I am reading Path Integrals and Quantum Anomalies by Kazuo Fujikawa and Hiroshi Suzuki. In chapter 4.2 they calculate the self-energy of photon for QED and say that the actual calculation is performed in Euclidean theory. The recipe they gave to change from Minkowski to Euclidean theory is
time: x0→−ix4contravariant vector component: V0→−iV4covariant vector component: A0→iA4metric: (+1,−1,−1,−1)→(−1,−1,−1,−1)gamma matrix: γ0=−iγ4.
Note that different from the first four transformation which I use "
→", in the last one I use "
=", since we really need the numerical value of
γ4, given the knowledge of
γ0.
I understand that this transformation allows us to compute the integral in Minkowski space in a convergent manner and still obtain *the same answer* since the mathematical meaning of changing from Minkowski to Euclidean theory is merely a *change of integration axis that preserves the answer, provided that we do not hit singularities when we change the integration axis* (correct me if I am wrong though).
I am totally fine with the first four transformations I list above. However, I have difficulties to understand the last one: γ0→−iγ4. Since γ0 by itself is really just a numerical matrix, it is hard to imagine why it should change in this way.
The following are some of my thoughts. In fact, it is not the γ0 that change, it is the
ˉψγμAμψ
that we want to preserve upon the change from Minkowski to Euclidean theory, where
Aμ is some covariant vector component. We are *assuming* that in the Lagrangian, whenever
γμ appears, it will always be contracted with some covariant vector component
Aμ, in order to make the Lagrangian a scalar. And therefore, we may roughly regard
ˉψγμψ
as some contravariant vector component
Vμ (actually indeed
ˉψγμψ transforms as a vector field) and then by requiring
VμAμ is preserved we certainly need to change
ˉψγ0ψ→−iˉψγ4ψ
which, together with the change of metric and
A0→iA4, will preserve the
ˉψγμAμψ before and after the transformation.
In short, we should properly say that
ˉψγ0ψ→−iˉψγ4ψ
which in literature they will just simply write
γ0=−iγ4.
Is my understanding correct? If not is there any suggested reference? It seems like in this book the authors didn't elaborate more on this. Thanks!