Complex Hamiltonian evolution equations and field theory

1998 ◽  
Vol 39 (11) ◽  
pp. 5700-5715
Author(s):  
Theodore Bodurov
Keyword(s):  
Field Theory ◽  
Evolution Equations ◽  
Hamiltonian Evolution

scholarly journals SHORT-TIME EXISTENCE FOR SCALE-INVARIANT HAMILTONIAN WAVES

2006 ◽  
Vol 03 (02) ◽  
pp. 247-267 ◽  
Author(s):  
JOHN K. HUNTER
Keyword(s):  
Rayleigh Waves ◽  
Evolution Equations ◽  
Hyperbolic Surface ◽  
Smooth Solutions ◽  
Scale Invariant ◽  
Weakly Nonlinear ◽  
Hamiltonian Evolution ◽  
Short Time ◽  
Nonlinear Scale ◽  
Time Existence

We prove short-time existence of smooth solutions for a class of nonlinear, and in general spatially nonlocal, Hamiltonian evolution equations that describe the self-interaction of weakly nonlinear scale-invariant waves. These equations include ones that describe weakly nonlinear hyperbolic surface waves, such as nonlinear Rayleigh waves in elasticity.

Operations involving momentum variables in non-Hamiltonian evolution equations

1988 ◽  
Vol 101 (3) ◽  
pp. 333-355 ◽  
Author(s):  
F. Benatti ◽  
G. C. Ghirardi ◽  
A. Rimini ◽  
T. Weber
Keyword(s):  
Evolution Equations ◽  
Hamiltonian Evolution

NONCOMMUTATIVE KdV HIERARCHY

2007 ◽  
Vol 19 (07) ◽  
pp. 677-724 ◽  
Author(s):  
FRANÇOIS TREVES
Keyword(s):  
Evolution Equations ◽  
Vries Equation ◽  
Kdv Equation ◽  
Completely Integrable Systems ◽  
Maximal Abelian Subalgebra ◽  
Abelian Subalgebra ◽  
Kdv Hierarchy ◽  
Noncommutative Version ◽  
Constants Of Motion ◽  
Hamiltonian Evolution

The noncommutative version of the Korteweg–de Vries equation studied here is shown to admit infinitely many constants of motion and to give rise to a hierarchy of higher-order Hamiltonian evolution equations, each one the noncommutative version of the commutative KdV equation of the same order. The noncommutative KdV polynomials span, topologically, a maximal Abelian subalgebra of the Lie algebra of noncommutative Bäcklund transformations. Two classes of examples of "completely integrable" systems of evolution equations to which the theory applies are described in the last two sections.

Analytic continuation of harmonic sums near the integer values

2020 ◽  
Vol 35 (33) ◽  
pp. 2050210
Author(s):  
V. N. Velizhanin
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Positive Integer ◽  
Small Parameter ◽  
Analytic Continuation ◽  
Evolution Equations ◽  
Quantum Field ◽  
Simple Method

We present a simple method for analytic continuation of harmonic sums near negative and positive integer numbers. We provide a precomputed database for the exact expansion of harmonic sums over a small parameter near these integer numbers, along with MATHEMATICA code, which shows the application of the database for actual problems. We also provide the FORM code that was used to obtain the database mentioned above. The applications of the obtained database for the study of evolution equations in the quantum field theory are discussed.

scholarly journals GROUPOID SYMMETRY AND CONSTRAINTS IN GENERAL RELATIVITY

2013 ◽  
Vol 15 (01) ◽  
pp. 1250061 ◽  
Author(s):  
CHRISTIAN BLOHMANN ◽  
MARCO CEZAR BARBOSA FERNANDES ◽  
ALAN WEINSTEIN
Keyword(s):  
General Relativity ◽  
Initial Data ◽  
Cotangent Bundle ◽  
Evolution Equations ◽  
Gauge Theories ◽  
Riemannian Metrics ◽  
Einstein Equations ◽  
Cauchy Hypersurface ◽  
Lagrangian Field Theory ◽  
Hamiltonian Evolution

When the vacuum Einstein equations are cast in the form of Hamiltonian evolution equations, the initial data lie in the cotangent bundle of the manifold [Formula: see text] of Riemannian metrics on a Cauchy hypersurface Σ. As in every Lagrangian field theory with symmetries, the initial data must satisfy constraints. But, unlike those of gauge theories, the constraints of general relativity do not arise as momenta of any Hamiltonian group action. In this paper, we show that the bracket relations among the constraints of general relativity are identical to the bracket relations in the Lie algebroid of a groupoid consisting of diffeomorphisms between space-like hypersurfaces in spacetimes. A direct connection is still missing between the constraints themselves, whose definition is closely related to the Einstein equations, and our groupoid, in which the Einstein equations play no role at all. We discuss some of the difficulties involved in making such a connection. In an appendix, we develop some aspects of diffeology, the basic framework for our treatment of function spaces.

scholarly journals Non-Equilibrium Quantum Electrodynamics in Open Systems as a Realizable Representation of Quantum Field Theory of the Brain

Entropy ◽  
2019 ◽  
Vol 22 (1) ◽  
pp. 43 ◽  
Author(s):  
Akihiro Nishiyama ◽  
Shigenori Tanaka ◽  
Jack A. Tuszynski
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Time Evolution ◽  
Central Region ◽  
Quantum Electrodynamics ◽  
Quantum Tunneling ◽  
Evolution Equations ◽  
Open Systems ◽  
Quantum Field ◽  
The Brain

We derive time evolution equations, namely the Klein–Gordon equations for coherent fields and the Kadanoff–Baym equations in quantum electrodynamics (QED) for open systems (with a central region and two reservoirs) as a practical model of quantum field theory of the brain. Next, we introduce a kinetic entropy current and show the H-theorem in the Hartree–Fock approximation with the leading-order (LO) tunneling variable expansion in the 1st order approximation for the gradient expansion. Finally, we find the total conserved energy and the potential energy for time evolution equations in a spatially homogeneous system. We derive the Josephson current due to quantum tunneling between neighbouring regions by starting with the two-particle irreducible effective action technique. As an example of potential applications, we can analyze microtubules coupled to a water battery surrounded by a biochemical energy supply. Our approach can be also applied to the information transfer between two coherent regions via microtubules or that in networks (the central region and the N res reservoirs) with the presence of quantum tunneling.

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