# Nonlocal Scalar Quantum Field Theory—Functional Integration, Basis Functions Representation and Strong Coupling Expansion

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## Abstract

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## 1. Introduction

## 2. General Theory

#### 2.1. Derivation of GF $\mathcal{Z}$ in Basis Functions Representation—Trivial Integration

#### 2.1.1. Inverse Propagator Splitting

#### 2.1.2. Primary Field Integration

#### 2.2. Derivation of GF $\mathcal{Z}$ in Basis Functions Representation—Nontrivial Integration

#### 2.2.1. 1-Norm and Product Integral

#### 2.2.2. 0-Norm and Replica-Functional Taylor Series

#### 2.2.3. Another Way to Derive 0-Norm

#### 2.2.4. Grand Canonical Partition Function—Like 0-Metric

## 3. Polynomial Theory ${\mathbf{\phi}}_{\mathbf{D}}^{\mathbf{4}}$

#### 3.1. GF $\mathcal{Z}$ of the ${\phi}_{D}^{4}$ Theory in Basis Functions Representation: 1-Norm

#### 3.2. GF $\mathcal{Z}$ of the ${\phi}_{D}^{4}$ Theory in Basis Functions Representation: 0-Norm

## 4. Nonpolynomial Theory ${\mathrm{Sin}\mathrm{h}}^{\mathbf{4}}{\mathbf{\phi}}_{\mathbf{D}}$

#### 4.1. GF $\mathcal{Z}$ of the ${\mathrm{Sin}\mathrm{h}}^{4}{\phi}_{D}$ Theory in Basis Functions Representation: 1-Norm

#### 4.2. GF $\mathcal{Z}$ of the ${\mathrm{Sin}\mathrm{h}}^{4}{\phi}_{D}$ Theory in Basis Functions Representation: 0-Norm

## 5. Continuous Lattice of Functions

#### 5.1. Derivation of GF $\mathcal{Z}$ in Basis Functions Representation

#### 5.2. Polynomial Theory ${\phi}_{D}^{4}$

#### 5.3. GF $\mathcal{Z}$ in Terms of the Parseval–Plancherel Identity

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

FRG | Functional Renormalization Group |

GF | Generating Functional |

GLT | Ginzburg–Landau Theory |

HS | Hilbert Space |

PT | Perturbation Theory |

QCD | Quantum Chromodynamics |

QED | Quantum Electrodynamics |

QFT | Quantum Field Theory |

QG | Quantum Gravity |

RG | Renormalization Group |

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**Figure 1.**A plot demonstrates how the function ${\int}_{-\infty}^{+\infty}d\xi {e}^{-{\xi}^{2n}+iw\xi}$ ($n=2,3,4$) evolves with a change of value w. The plotted function is not positive-definite.

**Figure 2.**The 1-norm in the nonpolynomial ${sinh}^{4}\phi $ theory plotted with respect to w for different values of the coupling constant g. The field scale ${\phi}_{0}=1$.

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**MDPI and ACS Style**

Bernard, M.; Guskov, V.A.; Ivanov, M.G.; Kalugin, A.E.; Ogarkov, S.L. Nonlocal Scalar Quantum Field Theory—Functional Integration, Basis Functions Representation and Strong Coupling Expansion. *Particles* **2019**, *2*, 385-410.
https://doi.org/10.3390/particles2030024

**AMA Style**

Bernard M, Guskov VA, Ivanov MG, Kalugin AE, Ogarkov SL. Nonlocal Scalar Quantum Field Theory—Functional Integration, Basis Functions Representation and Strong Coupling Expansion. *Particles*. 2019; 2(3):385-410.
https://doi.org/10.3390/particles2030024

**Chicago/Turabian Style**

Bernard, Matthew, Vladislav A. Guskov, Mikhail G. Ivanov, Alexey E. Kalugin, and Stanislav L. Ogarkov. 2019. "Nonlocal Scalar Quantum Field Theory—Functional Integration, Basis Functions Representation and Strong Coupling Expansion" *Particles* 2, no. 3: 385-410.
https://doi.org/10.3390/particles2030024