# S-Matrix of Nonlocal Scalar Quantum Field Theory in Basis Functions Representation

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. General Theory

#### 2.1. Derivation of the $\mathcal{S}$-Matrix in Terms of the Grand Canonical Partition Function

#### 2.2. Propagator Splitting

#### 2.3. Derivation of the $\mathcal{S}$-Matrix in the Representation of Basis Functions (Discrete Lattice of Functions)

#### 2.4. Majorant and Variational Principle

#### 2.5. Derivation of the $\mathcal{S}$-Matrix in the Representation of Basis Functions (Continuous Lattice of Functions)

#### 2.6. Redefinition of the $\mathcal{S}$-Matrix and Variational Principle

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

#### 3.1. $\mathcal{S}$-Matrix of the ${\phi}_{D}^{4}$ Theory in the Representation of Basis Functions (Discrete Lattice of Functions)

#### 3.2. Majorant of the $\mathcal{S}$-Matrix of the ${\phi}_{D}^{4}$ Theory (Discrete Lattice of Functions)

#### 3.3. Integral Equation with Separable Kernel (Discrete Lattice of Functions)

#### 3.4. Majorant of the $\mathcal{S}$-Matrix of the ${\phi}_{D}^{4}$ Theory (Continuous Lattice of Functions)

#### 3.5. Integral Equation with Separable Kernel (Continuous Lattice of Functions)

## 4. Nonpolynomial Sine-Gordon Theory

#### 4.1. Majorant of the $\mathcal{S}$-Matrix of the Sine-Gordon Theory (Discrete Lattice of Functions)

#### 4.2. Estimates for the Solution of the Variational Principle Equations (Discrete Lattice of Functions)

#### 4.2.1. Upper Bound

#### 4.2.2. Lower Bound

#### 4.3. Simple Majorant of $\mathcal{G}$ on the Discrete Lattice of Functions

#### 4.4. Majorant of the $\mathcal{S}$-Matrix of the Sine-Gordon Theory (Continuous Lattice of Functions)

#### 4.5. Estimates for the Solution of the Variational Principle Equations (Continuous Lattice of Functions)

#### 4.5.1. Upper Bound

#### 4.5.2. Lower Bound

#### 4.6. Simple Majorant of $\mathcal{G}$ on the Continuous Lattice of Functions

## 5. From D = 1 to D = 26 and Beyond

#### 5.1. Polynomial Theory ${\phi}_{1}^{4}$

#### 5.1.1. Analytics

#### 5.1.2. Plots

#### 5.2. Nonpolynomial Sine-Gordon Theory

#### 5.2.1. Analytics

#### 5.2.2. Plots

#### 5.3. Nonlocal QFT and Compactification Process

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

QFT | Quantum Field Theory |

QED | Quantum Electrodynamics |

QCD | Quantum Chromodynamics |

SM | Standard Model |

RG | Renormalization Group |

FRG | Functional Renormalization Group |

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**Figure 1.**Dependence of the variational functions ${q}_{s}$ on s for different values of the parameters of width w and $\epsilon $. At the same time, we chose the following system of units: the product $\sqrt{g\left(0\right)}{D}^{2}\left(0\right)=1$, as well as the width of the basis functions ${\psi}_{s}\left(x\right)$.

**Figure 2.**Dependence of the functions ${q}_{s}$ on s for different values of the amplitude of the coupling constant $g\left(0\right)$. The parameters w and $\epsilon $ are fixed and equal to $w=100$ and $\epsilon =0.5$, respectively. The amplitude of the propagator $D\left(0\right)=1$ (in this case, the variation of the product $\sqrt{g\left(0\right)}{D}^{2}\left(0\right)$ means the variation of the $g\left(0\right)$).

**Figure 3.**Dependence of the majorant ${F}_{\mathrm{sep}}$ on the width w of the coupling constant $g\left(x\right)$ for different values of the parameters $\epsilon $ and the amplitude of the coupling constant $g\left(0\right)$. (

**Left**) $\sqrt{g\left(0\right)}{D}^{2}\left(0\right)=1$. (

**Right**) $\epsilon =0.5$; $D\left(0\right)=1$.

**Figure 4.**Dependence of the majorant ${F}_{\mathrm{sep}}$ on the amplitude of the coupling constant $g\left(0\right)$ for different values of $\epsilon $. The width parameter $w=100$, the propagator amplitude $D\left(0\right)=1$ (which demonstrates the dynamics with respect to the value $g\left(0\right)$).

**Figure 5.**Dependence of the variational functions ${q}_{s}^{(+)}$ on s. (

**Left**) w changes, $\epsilon =0.5$. (

**Right**) $\epsilon $ changes, $w=200$. For both plots $g\left(0\right)=100$.

**Figure 6.**Dependence of the functions ${q}_{s}^{(+)}$ on s for different values of the amplitude of the coupling constant $g\left(0\right)$. The parameters w and $\epsilon $ are fixed and equal to $w=200$ and $\epsilon =0.5$, respectively.

**Figure 7.**Dependence of the variational functions ${q}_{s}^{(-)}$ on s. (

**Left**) w changes, $\epsilon =0.5$. (

**Right**) $\epsilon $ changes, $w=200$. For both plots $g\left(0\right)=100$.

**Figure 8.**Dependence of the functions ${q}_{s}^{(-)}$ on s for different values of the amplitude of the coupling constant $g\left(0\right)$. The parameters w and $\epsilon $ are fixed and equal to $w=200$ and $\epsilon =0.5$, respectively.

**Figure 9.**Dependence of the ratios ${q}_{s}/{q}_{s}^{(-)}$ on s. (

**Left**) w changes, $\epsilon =0.5$. (

**Right**) $\epsilon $ changes, $w=200$. For both plots $g\left(0\right)=100$.

**Figure 10.**Dependence of the ratios ${q}_{s}/{q}_{s}^{(-)}$ on s for different values of the amplitude of the coupling constant $g\left(0\right)$. The parameters w and $\epsilon $ are fixed and equal to $w=200$ and $\epsilon =0.5$, respectively.

**Figure 11.**Dependence of the majorant ${F}_{\mathrm{simple}}+g$ on the width of the coupling constant w for different values of the parameters $\epsilon $ and the amplitude of the coupling constant $g\left(0\right)$. (

**Left**) $g\left(0\right)=100$. (

**Right**) $\epsilon =0.5$.

**Figure 12.**Dependence of the majorant ${F}_{\mathrm{simple}}+g$ on the amplitude of the coupling constant $g\left(0\right)$ for different values of the parameter $\epsilon $. The width parameter $w=200$.

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**MDPI and ACS Style**

Chebotarev, I.V.; Guskov, V.A.; Ogarkov, S.L.; Bernard, M. *S*-Matrix of Nonlocal Scalar Quantum Field Theory in Basis Functions Representation. *Particles* **2019**, *2*, 103-139.
https://doi.org/10.3390/particles2010009

**AMA Style**

Chebotarev IV, Guskov VA, Ogarkov SL, Bernard M. *S*-Matrix of Nonlocal Scalar Quantum Field Theory in Basis Functions Representation. *Particles*. 2019; 2(1):103-139.
https://doi.org/10.3390/particles2010009

**Chicago/Turabian Style**

Chebotarev, Ivan V., Vladislav A. Guskov, Stanislav L. Ogarkov, and Matthew Bernard. 2019. "*S*-Matrix of Nonlocal Scalar Quantum Field Theory in Basis Functions Representation" *Particles* 2, no. 1: 103-139.
https://doi.org/10.3390/particles2010009