Superspace and superfield components

Add odd coordinates $\theta^\alpha$ and $\bar\theta^{\dot\alpha}$, which anticommute with one another and with fermionic component fields. With left derivatives define $\bar D_{\dot\alpha}=-\partial_{\bar\theta^{\dot\alpha}}-i\theta^\alpha\sigma^\mu_{\alpha\dot\alpha}\partial_\mu$. The shifted coordinate $y^\mu=x^\mu+i\theta\sigma^\mu\bar\theta$ is annihilated by this operator. The cancellation uses the minus sign acquired when the barred derivative passes an unbarred odd coordinate. Treating the derivative as an ordinary commuting derivative would fail precisely at this step.

The chiral constraint therefore has solutions depending only on y and theta. Since there are only two independent unbarred odd coordinates, the expansion stops at quadratic order. Its three independent coefficient fields are the scalar, Weyl spinor, and auxiliary already encountered. Reexpanding their y dependence in ordinary x introduces barred-coordinate terms containing spacetime derivatives of the same components. Those additional terms are not new independent fields. In particular, chirality does not mean simply deleting every barred coordinate from the ordinary-x expansion.

For one odd variable choose the integral of one to be zero and of the variable itself to be one. For several variables, integrate from the rightmost differential using left derivatives. In particular, integrating theta one theta two against d theta one d theta two gives minus one. Our spinor contraction is minus twice that ordered monomial, so the normalized two-theta measure includes one half. The dotted contraction has the opposite component sign and its corresponding measure includes an additional minus sign.

These choices give unit integrals of theta-theta and barred-theta barred-theta. The full measure is their product, with the unbarred block written first. A full superspace integral of a real function K selects its highest four-odd-coordinate component. A chiral integral selects the highest unbarred component and must be accompanied by its Hermitian conjugate for a real action. Supersymmetry changes these components by spacetime total derivatives; invariance of the integrated action assumes appropriate falloff, periodicity, or boundary terms compatible with the symmetry.

Multiplying two chiral superfields gives another chiral superfield. Its scalar component is the product of scalars, and its highest component is $\phi_1F_2+F_1\phi_2-\psi_1\psi_2$. The fermion term's minus sign comes from moving an odd coordinate past a fermionic coefficient and contracting the remaining coordinates. Since contracted Weyl bilinears are symmetric under exchanging the two complete spinors, this formula also has the required symmetry under exchanging the two even superfields. A naive product of commuting component polynomials would miss that sign.

In four dimensions theta has mass dimension minus one half, so the chiral and full measures have dimensions one and two. An action density therefore needs a dimension-three superpotential under the chiral measure and a dimension-two Kähler function under the full measure. A product of two canonical chiral fields has dimension two and needs a mass coefficient to serve as a renormalizable superpotential term. Superspace makes these constraints economical, but it does not remove dimensional analysis, gauge invariance, boundary assumptions, or the need to verify which components actually propagate.