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Journal of Mathematical Physics

Euclidean Majorana fermions, fermionic integrals, and relative Pfaffians

David Borthwick

Published Jul 1, 1993• 3 citations
Publisher

Abstract

The construction of Euclidean Majorana fermions on a two-dimensional cylindrical space–time is discussed, based on a given real-time theory. After a review of the basics of defining Euclidean fermions in the free theory, a functional integral formulation for the Euclidean theory is introduced. This functional integral connects the expectation values of the Euclidean fields with the theory of relative Pfaffians. A proof, which draws on these ideas, is given for the Feynman–Kac formula of a general interacting theory.

DOI
10.1063/1.530413

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The construction of Euclidean Majorana fermions on a two-dimensional cylindrical space–time is discussed, based on a given real-time theory. After a review of the basics of defining Euclidean fermions in the free theory, a functional integral formulation for the Euclidean theory is introduced. This functional integral connects the expectation values of the Euclidean fields with the theory of relative Pfaffians. A proof, which draws on these ideas, is given for the Feynman–Kac formula of a general interacting theory.
In Quantum Physics, the term “Euclidean” means that one, compared to the conventional formulation, works in “imaginary time.” It is a general fact that objects associated with Euclidean as opposed to Minkowski space are easier to deal with; this is seen for instance when comparing the usual Laplacian, an elliptic operator, with its Minkowski space counterpart, the hyperbolic d’Alembertian. In Quantum Mechanics, the term “Euclidean” is usually associated with approaches based on the so-called Feynman-Kac formula. The latter originates in attempts to give Feynman’s ideas [10, 11] on path integrals a solid mathematical founding.
Abstract Functional integrals are basic tools to study first quantum mechanics (QM), and quantum field theory (QFT). The path integral formulation of QM is well suited to the study of systems with an arbitrary number of degrees of freedom. It makes a smooth transition between nonrelativistic QM and QFT possible. The Euclidean functional integral also emphasizes the deep connection between QFT and the statistical physics of systems with short-range interactions near a continuous phase transition. The path integral representation of the matrix elements of the quantum statistical operator e-β H for Hamiltonians of the simple separable form p2/2m +V(q) is derived. To the path integral corresponds a functional measure and expectation values called correlation functions, which are generalized moments, and related to quantum observables, after an analytic continuation in time. The path integral corresponding to the Euclidean action of a harmonic oscillator, to which is added a time-dependent external force, is calculated explicitly. The result is used to generate Gaussian correlation functions and also to reduce the evaluation of path integrals to perturbation theory. The path integral also provides a convenient tool to derive semi-classical approximations.
We discuss a nonperturbative formulation of the Sachdev–Ye–Kitaev (SYK) model. The partition function of the model can be represented as a well-defined functional integral over Grassmann variables in Euclidean time, but it diverges after the transformation to fermion bilocal fields. We note that the generating functional of the SYK model in real time is well defined even after the transformation to bilocal fields and can be used for nonperturbative investigations of its properties. We study the SYK model in zero dimensions, evaluate its large-N expansion, and investigate phase transitions.
We review the applications of the integral over anticommuting Grassmann variables (nonquantum fermionic fields) to the analytic solutions and the field-theoretical formulations for the 2D Ising models. The 2D Ising model partition function QQQ is presentable as the fermionic Gaussian integral. The use of the spin-polynomial interpretation of the 2D Ising problem is stressed, in particular. Starting with the spin-polynomial interpretation of the local Boltzmann weights, the Gaussian integral for QQQ appears in the universal form for a variety of lattices, including the standard rectangular, triangular, and hexagonal lattices, and with the minimal number of fermionic variables (two per site). The analytic solutions for the correspondent 2D Ising models then follow by passing to the momentum space on a lattice. The symmetries and the question on the location of critical point have an interesting interpretation within this spin-polynomial formulation of the problem. From the exact lattice theory we then pass to the continuum-limit field-theoretical interpretation of the 2D Ising models. The continuum theory captures all relevant features of the original models near TcT_cTc​. The continuum limit corresponds to the low-momentum sector of the exact theory responsible for the critical-point singularities and the large-distance behaviour of correlations. The resulting field theory is the massive two-component Majorana theory, with mass vanishing at TcT_cTc​. By doubling of fermions in the Majorana representation, we obtain as well the 2D Dirac field theory of charged fermions for 2D Ising models. The differences between particular 2D Ising lattices are merely adsorbed, in the field-theoretical formulation, in the definition of the effective mass.