Renormalization: Dodging Infinities
by John G. Cramer
When I was in physics graduate school, we advanced graduate students were taught that there were serious problems with two important theories of particle physics: quantum electrodynamics (QED), the relativistic quantum theory of the electromagnetic force, and quantum field theory (QFT), the relativistic quantum theory of particles and fields. The problems arose because both formalisms contained spurious infinities associated with self-energy that needed to be subtracted away “by hand” using a procedure called “renormalization” before meaningful calculations and realistic predictions could be made. Our professors assured us that this flaw was only a temporary embarrassment that would soon be corrected by replacing the current “paste-up” theories with something better. At about the same time, Paul Dirac, one of the founders of quantum mechanics, declared that renormalization made QFT completely inconsistent and in need of immediate replacement.
That was many decades ago, but the offending theories are still actively used in particle physics, and no serious replacements have appeared. What has happened instead is that only the embarrassment has gone away. Particle theorists have learned to accommodate renormalization and the removal of infinities as just one of the rules of the game, they have ceased to worry about it or its implications, and some even regard renormalization as a virtue. Further, the formalisms of QED and QFT are now well known to be fundamentally incompatible with general relativity, our standard theory of gravitation. This is in part because QED and QFT need renormalization and in part because of their reliance on an underlying background space-time. However, these problems are not widely taken as a criticism of the quantum theories.
There is also another major problem with quantum field theory. The quantum vacuum, according to QFT, is filled with fireworks of variable-mass virtual particles that spring briefly into existence, interact to move energy, momentum, angular momentum, and quantum numbers around, and then disappear. As a side effect, these virtual processes give intrinsic energy to the quantum vacuum, and that has consequences. It has now been shown that QFT overestimates the energy present in the vacuum of the universe, as represented by Einstein’s cosmological constant L, by a factor of about 10120. That’s an enormous number: a one followed by one hundred and twenty zeroes.
In fact, this QFT calculation holds the record as the worst theoretical prediction in the history of physics. Somehow, these problems have not produced much concern in the theoretical physics community about the validity of the theory that produced it and that they routinely use. QED and QFT remain as central pillars of the Standard Model of Particle Physics, and they are actively taught today in physics graduate schools all over the world.
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So what is the alternative? We know that somewhere out there, beyond the limits of our present comprehension, there must be a better theory of quantum gravity (or perhaps even a Theory of Everything), which covers all the bases and unifies gravitation with quantum mechanics. Despite massive effort, not much progress has been made toward formulating such a theory. However, some theorists are have found ways to deduce some of its outlines and characteristics.
As an example, some years ago J. Gegelia and N. Kiknadze (G&K) of Tbilisi State University, Georgia (the nation) investigated the question of whether such an ultimate comprehensive Theory of Everything could possibly include and use renormalization. The issue G&K addressed was whether the presence of infinities and the need for renormalization are only passing artifacts arising from QED and QFT being low-energy reductions of the higher-level Theory of Everything. The alternative is that there is no such connection and that the higher-level theory permits no infinities or renormalizations, even in its low-energy limits. G&K showed that the latter is true, that the higher-level Theory of Everything must either be non-renormalizable or must contain no infinities at all, and that it cannot be reduced to an infinity-containing renormalizable theory like QED or QFT in any low-energy limit. In other words, quantum electrodynamics and quantum field theory, our present workhorse theories of particle physics, are wrong in some very fundamental way. They do not need incremental improvement; they need complete replacement from the ground up.
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Where does that leave us? A few decades so ago, the hope arose in the theoretical physics community that the development of string theory might resolve the quantum gravity problem and eliminate the need for renormalization. String theory, describing particles as standing waves on tiny compactified loops of multi-dimensional string, offered a new and rather mathematically elegant approach to the development of a Theory of Everything. It included gravitational physics, particle physics, and quantum physics under one roof, and at the same time made renormalization and the subtraction of infinities unnecessary. There were interesting connections with black holes as described by general relativity that were taken to be significant. String theory became very fashionable in the prominent physics graduate schools of the world. Several generations of the best and brightest young theory students at the top universities boldly climbed aboard the string theory bandwagon.
However, Harvard’s late great theoretical physicist Sidney Coleman, a lifelong enthusiast of science fiction, once commented that in conversations with the top-level string theorists that had been passing through Harvard, he always had the distinct feeling that they were “stringing him along.” Coleman was right. Regrettably, after a huge amount of theoretical effort, the dust has finally settled to reveal that: (a) string theory is incapable of making testable predictions that can be compared with experiment, thereby failing philosopher of science Karl Popper’s falsification criterion, which must be met before any formalism can legitimately be called a “scientific theory,” and (b) string theory has been shown to describe a nearly infinite “landscape” of different possible universes, with no assurance that our own universe, which seems to be unusual in having a relatively small cosmological constant, is even one of the hypothetical universes populating that stringy landscape. The rising consensus within the general physics community is that string theory is a mathematically elegant dead end.
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As an infinity-free replacement for QED and QFT, the leading alternative to string theory is called loop quantum gravity (LQG). Loop quantum gravity is less ambitious than string theory, in that it does not attempt to describe what particles are, but only aims to unify general relativity with quantum mechanics. Its approach is to quantize space and time into small indivisible loci that form a “spin foam network” and to build a theory of gravity on this quantized base. LQG has the advantage that unlike QED and QFT, it does not require a background space time, but instead produces its own space-time in the style of general relativity.
A question that immediately arises is whether LQG is able to fulfill the Popper criterion by making testable and potentially falsifiable predictions. In most situations LQG simply makes predictions that are identical to those of standard general relativity. The in-principle testable predictions unique to LQG, which distinguish it from general relativity, are concentrated in the areas of black hole behavior and cosmology. For example, LQG predicts that under some conditions a collapsing black hole should “bounce” and reverse its motion, becoming a “white hole” from which light and particles will flow outward. Similarly, in cosmology LQG predicts that the Big Bang was actually a “Big Bounce,” asserting that some predecessor universe collapsed to a critical point at which it reversed its motion and became our own Big Bang.
The problem with such assertions and predictions is that they do not overlap with most astronomical observations. It is not clear if LQG’s Big Bounce hypothesis could be connected in any way with existing measurements of the cosmic microwave background or other astronomical observables. There are a few observations that are attributable to black hole behavior, but most of these concern the accretion disk, not the black hole itself. Recent detections of gravitational waves arising from the merging of two massive black holes are in the right general area for such testing, but so far the details of the observed gravitational wave structure are well accounted for by standard general relativity.
Attempts have been made by LQG advocates to associate the white-hole prediction of the theory with the observed phenomenon of gamma ray bursts (see my columns 94 in the March 1999 and 142 in the May 2008 issues of Analog). However, there are already fairly good explanations of the origin of gamma ray bursts that do not require the white-hole hypothesis. While the predictions of LGQ are better than no predictions at all, it is presently not possible to falsify or confirm them with astronomical observations.
Further, LQG presently has no overlap with most of particle physics, and in particular does not provide an alternative to QED and QFT that can be used for making predictions in the domain of particle properties and interactions. It may perhaps show a path that might lead to a Theory of Everything, but if so, it stands at the very beginnings of that path.
Thus, contemporary theoretical physics finds itself in a difficult situation: it has become clear that the workhorse theories of QED and QFT, pillars of the Standard Model of Particle Physics, are fundamentally flawed, is some cases give ridiculously wrong predictions, and are in urgent need of replacement. However, no such replacement is on the horizon. New approaches are badly needed.
Critiques of Renormalization:
“Can the Fundamental Theory of Everything be Renormalizable?”, J. Gegelia and N. Kiknadze, ArXiv: 9507048v3
“Does renormalization make sense?”, P. A. M. Dirac, AIP Conference Proceedings 74, 129 (1981); https://doi.org/10.1063/1.33110
Karl Popper, Realism and the Aim of Science: From the Postscript to The Logic of Scientific Discovery (1st ed.). London and New York: Routledge (1983); ISBN 0-415-08400-8.
John G. Cramer’s 2016 nonfiction book describing his transactional interpretation of quantum mechanics, The Quantum Handshake—Entanglement, Nonlocality, and Transactions, (Springer, January 2016) is available online as a hardcover or eBook at: http://www.springer.com/gp/book/9783319246406.
SF Novels by John Cramer: eBook editions of hard SF novels Twistor and Einstein’s Bridge are available from the Book View Café co-op at: http://bookviewcafe.com/bookstore/?s=Cramer.
Alternate View Columns Online: Electronic reprints of 202 or more “The Alternate View” columns written by John G. Cramer and previously published in Analog are currently available online at: http://www.npl.washington.edu/av.
Copyright © 2019 John G. Cramer