Van der Waerden equation

This is the classical relativistic expression for the interaction of an electron or proton with the classical electromagnetic held. The quantized version of Eq. (275) is the van der Waerden equation [1] as described by Sakurai [68] in his Eq. (3.24). The RFR term in relativistic classical physics is contained within the term e2fit1 1 fit2 1, a result that can be demonstrated by expanding this term as follows... 

On the relativistic quantum level, the Einstein equation becomes the van der Waerden equation [ 1,68] with the usual operator rules... 

In our assumptions, system (27) has the finite number of roots (by Lemma 14.2 in Bykov et al., 1998), so that the product in Equation (26) is well defined. We can interpret formula (26) as a corollary of Poisson formula for the classic resultant of homogeneous system of forms (i.e. the Macaulay (or Classic) resultant, see Gel fand et al., 1994). Moreover, the product Res(R) in Equation (26) is a polynomial of R-variable and it is a rational function of kinetic parameters fg and Tg (see a book by Bykov et al., 1998, Chapter 14). It is the same as the classic resultant (which is an irreducible polynomial (Macaulay, 1916 van der Waerden, 1971) up to constant in R multiplier. In many cases, finding resultant allows to solve the system (21) for all variables. ... 

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