I'm having difficulty deciphering my notes which 'proove' that the commutor of two real free fields φ(x) and φ(y) (lets call it i∆) ie. i∆=[φ(x),φ(y)] are Lorentz invariant under an orthocronous Lorentz transformation. Not sure if it helps but φ(x)=∫d(adsbygoogle = window.adsbygoogle || []).push({}); ^{3}k[α(k)e^{-ikx}+α^{+}(k)e^{ikx}].

Now, apparently I have to 'observe' that:

i∆=∫d^{4}k.δ(k^{2}-m^{2})ε(k^{0})e^{-ikx}

where ε(k^{0}) = 1 if k^{0}>0 and -1 if k^{0}<0.

Firstly, I can't see at all, how this expression comes about: why there is a delta function δ(k^{2}-m^{2}) (this looks like the mass shell condition, but why is it relavent here?)

The "ε(k^{0}) keeps positive k^{0}positive and negative k^{0}negative". I guess this makes sense because an orthocronous LT maps future directed vectors to future directed vector and past to past.

But I still can't see why that expression proves that ∆(Lx)=∆(x).

Any thought?

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# Showing that commutator is invariant under orthchronous LTs

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