Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A pseudo-bosonic Klein-Gordon field with finite two-points function

Fabio Bagarello

TL;DR

This work addresses the divergence of equal-time two-point functions in standard quantum field theory by introducing a pseudo-bosonic Klein-Gordon field (PBKGF) in 1+1 dimensions, built from a continuous family of pseudo-bosonic modes. The author constructs the field using $A_θ(k)$ and $B_θ(k)$ derived from $c(k)$ and $c^2dagger(k)$, yielding a θ-dependent field $Φ_θ(x,t)$ and a Hamiltonian $H_θ$ that, while θ-dependent at the density level, reproduces the usual Klein-Gordon dynamics for the total energy. A key result is that, for θ=π/4, the equal-time two-point function $F_θ^{(2)}(x,0;x,0)$ is finite for $x≠0$, with $F_{π/4}^{(2)}(x,0;x,0)=(i/2π)K_0(2m|x|)$, and the conjugate momentum two-point function $G_θ^{(2)}$ can be handled distributionally. This suggests a novel, non-renormalization-like mechanism for regulating divergences via the pseudo-bosonic structure, motivating further exploration of PBs in QFT and their potential applicability to interacting theories and higher dimensions.

Abstract

We introduce a class of pseudo-bosonic Klein-Gordon fields in 1+1 dimensions and we discuss some of their properties. This work originates from non Hermitian quantum mechanics and deformed canonical commutation relations. We show that, within this class of fields, there exist a specific subclass with the interesting feature of having finite equal space-time two-points function, contrarily to what happens for {\em standard} Klein-Gordon fields. This, in our opinion, is a relevant aspect of our proposal which is a good motivation to undertake a deeper analysis of this (and related) quantum fields.

A pseudo-bosonic Klein-Gordon field with finite two-points function

TL;DR

This work addresses the divergence of equal-time two-point functions in standard quantum field theory by introducing a pseudo-bosonic Klein-Gordon field (PBKGF) in 1+1 dimensions, built from a continuous family of pseudo-bosonic modes. The author constructs the field using and derived from and , yielding a θ-dependent field and a Hamiltonian that, while θ-dependent at the density level, reproduces the usual Klein-Gordon dynamics for the total energy. A key result is that, for θ=π/4, the equal-time two-point function is finite for , with , and the conjugate momentum two-point function can be handled distributionally. This suggests a novel, non-renormalization-like mechanism for regulating divergences via the pseudo-bosonic structure, motivating further exploration of PBs in QFT and their potential applicability to interacting theories and higher dimensions.

Abstract

We introduce a class of pseudo-bosonic Klein-Gordon fields in 1+1 dimensions and we discuss some of their properties. This work originates from non Hermitian quantum mechanics and deformed canonical commutation relations. We show that, within this class of fields, there exist a specific subclass with the interesting feature of having finite equal space-time two-points function, contrarily to what happens for {\em standard} Klein-Gordon fields. This, in our opinion, is a relevant aspect of our proposal which is a good motivation to undertake a deeper analysis of this (and related) quantum fields.
Paper Structure (5 sections, 1 theorem, 53 equations)

This paper contains 5 sections, 1 theorem, 53 equations.

Table of Contents

  1. Introduction
  2. A short review on pseudo-bosons
  3. The pseudo-bosonic Klein Gordon field
  4. A finite two points function
  5. Conclusions

Key Result

Theorem 2

Let $\phi \in {\cal D}({\bf R})$ be a given function with supp $\phi \subseteq [-1,1]$ and $\int \phi (x) \, dx =1$. We call $\delta-$sequence the sequence $\delta_n,\, n\in {\bf N},$ defined by $\delta_n(x) \equiv n\, \phi(nx)$. Then, $\forall \, T \in {\cal D'}({\bf R})$, the set of distribution

Theorems & Definitions (2)

  • Definition 1
  • Theorem 2