Introduction to Noncommutative Field Theory

But what is it exactly? Let us motivate the problem in a mathematical way: consider the Gel’fand-Naimark duality theorem, a mathematical theorem which establishes a one-to-one correspondence between topological (Hausdorff) spaces X and commutative C∗-algebras A=C(X), where A is the algebra of continuous complex-valued functions f : X → C with the pointwise multiplication (f.g)(x) = f(x).g(x). Generalizing this result to non-commutative C∗-algebras, we obtain a relation between noncommutative algebras and noncommutative spaces. All geometry we know (differential, Riemannian, etc.) is done in a purely operator algebraic framework (originally by Connes, Woronowicz, Drinfel’d, etc., in the 1980’s) which enables one to define field theories (in particular Yang-Mills gauge theories) on large classes of noncommutative spaces. Note that in general, the tensor θ can depend on spacetime coordinates x, and even on momenta p (defining algebras of pseudo-differential operators).