An ultraviolet‐finite Hamiltonian approach on the noncommutative Minkowski space

Abstract

This is an exposition of joint work with S. Doplicher, K. Fredenhagen, and G. Piacitelli on field theory on the noncommutative Minkowski space [1]. The limit of coinciding points is modified compared to ordinary field theory in a suitable way which allows for the definition of so-called regularized field monomials as interaction terms. Employing these in the Hamiltonian formalism results in an ultraviolet finite S-matrix.

Quantum Field Theory on Quantum Spacetime

“…This feature might imply a better reguralization of interactions: indeed already the one implied by the Basic Model for interactions in the form originally introduced in [3], has been shown to be sufficient to reguralise the φ 3 interaction [17]; the present model might well go beyond.…”

DFR Perturbative Quantum Field Theory on Quantum Space Time, and Wick Reduction

“…More recent concerns about possible unitarity violation were the consequence of a too naive way of performing the time ordering prescription (see [10], and references therein; see also [11,12]). The ultraviolet regularity of a φ ⋆3 DFR model has recently been proved by Bahns [13], under a weaker prescription for averaging over σ.…”

Dispersion Relations in the Noncommutative $$\phi^3$$ and Wess–Zumino Model in the Yang–Feldman Formalism

“…stands for a term that is normal ordered and whose spectrum has no overlap with the positive or negative mass shell if the support ofĝ is chosen small enough. Thus, this term drops out in the first two terms in (6). As shown below,Σ can be identified with the inverse Fourier transform of the self-energy.…”

“…for the first two terms in (6) under the condition that Σ(k) = Σ(−k) in a neighborhood of the mass shell. Here Σ is the Fourier transform ofΣ and can be identified with the self-energy.…”

“…If then Σ(k) = Σ(−k) in a neighborhood of the mass shell, we use (8) as the sum of the first two terms in (6). Now Σ(k) is in general not only a function of k 2 , but also of (kσ) 2 .…”