It is not about the math! It is about getting a feeling for some of the words and methodologies underpinning the scientific advances of recent decades. It is those advances that have hailed the dawn of a new era of understanding.
We met many of the concepts that will be in this blog in the prior one on quantum mechanics. Quantum Field Theory is essentially just an extension of quantum mechanics.
In quantum mechanics we saw that mathematical operators function as observables. The Schrodinger wave equation gave the motion of a particle in one dimension and that structure can be extended to handle multiple particles.
No wave equation we have met so far, however, can be used in both relativity and quantum theory. Attempts to merge realtivity and quantum mechanics were attemped in the equations associated with the names of Klein-Gordon and Dirac, but it was the introduction of fields for wave equations that allowed the theory to truly advance.
The Dirac equation is like the Schrodinger with a modified Hamiltonian. Dirac fields satisfy the Klein-Gordon equation and relativistic relations among energy, mass and momentum.
Solutions to the Dirac equation can be found in momentum space using a Fourier expansion.
The Klein-Gordon equation emerges from a substitution of the quantum mechanical operators for energy and momentum into the Einstein special relativity energy, mass, and momentum relations. By viewing the equation in the context of a field rather than a particle, creation and annihilation operators are allowed to arise.
In classical mechanics, Lagrangians (Euler-Lagrange equations) are used in deriving equations of motion. In field theory they are used to derive field equations.
A Hamiltonian equation can be derived from the Lagrangian and momentum representations.
Symmetries leave the form of the Lagrangian and equations of motion invariant.
Path integrals allow for calculation of amplitudes of quantuim transitions. In classical mechanics, the path is deterministic. In quantum mechanics , there is no trajectory per se, only path integrals.
In classical physics, there were two components, objects and the fields which linked them. Quantum physics tells us that particles are mere manifestations of fields.
Like many findings in quantum physics, this has deep philosophical implications. At the core of reality is not a thing, but a connection. Everything is connected, and elements have no meaning in isolation.
In quantum theory, predictions involve calculating probability amplitudes. The S-Matrix is one of the tools used. Feynman Rules allow for fairly straightforward calculation of amplitude/probabilities. Feynman Diagrams show particle actions (scattering, decay...).
The mathematics of group theory formalizes symmetry structures. Unitary groups (U1) are important because unitary tansformations leave probabilities for transitions among states unaffected. Unitary operators commute with Hamiltonians.
SU(2) symmetries are important in electroweak interaction work, and SU(3) symmetries are important in strong force/quark study (QCD).
The standard model includes QED and QCD. Quantum Electrodynamics (QED) works with electromagnetic interactions. Electromagnetic forces arise from photon exchangeof electrons in an em field. Feynman Diagrams illustrate QED processes.
QED explains the behavior of charged particles in an em field within the framework of quantum theory. Chemistry rests on the behavior of the electrons surrounding the nucleus of the atom.
The number of particles which had been detected by physicists by the mid twentieth century was large, not unlike the number of elements which Mendeleev was able to classify for chemistry.
In the 1960's, Murray Gell-Mann and Israel Ne'eman were able to classify particles. into patterns based on qualities called charge, spin and strangeness.
Initially the particles were grouped into eight patterns, and the name given to the system was the Eightfold Way. Particles in the nucleus, neutrons and protons, were actually found to be made up of more fundamental entities called quarks.
After a good deal of mathematical gymnastics, involving symmetries, Yang-Mills equations, Abelian Theory, renormalization, and the Higgs Mechanism, we came up with Quantum Chromodynamuics(QCD).
In QED. gauge invariance (U1) involves a Lagrangian for the em field, a Dirac Lagrangian, and an interaction term.
The combination of QED and QCD gives rise to the standard model of physics which describes the entire particle world, the nature and interactions of fundamental particles. Remember that in particle physics, particles have wave as well as particle characteristics.
The three fundamental interactions (forces) in the standard model are electromagnetic (em), "weak", and "strong".
Each force is manifest in a "particle". For each interaction there is a field. The generators of the field come from the unitary group which describes the field symmetries.
Elementary particles develop mass in their interaction with a mathmatical formalism called a Higgs Field. While no such field has been found, it is needed for the math of the standard model to describe the fundamental entities in our universe and their interactions. It is often the case that mathematics predicts the existence of an element of reality before our experiments can detect it.
Adding gravity to the standard model has not been accomplished. Relativity and quantum mechanics are mathematically incompatible. String theory, which we will visit in the next blog, is one attempt to create a theory that encompasses all four forces.
In the next blog, on the mathematics of string theory, we will attempt to bring together the philosophical and mathematical implications of discoveries in particle physics.
Lee
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