# On Functional Hamilton–Jacobi and Schrödinger Equations and Functional Renormalization Group

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## Abstract

**:**

## 1. Introduction

## 2. Functional Hamilton–Jacobi Equation and its Hierarchy

#### 2.1. Functional Hamilton–Jacobi Equation

#### 2.2. Two-Particle Green Function Equation

#### 2.3. Translation-Invariant Solution for Green Function on Delta-Field Configuration

#### 2.3.1. Special Riccati Equation

#### 2.3.2. Self-Similar Riccati Equation

#### 2.4. Integration Formula for Functionals

#### 2.5. Translation-Invariant Functional Solution for Green Function

#### 2.6. Translation-Invariant Solution for Green Function on Constant-Field Configuration

#### 2.7. Separable Solution for Green Function on Delta-Field Configuration

## 3. Functional Schrödinger Equation and Semiclassical Approximation

#### 3.1. Functional Schrödinger Equation

#### 3.2. Derivation of Quantum Hamilton–Jacobi and Continuity Functional Equations and Semiclassics

#### 3.3. Hamilton–Jacobi and Continuity Functional Equations Hierarchies

#### 3.4. Translation-Invariant Solution for Green Functions on Delta-Field Configuration

#### 3.4.1. Hamilton–Jacobi Functional Equation Hierarchy

#### 3.4.2. Continuity Functional Equation Hierarchy and Optical Potential

#### 3.4.3. Open Quantum Field Systems

## 4. Wilson–Polchinski Functional Equation and Functional Renormalization Group

#### 4.1. Quantum and Classical Parts of the Wilson–Polchinski Equation

#### 4.2. Solution of the Approximated Wilson–Polchinski Equation: Two-Particle Green Function

#### 4.3. Solution of the Approximated Wilson–Polchinski Equation: Four-Particle Green Function

#### 4.4. Mode Coarse Graining Growth Functionals Rigorous Derivation

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AdS | Anti-de Sitter |

CFT | Conformal Field Theory |

NL | Newton–Leibniz |

HJ | Hamilton–Jacobi |

WP | Wilson–Polchinski |

GF | Green Function |

QM | Quantum Mechanics |

QFT | Quantum Field Theory |

RG | Renormalization Group |

LPA | Local Potential Approximation |

FRG | Functional Renormalization Group |

HRG | Holographic Renormalization Group |

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**MDPI and ACS Style**

Ivanov, M.G.; Kalugin, A.E.; Ogarkova, A.A.; Ogarkov, S.L. On Functional Hamilton–Jacobi and Schrödinger Equations and Functional Renormalization Group. *Symmetry* **2020**, *12*, 1657.
https://doi.org/10.3390/sym12101657

**AMA Style**

Ivanov MG, Kalugin AE, Ogarkova AA, Ogarkov SL. On Functional Hamilton–Jacobi and Schrödinger Equations and Functional Renormalization Group. *Symmetry*. 2020; 12(10):1657.
https://doi.org/10.3390/sym12101657

**Chicago/Turabian Style**

Ivanov, Mikhail G., Alexey E. Kalugin, Anna A. Ogarkova, and Stanislav L. Ogarkov. 2020. "On Functional Hamilton–Jacobi and Schrödinger Equations and Functional Renormalization Group" *Symmetry* 12, no. 10: 1657.
https://doi.org/10.3390/sym12101657