Unitarity condition

The two-loop slope was considered in the early pioneer works [17, 18], and for the first time the correct result was obtained numerically in [19]. This last work triggered a flurry of theoretical activity [20, 21, 22, 23], followed by the first completely analytical calculation in [24]. The same analytical result for the slope of the Dirac form factor was derived in [25] from the total e+e cross section and the unitarity condition. 

A remarkable feature of the set of equations (67) is that its solutions satisfy exactly the unitarity conditions (24)-(26) (although the coefficients q " 1 and rp"1 introduced via equation (28) have additional phase factors in comparison with the coefficients defined in Eq. (20), these phases do not affect the identities concerned), which can be rewritten as... 

Finally, the form of U and U22 is determined by the unitarity condition UU = 1, which is only satisfied if... 

Note that odd powers of W do not occur in this expansion because of the anti-hermiticity of W. With the requirement that different powers of W be linearly independent, we arrive at the following unitarity conditions for the coefficients ... 

Finally, we can take a sum over all states in the band, using the unitarity condition... 

Because of the unitarity conditions all other coefficients would automatically vanish as well, and U would be the identity transformation. The coefficient a-y has therefore necessarily to be chosen to be different from zero, and since defines only a simple scaling of W, we may choose = 1. 

The expansion coefficients have to satisfy the unitarity conditions (11.60)-(11.66). Transformation of the one-electron operator / now yields... 

From this it is easy to see that the unitarity condition is fiilfilled if... 

Analogous equations hold for the creation operators. The operators Cpq B) and drPQ S) are defined trough their Taylor expansions and are limited through the unitarity conditions... 

For the Taylor expansion of both adiabatic and nonadiabatic unitarity conditions see Eqs. 28.101-28.104 in Appendix 1. 

---