Noethers Theorem

In field theory, electric charge [6] is a symmetry of action, because it is a conserved quantity. This requirement leads to the consideration of a complex scalar field . The simplest possibility [U(l)] is that have two components, but in general it may have more than two as in the internal space of 0(3) electrodynamics which consists of the complex basis ((1),(2),(3)). The first two indices denote complex conjugate pairs, and the third is real-valued. These indices superimposed on the 4-vector give a 12-vector. In U(l) theory, the indices (1) and (2) are superimposed on the 4-vector, 4M in free space, so, 4M in U(l) electrodynamics in free space is considered as transverse, that is, determined by (1) and (2) only. These considerations lead to the conclusion that charge is not a point localized on an electron rather, it is a symmetry of action dictated ultimately by the Noether theorem [6]. 

Additive invariants were first studied by Pomeau [pomeau84] and Goles and Vich-niac [golesSb]. Although, as we shall see below, there are some techniques that can be used to extract a few invariants from jjarticular systems, no general methodology currently exists. A fundamental obstacle appears to be that there is no purely discrete analogue of Noether s Theorem. 

We recall, from elementary classical mechanics, that symmetry properties of the Lagrangian (or Hamiltonian) generally imply the existence of conserved quantities. If the Lagrangian is invariant under time displacement, for example, then the energy is conserved similarly, translation invariance implies momentum conservation. More generally, Noether s Theorem states that for each continuous N-dimensional group of transformations that commutes with the dynamics, there exist N conserved quantities. 

In thermodynamic equilibrium, the net result is zero in both cases, but locally, there may be a non-zero rate of doing work by these vacuum charges and currents on a device, creating thermal or mechanical energy. This process is unknown in the received view but conserves energy and is consistent with Noether s theorem [6]. 

Therefore, charge density and current density in the vacuum and in matter take the same form, [see Eqs. (732) and (733)]. This is a general result of assuming an 0(3) vacuum configuration as in Section I. Equations (736) are a form of Noether s theorem and charge/current enters the scene as the result of conservation and topology. Similarly, mass is curvature of the gravitational field. 

Base-change, 2-2, 2-4, 7-7 Bertini s theorem, 11-16,11-17 Brill-Noether map, 11-6,... 

The source of electric charge in this view is a symmetry of the action in Noether s theorem, a symmetry that means that ([> must be complex, that is, that there must be two fields ... 

For translation of the origin of space and time [46], Noether s theorem gives = -00 = 8M8V4 I 8C8a48a4 (271)... 

This type of transformation is not dependent on spacetime and is purely internal [46] in Noether s theorem. Under a global gauge transformation, Noether s theorem gives the conserved current... 

Within this context, ordinary differential equations are viewed as vector fields on manifolds or configuration spaces [2]. For example, Newton s equations are second-order differential equations describing smooth curves on Riemannian manifolds. Noether s theorem [4] states that a diffeomorphism,3 < ), of a Riemannian manifold, C, indices a diffeomorphism, D< >, of its tangent4 bundle,5 TC. If 4> is a symmetry of Newton s equations, then Dt(> preserves the Lagrangian o /Jc ) = jSf. As opposed to equations of motion in conventional... 

In the third section, we shall prove a Homomorphism Theorem and two Isomorphism Theorems for schemes of finite valency. All three of these results naturally generalize the finite versions of Emmy Noether s corresponding theorems for groups. 

The Homomorphism Theorem and the two Isomorphism Theorems were first proved in [35]. The thin case was already proved in 1929 by Emmy Noether cf. [32 I. 2],... 

The close connection between symmetry transformations and conservation laws was first noted by Jacobi, and later formulated as Noether s theorem invariance of the Lagrangian under a one-parameter transformation implies the existence of a conserved quantity associated with the generator of the transformation [304], The equations of motion imply that the time derivative of any function 3(p, q) is... 

By Noether s theorem, invariance of the Lagrangian under an infinitesimal time displacement implies conservation of energy. This is consistent with the direct proof of energy conservation given above, when L and by implication H have no explicit time dependence. Define a continuous time displacement by the transformation t = t + oi(t ) whereat/(,) = a(t ) = 0. subject to a —0. Time intervals on the original and displaced trajectories are related by dt = (1 + a )dt or dt = (1 — a )dt. The transformed Lagrangian is... 

As an extension of Noether s theorem to quantum mechanics, the hypervirial theorem [101] derives conservation laws from invariant transformations of the theory. Consider a unitary transformation of the Schrodinger equation, U(H — F)T = U(H — = 0, and assume the variational Hilbert space closed under a... 

Noether s theorem will be proved here for a classical relativistic theory defined by a generic field , which may have spinor or tensor indices. The Lagrangian density (, 9/x) is assumed to be Lorentz invariant and to depend only on scalar forms defined by spinor or tensor fields. It is assumed that coordinate displacements are described by Jacobi s theorem S(d4x) = d4x 9/xgeneral variation of the action integral, evaluated over a closed space-time region 2, is... 

Because the assumed hypervolume can be reduced to an infinitesimal, stationary or invariant action implies the local form of Noether s theorem, 3/27/2 = 0, an equation of continuity in space-time for the generalized current density determined by the field . 

The conserved current density determined by Noether s theorem is... 

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