The Ineffable Higgs
Surely its discovery meant something? The Standard Model (SM) of particle physics demanded its existence, after all; and the demand was met. If it took forty years and more than $16 billion to discover the thing, physicists could with satisfaction observe that the public got what it paid for, the first step, of course, in demanding that the public pay for more of what it got. Photographs of Peter Higgs staring tenderly into space, ses yeux perdus, conveyed an impression of appropriate intellectual satisfaction.
The discovery was announced; the story reported; and then there was silence. Physicists endeavoured, of course, to maintain the impression that they had discovered something of inestimable value. They were game. Writing in The Daily Beast, Sean Carroll predicted that the Higgs Boson would "revolutionize physics," and if this is what physicists always say, then at least they seem never weary of saying it.
Lawrence Krauss, writing in The Daily Beast as well, gave it his best. Many years ago, Leon Lederman had designated the Higgs Boson as the God particle. No one can today remember why. The God particle? "Nothing could be further from the truth," Krauss remarked. In this, of course, he was entirely correct: Nothing could be further from the truth.
In the end, Krauss, like Carroll before him, could do no better than an appeal to the revolution. The discovery of the Higgs Boson "validates an unprecedented revolution in our understanding of fundamental physics ..." Readers of The Daily Beast are always pleased to uphold the revolution, no matter how revolting. Yet, the Standard Model was completed in the early 1970s, the revolution it conveyed having begun in the late 1920s, circumstances that might suggest the gradual emergence of a soberly modulated consensus more than anything else -- the perfect truth, as it happens.
If the revolution is either far away or long ago, there is always God. The discovery of the Higgs Boson does nothing to confirm his existence, Krauss argued, therefore it must do everything to diminish his relevance. And so it does. The Higgs Boson, he wrote, brings "science closer to dispensing with the need for any supernatural shenanigans all the way back to the beginning of the universe -- and perhaps even before the beginning, if there was a before."
About this declaration, since it countenances a before before a beginning or a beginning before a before, all that one can say is that Krauss has covered his bases.
The Standard Model (SM) of particle physics was created to impose order on elementary particles that during the 1960s had seemed to multiply in proportion to the funding available to determine them. If there were very many particles before the advent of the SM, there remained very many particles after its advent ¬-- reason enough to wonder whether any of them were really elementary.
Whatever their ultimate nature, the elementary particles are today divided into fermions, hadrons and bosons. Fermions come in twelve flavors arranged in two families of leptons and quarks. The quarks form a three-fold family structure of their own: Up or down, charmed or strange, at the top or loitering in the bottom. These twelve particles give rise to twelve anti-particles.
The bosons comprise the photon, the gluon, and the electroweak bosons, of which the Higgs boson is now king.
And the hadrons are organized as families of bound quarks, and appear either as baryons, which are composed of three quarks, or as mesons in which quarks cohabit amicably with anti-quarks.
The SM has many virtues: Simplicity is not among them.
The SM is both a scheme of classification and a system of explanation, one encompassing quantum electrodynamics, and theories of the weak and strong nuclear forces. These theories are not by any means the same, but they are expressed in the common language of quantum field theory, and it is quantum field theory that the SM serves to make great.
Quantum mechanics was created in the 1920s to explain experimental results that defied classical explanations. Given the choice of passing through one of two experimental slits, a single photon somehow managed to pass through them both. The photon was thus a wave. Other experiments had before suggested that it was a particle. Quantum mechanics accommodated experience by defying common sense. The photon was both a wave and a particle.
Einstein's theory of special relativity and quantum mechanics, physicists understood, were not obviously in conflict, but neither were they the best of friends. Special relativity drew a tight connection between energy and mass; and quantum mechanics was professionally engaged in dissolving tight connections.
With respect to a particle's position and momentum, the axis on which the old world turned, classical physics had been absolute. The physical properties of a particle were accessible to measurement and they were accessible to measurement all the way down.
In quantum mechanics it is otherwise. In 1926, Werner Heisenberg argued that the formalism of quantum mechanics placed limits on measurement. The more certain a particle's position, the less certain its momentum. Measuring them both to the same level of accuracy was impossible and it was impossible in principle -- reason enough to suppose that the uncertainty principle was a principle of nature, and not an artifact of measurement.
The uncertainty principle and special relativity assigned to an otherwise empty region of space the power to produce new particles from the void. The greater the uncertainty about the void, the more it seethed, and since at short distances, it did a great deal of seething, it was at short distances that it seethed to most productive effect.
Any version of quantum mechanics incorporating special relativity would necessarily be a theory of many particles. They were potentially all over the place.
And for this, a field was required. It was this that Paul Dirac provided.
The date is 1927.
Quantum mechanics broke the distinction between particles and waves, quantum field theory the distinction between particles and fields. A field is something like the wind: It is everywhere in space and nowhere in particular. Within its ambit, particles appear as focused ripples, knots of a sort, some temporary but countable tightening of things.
Quantum field theories were by no means an immediate success because they were by no means consistent with themselves. Whenever they were applied, infinite magnitudes appeared, glum, glowering, irremovable. New mathematical techniques were required to remove them, and although they worked, physicists did not know why they worked and used them with the awkward sense that for every good reason, they were doing a very bad thing.
Quantum electrodynamics was completed in the late 1940s by Richard Feynman, Julian Schwinger and Sin-Itiro Tomonaga. Theoretical calculations of the electron's magnetic monopole agreed with experiments to one part in a billion. It is an agreement that may fairly be described as freakish. Having sinned to such good effect in the 1940s, physicists continued to sin thereafter. In the 1950s, Chen Yang and Robert Mills proposed a daring generalization of quantum electrodynamics; and in the 1960s, Steven Weinberg, Sheldon Glashow and Abdus Salaam took the Yang-Mills equations and ran like the wind. They stopped when they had a theory of the weak nuclear force in hand.
And more -- far more. The idea that there is a form of unity beneath the diversity of experience is one that has animated the imagination of physicists since the seventeenth century. It is a powerful but not an obvious idea. One of the achievements of the SM is the demonstration that at high energies, the weak and the electromagnetic force are the same. The W (W+ & W-) and Z bosons are the vehicles by which the weak nuclear force is mediated. They are nicely described in terms of a theory that establishes their underlying symmetry. Theoretical considerations suggested ¬-- they demanded -- that like the photon these particles should have no mass. The W and Z bosons were massive: They were simply huge. In the work that earned them their Nobel Prize, Weinberg, Salaam and Glashow argued that the symmetry controlling these bosons must have been broken: It must therefore have been broken. It is the Higgs mechanism that accounts for the symmetry breaking and so accounts for their mass.
There remained the strong force and its interactions. Although quarks were required in theory, they were impossible to observe in experiment. It was not until David Gross, David Politzer and Frank Wilczek proposed their theory of quark confinement that physicists were able to affirm that what could not be observed should not be observed because it could not be observed. Although Weinberg, Glashow and Salaam had demonstrated a form of unity between the electromagnetic and the weak force, the strong force remained like Achilles defiant in its isolation.
The SM accommodates three forces and it expresses three theories. It is obviously incomplete. It does not encompass the force of gravity. What it does, it does very well, but if tomorrow it would require renovation, no physicist would worry overmuch. Theories come and go. Quantum field theory is otherwise, a way of thought, irreplaceable. Among particle physicists, the conviction is almost universal that quantum field theory is the structure chosen by nature to conduct her affairs.
"Absolutely all phenomena," Ptolemy wrote in a slightly different context, "are in contradiction to any of the alternate notions that have been propounded."
And this raises a question at least as difficult as any raised by particle physics itself. Just how should claims of this sort be understood and if understood assessed? It makes no sense to take particle physicists at their word. It is their word that we are uncertain about taking.
There is in every human being something like a vacuum state, a lowest energy level in which physical judgments are spontaneously made, a zone of comfort. Evolutionary psychologists often describe this vacuum state as folk physics. I would prefer to see die Völker left in lederhosen but the idea is useful because it is inescapable. No one, of course, remotely suggests that the SM is much to be admired because it is gemütlich. Popular explanations are plainly absurd. "Here's the secret of the Higgs mechanism," Sean Carroll observed in Discover: "when you spontaneously break a gauge symmetry, the would-be Nambu-Goldstone boson gets 'eaten' by the gauge bosons!" The metaphor by which one boson eats another, although widely employed, means nothing whatsoever.
What is required of quantum field theory is not its translation into common sense. The physical sciences have been in retreat from common sense since the seventeenth century. There is nonetheless a distinction as important in the sciences as it is in military affairs. It is the distinction between a retreat and a rout. If the SM and quantum field theory are to have a claim on our intellectual allegiance, there must be some scheme by which the ideas that they embody may be recovered in a way that makes simple, persuasive and intuitive sense. It is not enough in this respect to say that certain ideas work because the predictions they make possible are accurate. That goes without saying. The question is why this is so.
It is in mathematics and not theoretical physics that judgments of this kind are made and then enforced. Across the vast range of arguments offered, assessed, embraced, deferred, delayed or defeated, it is only within mathematics that arguments achieve the power to compel allegiance because they are seen to command assent. And it is only by means of mathematics that the powerful ideas of an alien discipline such as theoretical physics may step by step be returned to the ordinary human power to grasp things without mediation and so to grasp things at once.
Quantum field theories are, of course, expressed in mathematical language. What else is there? But it is one thing to use a language, quite another to accept its discipline. Newtonian mechanics has in the twentieth century been formulated in exquisite mathematical detail. If Newton would not have understood a word of it, that is evidence only that physics and mathematics have different aims. What gives pause in the case of quantum field theory is just the odd fact that when quantum field theory is expressed in terms of the ancient strictures of mathematical rigor, the result is confusing.
Until the 1960s, particle physicists accepted the division of things into particles and fields. Which came first? In a series of superb lectures delivered during the 1970s, Richard Feynman was unequivocal. The photon? A particle, he said, Take my word for it. Particle accelerators are, after all, in the business of accelerating particles. It followed the particles were primary. The fields came afterwards.
The contrary view is equally compelling. "Since radiation is made only of fields," the physicist Art Hobson observed, "it would be surprising if matter were made of particles. Why should the universe be made of two such different building blocks?" It is fields that come first. The particles can look after themselves. "Particles are epiphenomena arising from real fields."
This is a view endorsed by Steven Weinberg: It is the majority view. "In its mature form," Weinberg writes, "the idea of quantum field theory is that quantum fields are the basic ingredients of the universe, and particles are just bundles of energy and momentum of the fields."
Nature evidently detests a dualism. "Quantum field theory hence led to a more unified view of nature than the old dualistic interpretation in terms of both fields and particles.''
What is curious about the confidence with which these views are expressed is just the fact that they have defied the best efforts of mathematicians to make good sense of them.
Some difficulties are of very long standing, so much so that theoretical physicists feel free to ignore them. A theorem published by Rudolf Haag more than sixty years ago, and generalized promptly by A.S. Wightman and D. Hall, demonstrated that to the extent that either free or interacting fields are wandering in space, there is no one space into which they may expand. A choice must be made among them. No one knows how the choice should be made.
Mathematicians have endeavoured to create something like an axiomatic quantum field theory since the late 1940s. Their theories, where they have been successful, have not been complete, and where they have been complete, they have not been successful. Mathematicians have not been able to capture lucidly the idea of a quantum field itself. The ontology of axiomatic quantum field theory is very different. It comprises an axiomatically specified net or a collection of abstract objects known as operator algebras and these contain the requisite information about the observable elements of a given theory.
The details are interesting, but not relevant. Axiomatic quantum field theory and quantum field theory are divided in their interpretation of reality.
"Quantum foundations are still unsettled," the melancholy Hobson remarks, "with harmful effects on science and society."
This is half false. Quantum physics vexes no man. If half false, then half true.
No one quite knows why mathematicians have been unable to settle even the simplest of questions about quantum field theory: What are they about?
And with this question, another, more insistent, and, indeed, more fundamental. Why are the methods of quantum field theory so astonishingly successful?
Questions of this sort ramify through particle physics. The experiments designed to confirm the theory of electroweak unification were bound to one state of affairs in a particle accelerator; that state of affairs was in turn interpreted to reflect the same state of affairs of the early universe. Something quite new in the history of physical thought was involved. It is something for which entirely appropriate standards of assessment are still lacking.
The chief aim of the experiments confirming electroweak symmetry was to discover conditions under which that symmetry was manifest, and not latent. The conclusion of the experiment was that those conditions, if they were present in the accelerator, were also present in the early universe. If the early universe is unrecoverable, particle accelerators represent an extraordinary anomaly in the ordinary conditions of matter. In the world as we find it, there is nothing like the electroweak symmetry. If these experiments are idealizations, of what are they idealizations?
What, then, is the connection between these experiments and experience itself?
There are only two possible answers. Either the connection is metaphysical, or there is none at all. Physicists such as Lawrence Krauss almost automatically offer a metaphysical explanation. "The properties of matter," he writes, "and the forces that govern our existence" are derived from the interaction among fields.
There is in this quotation a three-fold concourse between the Higgs field, the properties of matter, and the forces that govern our existence. The concourse must follow an inference: Given the SM, the properties of matter and the forces that govern our existence follow.
There is, of course, no way in which to test this inference experimentally. What conceivable experiment could coordinate matter in an immensely anomalous state within a particle accelerator and the ordinary facts of life?
If no experiment can do what cannot be done, there remains only standards of consistency and coherence, and since no one has ever assigned a clear meaning to coherence, there remains only consistency.
But consistency is a weak constraint. It is weak in the sense that consistent theories need not be satisfied in a single model or universe. The conclusion that follows from these observations is inevitable. There is no such thing as the world, or the universe to which the SM unequivocally points. And so there can be no large and general conclusions about what the study of the world, the universe, or of Nature reveal about the existence of God.
The image appropriate to physics in the 20th century is Brueghel's Tower of Babel. That the thing is incoherent, everyone can see. What is rarely noticed is that it remains standing.
This is its great achievement.