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59) As a result of gradient invariance the derivative ΘΛ^~)/δχμ can be omitted. e. that part of the operator of the electromagnetic field which provides the creation and annihilation of quanta with transverse polarization. We turn now to the construction of a set of vectors describing the state of a quantized field. We shall act on the vector Φ0 with the operators a^{k). 45) it follows that the operators a£(k) and a±(k) can appear here only in the combination a^(k)—a^(k). 45), with the operator az(k')—cu(k').
Flj(*i) Φ 0 . 65) Here the indices λΐ9 . . , Xm can assume the values 1 and 2. 65) describes the state with n(km, Xm) quanta with four-momentumfcOT and polarization eÀm(km)9n(km£1,Xm_1) quanta with four-momentum km_x and polarization ^m_1(fcw_i), and so forth. IFD5 57 CHAPTER 4 EXPANSION OF CHRONOLOGICAL PRODUCTS IN NORMAL PRODUCTS 8. The Interaction Hamiltonians. Normal and Chronological Products of the Field Operators Our next task is to calculate the matrix elements for the transition between the states which describe free particles with definite momenta.
23) Let us now consider what relationships between the ak(k) result from the Lorentz condition. 26) az(k) = a±{k). 26). 15) we find H= j at{k) axik) (e\k) e*(k)) k0dk = j £ at(k) a,(k)Vxk0dk. 26) and finally we obtain the following expression for the energy of the free electromagnetic field: H= $ Σ *Kk) ax(k) ko dk. 28) A = 1,2 Obviously H can assume only positive values. Note that the energy of the field becomes a positive definite quantity only after the additional Lorentz condition is applied.