- #1

Longstreet

- 98

- 1

The problem comes when trying to represent the field tensor in terms of the 4-potential. Here is the standard form:

[tex] F^{\mu\nu} = \partial^\mu A^{\nu} - \partial^\nu A^{\mu}[/tex]

[tex]\partial_\mu A^{\mu} = 0[/tex]

Here is the form presented in "Geometric Algebra for Physicists":

[tex] F = E + IB = \nabla A[/tex]

Here it uses the space time vectors [tex]\nabla = \gamma^\mu \partial_\mu , A = \gamma_\nu A^\nu: \mu, \nu= 0,1,2,3[/tex], the space time bi-vectors [tex]E = E^i \gamma_i \gamma_0 ; B = B^i \gamma_i \gamma_0: i = 1,2,3[/tex], and spacetime pseudoscalar [tex] I = \gamma_0 \gamma_1 \gamma_2 \gamma_3[/tex]

When I apply it to the 4-potential I get:

[tex]\nabla A = \gamma^\mu \gamma_\nu \partial_\mu A^\nu = \gamma^0 \gamma_0 \partial_0 A^0 + ... + \gamma^3 \gamma_3 \partial_3 A^3[/tex]

Now, at this point I make use of some identities: [tex]\gamma^0 = \gamma_0; \gamma_0 \gamma_0 = 1; \gamma^i = -\gamma_i; \gamma_i \gamma_i = -1: i = 1,2,3; \gamma_\mu \gamma_\nu = -\gamma_\nu \gamma_\mu[/tex]

When I do this to collect terms I should get a scalar, which represents the divergence of A = 0, and a set of bi-vectors which represents the field. But, instead I get numerous wronge signs.

[tex](\partial_0 A^0 + \partial_1 A^1 + \partial_2 A^2 + \partial_3 A^3) + (\partial_0 A^1 + \partial_1 A^0)\gamma_0 \gamma_1 + (\partial_0 A^2 + \partial_2 A^0)\gamma_0 \gamma_2 + ...[/tex]

The scalar is correct, but the bivectors

__should__be, for example: [tex](\partial_0 A^1 - \partial_1 A^0)\gamma_0 \gamma_1[/tex]

Thank you for anyone that can help point out my misunderstanding.