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I am trying to study Quantum Field Theory by myself from Mandl and Shaw second edition and I am having trouble understanding the section on covariant commutation relations. I understand the idea that field at equal times at two different points commute because they cannot "communicate" with each other. This is where the equal time commutation relation comes in. However, when one takes field at two different "space-time" points, things might be different. I am having trouble following the Mathematics in Mandl and Shaw. I can write the delta function as (3.43),

Δ (x) = (-c/(2π)

^{3}) ∫(dk

^{3}sin kx)/w

_{k}

I can also see how the equal time commutation relation comes from this. But how do they go from this to the four dimensional integral in k (3.45),

Δ (x) =(-i/(2π)

^{3})∫(dk

^{4}δ(k

^{2}- μ

^{2})ε(k

_{0})e

^{-ikx}

Also, how can I see that the four dimensional integral is Lorentz invariant and how does Lorentz invariance implies that integral is zero for any space-like interval.

Any discussion will be much appreciated.