Li and the wave equation gives
Note that there are both positive and negative frequency solutions. We will adopt the convention that un is always positive, and use -w„ for negative frequency solutions. The states with positive frequency are written
The negative frequency states have a time factor eluJnt, and since kn is both positive and negative, it is convenient to denote the negative frequency states by 0*(z,i). The normalization condition which these states satisfy is i:
However, by direct evaluation it is also true that rL
ro the states are not orthogonal in the usual sense. The most general real field can be expanded in normal modes as follows:
where an(0) are the coefficients of each normal mode in the expansion (1.10) and the real normalization factor c„ will be chosen later. It will sometimes be convenient to incorporate the time dependence of each normal mode into a generalized
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