A nonperturbative form of the spectral action principle in noncommutative geometry (original) (raw)
The spectral action principle in noncommutative geometry and the superstring
Physics Letters B, 1997
A supersymmetric theory in two dimensions has enough data to define a noncommutative space thus making it possible to use all tools of noncommutative geometry. In particular, we apply this to the N = 1 supersymmetric non-linear sigma model and derive an expression for the generalized loop space Dirac operator, in presence of a general background, using canonical quantization. The spectral action principle is then used to determine a spectral action valid for the fluctuations of the string modes.
Gravitational waves in the spectral action of noncommutative geometry
Physical Review D - PHYS REV D, 2010
The spectral triple approach to noncommutative geometry allows one to develop the entire standard model (and supersymmetric extensions) of particle physics from a purely geometry stand point and thus treats both gravity and particle physics on the same footing. The bosonic sector of the theory contains a modification to Einstein-Hilbert gravity, involving a nonconformal coupling of curvature to the Higgs field and conformal Weyl term (in addition to a nondynamical topological term). In this paper we derive the weak field limit of this gravitational theory and show that the production and dynamics of gravitational waves are significantly altered. In particular, we show that the graviton contains a massive mode that alters the energy lost to gravitational radiation, in systems with evolving quadrupole moment. We explicitly calculate the general solution and apply it to systems with periodically varying quadrupole moments, focusing in particular on the the well know energy loss formula for circular binaries.
This paper is a review of some interesting results that has been obtained in various sectors of noncommutative cosmology, string theory and loop quantum gravity. In the Section 1, we have described some results concerning the noncommutative model of the closed Universe with the scalar field. In the Section 2, we have described some results concerning the low-energy string effective quantum cosmology. In the Section 3, we have showed some results regarding the noncommutative Kantowsky-Sachs quantum model. In Section 4, we have showed some results regarding the spectral action principle associated with a noncommutative space and applied to the Einstein-Yang-Mills system. Section 5 is a review of some results regarding some aspects of loop quantum gravity. In Section 6, we've described some results concerning the dynamics of vector mode perturbations including quantum corrections based on loop quantum gravity. In Section 7, we've described some equations concerning matrix models as a non-local hidden variables theories. In Section 8, we have showed some results concerning the quantum supergravity and the role of a "free" vacuum in loop quantum gravity. In Section 9, we've described various results concerning the unifying role of equivariant cohomology in the Topological Field Theories. In conclusion, in Section 10 and in Appendix we have showed the possible mathematical connections between the arguments above mentioned and some relationship with some equations concerning some sectors of Number Theory.
Renormalization and Induced Gauge Action on a Noncommutative Space
Progress of Theoretical Physics Supplement, 2007
Field theories on deformed spaces suffer from the IR/UV mxing and renormalization is generically spoiled. In work with R. Wulkenhaar, one of us realized a way to cure this desease by adding one more marginal operator. We review these ideas, show the application to φ 3 models and use heat kernel expansion methods for a scalar field theory coupled to an external gauge field on a θ-deformed space and derive noncommutative gauge actions.
Quantum Gravity Boundary Terms from the Spectral Action of Noncommutative Space
Physical Review Letters, 2007
We study the boundary terms of the spectral action of the noncommutative space, defined by the spectral triple dictated by the physical spectrum of the standard model, unifying gravity with all other fundamental interactions. We prove that the spectral action predicts uniquely the gravitational boundary term required for consistency of quantum gravity with the correct sign and coefficient. This is a remarkable result given the lack of freedom in the spectral action to tune this term.
Quantum Gravity Boundary Terms from Spectral Action of Noncommutative Space
Eprint Arxiv 0705 1786, 2007
We study the boundary terms of the spectral action of the noncommutative space, defined by the spectral triple dictated by the physical spectrum of the standard model, unifying gravity with all other fundamental interactions. We prove that the spectral action predicts uniquely the gravitational boundary term required for consistency of quantum gravity with the correct sign and coefficient. This is a remarkable result given the lack of freedom in the spectral action to tune this term.
A novel approach to non-commutative gauge theory
Journal of High Energy Physics
We propose a field theoretical model defined on non-commutative space-time with non-constant non-commutativity parameter Θ(x), which satisfies two main requirements: it is gauge invariant and reproduces in the commutative limit, Θ → 0, the standard U(1) gauge theory. We work in the slowly varying field approximation where higher derivatives terms in the star commutator are neglected and the latter is approximated by the Poisson bracket, −i[f, g] ≈ {f, g}. We derive an explicit expression for both the NC deformation of Abelian gauge transformations which close the algebra [δ f , δ g ]A = δ {f,g} A, and the NC field strength F , covariant under these transformations, δ f F = {F , f }. NC Chern-Simons equations are equivalent to the requirement that the NC field strength, F , should vanish identically. Such equations are non-Lagrangian. The NC deformation of Yang-Mills theory is obtained from the gauge invariant action, S = F 2. As guiding example, the case of su(2)-like non-commutativity, corresponding to rotationally invariant NC space, is worked out in detail.
Canonical approach to noncommutative gauge theory
Physics Letters B, 2010
It is known that gauge transformation of the Kalb-Ramond field B μν with vector parameter Λ μ is symmetry of the closed string world-sheet action. It fails at the end points of the open string and can be restored by introducing Maxwell field A μ. We show that the same conclusion valid for space-time equations of motion, because they are conditions for conformal invariance of the world-sheet action. This is example how the symmetries of space-time theory can be investigated using properties of σ-model energy-momentum tensor. We also show that the reducible part of the closed string symmetry transformation, with Λ μ = ∂ μ λ, turns to irreducible part of the open string one. As well as Maxwell field A μ , the parameter λ is nontrivial only on the string endpoints. We show that after quantization the symmetry transformations of the background fields B μν and A μ turn to the modified symmetries defined in terms of Moyal star product. The modified Λ μ-transformation related different regularizations as well as Seiberg-Witten map.
UWThPh-2007-4 Induced Gauge Theory on a Noncommutative Space
2007
We consider a scalar φ4 theory on canonically deformed Euclidean space in 4 dimensions with an additional oscillator potential. This model is known to be renormalisable. An exterior gauge field is coupled in a gauge invariant manner to the scalar field. We extract the dynamics for the gauge field from the divergent terms of the 1-loop effective action using a matrix basis and propose an action for the noncommutative gauge theory, which is a candidate for a renormalisable model. PACS: 11.10.Nx, 11.15.-q
Noncommutative version of Borcherds' approach to quantum field theory
Proceedings of Frontiers of Fundamental Physics 14 — PoS(FFP14), 2016
Richard Borcherds proposed an elegant geometric version of renormalized perturbative quantum field theory in curved spacetimes, where Lagrangians are sections of a Hopf algebra bundle over a smooth manifold. However, this framework looses its geometric meaning when Borcherds introduces a (graded) commutative normal product. We present a fully geometric version of Borcherds' quantization where the (external) tensor product plays the role of the normal product. We construct a noncommutative many-body Hopf algebra and a module over it which contains all the terms of the perturbative expansion and we quantize it to recover the expectation values of standard quantum field theory when the Hopf algebra fiber is (graded) cocommutative. This construction enables to the second quantize any theory described by a cocommutative Hopf algebra bundle.
Gauge fields on noncommutative geometries with curvature
Journal of High Energy Physics, 2010
It was shown recently that the lagrangian of the Grosse-Wulkenhaar model can be written as lagrangian of the scalar field propagating in a curved noncommutative space. In this interpretation, renormalizability of the model is related to the interaction with the background curvature which introduces explicit coordinate dependence in the action. In this paper we construct the U 1 gauge field on the same noncommutative space: since covariant derivatives contain coordinates, the Yang-Mills action is again coordinate dependent. To obtain a two-dimensional model we reduce to a subspace, which results in splitting of the degrees of freedom into a gauge and a scalar. We define the gauge fixing and show the BRST invariance of the quantum action.
Field Theory on Curved Noncommutative Spacetimes
Symmetry, Integrability and Geometry: Methods and Applications, 2010
We study classical scalar field theories on noncommutative curved spacetimes. Following the approach of Wess et al. [Classical Quantum Gravity 22 (2005), 3511 and Classical Quantum Gravity 23 (2006), 1883], we describe noncommutative spacetimes by using (Abelian) Drinfel'd twists and the associated ⋆-products and ⋆-differential geometry. In particular, we allow for position dependent noncommutativity and do not restrict ourselves to the Moyal-Weyl deformation. We construct action functionals for real scalar fields on noncommutative curved spacetimes, and derive the corresponding deformed wave equations. We provide explicit examples of deformed Klein-Gordon operators for noncommutative Minkowski, de Sitter, Schwarzschild and Randall-Sundrum spacetimes, which solve the noncommutative Einstein equations. We study the construction of deformed Green's functions and provide a diagrammatic approach for their perturbative calculation. The leading noncommutative corrections to the Green's functions for our examples are derived.
Noncommutative geometry and theoretical physics
Journal of Geometry and Physics, 1989
The structure of amanifold can be encoded in the commutative algebra of functions on the manifold it sell-this is usual-. In the case of a non com.mut.ative algebra thereis no underlying manifold and the usual concepts and tools of diffe.rential geometry (differentialforms, De Rham cohomology, vector bundles, connections, elliptic operators, index theory.. .) have to be generalized. This is the subject of non commutative differential geometry and is believed to be of fundamental importance in our understanding of quantum field theories. The presentpaper is an introduction for the non specialist and a review oftheprincipal results on the field.
On the effective action of noncommutative Yang-Mills theory
Journal of Physics: Conference Series, 2008
We compute here the Yang-Mills effective action on Moyal space by integrating over the scalar fields in a noncommutative scalar field theory with harmonic term, minimally coupled to an external gauge potential. We also explain the special regularisation scheme chosen here and give some links to the Schwinger parametric representation. Finally, we discuss the results obtained: a noncommutative possibly renormalisable Yang-Mills theory.
SPACETIME SYMMETRIES IN NONCOMMUTATIVE GAUGE THEORY: A HAMILTONIAN ANALYSIS
Modern Physics Letters A, 2004
We study space-time symmetries in Non-Commutative (NC) gauge theory in the (constrained) Hamiltonian framework. The specific example of NC CP (1) model, posited in [9], has been considered. Subtle features of Lorentz invariance violation in NC field theory were pointed out in . Out of the two -Observer and Particle -distinct types of Lorentz transformations, symmetry under the former, (due to the translation invariance), is reflected in the conservation of energy and momentum in NC theory. The constant tensor θ µν (the noncommutativity parameter) destroys invariance under the latter.
Functional Integral Approach to Quantum Gauge Field Theories on a Noncommutative Space-time
2006
We discuss a functional integral approach to construction of Lorentz-covariant quantum gauge theories on a noncommutative space-time. There have been quite a number of work in this direction, mostly using various Moyal-type star products to construct Lagrangians. One of the most influential works was that of Seiberg and Witten [8], where they, among many other things, noted that simple problems of evolution of a string in a background force field invariably leads to some kind of noncommutativity of space-time coordinates. The type of noncommutativity they were using led to violation of the Lorentz covariance. There was a lot of works discussing these violations and attempting to fix this problem. In our paper [3] we proposed a version of the Moyal-type approach based on a group earlier used by Doplicher-Fredenhagen-Roberts [4] for other reasons. We came to the this group by contracting the group SO(4, 1) used by Snyder [9] to treat noncommutative space-time. Though our approach allo...