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The majorant method for the fermionic effective action

Wilhelm Kroschinsky, Domingos H. U. Marchetti, Manfred Salmhofer

Abstract

We revisit the problem of controlling Polchinski's equation by the solution of an associate Hamilton-Jacobi equation which determines a norm majorant for the fermionic effective action. This method, referred to as the majorant method, was first introduced by D. Brydges and J. Wright in 1988, but its original formulation contains a gap which has never been addressed. We overcome this gap and show that the majorant equation and its existence condition are analogous to the ones originally obtained by Brydges and Wright. As an application of the method, we investigate a fermion model with a local quartic interaction.

The majorant method for the fermionic effective action

Abstract

We revisit the problem of controlling Polchinski's equation by the solution of an associate Hamilton-Jacobi equation which determines a norm majorant for the fermionic effective action. This method, referred to as the majorant method, was first introduced by D. Brydges and J. Wright in 1988, but its original formulation contains a gap which has never been addressed. We overcome this gap and show that the majorant equation and its existence condition are analogous to the ones originally obtained by Brydges and Wright. As an application of the method, we investigate a fermion model with a local quartic interaction.
Paper Structure (15 sections, 12 theorems, 156 equations)

This paper contains 15 sections, 12 theorems, 156 equations.

Table of Contents

  1. Introduction
  2. Grassmann Algebras - An Overview
  3. Derivatives and Integrals on Grassman Algebras
  4. Grassmann Gaussian Integrals
  5. Norms and Bounds for the Correlation Functions
  6. The Renormalization Group Transformation and Polchinski's Equation
  7. The Majorant Method
  8. Application to Fermions with Local Quartic Interactions
  9. Regularized $\Psi^{4}_{d}$ Theory
  10. Setup of the renormalization group flow
  11. The determinant and decay bounds
  12. Scale-dependent bounds in $\Psi_{4}^{4}$ Theory
  13. Conclusion
  14. Proof of Proposition \ref{['Fseqlem']}
  15. Hamilton-Jacobi theory and the proof of Theorem \ref{['newmajorant']}

Key Result

Proposition 2.1

(a) If the weighted Laplacian operator $\Delta_{A}$ is defined by then holds.The exponential of $\Delta_{A}$ truncates to a polynomial by nilpotency of the derivatives, so the right hand side of (GA28.1) is well-defined and analytic in $A$. We refer to (GA28.1) as the Heat Kernel Formula. (b) If $A, B \in M_{2n}(\mathbb{C})$ are both skew-symmetric and invertible, then Relation (GA28.2) is also

Theorems & Definitions (26)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • Proposition 3.1
  • Definition 3.1
  • ...and 16 more