scholarly journals An exact mapping between loop-erased random walks and an interacting field theory with two fermions and one boson

2020 ◽  
Vol 9 (5) ◽  
Author(s):  
Assaf Shapira ◽  
Kay Joerg Wiese
Keyword(s):  
Field Theory ◽  
Random Walks ◽  
Path Integral ◽  
Lattice Model ◽  
Type Theory ◽  
Complex Field ◽  
Standard Formulation ◽  
Bosonic Field ◽  
Hypercubic Lattice ◽  
Formula Type

We give a simplified proof for the equivalence of loop-erased random walks to a lattice model containing two complex fermions, and one complex boson. This equivalence works on an arbitrary directed graph. Specifying to the dd-dimensional hypercubic lattice, at large scales this theory reduces to a scalar \phi^4ϕ4-type theory with two complex fermions, and one complex boson. While the path integral for the fermions is the Berezin integral, for the bosonic field we can either use a complex field \phi(x)\in \mathbb Cϕ(x)∈ℂ (standard formulation) or a nilpotent one satisfying \phi(x)^2 =0ϕ(x)2=0. We discuss basic properties of the latter formulation, which has distinct advantages in the lattice model.

scholarly journals AN ALGORITHMIC APPROACH TO QUANTUM FIELD THEORY

2006 ◽  
Vol 21 (03) ◽  
pp. 405-447 ◽  
Author(s):  
MASSIMO DI PIERRO
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Path Integral ◽  
Gauge Theories ◽  
Quantum Field ◽  
Algorithmic Approach ◽  
Main Emphasis ◽  
Lattice Computation ◽  
Theoretical Foundations ◽  
Minimal Residual

The lattice formulation provides a way to regularize, define and compute the Path Integral in a Quantum Field Theory. In this paper, we review the theoretical foundations and the most basic algorithms required to implement a typical lattice computation, including the Metropolis, the Gibbs sampling, the Minimal Residual, and the Stabilized Biconjugate inverters. The main emphasis is on gauge theories with fermions such as QCD. We also provide examples of typical results from lattice QCD computations for quantities of phenomenological interest.

scholarly journals A PATH-INTEGRAL APPROACH FOR BOSONIC EFFECTIVE THEORIES FOR FERMION FIELDS IN FOUR AND THREE DIMENSIONS

2000 ◽  
Vol 15 (05) ◽  
pp. 755-770 ◽  
Author(s):  
LUIZ C. L. BOTELHO
Keyword(s):  
Path Integral ◽  
Infrared Region ◽  
Three Dimensions ◽  
Integral Approach ◽  
Ultraviolet Region ◽  
Fermion Field ◽  
Massive Fermion ◽  
Bosonic Field ◽  
Fermion Fields ◽  
Effective Theories

We study four-dimensional effective bosonic field theories for (A) massive fermion field in the infrared region and (B) massive fermion in the ultraviolet region by using an appropriate fermion path integral chiral variable change and (C) Polyakov's Fermi–Bose transmutation in the 3D-Abelian Thirring model and its triviality as a quantum field theory.

scholarly journals COORDINATE-INVARIANT PATH INTEGRAL METHODS IN CONFORMAL FIELD THEORY

2006 ◽  
Vol 21 (17) ◽  
pp. 3525-3563 ◽  
Author(s):  
ANDRÉ VAN TONDER
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Path Integral ◽  
Direct Calculation ◽  
Conformal Anomaly ◽  
Operator Formalism ◽  
Conformal Field ◽  
Integral Methods ◽  
Change Of Variables ◽  
Definition Of

We present a coordinate-invariant approach, based on a Pauli–Villars measure, to the definition of the path integral in two-dimensional conformal field theory. We discuss some advantages of this approach compared to the operator formalism and alternative path integral approaches. We show that our path integral measure is invariant under conformal transformations and field reparametrizations, in contrast to the measure used in the Fujikawa calculation, and we show the agreement, despite different origins, of the conformal anomaly in the two approaches. The natural energy–momentum in the Pauli–Villars approach is a true coordinate-invariant tensor quantity, and we discuss its nontrivial relationship to the corresponding nontensor object arising in the operator formalism, thus providing a novel explanation within a path integral context for the anomalous Ward identities of the latter. We provide a direct calculation of the nontrivial contact terms arising in expectation values of certain energy–momentum products, and we use these to perform a simple consistency check confirming the validity of the change of variables formula for the path integral. Finally, we review the relationship between the conformal anomaly and the energy–momentum two-point functions in our formalism.

SUPERFLUID FIELD THEORY

1993 ◽  
Vol 08 (28) ◽  
pp. 5005-5021
Author(s):  
R.L. DAVIS
Keyword(s):  
Field Theory ◽  
Landau Theory ◽  
Fluid Model ◽  
Ginzburg Landau ◽  
Bosonic Field ◽  
Thermomechanical Effect ◽  
First Sound ◽  
Two Fluid Model ◽  
Two Fluid ◽  
Landau Criterion

The very low temperature dynamics of an isotropic superfluid is derived from a repulsive bosonic field theory. The field theory is a fully dynamical generalization of the Ginzburg-Landau theory, which at zero temperature has semiclassical superfluid solutions. It is shown that supercurrent quenching occurs above some intrinsic critical velocity. The speed of first sound is calculated and the Landau criterion for a maximum superfluid velocity is derived. At finite temperature, the thermodynamic potential is computed, the order parameter and gap equations are derived, the origin of the Landau two-fluid model is identified and the thermomechanical effect is explained. This theory successfully describes many of the features of 4He well below the critical temperature, as well as relativistic generalizations.

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