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The Ultraviolet Divergence

Calculations using Feynman rules give good agreement with experiment and can be carried out in much more depth that the simple instances of scattering considered on this site, but certain diagrams containing loops give divergent results which cannot be physically meaningful.

The cause of the divergence is the equal point multiplication between field operators, φ^†(x)φ(x). The divergence in this quantity was removed from the interaction density by normal ordering, but is not eliminated from the perturbation expansion. The integrals for Feynman diagrams are taken over all possible positions where an interaction might be detected (if such a measurement were possible). The integrand is not defined in the limit when the size of the loop goes to zero and for certain Feynman diagrams containing loops the calculation leads to an infinite result. This is known as the ultraviolet divergence of qed because small loop sizes correspond to high energies.

Although the theory appears to break down at this point, it turns out that it is possible to discard divergent quantities. When this is done the resulting corrections agree with experiment to 14 significant figures, and, with the exception of special relativity, make quantum electrodynamics the most empirically accurate theory known to science. The procedure for removing the divergent quantities is regularisation. A number of different methods for regularisation are used, together with justifications of different levels of validity. A detailed treatment goes beyond the scope of my immediate objectives for this site. I will concentrate on explaining how regularisation can be justified.

Regularisation

Causal perturbation theory, described in Finite Quantum Electrodynamics by G. Scharf, shows how loop diagrams can be calculated rigorously using the method of Epstein and Glaser. Propagators are written as a sum of advanced and retarded parts, separated using step functions, Θ, as shown in Wick’s Theorem. The step functions are replaced with a smooth, monotonic function, χ, and the advanced and retarded parts are calculated separately. Then a limit is taken in which χ becomes a step function and the advanced and retarded parts are described as distributions. When the distributions are combined loop integrals are finite and the usual Feynman rules are obtained for diagrams without loops. Thus the cause of the divergence is seen as the abuse of Wick’s theorem, combining the advanced and retarded parts prematurely and using a wrong order of taking limits.

The problem arises because quantum theory is conventionally set up as a continuum theory, using an infinite dimensional Hilbert space. In such a theory, is divergent and has no meaning. Mathematically the continuum only exists as a result of infinite limiting procedures. It strictly refers to a set of sequences indexed by n, such that taking n larger than some value, N, makes no practical difference to results. On this site, Hilbert space is regarded as strictly N-dimensional, for a large value of N. is well defined, but in terms of creation and annihilation operators, would mean that a particle is annihilated at the moment of its creation. Physically this is not sensible for a theory of particles. Regularisation simply means that amplitudes from this non-physical process must be excluded from Feynman diagrams. When this is done rigorously, the dependency on N is also removed.

This analysis of the origin of ultraviolet divergences is effectively the same as that given in causal perturbation theory, in that the limit is taken after removing the equal point multiplication. Similarly, lattice regularisation excludes the equal point multiplication between fields from the outset. The origin of the ultraviolet divergence is the incorrect use of Wick’s theorem. The analysis gives physical motivation to the method of Epstein & Glaser. The difference between the treatments is that I describe a limiting procedure using a discrete sum whereas Epstein and Glaser use a continuous switching function. While Scharf says (on p163 of Finite Quantum Electrodynamics) “the switching on and off the interaction is unphysical”, I regard the switching off and on of the interaction at as a physical constraint meaning that only one interaction takes place for each particle in any instant, or equivalently that a particle cannot interact again at the moment of its creation.

The removal of products describing the annihilation of a particle at the instant of its creation is most naturally done by normal ordering, but, whereas it is usual to normally order the interaction Hamiltonian, according to the argument above all equal time products of field operators must be normal ordered. In practice this is largely academic. It is seen in the evaluation of the propagator that the step functions strictly exclude the equal point multiplication from the perturbation expansion.

More ad hoc treatments can be justified by observing that integrals always stand for finite sums. The step functions can be replaced with and , for some small positive time interval, χ. This introduces an upper bound, or cut-off, in energy. The integrals are evaluated with a cut-off and cut-off dependent terms are discarded. Whichever method is used, regularisation removes divergent quantities, but leaves a degree of indeterminacy in the result. Indeterminacy is removed by requiring that mass, m, and charge, e, are the observable quantities.

Vacuum Fluctuations

Vacuum fluctuation diagrams contain no external lines. For any connected Feynman diagram one can draw another diagram which is the same but for the inclusion of a vacuum fluctuation diagram. The effect of adding a vacuum fluctuation to all diagrams is equivalent to multiplying the theory by a constant, and has no impact on the predictions of the theory. Vacuum fluctuation diagrams should therefore be ignored. It is not possible to say that the vacuum is in a state of constant activity, with the spontaneous creation annihilation of virtual particles, because any such particles do not interact with the observable matter of the universe, and should not be regarded as a part of physical reality.

Renormalisation

Older treatments justify regularisation using an argument based on renormalisation. It is said that the bare charge appearing in the interaction density, and the bare mass appearing in the Klein-Gordon and Dirac equations are not equal to the observable charge and mass. The mass of an electron is modified because it is surrounded by a cloud of “virtual” photons. This is wrong. Conservation of energy-momentum implies that the total energy and total momentum of a particle and a cloud of virtual particles in an external line is equal to the energy and momentum of a single free electron in an initial or final state. The derivations of Maxwell’s Equations and the Lorentz Force law show that the observable charge and mass are equal to the bare values. On external lines, amplitudes from the “virtual cloud” are zero and contribute nothing to the perturbation expansion.

In some treatments divergent quantities associated with loops are removed through the addition of counter terms to the interaction Hamiltonian. Often this is justified with a renormalisation argument. The addition of these terms is supposed to have the effect of replacing bare particles with particles “dressed” in clouds of virtual particles, which are thought to more accurately reflect underlying physics. The counter terms are equivalent to the removal of divergences arising from the equal point multiplication, but this is mathematically unsound. Counter terms are themselves not defined; one cannot remove an undefined quantity by subtracting another undefined quantity. It is necessary instead to formulate the theory in such a way that undefined quantities are excluded at the outset.

Methods such as these lead to the sharp criticisms of renormalisation from Feynman and Dirac, perhaps the two most important figures in the development of quantum electrodynamics. The central role of renormalisability in further developments of field theory has lead, not to further advances in physics but to apparently futile searches for Goldstone bosons, the Higgs particle, and to string theory which has yet to produce a useful prediction or show that it models Nature in any way. These methods are not required in causal perturbation theory, nor in the particle model described on this site.

The Renormalisation Group

For example,
is irreducible, but
is not. For each diagram containing an internal electron or photon line, there will be other diagrams found by adding loops to that line. Let iΣ(p) be the contribution from all one particle irreducible diagrams added to an internal photon line with momentum p.
A discussion of the summation of irreducible contributions to the electron propagator follows a similar pattern. The internal photon propagator together with the contribution from all additional internal loops is
Assuming the convergence of to a value with , this may be summed as a geometric progression. So, the photon propagator together with contributions from internal lines is
Thus adding all loop diagrams to the photon propagator is equivalent to multiplying the charge by a factor
This factor can be calculated in approximation and increases for large energies. The coupling constant is said to run, meaning that the apparent coupling constant is greater at high energies. The theory of the running coupling constants is known as the renormalization group.

The Landau Pole

It is found that, for internal lines with sufficiently high energy, , and the summation of contributions from internal loops breaks down. The point at which it breaks down is the Landau Pole. The importance of the Landau pole is a matter of some conjecture. The energy at the pole is far in excess of anything which can be achieved in practice, and the theory is regularised using a cut-off which can be taken at a much lower energy than the pole. Nonetheless, the energy integrals corresponding to loops are found from a contour integration, in which a limit is taken to infinity. When all contributions from internal lines are taken to the propagators, the integrand for the loop at the vertex becomes invalid at the Landau pole. Strictly, perturbative qed does not “exist”, in the mathematical sense meaning that it is not consistent. Often this is taken to mean that there must be some short distance modification to qed, corresponding to an energy scale less that that of the pole.

Relational quantum gravity will introduce a small effective interval between the interactions of a particle which does not allow the size of loops to go to zero. The geometric progression the expression for iΜ(p) is modified and terminates after a finite number of terms. As a result the modified theory does not have a Landau pole.

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