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In this wikipedia article https://en.wikipedia.org/wiki/Electromagnetic_tensor

we have the following invariant :

F^{αβ}F^{μη}ε_{αβμη}= 8E*B

However the determinant is the square of this quantity divided by 8, i.e. (E*B)^{2}.

Now from the definition of the determinant for a 4x4 matrix , we have :

M_{ia}M_{jb}M_{kc}M_{id}ε^{ijkl}= ε_{abcd}det(M) [D]

Now If I raise the expression 1/8 F^{αβ}F^{μη}ε_{αβμη}to the power of 2, I would get :

1/8 F^{αβ}F^{μη}ε_{αβμη}1/8 F^{θλ}F^{σω}ε_{θλσω}[E]

Now If I compare this expression with equation D above, I see that some indices do not fall into the right place and also in Equation D, we have the expression for the determinant , however in expression E, we see that we have the matrix F multiplied 4 times much like in expression D (or is it rows or columns that get multiplied).

We should also bear in mind that the magnetic field is the spatial part of F and thhat the electric field is the time part :

F^{i0}= E and F^{ij}ε_{ijk}= B_{k}

How can we reconcile expression E with D then ? or is this an error perhaps ?

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# Determinant of the electromagnetic matrix

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