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constantinou1
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Homework Statement
Consider the following scalar theory formulated in two-dimensional Euclidean space-time;
S=∫d2x ½(∂μφ∂μφ + m2φ2) ,
a) Determine the equations of motion for the field φ.
b) Compute the propagator;
G(x,y) = ∫d2k/(2π)2 eik(x-y)G(k).
Homework Equations
Euler-Lagrange equations for the field φ is;
∂L/∂φ - ∂μ(∂L/∂(∂μφ) = 0 .
2. The attempt at a solution
I can solve (a) fine, but its the integral that confuses me.
1.(a)
Well Euclidean space-time has the particular metric gμν = diag{1,1,1,1}, and the integrand in equation S yields the Lagrangian density L, whereby;
L=½(∂μφ∂μφ + m2φ2) .
By then using the Euler-Lagrange equations for the field φ, we find;∂μ∂μφ - m2φ = 0,
∴ (∂μ∂μ - m2) ⋅ φ = 0 .
∴ (∂μ∂μ - m2) ⋅ φ = 0 .
2.(a)
Since we are dealing with Euclidean space-time, then;
pμpμ = |p0|2 + |pi|2 = m2 + |p|2
The general idea is then to use G(k) = 1/(m2 + |k|2), so that;
G(x,y) = ∫d2k/(2π)2 eik(x-y)/(m2 + |k|2),
and then compute the integral. But how can I possibly solve this integral?Also, when its written d2k for a Fourier transform, it means dkx.dky right, not integrate twice over dk.dk.
Any pointers or assistance on how to solve this problem is greatly appreciated.