**Text: Nick Herbert ///// Metalwork: August O’Connor**

“The forces of nature are phase forces.”– C. N. Yang

“The notion of local gauge invariance provides a framework of almost comical simplicity

for the precise laws of physical interaction.”– Bruce Schumm

Physicists can describe the immense variety of the natural world by invoking only four fundamental forces. Until recently these four forces seemed to be arbitrary facts of nature but the discovery (outlined in UCSC physicist Bruce Schumm’s accessible book “Deep Down Things”) that the behavior of three of these forces can be derived from the operations of certain “symmetry groups” is one of the most exciting accomplishments in physics. Instead of accepting these forces as arbitrary facts of nature, physicists can now derive in precise detail three fundamental forces–electromagnetism, the weak and the strong nuclear forces (but not gravity)–from the transformation properties of three particular symmetry groups. Everything about the force follows from the theory (the number of “force particles”, for instance and whether these force particles self-interact or not) What needs to be supplied by experiment is one number for each force–the measured overall strength (or “charge”) of the interaction. Except for this one number every other feature of these forces is determined by the mathematical structure of its associated symmetry group–an extraordinary explanatory economy (in Schumm’s words) of “almost comical simplicity”.

The detailed behavior of three of nature’s fundamental forces is determined by the structure of three symmetry groups. But what exactly is a symmetry group?

A symmetry group is a class of operations on a system that leaves that system the same. It’s like balancing an automobile tire by spinning it. The tire is “invariant to rotations” when it doesn’t wobble when spun. To “balance the tire” you add weights at particular places until the wobble vanishes. Imagine your surprise if you found that, when you had balanced the tire, the position of the weights spelled out your name. Something similar happens in this new way of describing the three fundamental forces.

A system of free quantum particles left to themselves is not invariant under a particular rotation; the system “wobbles”. So the physicist adds certain mathematical “weights” to the system that exactly cancel the wobble. The position of the “weights” doesn’t spell out the physicist’s name but it does spell out (for the right choice of rotation) the exact nature of a particular physical force. So the simple requirement that the system “doesn’t wobble” brings about the existence of new forces in nature. In this scheme, as many new forces can be generated as there are kinds of rotations–including rotations in dimensions higher than 3. Surprisingly, three of the simplest kinds of rotations give rise to the three fundamental forces that govern the universe we live in. The electromagnetic force, for example, arises from the simplest imaginable rotation, a rotation in one dimension.

THE ELECTROMAGNETIC FORCE

The discovery of quantum theory–our current best description of the nature of the material world–was the most memorable accomplishment of the Twentieth Century. The central doctrine of quantum theory is that every physical object is wavelike. This means that every physical object possesses a “phase” that says in what part of its wave cycle the particle currently resides. Like the phase of the moon, a physical object’s phase can be mapped on to a circle where the phase varies from 0 to 360 degrees. (For concreteness you can imagine “a little clockface with arrow” attached to every object in the universe.) Altho phase is ubiquitous in our quantum description of nature, we use it without entirely understanding what we are doing. One of the oddest features of quantum theory is the coexistence of its immense practical success alongside a profound ignorance of its foundations. Einstein, dismayed by this fundamental ignorance at the heart of quantum physics, once exclaimed, “Who could have guessed that we would come to know so much and understand so little?”

Every physical object possess a phase (clockness); the mathematical operation that changes this phase is the Lie group U(1), or Unitary Group One. Since phase is like a clock, U(1) (pronounced “you won”) is just the collection of all possible rotations of that clock’s hand. The “1” means that this is a one-parameter group of transformation, the one parameter in this case being the object’s phase angle. “Lie”, by the way, refers to Sophus Lie (1842–1899), a Norwegian mathematician who invented these transformations, and his name is pronounced “Lee”.

Suppose we imagine a hypothetical interaction whose addition to a system makes it blind to any arbitrary change in an object’s phase anywhere in space. Such a phase-blinding force, in the language of symmetry groups, is said to make the system “locally invariant under transformation class U(1)” or more concisely “invariant under U(1)”.

The big surprise is that this hypothetical phase-blinding force possesses all the well-known features of electromagnetism, leaving unspecified only the overall strength of the interaction. In particular this phase-blinding force is mediated by one uncharged, massless spin-1 particle which we call the photon.

Our current model of the world is that matter particles (Fermions) interact by exchanging force particles (Bosons). The force particle responsible for electromagnetism is the photon. Photons link together all particles that possess electric charge (QE). Neutral particles (QE = 0) do not feel the electromagnetic force.

This new discovery means that electromagnetism can be seen as a “phase force” in Princeton physicist C.N. Yang’s words. Or, more precisely, a force whose properties derive from the fact that the presence of electromagnetism makes systems “not wobble” when all their particle’s “clocks” are reset.

Most of the phenomena we see around us–light, chemistry, biology, as well as static electricity and refrigerator magnets–are examples of electromagnetic forces. It is amusing to think that this immense variety arises from one particular force field’s confering obliviousness to quantum phase.

THE WEAK NUCLEAR FORCE

The decay of certain radioactive materials and some of the energetic reactions in the Sun whose energy production makes possible all life on Earth are governed by a force called the “weak interaction”. The properties of the weak force had been known for many years but were considered one quite arbitrary manifestation of nature’s whim.

However, if in the manner of electromagnetism, you imagine a force that confers obliviousness to the difference between an electron and a neutrino (plus their associated quantum phases) then this force can be described as confering “invariance under SU(2)”–a two-parameter rotation in an abstract space which mixes up electrons and neutrinos in the typical quantum way. In lieu of SchrÃ¶dinger’s Cat who could be alive and dead at the same time, an inhabitant of this space could be 1/2 electron and 1/2 neutrino–and a system interacting via this postulated force could not tell the difference.

“SU(2) (pronounced “ess-you-too”) means “Special Unitary Group Two”. What does a force that’s “invariant under SU(2)” look like?

For starts, it’s mediated by three massless spin-1 particles instead of the single photon of electromagnetic fame. Furthermore, these 3 force particles in addition to carrying the weak force between Fermions also weak-interact with each other.

If photons (the carrier of the EM force) interacted with one another as do the 3 carriers of the weak force, then optics would be impossible–two light beams crossing each other would strongly scatter. No images could ever form in such a fog of self-interacting light. Eyes evolved because light is invariant under U(1), not under some more complicated (non-Abelian) Lie group.

What makes the force particles associated with SU(2) self-interact is the mathematical fact that rotations in SU(2) space “do not commute”– that is, the result of a rotation by angle A followed by a rotation by angle B does not give the same result as the same operations carried out in opposite order. In the simple one-parameter clockface world of U(1) all rotations commute so this theory predicts (correctly) that photons don’t self-interact.

This hypothetical force (a force that can’t tell the difference between an electron and a neutrino) possesses all the properties observed experimentally for the weak force except for specifying the overall strength of the interaction. A measurement of one weak interaction–say the half-life of the muon–can specify this strength, known as “weak charge” QW, which then fixes the strength of all weak interactions everywhere. The property of invariance under Lie group SU(2) determines all the general properties of the weak force; the specific strength of the weak force–the magnitude of the weak charge–must be determined by experiment.

The three weak-force mediating particles–called W(plus), W(minus) and Z (zero)–have since been discovered and measuring their properties to high accuracy has become a major preoccupation of physicists at some of our large accelerator labs. These 3 weak-force-mediating particles are called “intermediate vector Bosons” or sometimes just “weakons.”

To just describe the electromagnetic force as a phase-blind interaction might have been considered a curiosity of science since the electromagnetic interaction was well understood before this new way of looking at forces was developed. To explain the weak interaction as a phase force, however, brought to light many new features (for instance the existence of 3 force particles) that were not anticipated and were later confirmed by experiment. Phase-force physics is now no mere curiosity. Phase force physics really works.

THE STRONG NUCLEAR FORCE

In recent years the force that binds together the nucleus of the atom has been traced to its source. It’s the result of the strong nuclear force acting between a class of Fermions called quarks. All quarks possess a property whimsically called “color” which can be either red, green or blue plus all possible quantum-phased combinations.

Now imagine a hypothetical force which makes a system (yes!) colorblind. This system cannot tell if a quark is red or green or blue. In the abstract quantum space that contains quark color, such a force is said to confer invariance under SU(3). The simple fact of SU(3) invariance fixes all the properties of this hypothetical force (except its overall strength).

For example, phase-force theory predicts that this force will be mediated by 8 massless, spin-1 particles. And because the elements of SU(3) do not commute, these eight particles will strong-interact between themselves as well as with quarks. By the way, simple groups, such as U(1), whose elements commute, are called Abelian (after Norwegian mathematician Niels Abel (1802-1829)). SU(3) (as well as SU(2)) are examples of non-Abelian Lie groups. If a group is non-Abelian, its associated force particles must self-interact.

This hypothetic phase force actually matches in exact detail the behavior of the strong force once one determines experimentally the magnitude of the strong charge QS. The eight force particles mediating the strong force are called “gluons”. Before the discovery of the quark/gluon structure of strong interactions, the entire field of high-energy physics was a real mess. Now every particle that we observe in our accelerators finds its place in the force structure characterized by the product of only three Lie Groups–U(1)SU(2)SU(3). There is more real knowledge encoded in this short string of symbols then can be found in a thousand text books.

CONCLUSION

In order to be considered part of our universe an object must interact by means of one of the four forces. The identity of a particle is, in a sense, determined by what forces it can respond to. Gravity acts on mass-energy which all particles (even the massless photon) possess. Everything responds to gravity. Quarks, the heaviest constituents of matter, respond to all of the remaining three forces. Quarks are defined as those particles that possess strong, weak and electromagnetic charge. Charged leptons (such as electrons and muons) interact both weakly and electromagnetically. Finally the interaction-impoverished neutrinos (neutral leptons) possess only weak charge; the neutrino interacts only via gravity and the weak force.

The universe as we know it is well described by the so-called Standard Model which contains 6 quarks and 6 leptons interacting thru three kinds of phase force: electromagnetism mediated by one photon, weak force mediated by 3 weakons, and strong force mediated by 8 gluons. Quarks and leptons (Fermions) are the “bricks” of the cosmos; The 12 force particles (Bosons) are the “mortar” that sticks it together. Twelve Bosons linking twelve fermions: that’s our physical universe in a nutshell!

Of course there are still problems. Physicists need jobs too. The biggest puzzle is still gravity. Despite almost a century of effort by really smart people, gravity has not yet been incorporated into the quantum picture. The recent discovery of Dark Matter which constitutes 90% of the universe and interacts, as far as we can tell, ONLY via gravitation, complicates the picture. On top of that, the recent discovery of Dark Energy which acts to accelerate our already expanding universe is another challenge to our ability to explain things with physics.

But leaving gravity aside, one big problem still exists within the Standard Model itself. The phase-force model of fundamental interactions unambiguously predicts that all force particles must be massless. Neither the 1 photon nor the 8 gluons possess mass (good!) but the 3 weakons appear very massive (about 100 times the mass of a proton)(bad!). New physics is being invented to fix this unpleasant feature of our otherwise splendid and wide-ranging new model of matter. The most hopeful fix postulates the existence of a brand-new massive spin-zero particle called the Higgs Boson whose main job is to interact with the 3 inherently massless weakons to give them a kind of “fake mass” which is the mass we observe. Luckily, the predicted mass of the hypothetical Higgs “fix-it particle” is low enough (in some models) that it should be easily observed in the upcoming Large Hadron Collider at CERN in Switzerland or perhaps glimpsed at the edge of the range of the Tevatron presently operating at FermiLab near Chicago.

One of the most exciting discoveries of recent years is the notion that three of the fundamental forces of nature can be exactly specified by simple rotations in three abstract “quantum phase spaces”. As well as contributing greatly to our practical understanding of the physical world this recent discovery has uncovered a deep and subtle beauty in the way our universe is put together–a splendid hidden beauty only now made visible. Praise Her!

NICK HERBERT is the author of “Quantum Reality”, “Faster Than Light”, “Elemental Mind” and a chapbook “Physics on All Fours”. He devised the shortest proof of Bell’s Theorem, had a hand in the Quantum No-Cloning Rule and is presently obsessed with Quantum Tantra.

AUGUST O’CONNOR is artist, musician, gardener and muralist as well as teacher of Celtic knots and the Irish frame drum (bodhran). She performs as part of the Celtic duo Dobhran (Gaelic for “otter”); their latest CD is “Otter’s Holt”. Her recent metalwork design for nature’s fundamental forces was inspired by patterns on Irish dolmens.

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