From Perturbative to Constructive Renormalization

Non-commutative Renormalization

“…Nowadays renormalization is considered the heart of QFT, and even much more [7]- [8]- [9]. But initially renormalization was little more than a trick, a quick fix to remove the divergences that plagued the computations of quantum electrodynamics.…”

“…Hence the mass term plays no significant role in the higher slices. It was remarked in [9] that because there are no overlapping problem associated to 1PI two point subgraphs, there is in fact no inconvenience to use the full renormalized m ren all the way from the bare to renormalized scales, with subtractions on 1PI two point subgraphs independent of their scale.…”

“…Then apart from strengthening the "vulcanization" procedure to get renormalizable non-commutative field theories, the Gross-Neveu model may also be useful for the study of the quantum Hall effect. It is also a good first candidate for a constructive study [9] of a non-commutative field theory as Fermionic models are usually easier to construct. Moreover its commutative counterpart being asymptotically free and exhibiting dynamical mass generation [92,93,94], a study of the physics of this model would be interesting.…”

“…The multiscale analysis allows to define the so-called effective expansion, in between the bare and the renormalized expansion, which is optimal, both for physical and for constructive purposes [9]. In this effective expansion only the subcontributions with all internal scales higher than all external scales have to be renormalized by counterterms of the form of the initial Lagrangian.…”

Constructive matrix theory

“…Constructive field theory build functions whose Taylor expansion is perturbative field theory [1,2]. Any formal power series being asymptotic to infinitely many smooth functions, perturbative field theory alone does not provide any well defined mathematical recipe to compute to arbitrary accuracy any physical quantity, so in a deep sense it is no theory at all.…”