Next Article in Journal
Which Side Looks Better? Cultural Differences in Preference for Left- or Right-Facing Objects
Next Article in Special Issue
Electron Symmetry Breaking during Attosecond Charge Migration Induced by Laser Pulses: Point Group Analyses for Quantum Dynamics
Previous Article in Journal
User Behavior on Online Social Networks: Relationships among Social Activities and Satisfaction
Previous Article in Special Issue
Violation of the Time-Reversal and Particle-Hole Symmetries in Strongly Correlated Fermi Systems: A Review
Due to scheduled maintenance work on our core network, there may be short service disruptions on this website between 16:00 and 16:30 CEST on September 25th.
Article

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

Moscow Institute of Physics and Technology (MIPT), Institutskiy Pereulok 9, 141701 Dolgoprudny, Moscow Region, Russia
*
Authors to whom correspondence should be addressed.
Symmetry 2020, 12(10), 1657; https://doi.org/10.3390/sym12101657
Received: 13 August 2020 / Revised: 27 September 2020 / Accepted: 29 September 2020 / Published: 10 October 2020
(This article belongs to the Special Issue Symmetry in Quantum Systems)
We consider the functional Hamilton–Jacobi (HJ) equation, which is the central equation of the holographic renormalization group (HRG), functional Schrödinger equation, and generalized Wilson–Polchinski (WP) equation, which is the central equation of the functional renormalization group (FRG). These equations are formulated in D-dimensional coordinate and abstract (formal) spaces. Instead of extra coordinates or an FRG scale, a “holographic” scalar field Λ is introduced. The extra coordinate (or scale) is obtained as the amplitude of delta-field or constant-field configurations of Λ. For all the functional equations above a rigorous derivation of corresponding integro-differential equation hierarchies for Green functions (GFs) as well as the integration formula for functionals are given. An advantage of the HJ hierarchy compared to Schrödinger or WP hierarchies is that the HJ hierarchy splits into independent equations. Using the integration formula, the functional (arbitrary configuration of Λ) solution for the translation-invariant two-particle GF is obtained. For the delta-field and the constant-field configurations of Λ, this solution is studied in detail. A separable solution for a two-particle GF is briefly discussed. Then, rigorous derivation of the quantum HJ and the continuity functional equations from the functional Schrödinger equation as well as the semiclassical approximation are given. An iterative procedure for solving the functional Schrödinger equation is suggested. Translation-invariant solutions for various GFs (both hierarchies) on delta-field configuration of Λ are obtained. In context of the continuity equation and open quantum field systems, an optical potential is briefly discussed. The mode coarse-graining growth functional for the WP action (WP functional) is analyzed. Based on this analysis, an approximation scheme is proposed for the generalized WP equation. With an optimized (Litim) regulator translation-invariant solutions for two-particle and four-particle amputated GFs from approximated WP hierarchy are found analytically. For Λ=0 these solutions are monotonic in each of the momentum variables. View Full-Text
Keywords: quantum field theory (QFT); scalar QFT; generating functionals; green functions (GFs); functional renormalization group (FRG); holographic renormalization group (HRG); functional Hamilton–Jacobi (HJ) equation; functional Schrödinger equation; Wilson–Polchinski (WP) equation; GFs equations hierarchy; functional Taylor series; Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence quantum field theory (QFT); scalar QFT; generating functionals; green functions (GFs); functional renormalization group (FRG); holographic renormalization group (HRG); functional Hamilton–Jacobi (HJ) equation; functional Schrödinger equation; Wilson–Polchinski (WP) equation; GFs equations hierarchy; functional Taylor series; Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence
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

Find Other Styles
Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.

Article Access Map by Country/Region

1
Back to TopTop