I understand that the Higgs Field grants mass to all the matter in the universe, and is equally distributed in all parts of it, but it was the Higgs Boson that was discovered at the LHC, and apparently, this boson has a wide field of applications in the standard model. My question is, if the Higgs Boson is generated by the Higgs field, but does not affiliate with giving mass, what exactly are the necessary applications of the Higgs Boson in physics and mathematics, and why was the theory of the Higgs created?
An introduction to the quantum field theory which is used to model elementary particle interactions in the standard model of particle physics.
It is based on the axiomatic assumption of the table, that these are the elementary particles that exist .
The hypothesis on which the mathematical standard model is based is that all particles in the table exist as a field in all space time, an electron field a neutrino field , a top quark field etc, for all the particles in the table. These are mathematically represented as the corresponding free particle wave function in the corresponding solution of the Dirac equation for fermions, the Klein Gordon for bosons and a quantized Maxwell for photons.
Differential operators on these fields, called creation operators, generate a measurable particle, a creation for an electron at (x,y,z,t) generates an electron at that space time point. A corresponding annihilation differential operator destroys an electron at that space time point if it exists. With a complex mathematical formulation particles propagate by creation and annihilation operators consecutively changing the position of the real particle ( which has to be described by a wavepacket but that is a further story).
The Higgs boson is an excitation of the Higgs field in exactly the same way that the electron is an excitation of the electron field. That is what quantum field theory is about, so that particle interactions can be described and calculated with the Feynman diagrams.
what exactly are the necessary applications of the Higgs Boson in physics and mathematics,
exactly the same as the application for the neutrino or any other particle in the table. It is an elementary particle necessary to be found so that the model would be complete mathematically.
and why was the theory of the Higgs created?
It became necessary when trying to unify weak and electromagnetic theories. In classical physics Maxwell united magnetic and electric fields in his electrodynamic equations. The objective of unifying all fundamental forces is still going strong in the theoretical studies of elementary particles.