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Regularized interacting scalar quantum field theories

Nicola Pinamonti

TL;DR

This work provides a rigorous, perturbative construction of interacting scalar quantum field theories in the pAQFT framework by introducing a two-parameter regularization of propagators and vertex operators, yielding convergent S-matrices that are represented as unitary operators in appropriate Hilbert space representations. By decomposing the interaction Lagrangian into vertex-operator form and carefully regulating ultraviolet and infrared aspects, the paper proves absolute convergence of the S-matrix in multiple dimensions, and it establishes convergence to the perturbative (BPHZ) predictions in 3D and 4D after removing regulators in controlled ways. It further develops the adiabatic limit via Mayer expansions and Kirkwood-Salsburg equations, providing bounds that guarantee convergence for sufficiently small coupling and clarifying the role of regulator removal in matching standard perturbation theory. The results illuminate how regularization, renormalization, and adiabatic limits can be coherently integrated in a pAQFT setting, offering a path toward constructing unitary, renormalized interacting QFTs in low dimensions and insights into the 4D case, with potential applicability to curved spacetimes via the same formalism.

Abstract

In this paper we consider self interacting scalar quantum field theories over a $d$ dimensional Minkowski spacetime with various interaction Lagrangians which are suitable functions of the field. The interacting field observables are represented as power series over the free theory by means of perturbation theory. The object which is employed to obtain this power series is the time ordered exponential of the interaction Lagrangian which is the $S$-matrix of the theory and thus itself a power series in the coupling constant of the theory. We analyze a regularization procedure which makes the $S$-matrix convergent to well defined unitary operators. This regularization depends on two parameters. One describes how much the high frequency contributions in the propagators are tamed and a second one which describes how much the large field contributions are suppressed in the interaction Lagrangian. We finally discuss how to remove the parameters in lower dimensional theories and for specific interaction Lagrangians. In particular, we show that in three spacetime dimensions for a $φ^4_3$ theory one obtains sequences of unitary operators which are weakly-$*$ convergent to suitable unitary operators in the limit of vanishing parameters. The coefficients of the asymptotic expansion in powers of the coupling constant of all the possible limit points coincide and furthermore agree with the predictions of perturbation theory. Finally we discuss how to extend these results to the case of a $φ^4_4$ theory were the final results turns out to be very similar to the three dimensional case.

Regularized interacting scalar quantum field theories

TL;DR

This work provides a rigorous, perturbative construction of interacting scalar quantum field theories in the pAQFT framework by introducing a two-parameter regularization of propagators and vertex operators, yielding convergent S-matrices that are represented as unitary operators in appropriate Hilbert space representations. By decomposing the interaction Lagrangian into vertex-operator form and carefully regulating ultraviolet and infrared aspects, the paper proves absolute convergence of the S-matrix in multiple dimensions, and it establishes convergence to the perturbative (BPHZ) predictions in 3D and 4D after removing regulators in controlled ways. It further develops the adiabatic limit via Mayer expansions and Kirkwood-Salsburg equations, providing bounds that guarantee convergence for sufficiently small coupling and clarifying the role of regulator removal in matching standard perturbation theory. The results illuminate how regularization, renormalization, and adiabatic limits can be coherently integrated in a pAQFT setting, offering a path toward constructing unitary, renormalized interacting QFTs in low dimensions and insights into the 4D case, with potential applicability to curved spacetimes via the same formalism.

Abstract

In this paper we consider self interacting scalar quantum field theories over a dimensional Minkowski spacetime with various interaction Lagrangians which are suitable functions of the field. The interacting field observables are represented as power series over the free theory by means of perturbation theory. The object which is employed to obtain this power series is the time ordered exponential of the interaction Lagrangian which is the -matrix of the theory and thus itself a power series in the coupling constant of the theory. We analyze a regularization procedure which makes the -matrix convergent to well defined unitary operators. This regularization depends on two parameters. One describes how much the high frequency contributions in the propagators are tamed and a second one which describes how much the large field contributions are suppressed in the interaction Lagrangian. We finally discuss how to remove the parameters in lower dimensional theories and for specific interaction Lagrangians. In particular, we show that in three spacetime dimensions for a theory one obtains sequences of unitary operators which are weakly- convergent to suitable unitary operators in the limit of vanishing parameters. The coefficients of the asymptotic expansion in powers of the coupling constant of all the possible limit points coincide and furthermore agree with the predictions of perturbation theory. Finally we discuss how to extend these results to the case of a theory were the final results turns out to be very similar to the three dimensional case.

Paper Structure

This paper contains 17 sections, 19 theorems, 143 equations.

Table of Contents

  1. Introduction
  2. Algebras and regularized propagators
  3. Regularized propagators and deformed algebras
  4. Regularized product rules among vertex operators
  5. Interaction Lagrangians, S-matrices and convergence
  6. The set of interaction Lagrangians and their S-matrices
  7. Representation as unitary operators on a suitable Hilbert space
  8. Limit $\Lambda \to 0$, renormalizability, counter terms and perturbation theory
  9. 2D generic interactions of compact (and small) support in the $a$ domain
  10. The three dimensional case for an interaction Lagrangian $\lambda \phi^4$
  11. Renormalization of $\lambda \phi^4_3$
  12. Restoring perturbation theory
  13. Extracting subsequences which converge in the weak-* topology
  14. The four dimensional case for an interaction Lagrangian $\lambda \phi^4$
  15. Towards the adiabatic limits: Mayer expansion in the regularized theory
  16. ...and 2 more sections

Key Result

Theorem 1.1

In a $d\geq 2$ dimensional Minkowski spacetime $M$, consider the power series in $\lambda$ defining $S(L_I)$, the $S$-matrix constructed with the regularized propagators for an interaction Lagrangian, $L_I$ in eq:LI-vertex described by a Schwartz function of the field and localized in a compact spac

Theorems & Definitions (35)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Definition 2.5
  • ...and 25 more