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## Main Question or Discussion Point

I'm having alot of problems with tensors. Here is what the professor in class told us in the lecture notes

In three spacetime dimensions (two space plus one time) an antisymmetric Lorentz tensor

F[itex]^{\mu\nu}[/itex] = -F[itex]^{\nu\mu}[/itex] is equivalent to an axial Lorentz vector, F[itex]^{\mu\nu}[/itex] = e[itex]^{\mu\nu\lambda}[/itex]F[itex]_{\lambda}[/itex]. Consequently, in 3D

one can have a massive photon despite unbroken gauge invariance of the electromagnetic

field A[itex]_{\mu}[/itex]. Indeed, consider the following Lagrangian:

L = -(1/2)*F[itex]_{\lambda}[/itex]F[itex]^{\lambda}[/itex] + (m/2)*F[itex]_{\lambda}[/itex]A[itex]^{\lambda}[/itex] (6)

where

F[itex]_{\lambda}[/itex](x) = (1/2)*[itex]\epsilon[/itex][itex]_{\lambda\mu\nu}[/itex]F[itex]^{\mu\nu}[/itex] = [itex]\epsilon[/itex][itex]_{\lambda\mu\nu}[/itex][itex]\partial[/itex][itex]^{\mu}[/itex]A[itex]^{\nu}[/itex],

or in components, F[itex]_{0}[/itex] = -B, F1 = +E[itex]^{2}[/itex], F[itex]_{2}[/itex] = -E[itex]^{1}[/itex].

In 2+1 dimension, [itex]\epsilon[/itex][itex]^{\alpha\beta\gamma}[/itex][itex]\epsilon[/itex][itex]_{\alpha}[/itex][itex]^{\mu\nu}[/itex] = g[itex]^{\alpha\mu}[/itex]g[itex]^{\beta\nu}[/itex] - g[itex]^{\alpha\nu}[/itex]g[itex]^{\beta\mu}[/itex]

That last part above may be a typo because I've never seen an epsilon without all of its indices either upstairs or downstairs

I'm having trouble with two things

1. Using the last part above, does that mean that

[itex]\epsilon[/itex][itex]_{\lambda\mu\nu}[/itex][itex]\epsilon[/itex][itex]^{\lambda\mu\nu}[/itex] = g[itex]^{\lambda\mu}[/itex]g[itex]^{\mu\nu}[/itex] - g[itex]^{\lambda\nu}[/itex]g[itex]^{\mu\mu}[/itex] = g[itex]^{\lambda\mu}[/itex]g[itex]^{\mu\nu}[/itex] - g[itex]^{\lambda\nu}[/itex]?

If so, would that give [itex]\epsilon[/itex][itex]_{\lambda\mu\nu}[/itex][itex]\epsilon[/itex][itex]^{\lambda\mu\nu}[/itex][itex]\partial[/itex][itex]^{\mu}[/itex]A[itex]^{\nu}[/itex][itex]\partial[/itex][itex]_{\mu}[/itex]A[itex]_{\nu}[/itex] = [itex]\partial[/itex][itex]^{\mu}[/itex]A[itex]^{\nu}[/itex][itex]\partial[/itex][itex]^{\nu}[/itex]A[itex]^{\mu}[/itex] - [itex]\partial[/itex][itex]^{\mu}[/itex]A[itex]^{\nu}[/itex][itex]\partial[/itex][itex]_{\mu}[/itex]A[itex]^{\lambda}[/itex] ?

2. But when I tried to write out the Lagrangian, I got

L = -(1/2)[itex]\epsilon[/itex][itex]_{\lambda\mu\nu}[/itex][itex]\partial[/itex][itex]^{\mu}[/itex]A[itex]^{\nu}[/itex][itex]\partial[/itex][itex]_{\mu}[/itex]A[itex]_{\nu}[/itex] + (m/2)[itex]\epsilon[/itex][itex]_{\lambda\mu\nu}[/itex][itex]\partial[/itex][itex]^{\mu}[/itex]A[itex]^{\nu}[/itex]A[itex]^{\lambda}[/itex]

so

[itex]\frac{\partial L}{\partial A^{\lambda}}[/itex] = (m/2)[itex]\epsilon[/itex][itex]_{\lambda\mu\nu}[/itex] [itex]\partial[/itex][itex]^{\mu}[/itex]A[itex]^{\nu}[/itex]

and

[itex]\frac{\partial L}{\partial (\partial_{\mu}A_{\nu}) }[/itex] = -[itex]\epsilon[/itex][itex]_{\lambda\mu\nu}[/itex][itex]\epsilon[/itex][itex]^{\lambda\mu\nu}[/itex][itex]\partial[/itex][itex]^{\mu}[/itex]A[itex]^{\nu}[/itex] + (m/2)[itex]\epsilon[/itex][itex]_{\lambda\mu\nu}[/itex]g[itex]^{\mu\nu}[/itex]A[itex]^{\lambda}[/itex]

In three spacetime dimensions (two space plus one time) an antisymmetric Lorentz tensor

F[itex]^{\mu\nu}[/itex] = -F[itex]^{\nu\mu}[/itex] is equivalent to an axial Lorentz vector, F[itex]^{\mu\nu}[/itex] = e[itex]^{\mu\nu\lambda}[/itex]F[itex]_{\lambda}[/itex]. Consequently, in 3D

one can have a massive photon despite unbroken gauge invariance of the electromagnetic

field A[itex]_{\mu}[/itex]. Indeed, consider the following Lagrangian:

L = -(1/2)*F[itex]_{\lambda}[/itex]F[itex]^{\lambda}[/itex] + (m/2)*F[itex]_{\lambda}[/itex]A[itex]^{\lambda}[/itex] (6)

where

F[itex]_{\lambda}[/itex](x) = (1/2)*[itex]\epsilon[/itex][itex]_{\lambda\mu\nu}[/itex]F[itex]^{\mu\nu}[/itex] = [itex]\epsilon[/itex][itex]_{\lambda\mu\nu}[/itex][itex]\partial[/itex][itex]^{\mu}[/itex]A[itex]^{\nu}[/itex],

or in components, F[itex]_{0}[/itex] = -B, F1 = +E[itex]^{2}[/itex], F[itex]_{2}[/itex] = -E[itex]^{1}[/itex].

In 2+1 dimension, [itex]\epsilon[/itex][itex]^{\alpha\beta\gamma}[/itex][itex]\epsilon[/itex][itex]_{\alpha}[/itex][itex]^{\mu\nu}[/itex] = g[itex]^{\alpha\mu}[/itex]g[itex]^{\beta\nu}[/itex] - g[itex]^{\alpha\nu}[/itex]g[itex]^{\beta\mu}[/itex]

That last part above may be a typo because I've never seen an epsilon without all of its indices either upstairs or downstairs

I'm having trouble with two things

1. Using the last part above, does that mean that

[itex]\epsilon[/itex][itex]_{\lambda\mu\nu}[/itex][itex]\epsilon[/itex][itex]^{\lambda\mu\nu}[/itex] = g[itex]^{\lambda\mu}[/itex]g[itex]^{\mu\nu}[/itex] - g[itex]^{\lambda\nu}[/itex]g[itex]^{\mu\mu}[/itex] = g[itex]^{\lambda\mu}[/itex]g[itex]^{\mu\nu}[/itex] - g[itex]^{\lambda\nu}[/itex]?

If so, would that give [itex]\epsilon[/itex][itex]_{\lambda\mu\nu}[/itex][itex]\epsilon[/itex][itex]^{\lambda\mu\nu}[/itex][itex]\partial[/itex][itex]^{\mu}[/itex]A[itex]^{\nu}[/itex][itex]\partial[/itex][itex]_{\mu}[/itex]A[itex]_{\nu}[/itex] = [itex]\partial[/itex][itex]^{\mu}[/itex]A[itex]^{\nu}[/itex][itex]\partial[/itex][itex]^{\nu}[/itex]A[itex]^{\mu}[/itex] - [itex]\partial[/itex][itex]^{\mu}[/itex]A[itex]^{\nu}[/itex][itex]\partial[/itex][itex]_{\mu}[/itex]A[itex]^{\lambda}[/itex] ?

2. But when I tried to write out the Lagrangian, I got

L = -(1/2)[itex]\epsilon[/itex][itex]_{\lambda\mu\nu}[/itex][itex]\partial[/itex][itex]^{\mu}[/itex]A[itex]^{\nu}[/itex][itex]\partial[/itex][itex]_{\mu}[/itex]A[itex]_{\nu}[/itex] + (m/2)[itex]\epsilon[/itex][itex]_{\lambda\mu\nu}[/itex][itex]\partial[/itex][itex]^{\mu}[/itex]A[itex]^{\nu}[/itex]A[itex]^{\lambda}[/itex]

so

[itex]\frac{\partial L}{\partial A^{\lambda}}[/itex] = (m/2)[itex]\epsilon[/itex][itex]_{\lambda\mu\nu}[/itex] [itex]\partial[/itex][itex]^{\mu}[/itex]A[itex]^{\nu}[/itex]

and

[itex]\frac{\partial L}{\partial (\partial_{\mu}A_{\nu}) }[/itex] = -[itex]\epsilon[/itex][itex]_{\lambda\mu\nu}[/itex][itex]\epsilon[/itex][itex]^{\lambda\mu\nu}[/itex][itex]\partial[/itex][itex]^{\mu}[/itex]A[itex]^{\nu}[/itex] + (m/2)[itex]\epsilon[/itex][itex]_{\lambda\mu\nu}[/itex]g[itex]^{\mu\nu}[/itex]A[itex]^{\lambda}[/itex]

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