By Y. Takahashi
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26) form a complete set of functions. *(t;k9j)A%\dl = 0, - * ) = 0. 28) are linear, the normalization of the wave functions is still arbitrary. For physical reasons, it is convenient to have the normalization condition independent of time. 29) is conserved. 30) k9 j) = - 1 . 31) Under the assumption that all the frequencies coj(k) are distinct we can prove that the wave functions belonging to different modes 38 3] Non-relativistic Fields are orthogonal. Thus, we have the orthogonality conditions taT(l; K j)r%'Xdt-dt)u(::Xt; ft, / ) = djj.
Otherwise, the energy and momentum do not form a four-vector. See the argument in Chapter VI, § 2. 25 An Introduction to Field Quantization [Ch. II independent of t. 39) with .
9 Ä ) - dtJ&Xx) · I#(JC)} = 0. 26) Here 9, denotes the time derivative, u — u" and v = Ό*. 13). The meaning and the motivation of these conditions will become clear in a later section where they will be discussed from a more general standpoint. 27) vï\x) = (2Vœ(k)yKir\k) e-ikxeioKk)t. 35) The statement can be verified in a straightforward manner. 34) is due to the so-called zero-point energy. The observed energy is the deviation from it. 32) allows an arbitrary c-number term to be added to H, we have chosen this term so that H can later be expressed in terms of A(x).
An Introduction to Field Quantization by Y. Takahashi