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A Majorana spinor equals its charge conjugation, ${\psi }^c=C{\bar{\psi }}^T=\psi$, with conventional phase choices absorbed into $C$. The two-component worldsheet Majorana fermion therefore has no independent complex conjugate components; in a Majorana representation its components can be real Grassmann variables. The same condition applies to the constant supersymmetry parameter. Grassmann statistics matter in the variation: replacing the spinors by commuting numerical vectors would give the wrong bilinear interchange signs.

The displayed rigid transformation is understood in flat conformal gauge. Take $h_{\mu \nu }={\eta }_{\mu \nu }=\mathrm{diag}(-1,1)$, $\{{\gamma }^{\mu },{\gamma }^{\nu }\}=2{\eta }^{\mu \nu }$, and chooseHere $C={\gamma }^0$ is a valid invariant charge-conjugation form: $C^T=-C$ and $({\gamma }^{\mu }{)}^{T}C=-C{\gamma }^{\mu }$. These concrete matrices make all signs checkable; equivalent Majorana conventions give the same result with their consistently transformed bars.

Let $A=\sqrt{{\alpha }^{\prime }/2}$ and $B=1/\sqrt{2{\alpha }^{\prime }}$, so ${\alpha }^{\prime }B=A$. Suppressing the common factor $-1/(4\pi {\alpha }^{\prime })$ in the action, the flat kinetic density isThe rigid variations have $\delta X^a=iA\overline{ϵ}{\psi }^a$ and $\delta {\psi }^a=B{\gamma }^{\nu }{\partial }_{\nu }{X}^aϵ$. The supersymmetry variation is even, so its product rule has no extra graded sign. From the Majorana Grassmann bilinear interchange identities,For example the last identity follows by moving the Grassmann-odd parameter through $\psi$, then using $C^T=-C$ and the transpose relations twice. No equation of motion has been used.

The two kinetic variations areThe Clifford algebra turns the first fermionic term plus the bosonic variation into $iA{\partial }_{\nu }{X}_a\overline{ϵ}{\gamma }^{\mu }{\gamma }^{\nu }{\partial }_{\mu }{\psi }^a$. In the second term, commute the partial derivatives and use the same algebra: the antisymmetric gamma product drops out, leaving $iA\square X_a\overline{ϵ}{\psi }^a$. Together they are precisely the rigid worldsheet supersymmetry boundary term:Therefore $\delta S=-(4\pi {\alpha }^{\prime }{)}^{-1}{\int }_{\partial \Sigma }d{\Sigma }_{\mu }{J}^{\mu }$. It vanishes on a closed or periodic worldsheet, for compactly supported changes, or under compatible supersymmetric endpoint conditions. The local total-derivative identity is off shell; the boundary assumptions are needed to call the integrated action invariant.

The flat-gauge qualification is substantive. On an arbitrary curved worldsheet, spinors need a zweibein and spin connection and a constant spinor parameter need not exist. A fully covariant locally supersymmetric action also involves the worldsheet gravitino; the isolated ordinary-derivative expression does not prove rigid invariance for arbitrary $h_{\mu \nu }$. We have proved the intended rigid symmetry of its flat gauge-fixed action, with the fermionic term inside the same integral and overall normalization. If the displayed last term were read as outside the integral, it would not even define an action.

For phenomenology, bosonic string theory has no spacetime fermions and its usual vacuum contains a tachyon. The spinning string has a Ramond sector, whose fermionic zero modes form a spacetime Clifford algebra and give spacetime spinor states, as well as a Neveu–Schwarz sector. In suitable consistent theories the GSO projection removes the tachyonic NS ground state and keeps the appropriate Ramond chirality. The critical dimension of the RNS superstring is 10 rather than 26: $D$ bosons and $D$ real fermions have matter central charge $3D/2$ per chiral sector, while the reparameterization and superconformal ghosts contribute $-26+11=-15$, so the total anomaly cancels at $D=10$.

Suitable GSO-projected superstrings admit spacetime fermions and a tachyon-free spectrum, making them more promising for particle physics than the bosonic string. Rigid worldsheet supersymmetry alone does not establish a tachyon-free spacetime theory or realistic phenomenology. The projection, consistent sectors and compactification in string theory are additional input; spacetime supersymmetry and a realistic four-dimensional spectrum are not automatic consequences of this classical action.

Use $\hslash =c=1$ and the Minkowski metric $\eta =\mathrm{diag}(1,-1,-1,-1)$ throughout. In the four-component Dirac spinor representation, the gamma matrices generate the Clifford algebraThus $({\gamma }^0{)}^2=I_4$, $({\gamma }^i{)}^2=-I_4$, and matrices with distinct indices anticommute. Indices are lowered with $\eta$, so ${\gamma }_{\mu }={\eta }_{\mu \nu }{\gamma }^{\nu }$ and $/p=p_{\mu }{\gamma }^{\mu }$.

First derive the needed gamma-matrix traces directly from the Clifford algebra. The cyclic property of the trace givesso $\mathrm{tr}({\gamma }^{\mu }{\gamma }^{\nu })=4{\eta }^{\mu \nu }$. For $T=\mathrm{tr}({\gamma }^{\mu }{\gamma }^{\nu }{\gamma }^{\rho }{\gamma }^{\sigma })$, move ${\gamma }^{\mu }$ successively through the other three matrices and cycle the final term back to the front:ThereforeThese are all the trace identities required.

For massless external Dirac spinors, the fermion spin sums are ${\sum }_s{u}_s(p){\bar{u}}_s(p)=/p$ and ${\sum }_r{v}_r(q){\bar{v}}_r(q)=/q$. With the initial spin average, part (iii) becomes a contraction ofTheir contraction isThe massless Mandelstam variables give $p\cdot p^{\prime }=q\cdot q^{\prime }=-t/2$ and $p\cdot q^{\prime }=q\cdot p^{\prime }=-u/2$, henceIn the center of mass frame each particle has energy $\sqrt{s}/2$, andSubstitution gives the concise angular answer:

The tree amplitude is, up to an overall convention-dependent sign,The Clifford algebra gives the momentum-space Dirac equation and its adjoint:ThereforeAlthough $\mu >2m$ permits the relativistic two-body decay kinematically, the matrix element vanishes. Thus

For metric ${\eta }^{\mu \nu }=\mathrm{diag}(1,-1,-1,-1)$, the defining Clifford algebra relation is

Average over the two initial electron spins and two initial photon polarizations, and sum over the final ones. The supplied fermion spin sum and photon polarization sum turn the result into gamma-matrix traces. The Clifford algebra impliesby anticommuting the outside matrix through the product. Together withthe two channel squares reduce, at $m=0$, to $-2e^{4}u/s$ and $-2e^{4}s/u$; the remaining cross terms cancel. Thus, in terms of the Mandelstam variables,Physical Compton kinematics has $s>0$ and $u<0$, so the displayed expression is nonnegative.

The quantized fermions obey a Clifford algebra, $\{{\hat{\psi }}^a,{\hat{\psi }}^b\}=g^{ab}$. When $M$ is a spin manifold, they act on the spinor bundle andthe Hilbert space of square-integrable spinor fields. The supercharge becomes the Dirac operator, the Hamiltonian is one half of its square, and because $M$ is even dimensional, $(-1{)}^F$ is the spinor chirality operator.

Define the spinor generatorsRepeated use of the Clifford algebra givesThe Spinor representation of the Lorentz group isThe adjoint relation for the Dirac matrices impliesExponentiating yields

The Dirac spinor representation is the four-dimensional representation $(\frac{1}{2},0)\oplus (0,\frac{1}{2})$ of the connected Lorentz group. It is characterized byForone may takewith the signs of ${\Omega }_{\rho \sigma }$ fixed by the displayed vector-representation convention. This formula follows by differentiating the covariance relation and using the Clifford algebra $\{{\gamma }^{\mu },{\gamma }^{\nu }\}=2{\eta }^{\mu \nu }$; moving both gamma indices upstairs changes the apparent parameter signs according to the metric.

Use the Weyl representation of the gamma matrices,

After using the Clifford algebra, integrations by parts and the Dirac equation, every allowed Lorentz-covariant symmetric rank-two expression with one derivative reduces, modulo terms vanishing on shell and identically conserved improvements, to an overall multiple ofThus the general nontrivial tensor in the stated class isIt is manifestly symmetric. Differentiating it, commuting partial derivatives and usingmakes the terms cancel pairwise, proving ${\partial }_{\mu }{\hat{T}}^{\mu \nu }=0$ on shell. The normalization $C$ remains free at this stage.

Choose $C=i/4$. Expandingwith the Clifford algebra, applying the product rule and then using the Dirac equation and its adjoint givesThis is the Belinfante-Rosenfeld stress-energy tensor written as an improvement of the canonical tensor.

The translation charges areFor spatial $\nu$, their difference is the integral of ${\partial }_i{S}^{i0\nu }-{\partial }^{\nu }(\bar{\psi }{\gamma }^0\psi )$. For $\nu =0$, use conservation of the Dirac current to replace ${\partial }^0(\bar{\psi }{\gamma }^0\psi )$ by a spatial divergence; also $S^{00\nu }=0$. Hence ${\hat{P}}^{\nu }-P^{\nu }$ is always a spatial boundary integral. Under the usual decay boundary condition it vanishes, so both currents generate the same four-momentum.

A pseudoclassical spinning particle supplements ordinary position and momentum by odd Grassmann variables. An even multiplier imposes $p^2=0$ and an odd multiplier imposes $p\cdot \zeta =0$. The odd symplectic term gives $\{{\zeta }^m,{\zeta }^n{\}}_D=-i{\eta }^{mn}$. Quantization represents these variables by a Clifford algebra, producing a spinor wavefunction.

The zero-mode anticommutator $\{d_0^I,d_0^J{\}}_{+}={\delta }^{IJ}$ becomes the Euclidean Clifford algebra after ${\gamma }^I=\sqrt{2}{d}_0^I$. It acts within the oscillator-vacuum space. Eight transverse Gamma matrices give a minimal complex representation of dimension sixteen, split into two eight-dimensional chirality sectors.

In flat conformal gauge, take $\mathcal{L}=\partial X\cdot \partial X+i{\alpha }^{\prime }\bar{\psi }{\gamma }^{\mu }{\partial }_{\mu }\psi$ and rigid transformations $\delta X=iA\overline{ϵ}\psi$, $\delta \psi =B{\gamma }^{\nu }{\partial }_{\nu }Xϵ$, with $A={\alpha }^{\prime }B$. The Majorana Grassmann bilinear interchange and Clifford algebra combine the variations intoThe identity is off shell. The integrated action is invariant when its boundary flux vanishes, for example on a closed worldsheet or with compatible supersymmetric endpoint conditions. Constant spinors and ordinary derivatives here refer to flat gauge; arbitrary curved worldsheets require covariant spinor derivatives and the locally supersymmetric completion.

The spin group $\mathrm{Spin}(n)$ is the connected double cover of $SO(n)$ constructed inside the units of a Clifford algebra.

Lift the Levi-Civita connection to the spinor bundle using the spin representation of the Lorentz algebra. Under a local Lorentz transformation, its matrix ${\Omega }_{\mu }=\frac{1}{4}{\omega }_{\mu AB}{\gamma }^{AB}$ obeys ${\Omega }_{\mu }^{\prime }=S{\Omega }_{\mu }{S}^{-1}-({\partial }_{\mu }S)S^{-1}$, so ${\nabla }_{\mu }^{\prime }{\psi }^{\prime }=S{\nabla }_{\mu }\psi$. The Clifford algebra identity $[{\gamma }^C,{\gamma }^{AB}]=2({\eta }^{CA}{\gamma }^B-{\eta }^{CB}{\gamma }^A)$ fixes the coefficient. The tetrad postulate then gives ${\partial }_{\mu }{\gamma }^{\nu }+{\Gamma }_{\mu \rho }^{\nu }{\gamma }^{\rho }+[{\Omega }_{\mu },{\gamma }^{\nu }]=0$, establishing compatibility with curved Clifford multiplication.

An irreducible complex Clifford algebra module in even spacetime dimension $2k$ has dimension $2^k$ and splits under the connected Spin group into two Weyl spinor modules of dimension $2^{k-1}$. In odd dimension $2k+1$, an irreducible spinor still has dimension $2^k$ but has no Weyl splitting. Reality conditions depend on signature and are additional to these complex counts.

Define ${\sigma }^{\mu \nu }=\frac{i}{2}[{\gamma }^{\mu },{\gamma }^{\nu }]$ using the gamma matrix Clifford algebra. Moving ${\gamma }^{\rho }$ through the two factors with $\{{\gamma }^{\mu },{\gamma }^{\nu }\}=2g^{\mu \nu }$ proves the displayed operator commutator. For real antisymmetric ${\omega }_{\mu \nu }$ and $S=1-\frac{i}{4}{\sigma }^{\mu \nu }{\omega }_{\mu \nu }$, it gives $S^{-1}{\gamma }^{\rho }S={\gamma }^{\rho }+{\omega }^{\rho }{}_{\nu }{\gamma }^{\nu }+O({\omega }^2)$. This is the infinitesimal Lorentz covariance identity required by the Dirac equation.

A spinor representation is a representation of a Spin group, or infinitesimally of a Special orthogonal Lie algebra, constructed from a Clifford algebra. In even dimensions it can split into two chiral spinor representations.