On nonperturbative quantum field theory and noncommutative geometry

Abstract

A general framework of non-perturbative quantum field theory on a curved background is presented. A quantum field theory is in this setting characterised by an embedding of a space of field configurations into a Hilbert space over R ∞ . This embedding, which is only local up to a scale that we interpret as the Planck scale, coincides in the local and flat limit with the plane wave expansion known from canonical quantisation. We identify a universal Bott-Dirac operator acting in the Hilbert space over R ∞ and show that it gives rise to the free Hamiltonian both in the case of a scalar field theory and in the case of a Yang-Mills theory. These theories come with a canonical fermionic sector for which the Bott-Dirac operator also provides the Hamiltonian. We prove that these quantum field theories exist non-perturbatively for an interacting real scalar theory and for a general Yang-Mills theory, both with or without the fermionic sectors, and show that the free theories are given by semi-finite spectral triples over the respective configuration spaces. Finally, we propose a class of quantum field theories whose interactions are generated by inner fluctuations of the Bott-Dirac operator. 1

Non‐perturbative Quantum Field Theory and the Geometry of Functional Spaces

“…where Δ is the Hodge-Laplace operator. This is very similar to the construction in [2] and [3], where 4…”

“…Here the series {s i } corresponds to the momentum of a free field on a compact manifold. Note that in this case the square of B is identical to the Hamiltonian operator of an infinite-dimensional harmonic oscillator, which we in [2] showed is identical to the Hamiltonian operator of the free sector of a gauge field coupled to a fermionic field.…”

“…The setup presented in this paper is similar to the construction proposed in [2]. There, a similar Bott-Dirac operator was shown to produce the free Hamiltonian of a Yang-Mills theory coupled to a fermionic field.…”

“…To understand how the two operators are related we replace the covariant Hodge-Laplace operator Δ A with the ordinary Hodge-Laplace operator Δ in (7). In this case 𝜉 i becomes a global orthonormal frame in T and we can define a global coordinate system and expand A according to A = A 0 + ∑ i x i 𝜉 i (for details see [2]). In this case the linear part of d i will be of the form s i x i , and the Bott-Dirac operator B will correspond to the one constructed by Higson and Kasparov in [16].…”

“…were used. In [2] it was proved that the representation of the QHD(M) algebra existed in dimension 3 when p > 5 4 . The current case of S(A) = ⟨A, ΔA⟩ would correspond to p = 1 2 , and we are therefore certain, that with this choice of S the representation of the QHD(M) algebra will not converge.…”