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By M. Srednicki
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It is now convenient to choose f (k) so that d3k/f (k) is Lorentz invariant. An integration measure that is manifestly invariant under orthochronous Lorentz transformations is d4k δ(k2 +m2 ) θ(k0 ), where δ(x) is the Dirac delta function, θ(x) is the unit step function, and k0 is treated as an independent integration variable. We then have +∞ dk0 δ(k2 +m2 ) θ(k0 ) = −∞ 1 . 17) where g(x) is any smooth function of x with simple zeros at x = xi ; in our case, the only zero is at k0 = ω. Thus we see that if we take f (k) ∝ ω, then d3k/f (k) will be Lorentz invariant.
8) in addition to h ¯ = 1. As with h ¯ , factors of c can restored, if desired, by dimensional analysis. We will adopt eq. 7) as the equation of motion that we would like ϕ(x) to obey. It should be emphasized at this point that we are doing classical physics of a real scalar field. We are not to think of ϕ(x) as a quantum wave function. Thus, there should not be any factors of h ¯ in this version of the Klein-Gordon equation. This means that the parameter m must have dimensions of inverse length; m is not (yet) to be thought of as a mass.
0 in an interacting quantum field theory. 56 5: The LSZ Reduction Formula Reference Notes Useful discussions of the LSZ reduction formula can be found in Brown, Itzykson & Zuber, Peskin & Schroeder, and Weinberg I. 5. Note that we must specify the type (a or b) of each incoming and outgoing particle. 1) where P and Q are operators obeying [Q, P ] = i. ) We wish to evaluate the probability amplitude for the particle to start at position q ′ at time t′ , and end at position q ′′ at time ′′ ′ t′′ .