# Metric convention conversions between expressions

I'm sure this has caused many people headaches.

First, is there a metric $(-+++)\leftrightarrow(+---)$ convention conversion chart where many common expressions are listed? Thank you in advance for sharing.

Second, as an example, I read that in $(+---)$ we have the fermion propagator $$\frac{i}{\gamma^ap_a-m},$$ and in $(-+++)$ we have $$\frac{-i(-\gamma^ap_a+m)}{p^2+m^2}.$$

Ok. How to go from the former to the latter? $$\frac{i}{\gamma^ap_a-m}=\frac{i(\gamma^bp_b+m)}{p^ap_a-m^2}=\frac{i(\gamma^bp_b+m)}{p^2-m^2}=(\text{change to }-+++)=\frac{-i(\gamma^bp_b+m)}{p^2+m^2}.$$

What did I do wrong? Upper-lower index expressions like $\gamma^ap_a$ shouldn't be affected by the metric convention, right?

And if I add in loop corrections as follows, in $(+---)$, $$\frac{i}{\gamma^ap_a-m_{Ph}-\Sigma(\gamma^ap_a)},$$ then what should I write for the $(-+++)$ case?

Finally, I like $(-+++)$ since I won't need to change signs when going from Euclidean to Minkowski, what are the benefits for $(+---)$? (I suppose the fermion propagator above does look easier in this convention...)

• All contractions change sign if you change the metric. – Count Iblis Aug 2 '14 at 2:47
• Just to be more explicit about Count's comment, any contraction is $g_{ab} p^a \gamma^b$, for example. So if the upper-index $p,\gamma$ components are the same in both conventions, which is often the case, there is still the explicit $g$ which is what changed the sign, so the product changes the sign, to! So the most usual rule is that all upper-index vectors and tensors are kept the same and one puts an extra minus sign for every lower index, visible or hidden, in the expressions. Of course, it's a matter of another convention to decide whether there is a hidden lower index in a scalar etc. – Luboš Motl Aug 2 '14 at 4:10