﻿ The Foundations of Quantum Field Theory
San José State University

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Thayer Watkins
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The Foundations of
Quantum Field Theory

Quantum Field Theory has a problem with computed quantities being infinite. Historically this problem was resolved by the so-called Renormalization Group Theory that does not technically involve a mathematical group. This is an investigation of whether the infinities might arise from fields that are generated by point charges. Such fields have an infinite amount of energy. It is difficult or impossible to maintain the principle of the conservation of energy if there is even one point charge field in the Universe.

The concept of a point particle served a role in the early theorizing of physics, but care must be taken that the concept is not pushed too far. The alternate model of a charged particle is that of spherical shell of positive radius, say R. Beyond R the field is the same as that of a point particle; within R the field is zero. At R there is a discontinuity. This field has a finite amount of energy proportional to Q/R, where Q is the charge of the particle.

## The Foundations of Quantum Field Theory

The opening line in Quantum Field Theory for the Gifted Amateur is

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.

## 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 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 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. 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. 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 one half of the initial profile moves to the right and the other 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. Thus static electrical or magnetic fields cannot exist in the absence of charged particles. .