I am working through a problem on the completeness relations for Dirac spinors and need help proving the following identity:
Given the Dirac spinors $ u^{(s)}(p) $ and $v^{(s)}(p)$, which satisfy the Dirac equations:
$$ (\not{p} - m) u^{(r)}(p) = 0, \quad (\not{p} + m) v^{(r)}(p) = 0, $$
and the orthogonality conditions:
$$ u^{(r) \dagger} u^{(s)} = 2E \delta_{rs}, \quad v^{(r) \dagger} v^{(s)} = 2E \delta_{rs}, $$
where $ r, s = 1,2 $, I want to show that:
$$ \sum_{s=1,2} u^{(s)}(p) \bar{u}^{(s)}(p) = \not{p} + m, \quad \sum_{s=1,2} v^{(s)}(p) \bar{v}^{(s)}(p) = \not{p} - m, $$
where $ \not{p} = \gamma^\mu p_\mu $.
How can I go about proving these relations without writing the spinors in their explicit form, simply using the conditions I've given?
