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The Foundations of
Quantum Field Theory

The Foundations of Quantum Field Theory

A standard exposition on Quantum Field Theory is Quantum Field Theory for the Gifted Amateur by Tom Lancaster and Stephen J. Blundell. The opening lines in that work are

What is quantum field theory?

Every particle and every wave in the Universe is simply an excitation of a quantum field that is defined over all space and time.

This is a reasonable theoretical goal for force-carrying particles such as photons but there is a major problem in it for charged particles such as electrons and positrons. The problem is that static electrical fields cannot exist without the presence of charged particles.

Free Fields and Maxwell's Equations

The behavior of electromagnetic fields is described by Maxwell's equations. The precise form of these equations depends upon the system of units used. Here the Gaussian system of J.D. Jackson's Classical Electrodynamics is used.

The Maxwell equations for electromagnetic fields in that system are:

∇·D = 4πρ
∇×H = (4π/c)J + (1/c)(∂D/∂t)
∇·H = 0
∇×E + (1/c)(∂B/∂t) = 0

where c is the speed of light in a vacuum, E and D are vector fields describing the electric field and B and H are vector fields for the magnetic field. The quantities ρ and J are the charge density and current density, respectively.

The relationships between E and D and B and H are

D = εE
B = μH

where ε is the dielectric constant of the material the fields are located in and μ is the permeability of that material.

Here only field configurations with ρ and J equal to zero everywhere will be considered; in effect free fields.

If such fields exist they change according to the dynamical equations

ε(∂E/∂t) = c∇×H
μ(∂H/∂t) = −c∇×E

If the first equation is differentiated once with respect to time the result is

ε(∂²E/∂t²) = c∇×(∂H/∂t)

The interchangeability of (∇×) and (∂/∂t) is utilized.

The second equation above can then be used to replace (∂H/∂t) which yields

ε(∂²E/∂t²) = c∇×[−(c/μ)(∇×E)]
which reduces to
(∂²E/∂t²) = −(c²/(εμ))[∇×(∇×E)]

The curl of the curl of E, ∇×(∇×E), can be expressed as

∇×(∇×E) = ∇(∇·E) − ∇²E

where ∇²E is the vector Laplacian of E. In Cartesian coordinates the i-th compponent of the vector Laplacian of a vector is equal to the scalar (ordinary) Laplacian of the i-th component of that vector.

For the case being considered in which there is no charge distribution ∇·E is equal to zero and ∇(∇·E) is everywhere equal to the zero vector. Thus

∇×(∇×E) = − ∇²E
and hence
(∂²E/∂t²) = (c²/(εμ))∇²E

This type of partial differential equation is known as a wave equation. It can have solution of a sinusoidal nature but the solution depends upon the initial conditions.

The speed of electromagnetic radiation in the material is equal to c/(εμ)½. Let that speed be designated as C. It is the speed of light in the material in which the fields exist. The wave equation is then of the form

(∂²E/∂t²) = C²∇²E

Suppose the space is one dimensional. Let f(x)=E(x,0). Then the solution, known as the d'Alembert solution, is of the form

E(x, t) = ½[f(x+Ct) + f(x-Ct)]

Thus half of the initial profile moves to the right and half moves to the left. This constitutes moving electric fields but not radiation. It is more in the nature of a whoosh rather than a wave.

If the above procedure was carried out with H rather than E the same wave equation would result and likewise for D and B.

The solutions can be construed to be an energy flow but that terminology is a bit misleading. It is the fields that flow and take their energy along with them. The analysis for a three dimensionally distributed field is covered elsewhere. Thus static electrical or magnetic fields cannot exist in the absence of charged particles. The notion of Quantum Field Theory of replacing all particles with fields is an aesthetic consideration rather than something derived from empirical fact or a general principle.


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