pellman
Sep11-10, 10:35 AM
Reading some QFT in which we are faced with inhomogeneous equation
(\partial^\mu \partial_\mu + m^2)\phi(x)=J(x)
The solution is given as
\phi(x)=\phi^{(+)}_{in}(x)+\phi^{(-)}_{out}(x)+i\int{d^4 x\Delta(x-x')J(x')
where \Delta is the appropriate Green's function. "in" means the solution for early times (when J vanishes) and "out" for late times (when J vanishes). The text states "where the superscripts (+) and (-) indicate the positive- and negative-frequency parts, respectively."
I don't understand why the in- and out-solutions are restricted to the positive- and negative-frequency parts.
(\partial^\mu \partial_\mu + m^2)\phi(x)=J(x)
The solution is given as
\phi(x)=\phi^{(+)}_{in}(x)+\phi^{(-)}_{out}(x)+i\int{d^4 x\Delta(x-x')J(x')
where \Delta is the appropriate Green's function. "in" means the solution for early times (when J vanishes) and "out" for late times (when J vanishes). The text states "where the superscripts (+) and (-) indicate the positive- and negative-frequency parts, respectively."
I don't understand why the in- and out-solutions are restricted to the positive- and negative-frequency parts.