Understanding the Discrepancy in Four-Vector Differentiation in QFT

[welcomeblack]

Hi all, I'm working on some QFT and I've run into a stupid problem. I can't figure out why my two methods for evaluating

$i\gamma^\mu \partial_\mu \exp(-i p \cdot x)$

don't agree. I'm using the Minkowski metric $g_{\mu\nu} = diag(+,-,-,-)$ and I'm using $\partial_\mu = (\frac{\partial}{\partial t},\vec{\nabla} )$

Method one (correct):

$i\gamma^\mu \partial_\mu \exp(-i p \cdot x) = i\gamma^\mu \frac{\partial}{\partial x^\mu} \exp(-i p_\mu x^\mu) \\ i\gamma^\mu \partial_\mu \exp(-i p \cdot x) =i\gamma^\mu (-i p_\mu) \exp(-i p_\mu x^\mu) \\ i\gamma^\mu \partial_\mu \exp(-i p \cdot x) =\gamma^\mu p_\mu \exp(-i p_\mu x^\mu) \\ i\gamma^\mu \partial_\mu \exp(-i p \cdot x) =[\gamma^0 E - \gamma^1 p_x - \gamma^2 p_y - \gamma^3 p_z] \exp(-i p_\mu x^\mu)$

Method 2 (incorrect):

$i\gamma^\mu \partial_\mu \exp(-i p \cdot x)=i[\gamma^0 \frac{\partial}{\partial t} - \gamma^1 \frac{\partial}{\partial x} - \gamma^2 \frac{\partial}{\partial y} - \gamma^3 \frac{\partial}{\partial z}] \exp(-iEt+ip_x x + i p_y y + i p_z z) \\ i\gamma^\mu \partial_\mu \exp(-i p \cdot x)=i[\gamma^0 (-iE) - \gamma^1 (ip_x) - \gamma^2 (ip_y) - \gamma^3 (ip_z)] \exp(-iEt+ip_x x + i p_y y + i p_z z) \\ i\gamma^\mu \partial_\mu \exp(-i p \cdot x)=[\gamma^0 E + \gamma^1 p_x + \gamma^2 p_y + \gamma^3 p_z] \exp(-i p_\mu x^\mu)$

What's going on? It feels like I'm going crazy.

 

[Bill_K]

Method 2 (incorrect):
$i\gamma^\mu \partial_\mu \exp(-i p \cdot x)=i[\gamma^0 \frac{\partial}{\partial t} - \gamma^1 \frac{\partial}{\partial x} - \gamma^2 \frac{\partial}{\partial y} - \gamma^3 \frac{\partial}{\partial z}] \exp(-iEt+ip_x x + i p_y y + i p_z z)$

Should be
$i\gamma^\mu \partial_\mu \exp(-i p \cdot x)=i[\gamma^0 \frac{\partial}{\partial t} + \gamma^1 \frac{\partial}{\partial x} + \gamma^2 \frac{\partial}{\partial y} + \gamma^3 \frac{\partial}{\partial z}] \exp(-iEt+ip_x x + i p_y y + i p_z z)$

You only need to insert minus signs when raising or lowering indices, or when the Minkowski metric is explicitly present. In this case, γ^μ is contravariant and ∂/∂x^μ is covariant, so everything's fine, and the sum over μ is just a sum.

 

[welcomeblack]

Ohhh okay I get it. Thanks for your help!