# 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 |

## References

- Zinn-Justin, J. Quantum Field Theory and Critical Phenomena; Clarendon: Oxford, UK, 1989. [Google Scholar]
- Vasiliev, A.N. The Field Theoretic Renormalization Group in Critical Behavior Theory and Stochastic Dynamics; Chapman and Hall/CRC: Boca Raton, FL, USA, 2004. [Google Scholar]
- Popov, V.N. Path Integrals in Quantum Field Theory and Statistical Physics; Atomizdat: Moscow, Russia, 1976. (In Russian) [Google Scholar]
- Kleinert, H. Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets; World Scientific Publishing Co. Pte. Ltd.: Singapore, 2009. [Google Scholar]
- Mosel, U. Path Integrals in Field Theory. An Introduction; Advanced Texts in Physics; Springer: Berlin/Heidelberg, Germany, 2004. [Google Scholar]
- Simon, B. Functional Integration and Quantum Physics; AMS Chelsea Publishing: Providence, RI, USA, 2005. [Google Scholar]
- Efimov, G.V.; Ivanov, M.A. The Quark Confinement Model of Hadrons; Taylor and Francis Group: New York, NY, USA, 1993. [Google Scholar]
- Dineykhan, M.; Efimov, G.V.; Ganbold, G.; Nedelko, S.N. Oscillator Representation in Quantum Physics; Lecture Notes in Physics; Springer: Berlin/Heidelberg, Germany, 1995. [Google Scholar]
- Brydges, D.C.; Martin, P.A. Coulomb Systems at Low Density: A Review. J. Stat. Phys.
**1999**, 96, 1163–1330. [Google Scholar] [CrossRef] [Green Version] - Rebenko, A.L. Mathematical Foundations of Equilibrium Classical Statistical Mechanics of Charged Particles. Russ. Math. Surv.
**1988**, 43, 65–116. [Google Scholar] [CrossRef] - Efimov, G.V. Strong Coupling in the Quantum Field Theory with Nonlocal Nonpolynomial Interaction. Commun. Math. Phys.
**1977**, 57, 235–258. [Google Scholar] [CrossRef] - Efimov, G.V. Vacuum Energy in g-Phi-4 Theory for Infinite g. Commun. Math. Phys.
**1979**, 65, 15–44. [Google Scholar] [CrossRef] - Efimov, G.V. Elastic Scattering and the Path Integral. Theor. Math. Phys.
**2014**, 179, 695–711. [Google Scholar] [CrossRef] [Green Version] - Efimov, G.V. Quantum Particle in a Random Medium. Theor. Math. Phys.
**2015**, 185, 1433–1444. [Google Scholar] [CrossRef] - Albeverio, S.A.; Høegh-Krohn, R.J.; Mazzucchi, S. Mathematical Theory of Feynman Path Integrals. An Introduction; Lecture Notes in Mathematics; Springer: Berlin/Heidelberg, Germany, 2008. [Google Scholar]
- Mazzucchi, S. Mathematical Feynman Path Integrals and Their Applications; World Scientific Publishing Co. Pte. Ltd.: Singapore, 2009. [Google Scholar]
- Cartier, P.; DeWitt-Morette, C. Functional Integration: Action and Symmetries; Cambridge Monographs on Mathematical Physics; Cambridge University Press: Cambridge, UK, 2006. [Google Scholar]
- Montvay, I.; Münster, G. Quantum Fields on a Lattice; Cambridge Monographs on Mathematical Physics; Cambridge University Press: Cambridge, UK, 1994. [Google Scholar]
- Johnson, G.W.; Lapidus, M.L. The Feynman Integral and Feynman’s Operational Calculus; Oxford Mathematical Monographs; Oxford University Press: Oxford, UK, 2000. [Google Scholar]
- Grosche, C.; Steiner, F. Handbook of Feynman Path Integrals; Springer Tracts in Modern Physics; Springer: Berlin/Heidelberg, Germany, 1998. [Google Scholar]
- Smolyanov, O.G.; Shavgulidze, E.T. Path Integrals; URSS: Moscow, Russia, 2015. (In Russian)
- 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, 9. [Google Scholar] [CrossRef] [Green Version] - 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, 24. [Google Scholar] [CrossRef] [Green Version] - Kopietz, P.; Bartosch, L.; Schütz, F. Introduction to the Functional Renormalization Group; Lecture Notes in Physics; Springer: Berlin/Heidelberg, Germany, 2010. [Google Scholar]
- Wipf, A. Statistical Approach to Quantum Field Theory; Lecture Notes in Physics; Springer: Berlin/Heidelberg, Germany, 2013. [Google Scholar]
- Rosten, O.J. Fundamentals of the Exact Renormalization Group. Phys. Rep.
**2012**, 511, 177–272. [Google Scholar] [CrossRef] [Green Version] - Igarashi, Y.; Itoh, K.; Sonoda, H. Realization of Symmetry in the ERG Approach to Quantum Field Theory. Prog. Theor. Phys. Suppl.
**2009**, 181, 1–166. [Google Scholar] [CrossRef] [Green Version] - Efimov, G.V. Nonlocal Interactions of Quantized Fields; Nauka: Moscow, Russia, 1977. (In Russian) [Google Scholar]
- Efimov, G.V. Problems of the Quantum Theory of Nonlocal Interactions; Nauka: Moscow, Russia, 1985. (In Russian) [Google Scholar]
- Petrina, D.Y.; Skripnik, V.I. Kirkwood–Salzburg Equations for the Coefficient Functions of the Scattering Matrix. Theor. Math. Phys.
**1971**, 8, 896–904. [Google Scholar] [CrossRef] - Rebenko, A.L. On Equations for the Matrix Elements of Euclidean Quantum Electrodynamics. Theor. Math. Phys.
**1972**, 11, 525–536. [Google Scholar] [CrossRef] - Fradkin, E.S. Selected Papers on Theoretical Physics; Papers in English and Russian; Nauka: Moscow, Russia, 2007. [Google Scholar]
- Lizana, J.M.; Morris, T.R.; Pérez-Victoria, M. Holographic Renormalisation Group Flows and Renormalisation from a Wilsonian Perspective. J. High Energy Phys.
**2016**, 2016, 198. [Google Scholar] [CrossRef] [Green Version] - Akhmedov, E.T. A Remark on the AdS/CFT Correspondence and the Renormalization Group Flow. Phys. Lett. B
**1998**, 442, 152–158. [Google Scholar] [CrossRef] [Green Version] - De Boer, J.; Verlinde, E.; Verlinde, H. On the Holographic Renormalization Group. J. High Energy Phys.
**2000**, 2000, 3. [Google Scholar] [CrossRef] [Green Version] - Verlinde, E.; Verlinde, H. RG-Flow, Gravity and the Cosmological Constant. J. High Energy Phys.
**2000**, 2000, 34. [Google Scholar] [CrossRef] [Green Version] - Fukuma, M.; Matsuura, S.; Sakai, T. Holographic Renormalization Group. Prog. Theor. Phys.
**2003**, 4, 489–562. [Google Scholar] [CrossRef] [Green Version] - Akhmedov, E.T. Notes on Multi-Trace Operators and Holographic Renormalization Group. arXiv
**2002**, arXiv:hep-th/0202055v3. [Google Scholar] - Akhmedov, E.T.; Gahramanov, I.B.; Musaev, E.T. Hints on Integrability in the Wilsonian/Holographic Renormalization Group. JETP Lett.
**2011**, 93, 545–550. [Google Scholar] [CrossRef] [Green Version] - Heemskerk, I.; Polchinski, J. Holographic and Wilsonian Renormalization Groups. J. High Energy Phys.
**2011**, 2011, 31. [Google Scholar] [CrossRef] [Green Version] - Maldacena, J.M. The Large-N Limit of Superconformal Field Theories and Supergravity. Int. J. Theor. Phys.
**1999**, 38, 1113–1133. [Google Scholar] [CrossRef] [Green Version] - Witten, E. Anti-de Sitter Space and Holography. Adv. Theor. Math. Phys.
**1998**, 2, 253–291. [Google Scholar] [CrossRef] - Gubser, S.S.; Klebanov, I.R.; Polyakov, A.M. Gauge Theory Correlators from Non-Critical String Theory. Phys. Lett. B
**1998**, 428, 105–114. [Google Scholar] [CrossRef] [Green Version] - Polchinski, J. String Theory; Cambridge University Press: Cambridge, UK, 1998. [Google Scholar]
- Doplicher, L. Generalized Tomonaga–Schwinger Equation from the Hadamard Formula. Phys. Rev. D
**2004**, 70, 064037. [Google Scholar] [CrossRef] [Green Version] - Breuer, H.-P.; Petruccione, F. The Theory of Open Quantum Systems; Oxford University Press: Oxford, UK, 2002. [Google Scholar]
- Baidya, A.; Jana, C.; Loganayagam, R.; Rudra, A. Renormalization in Open Quantum Field Theory. Part I. Scalar Field Theory. J. High Energy Phys.
**2017**, 2017, 204. [Google Scholar] [CrossRef] [Green Version] - Baidya, A.; Jana, C.; Rudra, A. Renormalization in Open Quantum Field Theory. Part II. Yukawa Theory and PV Reduction. arXiv
**2017**, arXiv:1906.10180v1. [Google Scholar] - Bogoliubov, N.N.; Shirkov, D.V. Introduction to the Theory of Quantized Fields; A Wiley-Intersciense Publication; John Wiley and Sons Inc.: New York, NY, USA, 1980. [Google Scholar]
- Bogoliubov, N.N.; Shirkov, D.V. Quantum Fields; Benjiamin/Cummings Publishing Company Inc.: San Francisco, CA, USA, 1983. [Google Scholar]
- Bogolyubov, N.N.; Logunov, A.A.; Oksak, A.I.; Todorov, I.T. General Principles of Quantum Field Theory. In Mathematical Physics and Applied Mathematics; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1990. [Google Scholar]
- Guskov, V.A.; Ivanov, M.G.; Ogarkov, S.L. A Note on Efimov Nonlocal and Nonpolynomial Quantum Scalar Field Theory. arXiv
**2017**, arXiv:1711.08829v5. [Google Scholar] - Felder, G. Renormalization Group in the Local Potential Approximation. Commun. Math. Phys.
**1987**, 111, 101–121. [Google Scholar] [CrossRef]

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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