Noncommutative Geometry in Quantum Field Theory
Summary
Noncommutative geometry generalises the notion of space by replacing the algebra of functions on a manifold with a noncommuting algebra, thereby encoding geometric information in algebraic and operator‐theoretic terms. Central to this framework is the spectral triple, comprising an algebra, a Hilbert space and a Dirac operator, which together capture metric and topological data. In quantum field theory, this approach offers a novel foundation for gauge theories and gravity, unifying internal symmetries with spacetime geometry. The spectral action principle translates physical Lagrangians into traces of functions of the Dirac operator, yielding natural ultraviolet cutoffs and insights into renormalisation. Applications range from reformulations of the Standard Model and grand unification schemes to toy models of quantum gravity on “fuzzy” spaces and random matrix ensembles. Interconnections between algebraic structures and field‐theoretic phenomena have led to fresh perspectives on space‐time singularities, phase transitions in spectral phases and the emergence of continuum behaviour from finite noncommutative geometries.
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Technical terms
Spectral triple: A trio of an algebra, a Hilbert space and a Dirac operator encoding geometric data in operator form.
Dirac operator: A self-adjoint operator generalising the classical Dirac matrix, defining the differential structure in noncommutative spaces.
Spectral action: An action functional expressed as the trace of a function of the Dirac operator, generating physical Lagrangians.
Fuzzy space: A finite-dimensional approximation of a manifold realised via noncommutative coordinates.
Functional renormalisation group: A method to study scale dependence of actions using flow equations in theory space.
References
- Noncommutativity and physics: a non-technical review. The European Physical Journal Special Topics (2023).
- From noncommutative geometry to random matrix theory. Journal of Physics A: Mathematical and Theoretical (2022).
- On Multimatrix Models Motivated by Random Noncommutative Geometry I: The Functional Renormalization Group as a Flow in the Free Algebra. Annales Henri Poincaré (2021).
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