1

This is a basic qft question. I am looking for the condition on a free scalar $\phi$ of mass $m$ in Euclidean space such that it satisfies the Klein-Gordon equation.

The Euclidean space Klein-Gordon operator is $(-\nabla^2+m^2)\phi(x)=0$. In momentum space, this becomes $(p^2+m^2)\tilde{\phi}(p)=0$, which seems to imply that $p^2=-m^2$ is the condition I'm looking for. However, this result seems nonsensical to me as a Euclidean vector can't have negative norm.

Is there some minus sign subtlety I'm messing up? Or is there a deeper reason to think that Euclidean space scalars can't satisfy the classical equation of motion?

Sam
  • 135
  • 6

1 Answers1

1

Yes, OP is right: In Euclidean (E) signature, the mass-shell condition is $-E_E^2=E^2_M={\bf p}^2+m^2$, and we're solving an elliptic boundary value problem, which is different from the hyperbolic boundary value problem in Minkowski (M) signature.

Qmechanic
  • 220,844