Abstract
We express connected Fermionic Green's functions in terms of two totally explicit tree formulas. The simplest and most symmetric formula, the Brydges–Kennedy formula is compatible with Gram's inequality. The second one, the rooted formula of Abdesselam and Rivasseau, respects even better the antisymmetric structure of determinants, and allows the direct comparison of rows and columns which correspond to the mathematical implementation in Grassmann integrals of the Pauli principle. To illustrate the power of these formulas, we give a ‘three lines proof’ that the radius of convergence of the Gross–Neveu theory with cutoff is independent of the number of colors, using either one or the other of these formulas.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abdesselam, A. and Rivasseau, V.: Trees, forests and jungles: a botanical garden for cluster expansions, in: Constructive Physics, Lecture Notes in Phys. 446, Springer-Verlag, New York, 1995.
Abdesselam, A. and Rivasseau, V.: An explicit large versus small field multiscale cluster expansion, Rev. Math. Phys. 9(2), (1997), 123 (and references therein).
Brydges, D.: Weak perturbations of the massless gaussian measure, in: Constructive Physics, Lecture Notes in Phys. 446, Springer-Verlag, New York, 1995 (and references therein).
Brydges, D. and Kennedy, T.: Mayer expansions and the Hamilton–Jacobi equation, J. Stat. Phys. 48 (1978) 19 2064.
Disertori, M. and Rivasseau, V.: hep-th/9802145.
Feldman, J., Knörrer, H. and Trubowitz, E.: A representation for fermionic correlation functions, ETH-Zürich preprint, June 1997.
Feldman, J., Lehmann, D., Knörrer, H. and Trubowitz, E.: Fermi liquids in two space dimensions, in: Constructive Physics, Lecture Notes in Phys. 446, Springer-Verlag, New York, 1995.
Feldman, J. Magnen, V. Rivasseau and E. Trubowitz, An infinite volume expansion for many fermion green's functions, Helv. Phys. Acta 65 (1992), 679.
Iagolnitzer, D. and Magnen, J.: Asymptotic completeness and multiparticle structure in field theories, II, Theories with renormalization, Comm. Math. Phys. 111 (1987), 81.
Salmhofer, M.: Continuous renormalization for fermions and Fermi liquid theory, condmat/ 9706188.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Abdesselam, A., Rivasseau, V. Explicit Fermionic Tree Expansions. Letters in Mathematical Physics 44, 77–88 (1998). https://doi.org/10.1023/A:1007413417112
Issue Date:
DOI: https://doi.org/10.1023/A:1007413417112