Calculating trace with slashed item

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Homework Statement

I was asked to find Tr [STRIKE]q[/STRIKE] ([STRIKE]p[/STRIKE] + m) [STRIKE]q[/STRIKE] ([STRIKE]p[/STRIKE] + m)

Homework Equations

Tr [STRIKE]p[/STRIKE] [STRIKE]q[/STRIKE] = 4pq

The Attempt at a Solution

If I expand it as Tr ([STRIKE]p[/STRIKE] [STRIKE]q[/STRIKE] [STRIKE]p[/STRIKE] [STRIKE]q[/STRIKE] + m [STRIKE]q[/STRIKE] [STRIKE]p[/STRIKE] [STRIKE]q[/STRIKE] + m [STRIKE]q[/STRIKE] [STRIKE]q[/STRIKE] [STRIKE]p[/STRIKE] + (m^2)([STRIKE]q[/STRIKE])^2 ), although Tr Π(odd number of gamma matrices) = 0, since [STRIKE]q[/STRIKE] [STRIKE]p[/STRIKE] [STRIKE]q[/STRIKE] and similar terms are not square matrices, trace has no meaning. If I treat ([STRIKE]p[/STRIKE] + m) as a covariant vector (adding m to each component of [STRIKE]p[/STRIKE], I can get a scalar*I, but a lot of [STRIKE]q[/STRIKE]_0,1,2,3 [STRIKE]p[/STRIKE]_0,1,2,3 terms will come out and get really messy. Can anyone help me please? Thank you.

 

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I thought [STRIKE]p[/STRIKE] was a vector, but now I realize that it is not. So the trace makes sense.