By Bozhidar Z. Iliev

The monograph is dedicated to in-depth research of the Langrangian process in momentum photo of movement to quantum unfastened scalar, spinor and vector fields. the most goal of the monograph is to indicate a brand new procedure and research of recognized of difficulties which ends up in reestablishment of identified proof and new effects. just about all concerns might be performed within the momentum photo of movement rather than the traditional illustration in Heisenberg photograph of movement. All calculations are given in complete information and the hypotheses underlying the idea are explicitly formulated on the issues after they are logically required.

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**Example text**

More details concerning the definition of a vacuum and construction of state vectors from it will be given in sections 7 and 8. 19)). 1, we have the following connection between the creation/annihilation operators in Heisenberg and momentum picture: ϕ˜ ± (k) = e± i 1 μ x0 kμ ± ϕ0 (k) k0 = m2 c2 + k 2 . 1. 10). (Hint: apply the Fourier decomposition from Sect. ) 5. The Klein-Gordon Equations in Terms of Creation and Annihilation Operators Let us return now to our main problem: to be found, if possible, the explicit form of the field operator ϕ0 (and momentum operator Pμ ).

Iliev which can, equivalently, be rewritten as ϕ0 = d4 kδ(k2 − m2 c2 ) f (k)ϕ0 (k). 8) f ± are complex-valued functions (resp. 3) (resp. 3)), and f is a complex-valued function (resp. 3)(resp. 3)). 3). 1. 8) for which k2 := kμ kμ = k02 − k 2 = m2 c2 . 9) holds. 2. s. 3). 10) where ϕ˜ 0 (x, k) := U −1 (x, x0 ) ◦ ϕ0 (k) ◦ U (x, x0 ). 6), this solution is described by m = 0 and, for example, f± (k) = ( 12 ± a)δ3 (k), a ∈ C or f (k) such that f (k)|k0=±|k| = (1 ± 2a)|k|δ3 (k), a ∈ C. 3. 5)), we have f ± (k)ϕ0 (k) k0 =± √ m2 c2 +k2 † = − f∓(−k)ϕ0 (−k) k0 =∓ √ m2 c2 +k2 .

9) is completely correct for the reasons given in loc. cit. 11b). The reader is referred for more detail on that item to [3], 3 The discussed non-uniqueness of the energy-momentum operator, as well as of other conserved currents, if any, is due to the accepted rules for differentiation with respect to non-commuting variables. A rigorous treatment of this kind of differentiation restores the uniqueness of the conserved currents – see [3] and the next paragraph. 28 Bozhidar Z. 1 in it. The only reason why we use a non-rigorous formalism is our intention to stay closer to the standard books on Lagrangian quantum field theory.