Klein-Goron equation describes the propagation of a boson in vacuum. How does it look inside a medium?
Dear Arbab,
he Klein–Gordon equation (Klein–Fock–Gordon equation or sometimes Klein–Gordon–Fock equation) is a relativistic version of the Schrödinger equation.
Its solutions include a quantum scalar or pseudoscalar field, a field whose quanta are spinless particles. It cannot be straightforwardly interpreted as a Schrödinger equation for a quantum state, because it is second order in time and because it does not admit a positive definite conserved probability density. Still, with the appropriate interpretation, it does describe the quantum amplitude for finding a point particle in various places, the relativistic wavefunction, but the particle propagates both forwards and backwards in time. Any solution to theDirac equation is automatically a solution to the Klein–Gordon equation, but the converse is not true.
The equation was named after the physicists Oskar Klein and Walter Gordon, who in 1926 proposed that it describes relativistic electrons. Other authors making similar claims in that same year were Vladimir Fock, Johann Kudar, Théophile de Donder and Frans-H. van den Dungen, and Louis de Broglie. Although it turned out that the Dirac equation describes the spinning electron, the Klein–Gordon equation correctly describes the spinless pion, a composite particle. On July 4, 2012 CERN announced the discovery of the Higgs boson. Since the Higgs boson is a spin-zero particle, it is the first elementary particle that is described by the Klein-Gordon equation. Further experimentation and analysis is required to discern whether the Higgs boson found is that of the Standard Model, or a more exotic form.
References:
[1] C Itzykson and J-B Zuber, Quantum Field Theory, McGraw-Hill Co., 1985, pp. 73–74. Eq. 2.87 is identical to eq. 2.86 except that it features j instead of ℓ.
[2] David Tong, Lectures on Quantum Field Theory, Lecture 1, Section 1.1.1
[3] S.A. Fulling, Aspects of Quantum Field Theory in Curved Space–Time, Cambridge University Press, 1996, p. 117
For more details I have attached the following publication:
Journal of Modern Physics, 2013, 4, 21-30
Published Online November 2013 (http://www.scirp.org/journal/jmp)
http://dx.doi.org/10.4236/jmp.2013.411A1004
Open Access JMP
A Classical Approach to the Modeling of Quantum Mass
Donald C. Chang
Div of LIFS, Hong Kong University of Science and Technology, Hong Kong, China
Email: [email protected]
Received September 1, 2013; revised September 29, 2013; accepted October 27, 2013
Copyright © 2013 Donald C. Chang. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
In modern physics, a particle is regarded as the quantum excitation of a field. Then, where does the mass of a particle come from? According to the Standard Model, a particle acquires mass through its interaction with the Higgs field. The
rest mass of a free particle is essentially identified from the Klein-Gordon equation (through its associated Lagrangian density). Recently it was reported that a key feature of this theory (i.e., prediction of Higgs boson) is supported by experiments
conducted at LHC. Nevertheless, there are still many questions about the Higgs model. In this paper, we would like to explore a different approach based on more classical concepts. We think mass should be treated on the same footing as momentum and energy, and the definition of mass should be strictly based on its association with the momentum. By postulating that all particles in nature (including fermions and bosons) are excitation waves of the vacuum medium, we propose a simple wave equation for a free particle. We find that the rest mass of the particle is associated with a “transverse wave number”, and the Klein-Gordon equation can be derived from the general wave equation if one considers only the longitudinal component of the excitation wave. Implications of this model and its comparison
with the Higgs model are discussed in this work.
Hoping this will be helpful,
Rafik
Depends on how the medium is described-either by modifying the metric, or by adding a source, or a chemical potential. The Klein-Gordon equation is a field equation, not the equation of a single particle.
The boson gets affected by the external field, formed by the medium. It is not, in general possible to calculate this external field, as it requires the exact solution of the relativistic many -body problem but it can be parametrized phenomenologically and can be added to the KG equation. In the case of fermions similar procedure leads to a very successful Dirac phenomenology. For further reading one can have a look at "fermions and bosons in external fields" by A.B.Migdal
It is a good question, but difficult to explain. For understanding the effect of a medium it is easier to consider the case of photons, (which are also bosons) and find out how the dielectric constant is affected by the medium. The easiest way is to assume the approximation of linear response and find out how electrons and nuclei affect the field of photons. In short, you have to consider a quantum field theory approach. An easier approach is to consider the additional field due to the charges of nuclei and electrons. For the case of metals you will find that there is a redistribution of the electron density so that the electromagnetic field can penetrate only a few angstroms inside the metal.
No, the mass term of the Klein-Gordon equation, realizes dispersion for one-particle states; however, since it is a linear term, doesn't induce variation of the number of excitations: as is well known, there isn't any net particle production by free fields-nor in integrable models, that are free fields in disguise.
"medium"? the free particle enters a constant potential. the result should be similar to that of a Dirac particle entering a constant "medium" = potential , which is described in some decent textbooks. Or take the Schrödinger eq., the nr. version of the KG eq.
- Badges
- Science topic
- Similar topics
- Physical Sciences
- Quantum Physics
- Quantum Mechanics