ABSTRACT
Fields whose quanta obey the Pauli exclusion principle have to be described by functions whose values are not normal numbers but anti- commuting, Grassmann, variables. Starting from the Fermi oscillator, the rules for constructing derivatives and integrals of Grassmann variables are spelled out and the construction of the generating functional for free-fields is illustrated. The construction of the Feynman Path Integral with anti commuting variables is discussed in detail. For later use, e.g. in Chapter 14, the calculation of Gaussian integrals over anti-commuting variables is illustrated, to compare with the result of the analogous integrals over commuting variables. The quantisation of the Dirac field is worked out explicitly including a derivation of the fermion propagator. The spin-statistics theorem, according to which particles of integer spin are described by commuting fields while those of half-integer spin require anti-commuting fields, is one of the few exact results in field theory. In the last section it is explicitly shown that the quantum theory of a free Dirac field necessarily requires that https://www.w3.org/1998/Math/MathML" display="inline"> ψ and ψ ¯ should be anti-commuting quantities, and that, conversely, a free scalar field, https://www.w3.org/1998/Math/MathML" display="inline"> ϕ , is necessarily a commuting quantity.
