By Kleinert.

It's the function of this e-book to debate an easy process through Feynman direction necessary formulation within which the transformation to collective fields quantities to merechanges of integration variables in useful integrals. After the transformation, the trail formula will back be discarded. The ensuing box conception is quantizedin the traditional model and the elemental quanta at once describe the collective excitations

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74) yielding 1 ± nξ(p) . In the operator language, these limits correspond to the expectation values T G (x, τ ; x, τ + η) = T G (x, τ ; x, τ − η) = † † 0|Tˆ ψˆH (x, τ )ψˆH (x, τ + η) |0 = ± 0|ψˆH (x, τ )ψˆH (x, τ )|0 † † (x, τ − η) |0 = 0|ψˆH (x, τ )ψˆH (x, τ )|0 0|Tˆ ψˆH (x, τ )ψˆH † = 1 ± 0|ψˆH (x, τ )ψˆH (x, τ ∓ η)|0 The function n(ξ(p)) is the thermal expectation value of the number operator † ˆ = ψˆH N (x, τ )ψˆH (x, τ ). 75) Also in the case of T = 0 ensembles, it is useful to employ a four-vector notation.

94) Since the Lorentz transformations Λ1 , Λ2 are elements of a group, the product Λ ≡ Λ2 Λ1 is again a Lorentz transformation. 95) so that under Λ = Λ2 Λ1 , Λ Λ 2 1 Ψ(x) −→ Ψ (x ) = D(Λ2 )D(Λ1 )Ψ(x). 96) But for Λ itself, the transformation matrix is D(Λ) and Ψ (x ) = D(Λ2 Λ1 )Ψ(x). 96) shows that the matrices D(Λ) which mix the spinor field components under the Lorentz group must follow a group multiplication law which has to be conmpatible with that of the group itself. 98) is a homomorphism and the D(Λ)’s form a matrix representation of the group.

Rev. 174, 227 (1968); Y. Kuroda, A. D. S. Nagi, J. Low. Temp. 23, 751 (1976). [31] D. Rainer, J. W. Serene, Phys. Rev. B13, 4745 (1976); Y. Kuroda, A. D. S. Nagi, J. Low Temp. Phys. 25, 569 (1976). 38 2 Relativistic Fields [32] For a review see: D. R. Bes, R. A. Broglia, Lectures delivered at “E. Fermi” Varenna Summer School, Varenna, Como. Italy, 1976. For recent studies: D. R. Bes, R. A. Broglia, R. Liotta, B. R. Mottelson, Phys. Letters 52B, 253 (1974); 56B, 109 (1975), Nuclear Phys. B A260, 127 (1976).