The Euclidean action 
of a physical system is the action integral obtained
by the Wick rotation 
from real Lorentzian time 
to imaginary time 
. In Euclidean
quantum field theory, a scalar field with potential-energy
density 
has the formal action
![S_E[phi]=int[1/2(del phi)^2+V(phi)]d^dx.](/images/equations/EuclideanAction/NumberedEquation1.svg) |
The Euclidean functional integral weight is 
, rather than the oscillatory
real-time weight 
.
This change is one reason Euclidean formulations connect quantum
field theory with probability and statistical
mechanics.
