/sci/ - Science & Math

>>11421379
All metrics on finite dimensional normed linear spaces are equivalent.
>>11421716
Subtract the two equations you get from permuting the indices. [eqn]0=\gamma_\mu\gamma_\nu\partial_\mu\partial_\nu \phi - \gamma_\nu \gamma_\mu\partial_\nu\partial_\mu\phi = (\gamma_\mu\gamma_\nu - \gamma_\nu\gamma_\mu)\partial_\mu\partial_\nu \phi. \qquad \ast[/eqn]
which states that the antisymmetric part vanishes. Generally, tensors satisfy the graded commutation relation [math]A\otimes B = (-1)^{ab}B\otimes A[/math] where [math]a,b[/math] are the degrees of [math]A,B[/math], respectively. Now since [math]\{\gamma_\mu,\gamma_\nu\} = 2\delta_{\mu\nu}[/math], [math]\gamma_\mu\gamma_\nu[/math] is antisymmetric while [math]\partial_\mu\partial_\nu[/math] is symmetric, so [math]\gamma_\mu\gamma_\nu\partial_\mu\partial_\nu[/math] is antisymmetric. Klein-Gordon then implies (*) on the spinors [math]\phi[/math].
>>11421871
Consider [math]z = y'[/math].