This article reviews the generalization of field theory to space-time with noncommuting coordinates, starting with the basics and covering most of the active directions of research. Such theories are now known to emerge from limits of M theory and string theory and to describe quantum Hall states. In the last few years they have been studied intensively, and many qualitatively new phenomena have been discovered, on both the classical and the quantum level.

Noncommutative field theory

Quantum field theory on noncommutative spaces

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Methods Of Modern Mathematical Physics

We propose uncertainty relations for the different coordinates of spacetime events, motivated by Heisenberg's principle and by Einstein's theory of classical gravity. A model of Quantum Spacetime is then discussed where the commutation relations exactly implement our uncertainty relations.

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The quantum structure of spacetime at the Planck scale and quantum fields

This paper introduces five notions, including π-algebras, π-ideals, Hopf π-algebras, π-modules and Hopf π-modules, verifies the fundamental isomorphism theorem of π-algebras and studies some algebraic properties of Hopf π-algebras as well.

Hopf π- Algebras

A formulation of supersymmetric quantum mechanics is given and superunitary and generalized canonical transformations are defined acting in a module. Next it is assumed that there are operators that give an irreducible representation of the canonical commutation and anticommutation relations, respectively, and it is proved that two such representations are connected by a uniquely determined superunitary transformation, under suitable domain assumptions. This extends the well‐known uniqueness theorem of von Neumann to canonical (anti−) commutation relations using anticommuting parameters.

A supersymmetric generalization of von Neumann’s theorem

Low-Dimensional Models in Statistical Physics and Quantum Field Theory

Quantum groups (or more precisely, function algebras on quantum groups), i.e. bialgebras with certain additional properties, can be gained by deforming an appropriate function algebra on a group. In a similar way, we show that a polynomial-like algebra on the (function algebra of the) Manin plane leads to a so-called trialgebra (as suggested by Crane and Frenkel), i.e. an algebraic structure possessing a coproduct and two products in a compatible way. We show how to deform this trialgebra to a noncocommutative and totally noncommutative (i.e. in both products) one. Trialgebras are of interest for various reasons, e.g. the search for topological invariants for four manifolds or the duality operation for non-Abelian lattice gauge theories, recently suggested by the authors.

On a Trialgebraic Deformation of the Manin Plane

It was a great pleasure for me (Harald Grosse) to be invited to talk at the meeting celebrating the 70th birthday of Prof. Julius Wess. I remember various interactions with Julius during the last years: At the time of my studies at Vienna with Walter Thirring, Julius left already Vienna, I learned from his work on effective chiral Lagrangians. Next we met at various conferences and places like CERN (were I worked with Andre Martin, an old friend of Julius), and we all learned from Julius' and Bruno's creation of supersymmetry, next we realized our common interests in noncommutative quantum field theory and did have an intensive exchange. Julius influenced our perturbative approach to gauge field theories were we used the Seiberg-Witten map after his advice. And finally I lively remember the sad days when during my invitation to Vienna Julius did have the serious heart attack. So we are very happy, that you recovered so well, and we wish you all the best for the forthcoming years. Many happy recurrences.

Noncommutative QFT and Renormalization

Take a one-parameter family of self-adjoint Fredholm operators {A(t)}t∈ℝ on a Hilbert space ℋ, joining endpoints A±. There is a long history of work on the question of whether the spectral flow along this path is given by the index of the operator DA = (d/dt) + A acting in L2(ℝ; ℋ), where A denotes the multiplication operator (Af)(t) = A(t)f(t) for f ∈dom(A). Most results are about the case where the operators A(⋅) have compact resolvent. In this article, we review what is known when these operators have some essential spectrum and describe some new results. Using the operators H1 = DA∗D A, H2 = DADA∗, an abstract trace formula for Fredholm operators with essential spectrum was proved in [23], extending a result of Pushnitski [35], although, still under strong hypotheses on A(⋅): trL2(ℝ;ℋ)((H2 − zI)−1 − (H 1 − zI)−1) = 1 2ztrL2(ℋ)(gz(A+) − gz(A−)), where gz(x) = x(x2 − z)−1/2, x ∈ ℝ, z ∈ ℂ\[0,∞). Associated to the pairs (H2,H1) and (A+,A−) are Krein spectral shift functions ξ(⋅; H2,H1) and ξ(⋅; A+,A−), respectively. From the trace formula, it was shown that there is a second, Pushnitski-type, formula: ξ(λ; H2,H1) = 1 π∫−λ1/2λ1/2 ξ(ν; A+,A−)dν (λ − ν2)1/2 for a.e. λ > 0. This can be employed to establish the desired equality, Fredholm index = ξ(0; A+,A−) = spectral flow. This equality was generalized to non-Fredholm operators in [14] in the form Witten index = [ξR(0; A+,A−) + ξL(0; A+,A−)]/2, replacing the Fredholm index on the left-hand side by the Witten index of DA and ξ(0; A+,A−) on the right-hand side by an appropriate arithmetic mean (assuming 0 is a right and left Lebesgue point for ξ(⋅; A+,A−) denoted by ξR(0; A+,A−) and ξL(0; A+,A−), respectively). But this applies only under the restrictive assumption that the endpoint A+ is a relatively trace class perturbation of A− (ruling out general differential operators). In addition to reviewing this previous work, we describe in this article some extensions using a (1 + 1)-dimensional setup, where A± are non-Fredholm differential operators. By a careful analysis we prove, for a class of examples, that the preceding trace formula still holds in this more general situation. Then we prove that the Pushnitski-type formula for spectral shift functions also holds and this then gives the equality of spectral shift functions in the form ξ(λ; H2,H1) = ξ(ν; A+,A−)for a.e. λ > 0 and a.e.ν ∈ ℝ, for the (1 + 1)-dimensional model operator at hand. This shows that neither the relatively trace class perturbation assumption nor the Fredholm assumption are required if one works with spectral shift functions. The results support the view that the spectral shift function should be a replacement for the spectral flow in certain non-Fredholm situations and also point the way to the study of higher-dimensional cases. We discuss the connection with summability questions in Fredholm modules in an appendix.

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Harald Grosse

Papers

A supersymmetric generalization of von Neumann’s theorem

Low-Dimensional Models in Statistical Physics and Quantum Field Theory

On a Trialgebraic Deformation of the Manin Plane

Noncommutative QFT and Renormalization

Trace Formulas for a Class of non-Fredholm Operators: A Review