Bare Parameters
The previous Feynman’s QED model also introduced a parameter, labelled [n], which appeared to ‘weight’ the probability amplitude associated with a given conceptual pathway along which a particle might move in spacetime. While this parameter is not really discussed in any detail, it was suggested that this parameter masked a fundamental problem at the heart of quantum electrodynamics, which was only later resolved through a process of ‘renormalisation’ to be discussed next. However, before discussing renormalisation, there may be some value in trying to outline a concept sometimes known as ‘bare parameters’ that are linked to the determination of mass and charge of a particle like the electron; although we shall start with another questioning quote:
“It is rather remarkable that the modern concept of electrodynamics is not quite 100 years old and yet still does not rest firmly upon uniformly accepted theoretical foundations. Maxwell’s theory of the electromagnetic field is firmly ensconced in modern physics, to be sure, but the details of how charged particles are to be coupled to this field remain somewhat uncertain, despite the enormous advances in quantum electrodynamics over the past 45 years. Our theories remain mathematically ill-posed and mired in conceptual ambiguities which quantum mechanics has only moved to another arena rather than resolve. Fundamentally, we still do not understand just what is a charged particle.” - Walter Gandy |
The only purpose of the previous quote is to act as a counterpoint to the apparent acceptance of the mathematical concepts being discussed having physical reality. While the quantum description might ultimately be proved correct, or possibly more correct in its predictions, there may still be some benefit in a little scepticism as, in practice, a ‘bare’ charge is never seen and we only ever measure an effective charge shielded by polarized virtual electron/positron pairs.
However, the existence of a larger ‘masked’ charge may be seen as a valid conclusion of high energy-momentum interactions in as much as they approach closer to the central charge, as illustrated in the diagrams above. However, in QED, the bare charge of an electron is actually considered to be infinite and due to the field-energy near this infinite central charge, the bare mass of the electron is also considered to be infinite. We might try to add to this description of the bare mass by making reference to the apparent infinite self-energy of the electron in connection to its own electromagnetic field. So, in this context, quantum field theory had to face up to some serious theoretical difficulties, which appeared to result in infinite contributions, when computed using the perturbation techniques in the early days of QED development.
Is there any parallel to the classical calculation of force as the radius approaches zero?
Actually, the issue known as the ‘electron self-energy problem’ was also recognised within classical electromagnetic field theory, linked to the electron being assigned a finite size. In this context, the charge on an electron generates an electromagnetic field, such that the forces between the charge and the field produce a type of electromagnetic tension around the particle. This ‘tension’ corresponds to the internal energy stored within the field, but according to Einstein’s most famous equation [E=mc^{2}], this energy must also have an equivalent mass. This process of ‘self-interaction’ between the electron's field and its own charge, then adds to the effective mass of the particle. The earliest models of the electron as a finite charged sphere had a classical radius [r_{0}]:
[1]
From [1], we might approximate the electromagnetic self-energy as follows:
[2]
Based on [2], it seemed that the rest mass of an electron was essentially electromagnetic in origin; although experiments suggested that the electron radius was much smaller than the value [r_{0}] shown in [1]. For example, recent high-energy electron–positron scattering experiments suggest an upper limit to the electron radius as:
[3]
So, over the years, a number of physicists have tried to develop schemes to avoid the apparent divergence of the electromagnetic self-energy of a point-particle electron, e.g. Dirac and Feynman to name just two. However, we might simply generalise the main arguments for the purposes of this discussion.
- Point charges are quantum ‘objects’ and, as such, classical
electrodynamics breaks down as the quantum domain is approached.
- Point charges do not exist as ‘real’ particles having a
finite size, such that the problem is one of mis-interpretation of the
real nature of the wave-particle.
- By a process of mathematical formalism, the radius of a charge can be reduced to zero without introducing infinities.
As a result, the early phase of post-war developments struggled to reconcile the ideas of classical electromagnetics with the quantum perspective of the electrodynamic properties of electrons, photons and other elementary particles. However, while the first two points above might have been a natural assumption in quantum field theory, there was still the practical problem connected to its calculations, which appeared to produce an infinite answer for the mass [n] of an electron, as used in Feynman’s QED model. Finally, in order to try and resolve this problem, the idea of ‘renormalization’ was introduced in which the electron was given an infinite ‘bare’ mass that would cancel out the infinity arising from its self-energy. The ‘measured’ mass was then defined as the difference between these two infinities, which returned a finite quantity and appeared to support the process of renormalisation, at least, in terms of the experimental results, if not the theory.