Bismillah. More from Dr. Sabbir Rahman:

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Assalamu `alaikum,

Some time ago the question was asked as to whether the “Standard Model” of physics (i.e. our current model of elementary particles such as electrons, quarks, neutrinos, photons, gauge bosons, etc) was “renormalisable”.

The question is an important because it is often the case that in quantum field theory, when we naively try to calculate the probability of some physical process taking place, we end up with the answer ‘infinity’ which is non-sensical. In particular, this happens in quantum electrodynamics (QED), which was invented by Feynman & Wheeler and others around 1945 in a series of famous papers. The problem is, that if we assume that the electron has a finite mass, and that the electromagnetic force has finite strength (i.e. “coupling”), then the integrals in the calculations are divergent and give rise to infinities because of self-interactions of the electron (e.g. where it emits and then re-absorbs a photon), which for example would cause the electron to appear to have infinite mass, as opposed to the finite mass that it is known to have.

Fortunately, a mathematical ‘trick’ was developed, called “renormalisation”, which basically involves manually introducing additional infinite ‘counterterms’ into the calculations, which are designed to precisely cancel out the divergences in the integrals. This is like setting the bare mass and couplings to infinity in just the right way to make the actual measured mass come out just right. None of this is really justified mathematically, but as it happens, the answers come out just riright, and indeed QED is the most accurate physically theory we have. In fact it is so accurate as to be astonishing. Although renormalisation works fantastically well, the physical interpretation of the resulting theory becomes something of a challenge. This is a strange situation, but it is a fair reflection of the current state of our understanding of theoretical physics.

While QED is ‘renormalisable,’ not all theories are – and the rules as to whether any given theory is renormalisable or not can be very complex. If a theory is not renormalisable, then the infinities in the calculations cannot be cancelled out by adding counterterms, and the theory is therefore generally considered inconsistent.

Now the Standard Model, which describes not only the electromagnetic interactions between electrons and photons, but also the weak and strong forces, has a much more complicated form than QED, with a rather complex Langrangian containing many parameters such as masses and coupling constants. According to the Wikipedia entry on ‘renormalization’, “… the Standard Model of particle physics contains only renormalizable operators.” While this means that the Standard Model may potentially be renormalisable, as far as I am aware, no-one has actually gone to the immense trouble of actually proving it. Rather, it is generally (and rather naively) merely assumed to be renormalisable.

Now this is quite a contentious issue, as it lays open the possibility that our current theory of elementary particles is actually inconsistent. This is not merely idle speculation – some years ago, while I was at MIT, a professor in the mathematics department actually held an emergency seminar in which he claimed that the standard model was actually non-renormalisable, based upon his (extremely complicated) calculations to third order in perturbation theory. This obviously sent shock waves through the Theoretical Physics department, and a fellow graduate student of mind was tasked with checking the calculation himself. It turned out that all as well to third order, and that the Mathematics professor had made a mistake somewhere in his calculations.

But on the other hand, it did highlight the fact that renormalisability of the theory had not actually been checked. A few years ago, I brought the matter up on the sci.physics.research newsgroup, and it appears that no-one really knows, even now, whether the Standard Model is renormalisable. Furthermore, our perspective of both the Standard Model (it is only a low energy approximation to the real ‘theory of everything’) and renormalisation has changed over time, and so it is no longer clear whether the question is even relevant or meaningful (for example due to the existence of ‘dualities’ between theories with large and small parameters – a theory with infinities may actually be equivalent to another theory that is finite, and so may be well-defined after all).

The same professor mentioned above went on to ‘prove’ in two further papers that the Standard Model was indeed nonrenormalisable. It is possible that these papers also contain calculational errors (though I would imagine that the professor would have taken significantly more care the second time around), and so the matter remains, as far as I am aware, unresolved.

This is the newsgroup thread to which I refer above (see the seventh message in the thread, submitted by myself, which initiated the subsequent fairly interesting discussion):

Was-salam,

Sabbir.

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