Classical and quantum dynamics of the Faraday lines of force

Abstract

We study the vacuum Maxwell theory by expressing the electric field in terms of its Faraday lines of force. This representation allows us to capture the two physical degrees of freedom of the electric field by means of two scalar fields. The corresponding classical canonical theory is constructed in terms of four scalar fields, is fully gauge invariant, has an attractive kinematics, but a rather complicated dynamics. The corresponding quantum theory can be constructed in a well-defined functional representation, which we refer to as the Euler representation. This representation turns out to be related to the loop representation. The resulting quantization scheme is, perhaps, of relevance for non-Abelian theories and for gravity. PACS number(s): 03.50.De, 03.70.+k

Space, Time, Matter in Quantum Gravity

“…We have limited the discussion here to quantum geometrodynamics. The main conclusions also hold for the path-integral approach and to loop quantum gravity.25 In loop quantum gravity, there are analogies with gauge theories, for example with Faradays's lines of forces, see Frittelli et al (1994). Still, it is not a gauge theory by itself, and many conceptual issues such as the semiclassical limit are much less clear than in quantum geometrodynamics.…”

Topological Electromagnetism with Hidden Nonlinearity

“…where q k ; p k ; v k ; u k are functions of spacetime that can also be chosen as coordinates of the field [62] and a is a constant with dimensions of action times velocity, introduced here in order for these functions to be dimensionless. Each of the two terms in these sums is a radiation field (i. e. verifies E Á B ¼ 0).…”

“…The functions (p; q) and (u; v) are the Clebsch variables of B and E, respectively [63,64]. They can be used as canonical variables [62]. As explained above, they are not uniquely defined, but may be changed by canonical transformations.…”

Background Independent Quantum Mechanics, Classical Geometric Forms and Geometric Quantum Mechanics—I

“…An appropriate example in the sight is the case of theory of Loop Quantum Gravity (LQG), where the states and therefore the scalar product [27,29] are time independent. In a modern perspective such Faraday loops are realised as the gauge invariant degrees of freedom for General Relativity and Yang-Mills theory in Loop Quantum Gravity (LQG) [14,22,27]. We give yet another interpretation to this idea as follows: The (Faraday) lines that make these loops are the locus of probability flow.…”