# Collective quantum fields by Kleinert.

By Kleinert.

It's the goal of this booklet to debate an easy procedure through Feynman course crucial 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 normal type and the basic quanta without delay describe the collective excitations

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100) where M is the mass of the spin-1/2 -particles described by ψ(x). 102) σ ˜ µ ≡ (1, −σ i ). 103) ψ¯ ≡ ψ † γ 0 . 104) and ¯ The symbol ψ(x) is short for As a historical note we mention that Dirac found his by considering the naive relativistic time independent Schr¨odinger equation of an electron ˆ Hψ(x) = ˆ 2 + M 2 ψ(x) = Eψ(x). 105) He asked the question whether the square root could be found explicitly if the equation were considered as a matrix equation acting on several components of ψ(x, t) to represent the spin degrees of freedom of the electron.

Hence, particles with a single fixed spin s can only follow the D (s,0) (Λ) or D(0,s) (Λ) transformation laws. For spin 1/2, the relativistic free-field which is invariant under parity has four components and is called the Dirac field . 100) where M is the mass of the spin-1/2 -particles described by ψ(x). 102) σ ˜ µ ≡ (1, −σ i ). 103) ψ¯ ≡ ψ † γ 0 . 104) and ¯ The symbol ψ(x) is short for As a historical note we mention that Dirac found his by considering the naive relativistic time independent Schr¨odinger equation of an electron ˆ Hψ(x) = ˆ 2 + M 2 ψ(x) = Eψ(x).

Phys. 11, 258 (1970). [43] M. J. Ablowitz, D. J. Kaup, A. C. Nevell, H. Segur, Phys. Rev. Letters 30, 1262 (1973). [44] K. Maki, H. Ebisawa, J. Low Temp. Phys. 23 351 (1976). [45] K. Maki, P. Kumar, Phys. Rev. 16, 174 (1977). [46] J. Schwinger, J. Math. Phys. 5, 1606 (1964). H. Kleinert, COLLECTIVE QUNATUM FIELDS Part II Plasmas and Superconductors 39 H. tex August 10, 2001 1 Introduction In this part we shall develop the theory of collective quantum fields by treating the collective phenomena in in two important many-electron systems.