[Paper Review] The Absence of Ultralocal Ginsparg-Wilson Fermions

Wolfgang Bietenholz|ArXiv.org|Jan 8, 1999

Quantum Chromodynamics and Particle Interactions3 citations

TL;DR

This paper proves that all Ginsparg-Wilson fermions—regardless of their specific form—cannot be ultralocal; instead, their couplings must decay exponentially with distance. The result resolves a long-standing question in lattice field theory by showing that ultralocality is fundamentally incompatible with the Ginsparg-Wilson relation, even in generalized forms.

ABSTRACT

It was shown recently by I. Horvath that lattice fermions obeying the standard form of the Ginsparg-Wilson relation cannot be ultralocal. However, there are more general forms of the Ginsparg-Wilson relation, which also guarantee the physical properties related to chirality, but which are not covered by Horvath's consideration. Here we present a proof which applies to all Ginsparg-Wilson fermions, demonstrating that they can only be local in the sense of an exponential decay of their couplings, but not ultralocal.

Motivation & Objective

• To resolve the question of whether Ginsparg-Wilson fermions can be ultralocal, a property desirable for computational efficiency and locality.
• To extend previous results—limited to the standard form of the Ginsparg-Wilson relation—to all possible forms of the relation.
• To establish a general mathematical proof that ultralocality is incompatible with the Ginsparg-Wilson condition across all fermion realizations.
• To clarify the fundamental locality properties of lattice fermions preserving chiral symmetry via the Ginsparg-Wilson relation.

Proposed method

• Formal analysis of the Ginsparg-Wilson relation in its most general form, without restricting to the standard version.
• Use of operator-theoretic techniques to analyze the structure of fermion kernels on the lattice.
• Proof by contradiction to show that ultralocality leads to a contradiction with the Ginsparg-Wilson condition.
• Demonstration that any solution to the Ginsparg-Wilson equation must exhibit exponential decay in couplings, not strict short-range (ultralocal) behavior.
• Generalization of earlier results by Horvath, which applied only to the standard form, to all possible forms of the relation.
• Use of spectral theory and lattice field theory formalism to derive constraints on the support of fermion kernels.

Experimental results

Research questions

• RQ1 Can Ginsparg-Wilson fermions be ultralocal, meaning their interactions are strictly short-ranged?
• RQ2 Does the absence of ultralocality persist across all forms of the Ginsparg-Wilson relation, not just the standard one?
• RQ3 What is the precise nature of the locality structure (e.g., exponential decay) of Ginsparg-Wilson fermion kernels?
• RQ4 Is there any mathematical obstruction preventing ultralocality while preserving the chiral symmetry properties encoded in the Ginsparg-Wilson relation?
• RQ5 How does the generalized Ginsparg-Wilson condition constrain the spatial structure of fermion interactions on the lattice?

Key findings

• All Ginsparg-Wilson fermions, regardless of the specific form of the relation, cannot be ultralocal.
• The couplings of Ginsparg-Wilson fermions must decay exponentially with distance, not vanish beyond finite range.
• The proof applies universally to all forms of the Ginsparg-Wilson relation, extending beyond the standard version previously considered.
• Ultralocality is fundamentally incompatible with the Ginsparg-Wilson condition, even in generalized forms.
• The result confirms that the locality of these fermions is strictly weaker than ultralocality, limiting their use in certain computational schemes.
• The conclusion resolves a foundational question in lattice field theory regarding the interplay between chiral symmetry and locality.