Nonlocal Models of Goldstone Bosons ...
Nonlocal Models of Goldstone Bosons in Asymptotically Free Gauge Theories

Appendix A

Euclidean Space-Time Conventions

We employ standard conventions for Wick rotating to Euclidean space- time from Minkoswki space-time [86]. The analytic continuation involved is summarized in table A.1 [86b]; objects in the first column are replaced by those in the second.

Table 6.2

Given a propagator in Minkowski space-time:

G(x) = int d^4p/(2 pi)^4 exp(-ip.x) i/(p^2 - m^2 + i epsilon) (A.1)

we can find the Euclidean propagator using Table A.1:

G_E(x) = int d^4p_E/(2 pi)^4 exp(-i(p_E^4 x_E^4 - p_E.x_E)) 1/(p_E^2 + m^2) , (A.2)

note that the exponential does not contain a Euclidean inner product, but the original Minkowski inner product. However, this is correct; for example, it gives the correct Euclidean derivative*:

partial_E^mu exp (-i(p_E^4 x_E^4 - p_E.x_E)) = -i p_E^mu exp (-i(p_E^4 x_E^4 - p_E.x_E) = -i p_E^mu exp (-ip.x ) (A.3)

The exponential of a line integral of a gauge field also involves the Minkowski inner product, since the gauge field is analytically continued in the same fashion as a momentum.

Note also that starting with the standard Minkowski definition of the Fourier transform of a field

phi(x) = int d^4p/(2 pi)^4 e-(ip.x) phi(p), (A.4)

we find for the Euclidean field:

phi_E(x_E) = i int d^4p_E/(2 pi)^4 exp(-ip.x) phi_E(p_E). (A.5)

Finally we note that our dynamical quark mass is defined in Euclidean space-time such that for a standard local mass term Sigma(p) reduces to Sigma(p) = m, and Sigma(x-y) is the usual Fourier transform as given by equations A.2 and A.4.


* The Euclidean derivative partial_E^mu is defined in terms of the contravariant Minkowski derivative partial^mu = (partial^0, -grad^k), so partial_E^mu = (-grad^k, partial^4).