Themes > Science > Physics > Elementary particle physics > Elementary particle physics Today > Quantum Field Theory > Historical Perspective: from Newton...Maxwell....Einstein....Dirac... Feynman...

Newton's laws of gravitation led to the Coulomb's law of electromagnetism which in turn led to Maxwell's equation. In this picture electromagnetic field is supposed to surround every electric charge and when the charge moves, the propagation of disturbance in the field was described by waves. The finite velocity of propagation of electromagnetic field was identified with the speed of light. In 1905,
Einstein proposed special theory of relativity which recognized the speed of light , as having a special significance in being the maximum velocity attainable by any physical entity. This led to a revision of the ideas of spacetime. Therefore, the special relativity explained why the Coulomb's law failed to describe all electromagnetism. The interaction in Coulomb's law is instantaneous:  i. e the interaction propagated from one particle to another with  infinite velocity. This conclusion led people to wonder whether the Newton's laws of gravitation was also wrong. Once we accept special relativity, we must object to Newton's laws of gravitation on the same grounds that apply to Coulomb's laws. In 1915, Einstein proposed a radical revision of the law of gravitation which is known as general theory of relativity.

Einstein approached the subject of gravitation by comparing the properties of gravitation with those of electromagnetic field. From the broad similarity of Newton's and Coulomb's laws, we might expect gravitation to be a field in the Maxwellian sense. However, the electromagnetic field is
capable of being switched on or off while ( in the absence of antigravity ), the gravitational field in a given region of spacetime is more permanent. Einstein expressed this result by relating gravitation to the geometrical properties of spacetime. He postulated that the presence of gravitation means the spacetime geometry is non-Euclidean.  Starting from the principle of relativity and finiteness of
the velocity of propagation of interaction emerges Einstein's relativity in terms of Lorentz invariance.  From the principle of equivalence of the motion between the noninertial and inertial system in a gravitational field will emerge the notion of curved space time, the general theory
of relativity.)

In classical physics, "particles" and "fields" are two different concepts: they are characterized by discreteness and continuity respectively.  A system may contain a vast number of particles - such as the molecules in a gas, but as long as they are denumerable, the basic theory is classical dynamics. But if the variables of the system are not denumerable - such as electric field or velocity field of a fluid, the
basic theory is called field theory. The fundamental theory in classical dynamics is the Newtonian dynamics which can be expressed in different mathematical forms, such as Lagrangian, Hamiltonian and the variational form.

The phenomena and the laws of fields are expressed in partial differential equations such as those of fluid dynamics and electromagnetic fields. But these equations can also be expressed in the form of Lagrange or Hamilton's equations which can be derived using a variational principle.  Therefore variational methods unify the description of particle and waves.

The quantum theory of radiation of Einstein in 1905, led us to seek a mathematical theory to describe the quantum properties of continuum fields. This is analogous to the quantization of atomic and molecular phenomena in quantum mechanics. To extend quantum mechanics  to the quantized properties of fields, the theory is called " Quantum Field Theory" or second quantization.

There are various kids of fields. One kind is the classical field such as the electromagnetic field. The other are the quantum fields described by Schrodinger equation, Klein Gordan equation and the Dirac equation. In these cases the quantum mechanics represents both the particle and wave properties of what have been described as particles in classical mechanics. In these equations, the particle and wave properties are described by Einstein-de-Broglie relations What we seek in quantum field theory is to extend the method of quantization for particle dynamics to the continuous fields. That is why it is called second quantization.
Therefore, starting from a particle point of view, one seeks a theory of particles that satisfies the relativity principle and finds that the appropriate theory is a many-body theory and the appropriate representation of the many-body system is "field".

The method consists of (1)defining the field variables  and finding their canonical momenta.(2) Quantize the field variables by the commutation rules.  If we quantize emf, the quantized field is represented by photons. For Klein-Gordan equation, the quantized field describes the pions. Dirac equation, the uantized field describes fermions. In the second quantization theory, we extend the theory of single particle to a theory describing a system of particles in which particles can be created or annihilated.

The theory describing system of electron coupled to electromagnetic field is known as "Quantum Electrodynamics (QED)". This theory presents serious deep-rooted difficulties, namely the presence of infinities in the results of calculations of physical quantities. The breakthrough came in mid 1940's with the work of Tomonaga, Schwinger, Feynman and Dyson. Although the infinities remain, theory succeeds in subtracting them in a consistent way to that finite result can be obtained.


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