Transcript Document

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
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?
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…
The Standard Model of Particle Physics (1968) provides 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.
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.
A Basic Example: Clock Arithmetic
A good example is “clock arithmetic” on the set of
four elements:
0
3
1
2
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
In fact, this set
of elements …
{
}
“2”
“+”
with this operation …
“3”
“3+2=1”
is the same group – clock arithmetic with four elements
(MOD{4})  MATHEMATICAL ABSTRACTION!!
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,
 a b  A B   A B  a b 





 c d  C D   C D  c d 
Rotation (Lie) Groups
In the 1870’s, Norwegian mathematician Sophus Lie realized that sets of
possible rotations form groups.
Elements: All the various possible
rotations (infinite number!)
Sophus Lie
Operation: Successive combination of
two rotations  may not
commute (order matters)!!
Lie found that rotation groups could be characterized by:
1) The number of dimensions of the space in which you’re
rotating;
2) The precise manner in which the ordering of the
elements in the operation matters (the “Lie Algebra”)
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
c) He figured if he could patent the notion of a
rotation, he would become a rich man
d) 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.
PHYSICS
In 1924, Count Louis-Victor de Broglie launched quantum
mechanics with the conjecture that particles have wavelike 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 overall phase of the wavefunction is immaterial.
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.
Particle physics is the quantum mechanics of the most
fundamental level. What are the fundamental constituents of matter?
Quarks and Leptons
u 
 
d 
c
 
s
t 
 
b
 e 
 
e 
  
 
 
  
 
 
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.
Antimatter
Antimatter is not a fiction! It was a prediction that arose
in the late 1920’s from P.A.M. Dirac’s attempts to reconcile
quantum mechanics with Eistein’s relativity. The antimatter electron – the positron, or e+ - was discovered by Carl
Anderson of Caltech in 1933.
u 
 
d 
c 
 
s 
t 
 
b 
Antiquarks
 e 
 
e 
  
 
 
  
 
 
Antileptons
When matter and antimatter of the same particle type
meet, the result is annihilation to pure energy.
The Modern View of Causation (Relativistic
Quantum Field Theory)
t (time)
Quark #2
Quark #1
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.
gluon
x (position)
Quark #1
Quark #2
For the Strong Nuclear
Force, this quantum is
know as a gluon.
Diagram: Think of a u and d quark bound in a proton.
The Electromagnetic Interaction
For the electromagnetic force, the exchanged 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 (uu) pair.
u
u
e+
e-
This makes use of the same underlying ingredients (matter
and/or antimatter connecting with photons) but the resulting phenomenon is quite different! Thus, QFT generalizes
the notion of force to that of an interaction.
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 for any of the quarks,
but none of the leptons.
u
u
e+
e-
Conjecture: There are three, not one, of each type of
quark – each quark comes in three different “colors”.
And, paradoxically:
1) This color property must be associated with the Strong
Nuclear Interaction (since leptons don’t have it).
2) 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!).
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 colorblind (it doesn’t care what color the
quark is), this is a symmetry space of
the Strong Interaction.
green
… and Color Blindness
red
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”).
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
P2
Some Wave at 12:00 Noon on 4/5/05
Top of
wave
“Global”
phase
change
Bottom
of wave
Top of
wave
Top of
wave
Bottom
of wave
Bottom
of wave
Top of
wave
Bottom
of wave
“Local”
phase
change
Global phase change: Same wavelength
Local phase change: Different wavelength – different physics!
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 cheating to make it so.
Original Wavefunction
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?
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 between fundamental particles.
B
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 inconsistent with a sterile universe – with a
universe devoid of causation.
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 have derived the
quantum theory of electromagnetism via a process of pure
thought.
Although this reshapes our understanding of electromagnetism, it doesn’t extend our understanding of the universe.
However, recall that for the Strong Nuclear Interaction, both phase and orientation in the 3-d (SU(3)) space of color
are irrelevant! This requires a substantially
different cheating term, and thus introduces an entirely different interaction!
P1
P2
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.
t
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
gluon
q
q
x
Furthermore, these properties should be
very definitively specified by this theory
of Quantum Chromodynamics.
Later that year, Gross and Wilczek (Princeton) and
Politzer (Harvard) set about exploring this conjecture.
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 Interaction 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.
Confinement and You
The Strong Interaction bears the name for good reason:
it’s about 100x as strong as the electromagnetic interaction 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 predicated 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.
Wow!!
Parting Thoughts
To no one’s greater surprise than the
mathematician’s, 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, 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 wonderful the Universe appears. Make the most of it!