Higgs mechanism

Superconductivity provides an illustration of the Higgs mechanism. It is the property of materials that show no electrical resistance, usually at low temperatures. Such materials are capable to carry persistent currents. These currents effectively screen out magnetic flux, which is therefore zero in a superconductor (the Meisner effect). Another way of describing the Meisner effect is to say that the photons are effectively massive, as in the Higgs phenomenon. These conclusions can be shown to follow from the Lagrangian (46). In this instance it is sufficient to consider a static situation, i.e. d4 = 0, etc, leading to the Lagrangian... 

Recall that the Higgs mechanism depends on local gauge symmetry, which is only defined within general relativity theory. 

The physical vacuum is assumed to be defined by the Higgs mechanism, and the SU(2) x SU(2) covariant derivative is... 

The theory so far is incomplete, however, because it has two SU(2) algebras that both act on the same Fermi spinor fields, and only one Higgs mechanism is used to compute the vacuum expectations for both fields. To improve the theory, consider that each SU(2) acts on separate spinor field doublets and that there are two Higgs fields that compute separate physical vacua for each SU(2) sector independently. The Higgs fields will give 2x2 vacuum diagonal expectations. If two entries in each of these matrices are equal, the resulting massive fermions in each of the two spinor doublets are identical. If the spin in one doublet... 

The second Higgs field acts in such a way that if the vacuum expectation value is zero, ( ) = 0, then the symmetry breaking mechanism effectively collapses to the Higgs mechanism of the standard SU(2) x U(l) electroweak theory. The result is a vector electromagnetic gauge theory 0(3)/> and a broken chiral SU(2) weak interaction theory. The mass of the vector boson sector is in the A(3) boson plus the W and Z° particles. 

The prediction of a heavy boson has received preliminary empirical support [92,96] from an anomaly in Z decay widths that points toward the existence of Z bosons with a mass of 812 GeV 1 33j [92,96] within the SO(l) grand unified field model, and a Higgs mechanism of 145 GeV4gj3. This suggests that a new massive neutral boson has been detected. Analysis of the hadronic peak cross sections obtained at LEP [96] implies a small amount of missing invisible width in Z decays. The effective number of massless neutrinos is 2.985 0.008, which is below the prediction of 3 by the standard model of electroweak interactions. The weak charge Qw in atomic parity violation can be interpreted as a measurement of the S parameter. This indicates a new Qw = 72.06 0.44, which is found to be above the standard model pre-... 

From the foregoing, it becomes clear that fields and potentials are freely intermingled in the symmetry-broken Lagrangians of the Higgs mechanism. To close this section, we address the question of whether potentials are physical (Faraday and Maxwell) or mathematical (Heaviside) using the non-Abelian Stokes theorem for any gauge symmetry ... 

It is seen that the acquisition of mass by the photon is the result of an equation of superconductivity, and this is, of course, the basis of spontaneous symmetry breaking and the Higgs mechanism (Section XIV). Beltrami equations account for all these phenomena, and are foundational in nature. Note that the London equation (919) is not gauge-invariant on the U(l) level because aphysical gauge-invariant current is proportional to the vector potential, which, in the received view, is gauge-noninvariant. This is another flaw of U(l) electrodynamics in the... 

By Ohm s law, the resistance of the conducting medium vanishes, and the medium becomes a superconductor. The Higgs mechanism and spontaneous symmetry breaking were derived using the properties of superconductors. 

M. W. Evans, P. K. Anastasovski, T. E. Bearden, et al., Schrodinger equation with a Higgs mechanism Inherentvacuum energy, Found. Physics (in press). 

V. Schrodinger Equation with a Higgs Mechanism Effect on the Wave Functions... 

It can be seen that the photon mass is carried by, 4Vl 1 and Av(2 but not by 4Vl 1 . This result is also obtained by a different route using the Higgs mechanism in Ref. 42, and is also consistent with the fact that the mass associated with 4Vl 3 corresponds with the superheavy boson inferred by Crowell [42], reviewed in... 

In contemporary thought, the Higgs mechanism has acted in such a way as to produce a field component Bi with mass, specifically, a scalar field with mass that is gauge-invariant. Therefore, spontaneous symmetry breaking of the vacuum introduces fields with effective mass. 

Therefore, a check for self-consistency has been carried out for indices p 2 and v = 1. It has been shown, therefore, that in pure gauge theory applied to electrodynamics without a Higgs mechanism, a richly structured vacuum charge current density emerges that serves as the source of energy latent in the vacuum through the following equation ... 

The above is a pure gauge field theory. The Higgs mechanism on the U(l) level provides further sources of vacuum energy as discussed already. On the 0(3) level, the Higgs mechanism can also be applied, resulting in yet more sources of energy. 

It has been demonstrated already that local gauge transformation on this Lagrangian leads to Eq. (153), which contains new charge current density terms due to the Higgs mechanism. For our present purposes, however, it is clearer to use the locally invariant Lagrangian obtained from Eq. (325), specifically... 

The effect of the Higgs mechanism can be seen most clearly by minimizing the Lagrangian (251) with respect to A ... 

V. SCHRODINGER EQUATION WITH A HIGGS MECHANISM EFFECT ON THE WAVE FUNCTIONS... 

The starting Lagrangian on the U(l) level for a free particle, such as an electron, is the standard Lagrangian for the Higgs mechanism ... 

The Higgs mechanism has produced an additional rest energy ... 

The left-hand side is the nonrelativistic kinetic energy of one particle. It can be seen that the Higgs mechanism changes the classical nonrelativistic expression... 

The Schrodinger equation in the presence of the Higgs mechanism is therefore... 

However, there remains the problem of how to obtain a locally gauge-invariant Proca equation. To address this problem rigorously, it is necessary to use a non-Abelian Higgs mechanism applied within gauge theory. 

The starting point of our derivation is the globally invariant 0(3) Lagrangian of the Higgs mechanism... 

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