gptejms said:
What I understand of positive and negative frequency solutions is this:
If
f(t)=\int_{-\infty}^{\infty}f(\omega)e^{-\iota \omega t}d\omega,
then if f(t) is real it can be written as
f(t)=\int_{0}^{\infty}f^{*}(\omega)e^{\iota \omega t}d\omega +<br />
\int_{0}^{\infty}f(\omega)e^{-\iota \omega t}d\omega,
where f^{*}(\omega)=f(-\omega)
The first term on RHS may be called the negative frequency term and the second the positive frequency term.Now in your posts you don't seem to be doing this.Your range of integration seems to be from -infinity to infinity always--there would be a problem if you were to show your results with the above definition.I don't know what limits the (authors of) books have in mind when they talk of +ve and -ve freq. terms(but I guess they take your definition).
The plane wave solutions of Klein-Gordon equations are
e^{-i(E_p t-p\cdot x)} (positive frequency)
and
e^{i(E_p t + p\cdot x)} (negative frequency)
(Actually I'm not sure about the sign convention for the term p\cdot x here. Perhaps there is a minus sign. It isn't important anyway, it leads only into a redefinition of \phi^-_p)
An arbitrary wave packet may be written as superposition of these like this
<br />
\phi(t,x) = \int\frac{d^3p}{(2\pi)^3} \Big( \phi^+_p e^{-iE_p t} + \phi^-_p e^{iE_p t}\Big) e^{ip\cdot x}<br />
Where the p is integrated over the space \mathbb{R}^3, and \phi^+_p and \phi^-_p are functions of p (I prefer this notation for Fourier transformations). If we are only interested in the positive frequency solutions, then the wave packet is of form
<br />
\phi(t,x) = \int\frac{d^3p}{(2\pi)^3} \phi^+_p e^{-i(E_p t - p\cdot x)}<br />
Notice that E_p=\sqrt{|p|^2+m^2} is getting only positive values, when p is integrated over the three dimensional real space. At time t=0 this becomes a Fourier transform of the \phi^+_p, so for a desired initial wave packet \phi(0,x) we can substitute an inverse transform
<br />
\phi^+_p = \int d^3x'\; \phi(0,x') e^{-ip\cdot x'}<br />
and then we get to the solution I wrote earlier.
Trying to write wave packets by integrating over the energy (or frequency) doesn't look very practical, because energy doesn't define the momentum uniquely.
