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The Explanation and History
of the Higgs Boson Concept

Methodology of
Mathematical
Particle Physics
Yang-Mills
Fields
The Goldstone
Theorem
The Higgs
Mechanism
Lie Algebras and
Subatomic Particles
Gauge
Invariance

For Warren

The Higgs Boson in Simplest Terms

According to the theory there is a field that prevades the universe. A vibration in that field is what has been called the Higgs boson. However, it is is the Higgs field that is the important matter. Higgs bosons are primarily important only as evidence that the Higgs field exists. It is the Higgs field that gives mass to certain particles. Other particles such as photons, the vibrations in the electomagnetic field, have no mass.

The relationship between mass and the Higgs field may be thought of as follows. Mass is the resistance to acceleration. The Higgs field is like a layer of water some particles have to move in. The water results in a resistance to acceleration for those particles. The massless particles are like objects floating on the surface which move without resistance.

The Dark Underside of the Higgs Boson

The article of Higgs' which is cited in the literature is a note of less than two pages appearing in the Physical Review Letters issue of October 19, 1964.. In that note Peter Higgs cites an article of his which is "to be published." No such article was published and that suggests that it had not been written at that point. The gist of Higgs' note is a mathematical result which requires rigorous proof that is not there. An article dealing with the topic of Higgs' note was published the day Higgs submitted his note for publication in the Physical Review Letters. The authors of that article, Englert and Brout, were associated with a university in Belgium while Higgs was on the faculty at the University of Edinburgh.

Higgs did eventually publish a full length article on the topic but only after several years delay. It appears that the intellectual community of Britain awarded Higgs fame on the basis that he was a Brit whereas the others publishing articles on the topic were not. Or, perhaps it was just that Higgs had a convenient monosyllable name.

Methodology of Mathematical Particle Physics

In the Lagrangian scheme for mechanics a Lagrangian or Lagrangian density function is formulated for a system. The Lagrangian function is the difference between the kinetic energy and potential energy for the system. The integral of the Lagrangian function over time is called the action of the system. The trajectory taken by the system is such as to make its action an extreme. An extreme is a minimum, a maximum or an inflection point. This requires that system's development satisfy its Euler-Lagrange equation.

If the Lagrangian is a function of a set of variables qj(t), j=1,…n, and their time derivates (rates of change) vj(t)=dqsup>i/dt then the Euler-Lagrange equation is

L/∂qj − d/dt (∂L/∂vj) = 0 for j=1,…,n.

Particle physics is concerned with fields which are spread over space, therefore the appropriate Lagrangian is a volume density and the action is the integral of that density over space and time. Theorizing in particle physics consists of formulating a Lagrangian density function for a system of particles and deriving its implications.

The Group Structures

The state variables can include more than the space-time variables. They can include elements of groups; i.e., elements for which there is a binary operation satisfying certain conditions analogous to the properties of addition of numbers.

To see how there could be hidden dimensions to a world consider the case of the surface of a long thin tube. This world appears to be one dimensional but it is actually two dimensional. One dimension is the distance along the tube; the other dimension is the angle of a position on the circular cross section. The angular dimensions can be added and subtracted.

A mathematical group is a set of elements and a binary operation (function) which is associative. Let (*) denote the binary operation. the operation is associative if for any elements a, b and c of the group

(a*b)*c = a*(b*c)

Additionally there must be an identity element e of the group such that for any element of the group

e*a = a

Further more there must exist an inverse for each element. This means that for any element a of the group there exists a b in the group such that

a*b = e, the identity element

In physics the elements of the group are square matrices, said to be group representations.

A Lie group is a group that has geometric structure like a circle or sphere as well as an algebraic structure. The group elements are associated with continuous parameters.

(To be continued.)

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