Buzz Lightyear‘s tag line “To infinity … and beyond” reminds me of the mantra in modern physics that the appearance of infinities in equations tells us that we don’t understand something, that our computational model has hit a wall, so to speak.
Wiki: Sometimes an infinite result for a physical quantity may mean that the theory being used to compute the result may be approaching the point where it fails. This may help to indicate the limitations of a theory. … it may mean simply that the theory is incapable of describing the situation properly.
The Feynman loops present us with a new problem that has bothered physicists in many different ways for many years: put simply, when we compute the loop sums for certain loop diagrams, we get infinity! The processes we compute seem to become nonsensical. The theory seems to crash and burn. — Lederman, Leon M.; Hill, Christopher T.. Symmetry and the Beautiful Universe (p. 253). Prometheus Books. Kindle Edition.
See other citations below.1
Also, often cited is the refrain that any attempt at a GUT (Grand Unified Theory) must deal with the infinities that arise when trying to unify quantum mechanics with general relativity. Aside from the equations, the reality of any such unification is at a scale which is experimentally unverifiable, namely, the Planck scale.
Even more problematic is that we can’t even predict what a merged theory of quantum gravity would look like. Physicists have often been able to predict particles that were later discovered experimentally (like the top quark or the Higgs boson), but so far all the theories we have that try to merge gravity and quantum mechanics fail; they keep giving nonsense results, like “infinity.” — Cham, Jorge; Whiteson, Daniel. We Have No Idea: A Guide to the Unknown Universe (p. 86). Penguin Publishing Group. Kindle Edition.
So, methods have, at least in QM, been developed to achieve practical results despite infinities. Perturbation theory and renormalization. In some cases, infinities on opposite sides of equations cancel, leaving a physically meaningful result.
 Feynman persevered, attended the ceremony, and prepared a beautiful Nobel lecture giving a truly personal history of his journey to discover how to tame the infinities of QED. And even as late as 1965, Feynman still felt the program of renormalization that he had spearheaded was merely a way of sweeping problems of infinities under the rug, not curing them. — Krauss, Lawrence M.. Quantum Man: Richard Feynman’s Life in Science (Great Discoveries) (p. 231). W. W. Norton & Company. Kindle Edition.
When we calculate certain things in quantum theories, such as certain properties of electrons and photons, we get divergent results. Even though almost all the things we compute get sensible answers-in fact, answers that are consistent with experiment-a few simply yield mathematical nonsense. As we have just seen, when we are calculating the vacuum energy density in quantum theory, we get a divergent series. If we didn’t halt the sum at some cutoff point, our result would be minus infinity for the vacuum energy density, and that’s just plain nonsense. If the vacuum had such an energy density, our universe would crumple up into an infinitesimal pinpoint, crushing everything into nothingness, including you, me, and all our literature. Getting this nonsensical answer of a negative, infinite vacuum energy density tells us that something fundamental is wrong, or at least missing, in our theory. — Leon M. Lederman; Christopher T. Hill. Quantum Physics for Poets (Kindle Locations 2612 – 2616). Kindle Edition.
You see, in order to make quantum mechanics work in the first place, physicists have to apply a special mathematical trick called renormalization. It’s what allows quantum mechanics to deal with strange infinities, like the infinite charge density of a point particle electron or the infinite number of very low-energy photons that an electron can radiate. By using renormalization, physicists can sweep all of these infinities under the rug and pretend there aren’t any dead bodies hiding under there. — Cham, Jorge; Whiteson, Daniel. We Have No Idea: A Guide to the Unknown Universe (pp. 296-297). Penguin Publishing Group. Kindle Edition.
One thought on “Infinity and beyond … under the rug”
I’ve encountered the term quantum correction a lot while reading Joseph Polchinski’s Memories of a Theoretical Physicist. My only connection is with Feynman diagrams, as referenced by Lederman and Krauss and a stub Wiki article (see below). Mathematical tools to deal with mind-boggling complexity, eh.
Feynman then proceeded to use precisely the kind of analysis that had stood him in such good stead with QED to calculate quantum corrections to gravitational processes. The effort was not so simple however. General relativity is a far more complex theory than QED because while photons interact with charges in QED, they do not interact directly with each other. However, because gravitons interact with any distribution of mass or energy, and since gravitons carry energy, gravitons interact with other gravitons as well. This additional complexity changes almost everything, or at least makes almost everything harder to calculate. Needless to say, Feynman did not find that a consistent quantum theory of gravity interacting with matter, without any nasty infinities, could be derived by simply treating general relativity as he had electrodynamics. There still is not such a definitive theory, though candidates have been proposed, including string theory. Nevertheless, every major development that has taken place in the fifty-odd years since Feynman began his work in this area, involving a line of scientists from Feynman to Weinberg to Stephen Hawking and beyond, has built on his approach and on the specific tools he developed along the way. — Krauss, Lawrence M.. Quantum Man: Richard Feynman’s Life in Science (Great Discoveries) (p. 249). W. W. Norton & Company. Kindle Edition.
The real power of Feynman diagrams is that we can systematically compute physical processes in relativistic quantum theories to a high degree of precision. This comes from what we call the quantum corrections, or the so-called higher-order processes. In figure 29, we show the second-order quantum corrections to the scattering problem of two electrons. This is a set of diagrams, each of which must be computed in detail and then added together including the previous diagram in figure 27, to get the final total result for the T-matrix. (The T-matrix, as noted above, is essentially the quantum version of the potential energy between the electrons and describes the scattering process.) This gives the total T-matrix to a precision of about 1/ 10,000. We can then go to the third order of higher quantum corrections to try to get even more precise agreement with experiment. Third-order calculations, however, are extremely difficult and very tiring for theoretical physicists. Only the brave and most energetic try them. — Lederman, Leon M.; Hill, Christopher T.. Symmetry and the Beautiful Universe (p. 252). Prometheus Books. Kindle Edition.
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