The Wick Theorem and Feynman Diagrammatic Expansion

D. J. Kim

Lode Pollet, Mikhail N. Kiselev, Nikolay V. Prokof'ev +1 more

Applying Feynman diagrammatics to non-fermionic strongly correlated models with local constraints might seem generically impossible for two separate reasons: (i) the necessity to have a Gaussian (non-interacting) limit on top of which the perturbative diagrammatic expansion is generated by Wick’s theorem, and (ii) Dyson’s collapse argument implying that the expansion in powers of coupling constant is divergent. We show that for arbitrary classical lattice models both problems can be solved/circumvented by reformulating the high-temperature expansion (more generally, any discrete representation of the model) in terms of Grassmann integrals. Discrete variables residing on either links, plaquettes, or sites of the lattice are associated with the Grassmann variables in such a way that the partition function (as well as all correlation functions) of the original system and its Grassmann-field counterpart are identical. The expansion of the latter around its Gaussian point generates Feynman diagrams. Our work paves the way for studying lattice gauge theories by treating bosonic and fermionic degrees of freedom on equal footing.

Giancarlo Calvanese Strinati

This chapter applies the Wick’s theorem to the contour single-particle Green’s function. The corresponding average is represented in the interaction picture, with no need to specify at the outset the kind of contour that is used. This procedure is summarized in a set of Feynman diagrammatic rules, which are reported schematically. Here, only the normal phase is considered, while, in Part II, the Feynman rules are extended to superfluid Fermi systems.

J. Schirmer

Josip Hendeković, Milica Pavlović, Franjo Sokolić

Shridhar Vinayak

The machinery of computing vacuum expectation values of a time-ordered sequence of position operators of the simple harmonic oscillator is already well established. It rests on a Wick theorem, which enables one to decompose such a quantity in terms of products of \emph{pairwise contractions}, which are vacuum expectation values of a time-ordered sequence of position operators taken two at a time. This result naturally leads to a diagrammatic approach of computing such correlators, and is already well known in the form of Feynman diagrams. We generalise this setup to encompass expectation values of a general ordered sequence of position operators (Wightman sequences) in general density matrices of the simple harmonic oscillator. A Wick theorem is first developed for this situation and consequently a diagrammatics is laid down. Wightman correlators in general density matrices of the \emph{anharmonic} oscillator are also analysed and a diagrammatic formalism is developed for them too.