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

1
Deep Down Beauty Particle Physics, Mathematics,
and the World Around Us
UCSC Science Library SYNERGY Lecture May 2,
2006 Line Drawings Thanks to Bill Rowe,
Illustrator of Deep Down Things
2
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 arent four of them.
Nor is the term force is general enough to
specify their role in nature
3
The Standard Model of Particle Physics (1968)
pro-vides a strikingly accurate, unified
description of electromagnetism and the weak
nuclear interaction (so, were down to three
forces at most!). Ideally, Id talk about this
aspect of the Standard Model, but its a little
to intricate to treat in a 50 minute talk
(spontaneous symmetry breaking, Higgs Boson,
etc.). Instead, Ill 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.
4
2006 SYNERGIZERTM AUDIENCE APPTITUDE ASSESSMENT
Please work through the following problem without
assistance from your neighbor.
If 5 7 12, then 7 5 ___ ?
5

If you followed the demonstration with the box,
then

• You got a whiff of what abstract mathematics is
  all about (rotation mathematical operation).

• You came one step closer to understanding why it
  is that the universe can support life.

How could this be?
6
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.
7
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., 321 Identity 0 Inverse Whatever
you need to add to get back to 0
8
In fact, this set of elements
with this operation
is again clock arithmetic with four elements
(MOD4) ? MATHEMATICAL ABSTRACTION!! ?
If we free our mind to think of math in these
abstract terms, we shall see we can make great
headway in physics
9
Commute Issues
Groups fall into two categories those for which
order doesnt matter, and those for which it does.
For clock arithmetic, the order in which you
combine elements doesnt matter 2 3 3
2 This operation is said to commute. Groups
whose operations commute are said to be
Abelian. But dont all operations commute
(addition, subtraction, multiplication, etc.?)
No. For example,
10

Rotation (Lie) Groups
In the 1870s, 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
  youre rotating
• The precise manner in which the ordering of the
  elements in the operation matters (the Lie
  Algebra)

11

• 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.
12
PHYSICS
In 1924, Count Louis-Victor de Broglie launched
quantum mechanics with the conjecture that
particles have wave-like properties.

• If youre 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.
13
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 (generalized) quantum
mechanical phase.
The notion of symmetry plays a deep role in the
organizing principles of the universe, in many
different contexts.
14
Particle physics is the quantum mechanics of the
most fundamental level of nature. What are the
basic constituents 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.
15
Antimatter
Antimatter is not a fiction! It was a prediction
that arose in the late 1920s from P.A.M. Diracs
attempts to reconcile quantum mechanics with
Eisteins 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.
16

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.
17

The Electromagnetic Interaction
For the electromagnetic force, the ex-changed
field quantum is the photon (?), the quantum of
light.
But in Quantum Field Theory (QFT), 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.
18

Color
Interestingly enough, when experiments like this
were done in the 1960s, the rate of
up-quark/up-antiquark production was three times
that expected from QFT. In fact, this was true
for any 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 dont
  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!).

19

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 doesnt 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, its 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
(generalized quantum mechanical phase).
20

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
21

Some Wave at 1200 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!
22

Yang Mills Just Fix the Darned Thing
YM were so convinced that phase invariance
needed to be local that they were willing to
commit the arch sin of cheating to make it so.
Original Wave-function
After local change of phase
Local phase change plus YM 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?
23

Yang and Mills Revelation (Gauge Theory)
Perhaps as much to their surprise as anyones,
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 YMs 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.
24

Mathematics and The Relevance of Irrelevance
But what interaction does this B particle
mediate? If were just concerned about the
irrelevance of phase, then B behaves just like a
photon (?) ? we have derived the quantum theory
of electromagnetism via a process of pure
thought. Although this reshapes our
understanding of electromag-netism, it doesnt
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 color are irrelevant! This
requires a substantially different cheating term,
and thus intro-duces an entirely different
interaction!
25

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 the
abstract 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.
26

The 2004 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.
27
How to Neutralize Quarks the SU(3) Way
28

Confinement and You
The Strong Interaction bears the name for good
reason its about 100x as strong as the
electromagnetic inter-action thats 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.
Its 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 thats
associated with the Strong Nuclear Interaction.
29

Wow!!
Parting Thoughts
To no ones greater surprise than the
mathematicians, abstract mathematical 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, weve 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!