scholarly journals GENERALIZED SCHRÖDINGER EQUATION IN EUCLIDEAN FIELD THEORY

2004 ◽  
Vol 19 (24) ◽  
pp. 4037-4068 ◽  
Author(s):  
FLORIAN CONRADY ◽  
CARLO ROVELLI
Keyword(s):  
Field Theory ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Boundary Surface ◽  
General Boundary ◽  
Classical Counterpart ◽  
Arbitrary Space ◽  
Euclidean Field Theory ◽  
Evolution Kernel ◽  
Free Scalar

We investigate the idea of a "general boundary" formulation of quantum field theory in the context of the Euclidean free scalar field. We propose a precise definition for an evolution kernel that propagates the field through arbitrary space–time regions. We show that this kernel satisfies an evolution equation which governs its dependence on deformations of the boundary surface and generalizes the ordinary (Euclidean) Schrödinger equation. We also derive the classical counterpart of this equation, which is a Hamilton–Jacobi equation for general boundary surfaces.

scholarly journals Time in Quantum Mechanics and the Local Non-Conservation of the Probability Current

Mathematics ◽  
2018 ◽  
Vol 6 (9) ◽  
pp. 155 ◽  
Author(s):  
Giovanni Modanese
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Quantum Mechanics ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Relativistic Quantum ◽  
Quantum Field ◽  
Relativistic Quantum Field Theory ◽  
Non Local ◽  
Time In Quantum Mechanics

In relativistic quantum field theory with local interactions, charge is locally conserved. This implies local conservation of probability for the Dirac and Klein–Gordon wavefunctions, as special cases; and in turn for non-relativistic quantum field theory and for the Schrödinger and Ginzburg–Landau equations, regarded as low energy limits. Quantum mechanics, however, is wider than quantum field theory, as an effective model of reality. For instance, fractional quantum mechanics and Schrödinger equations with non-local terms have been successfully employed in several applications. The non-locality of these formalisms is strictly related to the problem of time in quantum mechanics. We explicitly compute, for continuum wave packets, the terms of the fractional Schrödinger equation and the non-local Schrödinger equation by Lenzi et al. that break local current conservation. Additionally, we discuss the physical significance of these terms. The results are especially relevant for the electromagnetic coupling of these wavefunctions. A connection with the non-local Gorkov equation for superconductors and their proximity effect is also outlined.

scholarly journals EQUIVALENCE BETWEEN DIFFERENT CLASSICAL TREATMENTS OF THE O(N) NONLINEAR SIGMA MODEL AND THEIR FUNCTIONAL SCHRÖDINGER EQUATIONS

2003 ◽  
Vol 18 (05) ◽  
pp. 755-766 ◽  
Author(s):  
A. A. DERIGLAZOV ◽  
W. OLIVEIRA ◽  
G. OLIVEIRA-NETO
Keyword(s):  
Field Theory ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Sigma Model ◽  
Hamiltonian Formalism ◽  
Original Version ◽  
Nonlinear Sigma Model ◽  
Schrödinger Equations ◽  
Schrödinger Representation ◽  
The Way

In this work we derive the Hamiltonian formalism of the O(N) nonlinear sigma model in its original version as a second-class constrained field theory and then as a first-class constrained field theory. We treat the model as a second-class constrained field theory by two different methods: the unconstrained and the Dirac second-class formalisms. We show that the Hamiltonians for all these versions of the model are equivalent. Then, for a particular factor-ordering choice, we write the functional Schrödinger equation for each derived Hamiltonian. We show that they are all identical which justifies our factor-ordering choice and opens the way for a future quantization of the model via the functional Schrödinger representation.

scholarly journals GEOMETRIC PHASE AND CHIRAL ANOMALY: THEIR BASIC DIFFERENCES

2008 ◽  
Vol 23 (31) ◽  
pp. 4933-4944
Author(s):  
KAZUO FUJIKAWA
Keyword(s):  
Field Theory ◽  
Schrödinger Equation ◽  
Geometric Phase ◽  
Schrodinger Equation ◽  
Local Symmetry ◽  
Chiral Anomaly ◽  
Geometric Phases

All the geometric phases are shown to be topologically trivial by using the second quantized formulation. The exact hidden local symmetry in the Schrödinger equation, which was hitherto unrecognized, controls the holonomy associated with both of the adiabatic and nonadiabatic geometric phases. The second quantized formulation is located in between the first quantized formulation and the field theory, and thus it is convenient to compare the geometric phase with the chiral anomaly in field theory. It is shown that these two notions are completely different.

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