[Paper Review] Space and time dimensions of algebras with applications to Lorentzian noncommutative geometry and quantum electrodynamics

Nadir Bizi, Christian Brouder+1 more|arXiv (Cornell University)|Nov 21, 2016

Quantum Mechanics and Non-Hermitian Physics15 references3 citations

TL;DR

This paper introduces a novel assignment of space and time dimensions modulo 8 to complex *C*-algebras equipped with two self-adjoint involutions and an anti-unitary operator satisfying specific commutation relations, generalizing the $KO$-dimension concept. The framework enables a pseudo-Riemannian extension of noncommutative geometry via Krein spaces, successfully constructing the Lorentzian spectral triple for quantum electrodynamics and deriving its physical Lagrangian without fermion doubling.

ABSTRACT

An analogy with real Clifford algebras on even-dimensional vector spaces suggests to assign a couple of space and time dimensions modulo 8 to any algebra (represented over a complex Hilbert space) containing two self-adjoint involutions and an anti-unitary operator with specific commutation relations. It is shown that this assignment is compatible with the tensor product: the space and time dimensions of the tensor product are the sums of the space and time dimensions of its factors. This could provide an interpretation of the presence of such algebras in PT-symmetric Hamiltonians or the description of topological matter. This construction is used to build an indefinite (i.e. pseudo-Riemannian) version of the spectral triples of noncommutative geometry, defined over Krein spaces instead of Hilbert spaces. Within this framework, we can express the Lagrangian (both bosonic and fermionic) of a Lorentzian almost-commutative spectral triple. We exhibit a space of physical states that solves the fermion-doubling problem. The example of quantum electrodynamics is described.

Motivation & Objective

• To generalize the $KO$-dimension concept from Riemannian to pseudo-Riemannian (Lorentzian) noncommutative geometry by assigning space and time dimensions modulo 8 to algebras.
• To resolve the fermion-doubling problem in Lorentzian spectral triples by defining a physical state space using a fundamental symmetry.
• To construct a consistent indefinite (Krein space-based) spectral triple for Lorentzian almost-commutative geometry.
• To derive the physical Lagrangian of quantum electrodynamics (QED) within this framework, matching the standard physical form.
• To establish compatibility of the dimension assignment with the graded tensor product of algebras, ensuring consistency in composite systems.

Proposed method

• Assign space and time dimensions modulo 8 to algebras via three signs derived from commutation relations between a chirality operator $\chi$, a fundamental symmetry $\eta$, and a charge conjugation $J$.
• Define an indefinite spectral triple using Krein spaces instead of Hilbert spaces, allowing for non-positive definite inner products essential for Lorentzian signature.
• Construct the Dirac operator $D(A)$ in the Lorentzian case as $D - q\gamma^\mu A_\mu \otimes \varpi$, incorporating the gauge potential via a representation $\pi$.
• Define the Connes-Lott-Elsner (CLE) Lagrangian as $\mathcal{L}_{\mathrm{CLE}} = \mathrm{Tr}_z(\theta^\times \theta) + \frac{1}{2}(\Psi, D(A)\Psi)$, with a positive definite weight $z$ to resolve trace ambiguities.
• Use the fundamental symmetry $\eta$ and charge conjugation $J$ to ensure particles and antiparticles have equal mass and opposite current, without requiring anticommutation relations.
• Verify that the bosonic Lagrangian $\mathcal{L}_b = -8\rho q^2 F_{\mu\nu}F^{\mu\nu}$ becomes physical after choosing $\rho = 1/(32q^2)$, matching standard QED.

Experimental results

Research questions

• RQ1 Can a generalized $KO$-dimension concept be defined for indefinite (pseudo-Riemannian) spectral triples using algebraic structures?
• RQ2 How can the fermion-doubling problem be resolved in Lorentzian noncommutative geometry?
• RQ3 Is it possible to construct a consistent Lorentzian version of the Connes-Lott spectral action for quantum electrodynamics?
• RQ4 Does the dimension assignment modulo 8 for algebras remain compatible under the graded tensor product of algebras?
• RQ5 Can the physical QED Lagrangian be derived from a Krein space-based spectral triple without relying on anticommutation relations?

Key findings

• The space and time dimensions modulo 8 are consistently assigned to any algebra containing two self-adjoint involutions and an anti-unitary operator with specific commutation relations.
• The dimension assignment is compatible with the graded tensor product: the dimensions of $A_1 \hat{\otimes} A_2$ are the sum modulo 8 of the dimensions of $A_1$ and $A_2$.
• A physical state space is constructed that solves the fermion-doubling problem in Lorentzian spectral triples via the fundamental symmetry $\eta$.
• The Lorentzian spectral triple for QED is successfully defined, and its Lagrangian matches the standard physical form after trace regularization with a positive weight $z$.
• The fermionic and gauge Lagrangians are derived without assuming anticommutation relations; particle-antiparticle symmetry arises from the algebraic structure of $J$ and $\eta$.
• The bosonic Lagrangian becomes $\mathcal{L}_b = -8\rho q^2 F_{\mu\nu}F^{\mu\nu}$, and setting $\rho = 1/(32q^2)$ recovers the physical $-F_{\mu\nu}F^{\mu\nu}$ form.