Editing Hodge star operator (section)

=== Four dimensions ===
In case <math>n=4</math>, the Hodge star acts as an [[endomorphism]] of the second exterior power (i.e. it maps 2-forms to 2-forms, since {{math|1=4 − 2 = 2}}). If the signature of the [[metric tensor]] is all positive, i.e. on a [[Riemannian manifold]], then the Hodge star is an [[involution (mathematics)|involution]]. If the signature is mixed, i.e., [[Pseudo-Riemannian_manifold|pseudo-Riemannian]], then applying the operator twice will return the argument up to a sign – see ''{{section link|#Duality}}'' below. This particular endomorphism property of 2-forms in four dimensions makes [[Yang–Mills equations#Anti-self-duality equations|self-dual and anti-self-dual two-forms]] natural geometric objects to study. That is, one can describe the space of 2-forms in four dimensions with a basis that "diagonalizes" the Hodge star operator with eigenvalues <math>\pm 1</math> (or <math>\pm i</math>, depending on the signature).

For concreteness, we discuss the Hodge star operator in Minkowski spacetime where <math>n=4</math> with metric signature {{math|(− + + +)}} and coordinates <math>(t,x,y,z)</math>. The [[volume form]] is oriented as <math>\varepsilon_{0123} = 1</math>. For [[one-form]]s,
<math display="block">\begin{align}
  {\star} dt &= -dx \wedge dy \wedge dz \,, \\
  {\star} dx &= -dt \wedge dy \wedge dz \,, \\
  {\star} dy &= -dt \wedge dz \wedge dx \,, \\
  {\star} dz &= -dt \wedge dx \wedge dy \,,
\end{align}</math>
while for [[2-form]]s,
<math display="block">\begin{align}
  {\star} (dt \wedge dx) &= - dy \wedge dz \,, \\
  {\star} (dt \wedge dy) &= - dz \wedge dx \,, \\
  {\star} (dt \wedge dz) &= - dx \wedge dy \,, \\
  {\star} (dx \wedge dy) &=   dt \wedge dz \,, \\
  {\star} (dz \wedge dx) &=   dt \wedge dy \,, \\
  {\star} (dy \wedge dz) &=   dt \wedge dx \,.
\end{align}</math>

These are summarized in the index notation as
<math display="block">\begin{align}
  {\star} (dx^\mu) &= \eta^{\mu\lambda} \varepsilon_{\lambda\nu\rho\sigma} \frac{1}{3!} dx^\nu \wedge dx^\rho 
\wedge dx^\sigma \,,\\
  {\star} (dx^\mu \wedge dx^\nu) &= \eta^{\mu\kappa} \eta^{\nu\lambda} \varepsilon_{\kappa\lambda\rho\sigma} \frac{1}{2!} dx^\rho \wedge dx^\sigma \,.
\end{align}</math>

Hodge dual of three- and four-forms can be easily deduced from the fact that, in the Lorentzian signature, <math>{\star}^2=1</math> for odd-rank forms and <math>{\star}^2=-1</math> for even-rank forms. An easy rule to remember for these Hodge operations is that given a form <math>\alpha</math>, its Hodge dual <math>{\star}\alpha</math> may be obtained by writing the components not involved in <math>\alpha</math> in an order such that <math>\alpha \wedge ({\star} \alpha) = dt \wedge dx \wedge dy \wedge dz </math>.{{verify source|date=September 2019}} An extra minus sign will enter only if <math>\alpha</math> contains <math>dt</math>. (For {{math|(+ − − −)}}, one puts in a minus sign only if <math>\alpha</math> involves an odd number of the space-associated forms <math>dx</math>, <math>dy</math> and <math>dz</math>.)

Note that the combinations
<math display="block"> (dx^\mu \wedge dx^\nu)^{\pm} := \frac{1}{2} \big( dx^\mu \wedge dx^\nu \mp i {\star} (dx^\mu \wedge dx^\nu) \big)</math>
take <math>\pm i</math> as the eigenvalue for Hodge star operator, i.e.,
<math display="block"> {\star} (dx^\mu \wedge dx^\nu)^{\pm} = \pm i (dx^\mu \wedge dx^\nu)^{\pm} , </math>
and hence deserve the name self-dual and anti-self-dual two-forms. Understanding the geometry, or kinematics, of Minkowski spacetime in self-dual and anti-self-dual sectors turns out to be insightful in both [[Low-dimensional topology#Four dimensions|mathematical]] and [[Chirality (physics)#Chirality and helicity|physical]] perspectives, making contacts to the use of the [[Weyl equation#Weyl spinors|two-spinor]] language in modern physics such as [[spinor-helicity formalism]] or [[twistor theory]].