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By Hitchcock F.L.

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68) equal to zero. Let us consider again, as in the bosonic case, the basic example of a spinor field interacting with the external sources η(x), η(x). ¯ Note that for fermionic systems, the external sources are also chosen to be Grassmann variables. The Hamiltonian of this system reads as ¯ d 3 iψ(r) H [b∗, b; c∗ , c] = 3 ¯ ¯ γ j ∂ j ψ(r) + m ψ(r)ψ(r) + ψ(r)η(x) + η(x)ψ(r) ¯ j =1 2 = d 3 k [ωk (bi∗ (k)bi (k) + ci∗ (k)ci (k)) i=1 + ξi∗ (t, k)bi (k) + bi∗ (k)ξi (t, k) + ζi∗ (t, k)ci (k) + ci∗ (k)ζ(t, k)].

Itzykson and Zuber (1980)). Note that the functional W [ J ] is insensitive to normalization factors multiplying the functional [ J ] (an additive constant is absolutely inessential for generating functionals). 4, page 40). ♦ Variational equations for Green functions from path integrals The so-called Dyson–Schwinger equations (Dyson 1949, Schwinger 1951) are exact relations between different Green functions. All these relations can be presented as one equation with variational derivatives for the generating functional [ J ].

Expanding this functional as a power series in ϕ0 : ∞ S[ϕ0 ] = n=0 1 n! d 4 x 1 d 4 x 2 · · · d 4 x n Sn (x 1 , x 2 , . . 112) we obtain the so-called coefficient functions Sn (x 1 , x 2 , . . , x n ) of the S-matrix. In the operator approach, these appear in the process of expanding the S-matrix in a series over normal products of free fields. Convolution of these coefficient functions with the initial ψ1 (x 1 ), ψ2 (x 2 ), . . , ψl (xl ) and final ψl+1 (xl+1 ), ψl+2 (xl+2 ), . . , ψn (x n ) wavefunctions of the particles participating in the scattering gives the corresponding probability amplitude: ψl+1 , ψl+2 , .

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[Article] A Thermodynamic Study of Electrolytic Solutions by Hitchcock F.L.

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