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Causal Perturbative QFT and Space-time Geometry

Jaroslaw Wawrzycki

Abstract

This work is devoted to the causal perturbative Quantum Field Theory (QFT) due to Bogoliubov, including QED and other realistic QFT. It is given the white noise formulation of this theory. The white noise analysis and the Hida operators as the creation and annihilation operators for free fields are used. The whole Bogoliubov method is unchanged. Causal axioms of such QFT make sense on any globally causal space-times. Perturbative QFT with Hida operators is analysed on the flat Minkowski space-time and on the static Einstein Universe (EU). On the flat Minkowski space-time this allowed us to go a step further in the analysis of the adiabatic limit problem, the existence of which has a much wider scope than in the theory based on Wightman's operator valued distributions. The natural condition of existence of the adiabatic limit (as generalized integral kernel operators of white noise calculus) for higher-order contributions to the interacting fields, together with the remaining assumptions of causal QFT (including the natural invariance conditions, e.g. gauge invariance), allowed us to reduce the freedom in choice of renormalization and imposed nontrivial conditions on the masses of elementary particles. Obtained results on the flat space-time are confirmed by the analogous results obtained for QFT on UE, where, moreover, the operators of the interacting fields are much more regular, so, e.g. a consequent treatment of the bound state problem becomes possible on EU. Our approach also enables the analysis of the infrared asymptotics of QED on flat Minkowski spacetime, which we also present.

Causal Perturbative QFT and Space-time Geometry

Abstract

This work is devoted to the causal perturbative Quantum Field Theory (QFT) due to Bogoliubov, including QED and other realistic QFT. It is given the white noise formulation of this theory. The white noise analysis and the Hida operators as the creation and annihilation operators for free fields are used. The whole Bogoliubov method is unchanged. Causal axioms of such QFT make sense on any globally causal space-times. Perturbative QFT with Hida operators is analysed on the flat Minkowski space-time and on the static Einstein Universe (EU). On the flat Minkowski space-time this allowed us to go a step further in the analysis of the adiabatic limit problem, the existence of which has a much wider scope than in the theory based on Wightman's operator valued distributions. The natural condition of existence of the adiabatic limit (as generalized integral kernel operators of white noise calculus) for higher-order contributions to the interacting fields, together with the remaining assumptions of causal QFT (including the natural invariance conditions, e.g. gauge invariance), allowed us to reduce the freedom in choice of renormalization and imposed nontrivial conditions on the masses of elementary particles. Obtained results on the flat space-time are confirmed by the analogous results obtained for QFT on UE, where, moreover, the operators of the interacting fields are much more regular, so, e.g. a consequent treatment of the bound state problem becomes possible on EU. Our approach also enables the analysis of the infrared asymptotics of QED on flat Minkowski spacetime, which we also present.

Paper Structure

This paper contains 101 sections, 198 theorems, 6452 equations.

Table of Contents

  1. Introduction
  2. Adiabatic Limit Problem and its solution
  3. Ultraviolet and infrared asymptotics of the interacting field
  4. An algebra of operators in the Fock space closely related to the space-time
  5. On the relation between the space-time geometry and the interacting quantum field
  6. General remark on the notation
  7. Krein-isometric representations concentrated on single orbits and the transform $V_\mathcal{F}$
  8. Example 1: Representation $2U^{{}_{(m,0,0,0)} L^{{}^{1/2}}} =U^{{}_{(m,0,0,0)} 2L^{{}^{1/2}}}$ (spin $1/2$) and the Dirac equation
  9. Example 2: Representation $U^{{}_{(0,m,0,0)} [2L^{{}^{1/2}}]_{\textrm{Ass}}}$ associated to $U^{{}_{(m,0,0,0)} 2L^{{}^{1/2}}}$ (spin $1/2$) and concentrated on the orbit $\mathscr{O}_{(0,0,m,0)}$
  10. Spin 1/2 and the transform $V_\mathcal{F}$
  11. Example 3: Representation $U^{{}_{(m,0,0,0)} 4(L^{{}^{1/2}} \otimes L^{{}^{1/2}})}$ (spin 0 and 1) and the generalized Dirac equation
  12. Example 4: Representation $U^{{}_{(m,0,0,0)} [4(L^{{}^{1/2}})^{\otimes 2}]_\textrm{Ass}}$ associated to $U^{{}_{(m,0,0,0)} 4(L^{{}^{1/2}} \otimes L^{{}^{1/2}})}$ (spin 0 and 1) and concentrated on the orbit $\mathscr{O}_{(0,m,0,0)}$
  13. Spin 0 and 1 and the transform $V_\mathcal{F}$
  14. Direct integrals of higher spin representations and the construction of $V_\mathcal{F}$
  15. Construction of $V_\mathcal{F}$ and of the Dirac operator in the (Krein-) Hilbert space of free fields
  16. ...and 86 more sections

Key Result

Theorem 1

Let the extension $V$ of the representation $L(\cdot) = Q(\cdot, \bar{p})$ be Krein-unitary. Then the second transformation $W"$ defined by (W2), which transforms $\widetilde{\psi}$ belonging to the Krein space $(\mathcal{H}, \mathfrak{J})$ of the initial representation, equal to the Krein space of

Theorems & Definitions (247)

  • Theorem
  • Definition
  • Remark
  • Remark
  • Proposition
  • Definition
  • Definition
  • Definition
  • Remark 1
  • Remark 2
  • ...and 237 more