Deep Down Beauty: Particle Physics, Mathematics, and the World Around Us

1. Deep Down Beauty: Particle Physics, Mathematics, and the World Around Us “What Physicists Do” Sonoma State University April 5, 2005 Bruce Schumm UC Santa Cruz

2. If you followed the demonstration with the box, then • You got a whiff of what it is that gets abstract mathematicians excited (and employed!). • You came one step closer to understanding why it is that the universe can support life. • How could this be?

3. THE “FOUR FORCES OF NATURE” • The Universe is only an interesting place because of causation – the capability of objects to exert influence on one another. • Current evidence tells us that this influence is brought about through four modes of interaction: • Gravity – that persistent tug • Electromagnetism – pretty much everything we sense • Nuclear interaction (weak) – nuclear -decay (obscure) • Nuclear interaction (strong) – holds together nuclei Why the quotes? There really aren’t four of them. Nor is the term “force” is general enough to specify their role in nature…

4. The Standard Model of Particle Physics(1968)pro-vides a strikingly accurate, unified description of electromagnetism and the weak nuclear interaction (so, we’re down to three forces – at most!). Ideally, I’d talk about this aspect of the Standard Model, but it’s a little to intricate to treat in a 50 minute talk (spontaneous symmetry breaking, Higgs Boson, etc.). Instead, I’ll focus on the Strong Nuclear Interaction, which has an independent description within the Standard Model, and which is unencumbered by the above complications, getting more directly to the role of abstract mathematics in the physical Universe. Shameless Plug: If your appetite is whet, get ahold of a copy of Deep Down Things and learn about the electroweak component of the Standard Model.

5. Algebra 101: Group Theory To a mathematician, a group is a collection of elements Think: whole numbers …-3, -2, 1, 0, 1, 2, 3… together with an operation that combines elements within the collection Think: addition 2 + 5 = 7 that includes an identity element Think: zero, as in 1 + 0 = 1, 2 + 0 = 2, 3 + 0 = 3, etc. and an inverse for each element: Think: 3 + (-3) = 0, 8 + (-8) = 0, etc.

6. 0 0 3 1 0 3 1 2 3 1 2 2 A Basic Example: Clock Arithmetic A good example is “clock arithmetic” on the set of four elements: Elements: 0,1,2,3 Operation: clock addition, e.g., 3+2=1 Identity: 0 Inverse: Whatever you need to add to get back to 0

7. { } “2” “+” “3” “3+2=1” In fact, this set of elements … with this operation … is the same group – clock arithmetic with four elements (MOD{4}) MATHEMATICAL ABSTRACTION!!

8. Commute Issues Groups fall into two categories: those for which order doesn’t matter, and those for which it does. For clock arithmetic, the order in which you combine elements doesn’t matter: 2 + 3 = 3 + 2 This operation is said to commute. Groups whose operations commute are said to be Abelian. But don’t all operations commute (addition, subtraction, multiplication, etc.?) No. For example,

9. Sophus Lie Rotation (Lie) Groups In the 1870’s, Norwegian mathemat-ician Sophus Lie realized that sets of possible rotations form groups. Elements: All the various possible rotations (infinite number!) Operation: Successive combination of two rotations  may not commute (order matters)!! • Lie found that rotation groups could be characterized by: • The number of dimensions of the space in which you’re rotating; • The precise manner in which the ordering of the elements in the operation matters (the “Lie Algebra”)

10. Why was Lie compelled to think about this? • a) He knew that if he could just solve this problem, he would understand how to build a better light bulb • b) He was under military contract from the King of Norway • He figured if he could patent the notion of a rotation, he would become a rich man • He had an abstract curiosity about the underlying nature of rotations, and how the nature of everyday rotations might extend to less concrete mathematical systems. Certainly, he had no idea that his work would lie at the heart of the 20th century view of how the universe works.

11. PHYSICS In 1924, Count Louis-Victor de Broglie launched quantum mechanics with the conjecture that particles have wave-like properties. • If you’re at sea, you are concerned about • Wavelength • Wave height • Wave frequency but the phase(exact time you find yourself on top of a crest) is immaterial. Fundamental tenet of quantum mechanics: the over-all phase of the wavefunction is immaterial.

12. The Notion of Symmetry (or Invariance) Since no physical property can depend upon phase, we say that quantum mechanics is invariant, or symmetric, with respect to changes in overall phase. Usually, when we think of symmetry, we think of actions in everyday space (a sphere is rotationally symmetric). In this case, though, the symmetry is with respect to changes within the abstract mathematical space of quantum mechanical phase. The notion of symmetry plays a deep role in the organizing principles of the universe, in many different contexts.

13. Particle physics is the quantum mechanics of the most fundamental level. What are the fundamental constit-uents of matter? Quarks and Leptons Quarks: Do participate in Strong Nuclear Force (compose nuclear matter) Leptons: Do not participate in Strong Nuclear Force (do not compose nuclear matter) Ordinary Matter is composed of protons and neutrons (uud and udd quark combinations) and electrons (e-). Electron neutrinos (e) from the sun traverse out bodies at a rate of about 1013 per second.

14. Antimatter Antimatteris not a fiction! It was a predictionthat arose in the late 1920’s from P.A.M. Dirac’s attempts to reconcile quantum mechanics with Eistein’s relativity. The antimat-ter electron – the positron, or e+- was discovered by Carl Anderson of Caltech in 1933. Antiquarks Antileptons When matter and antimatter of the same particle type meet, the result is annihilation to pure energy.

15. Quark #2 t (time) Quark #1 gluon x (position) Quark #1 Quark #2 The Modern View of Causation (Relativistic Quantum Field Theory) Example:The interaction of two quarks (repulsion or attraction) via the Strong Nuclear Force In Quantum Field Theory, forces are “mediated” through the exchange of a quantum of the force-field. For the Strong Nuclear Force, this quantum is know as a gluon. Diagram: Think of a u and d quark bound in a proton.

16. e+ e- The Electromagnetic Interaction For the electromagnetic force, the ex-changed field quantum is the photon (), the quantum of light. But: in Quantum Field Theory, we can also take the photon and use it to mediate electron-positron annihilation (e.g., to a photon, which then turns into an up-quark, up-antiquark ( ) pair. This makes use of the same underlying ingredients (matter and/or antimatter connecting with photons) but the result-ing phenomenon is quite different! Thus, QFT generalizes the notion of force to that of an interaction.

17. e+ e- Color… Interestingly enough, when experiments like this were done in the 1960’s, the rate of up-quark/up-antiquark production was three times that expected from QFT. In fact, this was true forany of the quarks, but none of the leptons. • Conjecture: There are three,not one, of each type of quark – each quark comes in three different “colors”. And, paradoxically: • This color property must be associated with the Strong Nuclear Interaction (since leptons don’t have it). • But… the properties of the strong nuclear interaction must not depend on the color of the quark (there is only one proton, or uud quark combo, not three!).

18. green blue red … and Color Blindness One (very helpful) way to view this: Color is associated with some abstract space. Rotations in this abstract space change quarks from one color to another. Since the Strong Interaction is color-blind (it doesn’t care what color the quark is), this is a symmetry space of the Strong Interaction. This set of “symmetry transformations” (rotations) is mathematically equivalent to the set of rotations in three dimensions (of color, but abstractly, it’s all the same!). In fact, we need to worry about quantum mechanical phase also, so this is really the group SU(3) of rotations in three complex dimensions (but don’t worry about the “complex”).

19. P2 This sounds rather intriguing, but something about it really bugged C.N. Yang and R.L. Mills, because quantum mechanics is invariant with respect to overall changes in color and phase, but not changes that vary from point to point. From a 1954 article in the Physical Review : “... As usually conceived, however, this arbitrariness is subject to the following limitation: once one chooses [the color and phase of the wavefunction] at one space-time point, one is then not free to make any choices at other space-time points. It seems that this is not consistent with the localized field concept that underlies the usual physical theories. In the present paper we wish to explore the possibility of requiring all interactions to be invariant under {\it independent} [choices of phase] at all space-time points ..." In other words: If you change color by rotating in SU(3) color-space at P1, how is P2 to know of it, so the same change can be made there? P1

20. Some Wave at 12:00 Noon on 4/5/05 Top of wave Top of wave Top of wave Top of wave Bottom of wave Bottom of wave Bottom of wave Bottom of wave “Global” phase change “Local” phase change Global phase change: Same wavelength Local phase change: Different wavelength – different physics!

21. Yang & Mills: Just Fix the Darned Thing Y&M were so convinced that phase invariance needed to be local that they were willing to commit the arch sin of cheatingto make it so. Original Wave-function After local change of phase Local phase change plus Y&M cheating function This cheating function was just whatever function was needed to get the wavefunction back to its original form. Great… how could that possibly help us solve this problem?

22. B Yang and Mill’s Revelation (“Gauge Theory”) Perhaps as much to their surprise as anyone’s, what Yang and Mills found was that the cheating term had precisely the form of an interaction within quantum field theory. In other words, the cheating term introduced some new particle (call it “B”) that mediates interactions be-tween fundamental particles. In order to satisfy Y&M’s concerns, you need at least one such interaction. Thus, it seems that, at its most fundamental level, quantum mechanics is inconsis-tent with a sterile universe – with a universe devoid of causation.

23. P2 P1 The Relevance of Irrelevance But what interaction does this B particle mediate? If we’re just concerned about the irrelevance of phase, then B behaves just like a photon () we havederived the quantum theory of electromagnetism via a process of pure thought. Although this reshapes our understanding of electromag-netism, it doesn’t extend our understanding of the universe. However, recall that for the Strong Nuc-lear Interaction, both phase and orient-ation in the 3-d (SU(3)) space of colorare irrelevant! This requires a substantially different cheating term, and thus intro-duces an entirely different interaction!

24. gluon Quantum Chromodynamics In 1973, Fritzsch and Gell-Mann (CalTech) proposed that the B particle associated with making phase and color irrelevant to the wavefunction might just be the gluon of the Strong Nuclear Interaction. If so, the properties of the Strong Nuclear interaction should depend intimately on theabstract mathematical properties of the Lie Group SU(3) of rotations that change the color of quarks. q q t Furthermore, these properties should be very definitively specified by this theory of Quantum Chromodynamics. x q q Later that year, Gross and Wilczek (Princeton) and Politzer (Harvard) set about exploring this conjecture.

25. Last Year’s Nobel Prize in Physics Gross, Wilczek, and Politzer found that the very fact that SU(3) is non-Abelian leads to a very curious property: The strength of the force grows as the quarks get farther apart. Two quarks on opposite sides of the universe would contain an all-but-infinte amount of energy in the Strong-Interaction field between them. Instead, quarks must gang together in clumps that are seen as neutral by the Strong Inter-action just as atoms are electrically neutral. Protons (uud) and neutrons (udd) are two such clumps. This explanation of why quarks are confined in Strong-Interaction neutral clumps won them the 2004 Nobel Prize in Physics.

26. Confinement and You The Strong Interaction bears the name for good reason: it’s about 100x as strong as the electromagnetic inter-action that’s responsible for holding atoms together. Were quarks not confined into Strong Interaction neutral clumps, chemistry would be dominated by the Strong Nuclear Interaction. Chemical reactions would be catalyzed by X-rays and -rays rather than visible light. It’s hard to imagine life evolving in such an environment. In a very deep yet direct way, life seems to be predi-cated on the fact the Lie Groups are non-Abelian – that ordering matters in the abstract mathematical space of “color” that’s associated with the Strong Nuclear Interaction.

27. Wow!! Parting Thoughts To no one’s greater surprise than the mathematician’s, abstractmathematical principles lie at the heart of what makes the Universe vibrant and alive. The ever-deepening connection between math and science is a continual source of wonder and amazement for those who are in a position to appreciate it. In this talk, we’ve only touched on one facet of the full (and evolving) contemporary conception of the workings of nature. An increasingly broad popular literature addresses our current thinking on these questions. The deeper you view it, the stranger and more won-derful the Universe appears. Make the most of it!