San José State University
Thayer Watkins
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The Infinities Arising in
Quantum Field Theory

The energy density of a static electric field is proportional to the square of its electric field intensity E. The electrical field intensity at a distance r due to a charge Q uniformly distributed over a spherical surface of radius R is proportional to Q/r² for r>R and 0 for r<R. Therefore the total energy V of the field is given by

V = (Q²/(6πc²)(1/R)

where c is the speed of light. This is often called the self-energy of the particle.

Note that V→∞ as R→0. Thus, an electrical field emanating from a point, such as that generated by a point charge, has infinite energy. Therefore theorizing with point particles, as in classical analysis, may produce valid results, but when the theorizing involves manipulations of the particles' fields the results may be affected by the infinite energy of the field of a point charge.

Werner Heisenberg and Wolfgang Pauli 1929 and 1930 noted the problem of the infinite energy of the fields of point charge. Robert Oppenheimer in 1930 confirmed this problem of infinite energy for bound electrons and also in 1930 Ivar Waller did the same for a free electron. Physicists considered more sophisticated models involving photons as well as electrons. It was found that if all photons with wave numbers greater than 1/R are eliminated then the total self-energy of an electron asymptotically approaches a quantity proportional to 1/R² as R→0. This tendency of field energy to approach 1/R² is called an ultraviolet divergency.

Viktor Weisskopf found in a model involving electrons and photons that eliminating photons with wave numbers greater than 1/R resulted in a self-mass me of the following form

me = (3α/2π)m·ln(h/(mcR))

where α is the electromagnetic fine structure constant, m is the mass of an electron and h is Planck's constant divided by 2π.

The lesson in all of this that cannot be avoided is that point particles and the corresponding infinities are physical impossibilities. The charge radii may be small but they cannot be zero.

Physicists were looking for ways to cancel infinities even though ∞−∞ can be anything.

P.A.M. Dirac also found infinities arising in his theorizing on the effect of charges on the vacuum states.

Infinities occurred also in the analysis of the scattering of light by light.

In 1937 Felix Bloch and Arne Nordsieck found that infinities arising in the lower part of the range of integration could be eliminated by including processes in which any number of low energy photons are produced.

Werner Heisenberg in 1938 suggested that there is a quantum of length so no point particle can exist and hence the infinity associated with a point particle cannot arise.

There then developed the notion that problem of infinities in models could be dealt with by a systematic revision of the parameters of the model. This was called the renormalization of the parameters. This took the form of positing a bare mass for particles that should be used the models instead of the measured mass. Viktor Weisskopf in 1936 argued in an article that such renormalizations would generally solve the problem of infinities in quantum field theory. It was the work of Kenneth G. Wilson with what was called the Renormalization Group which finally resolved the problem of infinities in Quantum Field Theory. Wilson was awarded the Nobel Prize in 1982 in Physics for this work.

(To be continued.)


A.I. Miller (ed.), Early Quantum Electrodynamics, Cambridge University Press, 1994.

Steven Weinberg, The Quantum Theory of Fields, Vol.1, Cambridge University Press, 1995.

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