An explicit representation check

No gamma-matrix representation is needed for the final identities. It is nevertheless useful to verify every normalization once in the Dirac basis. For orthonormal two-spinors $\chi_r$ and $\eta_r$,

$$u_r(p) = \sqrt{E_{\mathbf p}+m} \begin{pmatrix} \chi_r\\[2pt] \dfrac{\boldsymbol{\sigma}\cdot\mathbf p} {E_{\mathbf p}+m}\chi_r \end{pmatrix}, \qquad v_r(p) = \sqrt{E_{\mathbf p}+m} \begin{pmatrix} \dfrac{\boldsymbol{\sigma}\cdot\mathbf p} {E_{\mathbf p}+m}\eta_r\\[2pt] \eta_r \end{pmatrix}.$$

At rest these become

$$u_r(m,\mathbf0) = \sqrt{2m} \begin{pmatrix}\chi_r\\0\end{pmatrix}, \qquad v_r(m,\mathbf0) = \sqrt{2m} \begin{pmatrix}0\\\eta_r\end{pmatrix}.$$

The rest spinors lie in the $+1$ and $-1$ eigenspaces of $\gamma^0$, respectively, so their barred norms are immediately $\pm2m$. Direct substitution and $(\boldsymbol{\sigma}\cdot\mathbf p)^2=\mathbf p^2$ give the stated $2E_{\mathbf p}$ dagger norm. These formulas are a check, not a definition: a simultaneous similarity transformation of the gamma matrices and spinors changes their components but not any bilinear or complete spin sum. Equivalent rest and boosted solutions are given in Schwartz 2014, § 11.2, pp. 188–190. That source writes the displayed mode exponentials with the opposite Fourier sign, so its phase and on-shell Dirac equation must be translated together; the component spinors and covariant sums are unchanged.

Completeness and the actual projectors

Completeness of the two-spinor bases at rest gives

$$\begin{aligned} \sum_r u_r(m,\mathbf0)\bar u_r(m,\mathbf0) &=m(1+\gamma^0),\\ \sum_r v_r(m,\mathbf0)\bar v_r(m,\mathbf0) &=m(\gamma^0-1). \end{aligned}$$

Under a spinor boost, the left sides transform by conjugation while $m\gamma^0$ becomes $p\!\!\!/$. Therefore

$$\boxed{ \begin{aligned} \sum_{r=1}^{2}u_r(p)\bar u_r(p)&=p\!\!\!/+m,\\ \sum_{r=1}^{2}v_r(p)\bar v_r(p)&=p\!\!\!/-m. \end{aligned}}$$

There is also a representation-independent derivation. Let $C_u=\sum_r u_r\bar u_r$ and $C_v=\sum_r v_r\bar v_r$. The two $u$ spinors and two $v$ spinors form a basis of the four-component spinor space. Their normalizations and mixed orthogonality give

$$\begin{array}{c|cc} &u_s&v_s\\ \hline C_u&2m\,u_s&0\\ C_v&0&-2m\,v_s . \end{array}$$

On shell,

$$(p\!\!\!/-m)(p\!\!\!/+m)=p^2-m^2=0,$$

and the on-shell equations show that $(p\!\!\!/+m)/(2m)$ has eigenvalue $1$ on the $u$ kernel and $0$ on the $v$ kernel, while $(m-p\!\!\!/)/(2m)$ does the reverse. Equality on a basis therefore gives $C_u=p\!\!\!/+m$ and $C_v=p\!\!\!/-m$. This construction and its standard normalization are developed in Schwartz 2014, § 11.2.1, pp. 190–191.

The spin sums are completeness matrices, but they are not both projectors. The idempotent projectors onto the two kernels are

$$\Pi_u(p)=\frac{p\!\!\!/+m}{2m}, \qquad \Pi_v(p)=\frac{m-p\!\!\!/}{2m},$$

and hence

$$\sum_r v_r(p)\bar v_r(p)=-2m\,\Pi_v(p).$$

Clifford factorization gives the complete set of checks

$$\begin{aligned} \Pi_u^2&=\Pi_u, & \Pi_v^2&=\Pi_v, & \Pi_u\Pi_v&=0,\\ \Pi_u+\Pi_v&=1, & \operatorname{tr}\Pi_u &=\operatorname{tr}\Pi_v=2. \end{aligned}$$

Two further traces detect normalization errors quickly:

$$\begin{aligned} \operatorname{tr}(p\!\!\!/+m)&=4m, & \operatorname{tr}(p\!\!\!/-m)&=-4m,\\ \operatorname{tr}\!\left[\gamma^0(p\!\!\!/\pm m)\right] &=4E_{\mathbf p}. \end{aligned}$$

The last line recovers the sum of the two $2E_{\mathbf p}$ dagger norms in either sector.

Spin labels, helicity, and polarization

A massive spin label begins with a unit direction $\boldsymbol{\zeta}$ in the rest frame and a choice of standard boost to $p$. The corresponding polarization four-vector is

$$s^\mu(p,\boldsymbol{\zeta}) = \left( \frac{\mathbf p\cdot\boldsymbol{\zeta}}{m}, \; \boldsymbol{\zeta} + \frac{\mathbf p(\mathbf p\cdot\boldsymbol{\zeta})} {m(E_{\mathbf p}+m)} \right),$$

which satisfies

$$s^2=-1, \qquad p\cdot s=0.$$

On this page the same $s^\mu$ denotes the physical spin direction in both frequency sectors. The polarized states are defined by

$$\gamma_5s\!\!\!/\,u(p,s)=u(p,s), \qquad \gamma_5s\!\!\!/\,v(p,s)=v(p,s).$$

In the explicit Dirac-basis check, this convention may be implemented with $\eta_r=i\sigma^2\chi_r^*$ up to phases. The lower two-spinor of $v$ then has the opposite Pauli polarization to $\chi_r$. Calling the lower-component Pauli label itself “the antiparticle spin” would reverse several $v$-sector axial signs.

The rank-one polarized completeness relations are

$$\begin{aligned} u(p,s)\bar u(p,s) &= \frac12(p\!\!\!/+m)(1+\gamma_5s\!\!\!/),\\ v(p,s)\bar v(p,s) &= \frac12(p\!\!\!/-m)(1+\gamma_5s\!\!\!/). \end{aligned}$$

The factors commute because $p\cdot s=0$. Reversing $s$ and adding the two polarizations removes the $\gamma_5s\!\!\!/$ term and recovers the unpolarized spin sums.

Helicity is the spin projection along the momentum. For $\mathbf p\ne0$, set $\boldsymbol{\zeta}=\lambda\widehat{\mathbf p}$ with $\lambda=\pm1$; then

$$s^\mu_\lambda = \lambda\left( \frac{|\mathbf p|}{m}, \frac{E_{\mathbf p}}{m}\widehat{\mathbf p} \right), \qquad h=\frac{\lambda}{2}.$$

For $m>0$, a boost can overtake the particle and reverse helicity, and helicity is undefined at rest. A rest-frame spin label is well defined, but different choices of standard boost are related by momentum-dependent Wigner rotations. In the massless theory, helicity becomes invariant under proper orthochronous Lorentz transformations because no rest frame exists and no allowed boost can overtake a lightlike momentum.

For a derivation using the opposite, mostly-plus metric convention, see Srednicki 2007, § 38, pp. 242–244. Translating that source’s slash sign into the site’s convention turns its completeness and polarized-projector formulas into the expressions displayed above.

Bilinear covariants and their on-shell values

The gamma-matrix basis organizes spinor bilinears into five proper-Lorentz types:

$$\begin{array}{ccl} S&=&\bar w'w,\\ P&=&i\bar w'\gamma_5w,\\ V^\mu&=&\bar w'\gamma^\mu w,\\ A^\mu&=&\bar w'\gamma^\mu\gamma_5w,\\ T^{\mu\nu}&=&\bar w'\sigma^{\mu\nu}w, \end{array} \qquad \sigma^{\mu\nu}=\frac{i}{2}[\gamma^\mu,\gamma^\nu].$$

These $1+1+4+4+6=16$ matrices form a basis of the complex $4\times4$ matrices; trace orthogonality proves their linear independence. For a spin lift $S(\Lambda)$,

$$S(\Lambda)^{-1}\gamma^\mu S(\Lambda) = \Lambda^\mu{}_\nu\gamma^\nu,$$

so conjugation by $S(\Lambda)$ makes the identity a scalar, $\gamma^\mu$ a vector, and the commutators $\sigma^{\mu\nu}$ an antisymmetric rank-two tensor. For proper transformations $\gamma_5$ is invariant, so $\gamma_5$ and $\gamma^\mu\gamma_5$ give the second scalar and vector types; orientation reversal distinguishes them as pseudoscalar and axial vector.

Under the connected Lorentz group, $S$ and $P$ are both scalars, while $V^\mu$ and $A^\mu$ are both vectors. Their “pseudo” and “axial” qualifications refer to orientation-reversing transformations such as parity. The scalar, Hermitian pseudoscalar, vector, and axial cases are worked through in Srednicki 2007, § 40, pp. 259–261; that source does not supply the tensor completion. The general algebraic classification and Fierz rearrangements belong to the mathematical spinor page; here the spinors also obey their on-shell equations.

For equal momenta, those equations imply

$$\begin{aligned} \bar u_r u_s&=2m\,\delta_{rs}, & \bar v_r v_s&=-2m\,\delta_{rs},\\ \bar u_r\gamma^\mu u_s&=2p^\mu\delta_{rs}, & \bar v_r\gamma^\mu v_s&=2p^\mu\delta_{rs},\\ \bar u_r\gamma_5u_s&=0, & \bar v_r\gamma_5v_s&=0. \end{aligned}$$

For example, insert $\{\gamma^\mu,p\!\!\!/\}=2p^\mu$ between two on-shell spinors. In the $u$ sector the two terms containing $p\!\!\!/$ each produce $m$; in the $v$ sector each produces $-m$, which cancels the minus sign in $\bar v_rv_s$. Both vector currents are therefore $2p^\mu\delta_{rs}$. Anticommutation of $\gamma_5$ with $p\!\!\!/$ similarly forces the pseudoscalar matrix element to vanish when $m>0$.

For a definite physical spin vector $s^\mu$, the remaining diagonal identities are

$$\begin{aligned} \bar u\gamma^\mu\gamma_5u&=2m\,s^\mu, & \bar v\gamma^\mu\gamma_5v&=-2m\,s^\mu,\\ \bar u\sigma^{\mu\nu}u &=-2\epsilon^{\mu\nu\rho\sigma}p_\rho s_\sigma, & \bar v\sigma^{\mu\nu}v &=-2\epsilon^{\mu\nu\rho\sigma}p_\rho s_\sigma. \end{aligned}$$

These signs use the inherited orientation and the physical antiparticle-spin convention stated above. They can be checked without component spinors by taking traces of the polarized completeness matrices. For unequal momenta or different spin axes, bilinears retain the same Lorentz type but are no longer fixed by $p^\mu$ and one polarization vector alone.

Convention dependencies and sign consequences

The following semantic table makes visible which declarations are inherited, which are local to this page, and which are intentionally deferred. In the table, $\chi_r$ and $\eta_r$ refer only to the Dirac-basis check, whereas $s^\mu$ is the covariant physical spin vector. The table is textual rather than an image so that every relation remains selectable, zoomable, and available to assistive technology.

Convention dependencies for on-shell Dirac spinors

Item	Declaration	Scope	Consequence on this page and downstream
Metric and orientation	η = diag(+1, −1, −1, −1); upper-index ε with 0123 component +1	Inherited	Sets p² = E² − |p|² and fixes the displayed tensor-bilinear sign.
Frequency labels	u(p) multiplies exp(−ip·x); v(p) multiplies exp(+ip·x); p⁰ is positive in both labels	Inherited Fourier sign; local labeling choice	Gives the kernels slash(p) − m and slash(p) + m without calling p a negative-energy label.
Clifford algebra and slash	{γμ, γν} = 2ημν; slash(p) means γμpμ; (γμ)† = γ⁰γμγ⁰	Inherited	Factors p² − m², makes γ⁰ Hermitian and spatial γ matrices anti-Hermitian, and fixes the numerators slash(p) ± m.
Dirac adjoint	ψ̄ = ψ†γ⁰	Inherited	Makes ψ̄ψ covariant; the negative value of v̄v is not a negative Hilbert norm.
Chirality operators	γ₅ = iγ⁰γ¹γ²γ³; PL = (1 − γ₅)/2 and PR = (1 + γ₅)/2	Inherited	Fixes axial and polarized-projector signs; the projectors become helicity selectors only in the appropriate massless sector.
Massive normalization	ūrus = 2mδrs; v̄rvs = −2mδrs; u†rus = v†rvs = 2Epδrs; dΠp = d³p/[(2π)³2Ep]	Local	Fixes spin-sum coefficients and the normalization that later mode expansions must match; the barred scalar normalization degenerates at m = 0.
Completeness	Σrurūr = slash(p) + m; Σrvrv̄r = slash(p) − m	Derived; trace and idempotence checked	Removes spin-basis phases and supplies the representation-independent matrices used in residues and unpolarized sums.
Antiparticle spin label	The same physical s labels u and v; in the component check ηr = iσ²χr* up to phases	Local	Gives v̄γμγ₅v = −2msμ; labeling v by its lower-component Pauli polarization would reverse that convention.
Charge-conjugation matrix	A basis-dependent C may satisfy C(γμ)TC−1 = −γμ; its phase is conventional	Structural declaration only	Relates convenient u and v bases but does not by itself assert that charge conjugation is a symmetry of an interacting theory.
Propagator numerator	The algebraic inverse away from shell is [slash(p) + m]/(p² − m²)	Derived; causal prescription deferred	On shell, its numerator matches the u spin sum; the overall i, pole displacement, time ordering, and equal-time contact term remain downstream.
Grassmann source ordering	No Grassmann sources or functional derivatives occur on this c-number wavefunction page	Deferred	Spin sums acquire no source-order sign here; the functional-integral page must declare source order and left/right differentiation, reproduce the same regulated inverse kernel, and track every exchange sign.

The last row is a real boundary, not an omitted convention. A choice such as the order of $\bar\eta,\eta,\bar\psi,\psi$ matters only once Grassmann sources and functional derivatives are introduced. Guessing that order here would make the later generating-functional signs less, rather than more, transparent.

The massless limit changes the projectors

Taking $m\to0$ in the complete sums gives

$$\sum_r u_r(p)\bar u_r(p) \longrightarrow p\!\!\!/, \qquad \sum_r v_r(p)\bar v_r(p) \longrightarrow p\!\!\!/.$$

The two frequency sectors have the same matrix numerator in this limit, but they remain distinguished by their exponentials. At the same time, $\bar uu$ and $\bar vv$ vanish, the projectors with $1/(2m)$ cease to exist, and the rest-frame polarization vector becomes singular. None of these facts means that the solution space disappears. One instead keeps the positive dagger normalization $w_h^\dagger w_{h'}=2E_{\mathbf p}\delta_{hh'}$ and uses helicity.

With the physical charge-conjugate spin labels chosen above,

$$\gamma_5u_h(p)=2h\,u_h(p), \qquad \gamma_5v_h(p)=-2h\,v_h(p), \qquad h=\pm\frac12.$$

The relation between chirality, helicity, particle labels, and antiparticle labels is developed on Weyl Fields and Chirality. The safe limiting statement here is the spin sum $p\!\!\!/$; formulas divided by $m$ must be rebuilt in a massless basis rather than evaluated at $m=0$.

Why the propagator has the same numerator

Away from the mass shell, Clifford factorization gives the matrix inverse

$$(p\!\!\!/-m)^{-1} = \frac{p\!\!\!/+m}{p^2-m^2}.$$

On shell, the numerator becomes the positive-frequency spin sum and maps arbitrary spinors into the Dirac kernel. Off shell it is the same matrix polynomial, while the scalar denominator measures the failure to be on shell. The negative-frequency sum supplies the corresponding numerator when a residue is parameterized with a future-directed antiparticle momentum.

This algebra does not choose a Green function. The overall factor of $i$, the $i\epsilon$ prescription, time ordering, the equal-time contact term, and the interpretation of the two poles are derived on The Fermion Propagator. Likewise, converting external spin sums into traces is amplitude technology and belongs in the scattering volume.

Common pitfalls

Treating the label on $v(p)$ as negative energy. The mode $v(p)e^{+ip\cdot x}$ has wave four-vector $-p$, but its label $p$ is chosen future directed. Mixing those two statements changes signs in both the on-shell equation and later momentum assignments.

Calling the negative-frequency spin sum a projector. $(p\!\!\!/-m)/(2m)$ squares to its negative. The actual projector is $(m-p\!\!\!/)/(2m)$, while the spin sum is $-2m$ times that projector because $\bar vv=-2m$.

Reading $\bar vv<0$ as a negative state norm. The positive norm uses $v^\dagger v=2E_{\mathbf p}$. The barred contraction is a Lorentz scalar with an indefinite $\gamma^0$, not the Hilbert-space norm.

Using a Pauli label without specifying what it means for $v$. The lower two-spinor polarization is opposite to the physical spin label adopted here. Changing that labeling is allowed, but every polarized $v$ identity must be changed consistently.

Substituting $m=0$ into massive spin projectors. The factors $1/(2m)$ and the rest-frame spin vector are not massless observables. Keep the finite spin sum and rebuild the basis with helicity.

Check your understanding

Check 1: verify the complementary projectors

Show directly that $\Pi_u+\Pi_v=1$ and that both matrices are idempotent on $p^2=m^2$.

Solution

The sum is immediate:

$$\Pi_u+\Pi_v = \frac{p\!\!\!/+m+m-p\!\!\!/}{2m} =1.$$

For the positive-frequency projector,

$$(p\!\!\!/+m)^2 =p^2+2mp\!\!\!/+m^2 =2m(p\!\!\!/+m).$$

Division by $4m^2$ gives $\Pi_u^2=\Pi_u$. Replacing $p\!\!\!/\mapsto-p\!\!\!/$ gives the same calculation for $\Pi_v$.

Check 2: derive the vector current without components

Starting from $\{\gamma^\mu,p\!\!\!/\}=2p^\mu$, derive $\bar u_r\gamma^\mu u_s=2p^\mu\delta_{rs}$.

Solution

Insert the anticommutator:

$$\begin{aligned} 2p^\mu\bar u_ru_s &= \bar u_r \left(\gamma^\mu p\!\!\!/+p\!\!\!/\gamma^\mu\right) u_s\\ &= 2m\,\bar u_r\gamma^\mu u_s. \end{aligned}$$

Since $\bar u_ru_s=2m\delta_{rs}$ and $m>0$, division by $2m$ gives the result. In the $v$ sector the two on-shell factors and the scalar normalization are both negative, so the same vector current follows.

Check 3: recover an axial sign at rest

Take physical spin along $+\hat{\mathbf z}$. Explain why the upper Pauli spinor of $u$ is a $+1$ eigenvector of $\sigma^3$, while the lower Pauli spinor of $v$ is a $-1$ eigenvector, even though both states carry the same physical spin label.

Solution

At rest, $\gamma_5s\!\!\!/$ acts as $\sigma^3$ on the upper two components and as $-\sigma^3$ on the lower two components in the Dirac basis. The condition $\gamma_5s\!\!\!/\,w=w$ therefore selects $\sigma^3\chi=+\chi$ for $u$ and $\sigma^3\eta=-\eta$ for $v$. This is why the physical-spin convention yields $\bar u\gamma^3\gamma_5u=+2m$ but $\bar v\gamma^3\gamma_5v=-2m$.

Where the calculation continues

• Canonical Quantization of the Free Dirac Field attaches creation and annihilation operators to these modes and fixes their anticommutators.
• The Fermion Propagator turns the same Clifford numerator into a vacuum time-ordered distribution with a pole prescription.
• Weyl Fields and Chirality rebuilds the massless solution space in irreducible chiral sectors.
• Grassmann Functional Integrals for Free Fermions declares source ordering and derives the inverse kernel with its Grassmann signs.

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.