[Paper Review] Leibniz rule, locality and supersymmetry on lattice

Mitsuhiro Kato, Makoto Sakamoto+1 more|arXiv (Cornell University)|Dec 7, 2012

Quantum chaos and dynamical systems2 references3 citations

TL;DR

This paper proves a no-go theorem for the Leibniz rule in finite-volume lattice field theories, showing that a local difference operator necessitates a non-local product rule. It identifies a new solution class where supersymmetry can be realized via a non-local product, resolving a long-standing conflict between finite-flavor systems and matrix representations of infinite-flavor theories through discrete momentum-space analysis and flavor matrix formalism.

ABSTRACT

In a finite volume system, we prove a no-go theorem on a Leibniz rule with a care of locality argument on latttice. The new possibility on the Leibniz rule solutions on lattice is discussed. Although the new solution admits a local difference operator, a non-local product rule is needed. In the case, a supersymmetric interacting theory is simply realized. The difference between finite flavor systems and matrix representations of infinite flavor systems is explained based on a finite volume system analysis including the no-go theorem.

Motivation & Objective

• To resolve the inconsistency between finite-flavor lattice systems and matrix representations of infinite-flavor systems in the context of supersymmetry.
• To establish a finite-volume lattice framework for locality and translational invariance using discrete bounded functions instead of holomorphic functions.
• To clarify the conditions under which the Leibniz rule can be satisfied on a lattice while preserving locality and supersymmetry.
• To analyze the distinction between multi-flavor systems and matrix representations in terms of Leibniz rule compliance and locality.

Proposed method

• Uses discrete momentum representations via Nth roots of unity to define lattice operators and product rules.
• Applies a complex extension of momentum-space functions (with small imaginary parts) to define locality conditions via exponential decay.
• Introduces a flavor matrix formalism to classify solutions into type-A (trivial difference operator) and type-B (trivial product rule).
• Derives a modified Leibniz rule in matrix form involving a non-zero remainder term R_L,M^pqr(ε) for infinite-flavor matrix representations.
• Applies a finite system no-go theorem to show that non-trivial local difference operators force trivial product rules in finite-flavor systems.
• Uses Jordan normal form to parametrize the difference operator in flavor space, enabling explicit classification of solutions.

Experimental results

Research questions

• RQ1 Can a local difference operator coexist with a non-trivial product rule on a finite lattice while satisfying the Leibniz rule?
• RQ2 Why does the matrix representation of an infinite-flavor system evade the finite-flavor no-go theorem for the Leibniz rule?
• RQ3 What is the role of the discrete momentum-space function with complex shift in defining locality on a finite lattice?
• RQ4 How do the solutions to the Leibniz rule differ between finite-flavor systems and matrix representations of infinite-flavor systems?
• RQ5 Can supersymmetry be consistently realized on a lattice when the Leibniz rule is satisfied only via a non-local product rule?

Key findings

• A no-go theorem proves that in a finite-volume lattice system, a non-trivial local difference operator forces the product rule to be trivial or non-local.
• A new solution class exists where the difference operator is local but the product rule is non-local, enabling the construction of a supersymmetric interacting theory.
• The matrix representation of an infinite-flavor system evades the no-go theorem due to a non-vanishing remainder term R_L,M^pqr(ε) in the Leibniz rule, which arises from boundary conditions in flavor space.
• In finite-flavor systems, the Leibniz rule can only be satisfied if either the difference operator or the product rule is trivial, leading to a classification into type-A and type-B flavors.
• The discrete momentum-space analysis with complex shifts provides a sufficient and necessary condition for locality, expressed as O(N⁰) bounds on the transformed operators.
• The non-local product rule in the second solution class allows for a consistent supersymmetric action with interactions, offering a viable path to lattice supersymmetry.