Sn

1
From Yang-Mills to Asymptotic Freedom to
Quantum Chromodynamics
Jirí Chýla, Institute of Physics, Prague
2
From Yang-Mills to Asymptotic Freedom to
Quantum Chromodynamics
Jirí Chýla, Institute of Physics, Prague
The story of the emergence of the concept of
gauge invariance and its importance for the
formulation of physical laws show that Dirac was
right to expect that
Physical laws should have mathematical beauty
3
From Yang-Mills to Asymptotic Freedom to
Quantum Chromodynamics
Jirí Chýla, Institute of Physics, Prague
The story of the emergence of the concept of
gauge invariance and its importance for the
formulation of physical laws show that Dirac was
right to expect that
Physical laws should have mathematical beauty
but the converse is not true as
mathematical beauty does not necessarily imply
physical relevance.
4
There are many excellent texts covering various
aspects of the emergence and application of
nonabelian gauge theories. My recommendations
D. Gross Twenty five years of asymptotic
freedom

• C.N. Yang Interview in The Mathematical
  Intelligencer 15/4
• N. Straumann Early Histrory of Gauge Theories
• S. Weinberg The Making of the Standard Model
• H. Lipkin Quark model and quark
  phenomenology
• O. Greenberg From Wigners supermultiplet theory
  to QCD
• G. t Hooft When was the asymptotic
  freedom discovered?
• de Rujula Fifty years of Yang-Mills theories
  a phenomeno-
• logical point of view
• D. Gross Oscar Klein and gauge theory

5
Premature burial
From Nambus book Quarks
The renormalization procedure, developed by
Dyson, Feynman, Schwinger and Tomanaga was
spectacularly successful in QED. The physical
meaning of renormalization was, however, not
truly understood and the renormalization was
considered by most physicists, including Dirac
and Wigner a trick.
The prevalent feeling was that renormalization
simply swept the infinities under the rug, but
that they were still there.
6
In middle 1950s Landau and Pomeranchuk
attempted to give the renormalization procedure
in QED good physical meaning and mathematical
sense. They put a finite bare electric charge
e0e(r0) on a sphere of radius r0 , placed it in
the QED vacuum and calculated how it appears at a
finite distance rgtr0.
bare charge e0 must be a function of the radius
r0!
i.e. the QED vacuum screens the bare electric
charge!
Sending the radius of bare electron to zero and
keeping the bare charge e0 constant, the
effective charge e2(r) vanishes for any fixed
distance r! This is the famous problem of zero
charge, which for Landau implied that QED is
incomplete
We reach the conclusion that within the limits of
formal electrodymics a point interaction is
equivalent to no interaction at all.
7
Landau pole in QED ...
Turning the argument around, they could have
asked how would the bare charge e0e(r0) or
rather a(r0) have to depend on r0 to yield a
finite effective electric coupling a(r) at
distance r when r0 vanishes.

The second formula suggests that it would have to
grow to infinity at finite distance rL defining
the so called Landau pole.
In fact, the problem with the renormalization
proce-dure in QED is not the fact that bare
electric charge diverges, but that it does so at
a finite (though very small) distance!
8
... is absent in QCD!
One can only wonder whether Landau and
Pomeranchuk asked themselves this natural
question. Had they done it, they might be led to
the concept of asymptotic freedom because it
suffices to change the sign of ß0 for the bare as
well as effective charges to be well-defined, and
actually vanish, at small distances
Modern, inherently nonperturbative, approach to
the renorma- lization, which lies at the heart of
lattice gauge theory, is just to construct the
dependence a0a(r0) in such a way to yield
finite values of physical quantities in the
limit of vanishing r0.
9
Dirac on renormalization of QED in 1974
Hence most physicists are very satisfied with the
situation. They say Quantum electrodynamics is
a good theory, and we do not have to worry about
it any more.
I must say that I am very dissatisfied with the
situation, because this so-called good theory
does involve neglecting infinities which appear
in its equations, neglecting them in an arbitrary
way. This is just not sensible mathematics.
Sensible mathematics involves neglecting a
quantity when it turns out to be small not
neglecting it just because it is infinitely great
and you do not want it!
10
Dirac draw uncompromising conclusion
Of course, the proper inference from this work is
that the basic equations are not right.... There
must be some drastic change introduced into them
so that no infinities occur in the theory at all
and so that we can carry out the solution of the
equations sensibly, according to ordinary rules.

• Dirac criticism of the renormalization procedure
• was justified for QED, but
• does not apply to Yang-Mills gauge theories.
• For these theories Dirac was thus wrong!

11
The beginning of all
starting point isotopic dublet of nucleons
In the present paper we wish to exlore the
possibility of requiring all interactions to be
invariant under independent rotations of the
isotopic spin at all space-time points,..
We then propose that all physical processes (not
involving electromagnetic field) be invariant
under the isotopic gau- ge transformation
12
this requirement lead them to the following
Lagrangian density
gauge bosons
Three electrically charged gauge bosons and their
selfcoupling ensued automatically
The quanta of the b-field clearly have spin unity
and iso- spin unity. We know their electric
charge too because all the interactions that we
propose must satisfy the law of conservation of
the electric charge, which is exact.
13
but question remained about the mass of the
b-quantum
We next come to the question of the mass of the
b-quantum, to which we do not have a satisfactory
answer. One may argue that without a nucleon
field the lagrangian would contain no quantity of
the dimension of a mass and that therefore the
mass of the b-quantum in such a case is zero. The
argument is how- ever subject to the criticism
that, like all field theories, the b-field is
beset with divergences and dimensional arguments
are not satisfactory.
b
b
mass of the gauge boson to be determined by its
full propagator
YM considered seriously the possibility that
their gauge bosons will eventually be massive
A conclusion about the mass of the b-quantum is
of course very important in deciding whether the
proposal of the existen- ce of the b-field is
consistent with experimental information.
14
Under the spell of gauge principle
Since around 1960 Sakurai, Salam, Ward, Neeman
and others started considering local gauge
invariance as guiding principle in constructing
theories of strong, weak as well as
electromagnetic interactions.
Abdus Salam John Ward in On a Gauge Theory of
Elementary particles
Nuovo Cimento 11 (1960), 165
Our basic postulate is that it should be possible
to generate strong, weak and electromagnetic
inter- action terms by making local gauge
transformations on the kinetic terms in the free
Lagrangian for all particles. This is the
statement of ideal, which in this paper at least,
is only very partially realized.
15
The straightforward generalization of the
original Yang-Mills was proposed in 1961 by Salam
and Ward who extended isospin symmetry to SU(3)
version of the Sakata model
by gauging the fundamental triplet of baryons
8-parameter traceless hermitian
matrix
they got the octet of selfinteracting gauge
vector mesons
infinitesimal gauge transformation
16
kinetic term invariant
full YM
these break GI!
Close to Eightfold way but different in basic
multiplet and no discussion of baryons beyond the
fundamental triplet p,n,?
17
As an alternative to the Sakata model based on
the relation
Y. Neeman and M. Gell-Mann proposed in early 1961
the Eightfold Way
which starts with the product of three SU(3)
triplets
and leads to different set of multiplets. At the
beginning of 1961 it was still not quite clear
which scenario was correct.
The stories of their discoveries are quite
different as are their professional careers and
whole lives.
18
Eightfold way according to Y. Neeman
Derivation of Strong interactions from a Gauge
invariance
Y. Neeman, Nucl. Phys. 26 (1961), 222
contains a full-fledged Yang-Mills gauge
theory of strong interactions extending the
original YM theory to SU(3) unitary symmetry.
Baryons are assigned to octets as are the
pseudoscalar mesons. Octet of selfinteracting
vector bosons is predicted, though no vector
meson was known at the end of 1960.
But no interpretation of the fundamental triplet
attempted.
19
(No Transcript)
20
Discovery of vector mesons
proceeding as quasi two-body process
followed by ? in May, F in July and ? in August
1961
21
Who was afraid of gauge theory?
MGMs preprint
is truly fantastic for the straightforwardness
with which the idea is presented.
22
The vector mesons are introduced in a very
natural way, by extension of the the gauge
principle of Yang and Mills. Here we have a
supermultiplet of eight mesons. In the limit of
unitary symmetry we have completely
gauge-invariant and minimal theory like
electromagnetism.
and on another place
Now the vector mesons themselves carry F spin and
there- fore contribute to the current which is
their source. The prob- lem of constructing a
nonlinear theory of this kind has been
completely solved in the case of isotopis spin
by Yang and Mills and by Shaw. We have only to
generalize their result (for three vector
mesons) to the case of F spin and eight vector
mesons.
23
leptons played the role of quarks
gauge transformations on all particles involved
24
full Yang-Mills Lagrangian written out
unique coupling
noting that
There are trilinear and quadrilinear
interactions amongst the vector mesons, as usual
...
But this preprint has never been published!!
25
instead we read in Symmetries of Baryons and
Mesons
In Section VIII we propose, as an alternative to
the symmetrical Sakata model, another scheme with
the same group, which we call eightfold way''.
Here the baryons, as well as mesons, can form
octets and singlets, and the baryons N, ?, ? and
? are supposed to constitute an approximately
degenerate octet. Nowhere does our work conflict
with the program of the Chew et al. of dynamical
calculation of the S-matrix from strong
interactions using dispersion relations. If
there are no fundamental fields . all baryons
and mesons being bound or resonant states of one
another, models like Sakata will fail the
symmetry properties we have abstracted can still
be correct, however.
Remarkably, this paper does not mention the gauge
principle and does not refer to Yang-Mills paper
at all!
26
S-matrix and bootstrap Theory of everything?
From G. Chew S-Matrix Theory, (W.A. Benjamin
Inc, 1963).
I believe the conventional association of fields
with strong interacting particles to be empty. It
seems to me that no aspect of strong interactions
has been clarified by the field concept. Whatever
success theory has achieved in this area is based
on the unitarity of the analytically continued
S-matrix plus symmetry principles.
I do not wish to assert (as does Landau) that
conventional field theory is necessarily wrong,
but only that it is sterile with respect to the
strong interactions and that, like an old
soldier, it is destined not to die but just to
fade away The notion, inherent in conventional
Lagrangian field theory, that certain particles
are fundamental while others are complex, is
becoming less and less palatable
27
For application of YM theories to strong
interactions the identification of
correct space to gauge was crucial. This sounds
trivial, but was not. It took 20 years to come
to the conclusion that the fundamental object of
nonabelian theory of strong interactions are
colored quarks
and that forces acting between them follow from
gauging the color degree of freedom.
28
Quark model according to Zweig
Zweig
Both mesons and baryons are constructed from a
set of three fundamental particles, called aces.
Each ace carries baryon number 1/3 and is
fractionally charged. SU(3) is adopted as a
higher symmetry for the strong inte-
interactions. Extensive space-time and group
theoretic structure is then predicted for both
mesons and baryons An experimental search for
the aces is suggested.
29
Quark model according to Gell-Mann
PL 8 (1964), 214
A formal mathematical model based on field theory
can be built up for the quarks exactly as for p,
n and ? in the old Sakata model, for example with
all strong interactions ascribed to a neutral
vector meson field interacting symmetrically with
the three particles. Within such a framework the
electromagnetic currents is just
30
MGMs view of the role of quarks (Physics 1
(1964), 63)
In order to obtain such relations that we
conjecture to be true, we use the method of
abstraction from a Lagrangian field theory model.
In other words, we construct a mathematical
theory of the strongly interacting particles,
which may or may not have anything to do
with reality, find suitable algebraic relations
that hold in the model, postulate their validity
and then throw away the
model. We may compare this process to a method
some-times employed in French cuisine a piece of
phea-sant meat is cooked between two slices of
veal, which are then
discarded.
31
Confinement consequence or source of nuclear
democracy?
M. Gell-Mann at 1992 ICHEP
I was reflecting that if those objects (i.e.
quarks) could not emerge to be seen
individually, then all observable hadrons could
still have integral charge and also the principle
of nuclear democracy could be preserved
unchanged for observable hadrons. With this
proviso, the scheme appealed to me.
For MGM nuclear democracy was fundamental
principle of strong interactions and confinement
its consequence
Since I was always convinced that quarks would
not emerge to be observed as single particles
(real quarks), I never paid much attention to
the Hahn-Nambu model in which their emergence was
supposed to be made possible by giving them
integral charges.
32
The concept of colored quarks
has been introduced in late 1964 primarily in
order to explain the apparent problem of quark
statistics implied by the success of SU(6)
symmetric quark model. To reconcile this model
with Pauli principle Greenberg proposed to
interpret quarks as parafermions of rank 3.
It soon became clear that this assumption is
equivalent to assigning to each quark flavor
another internal quantum number, which could take
three different values and which, following Pais
suggestion at 1965 Erice Summer School, has been
called color.
33
While for most of theorists color was introduced
to solve the quark statistics problem
Nambu had used it since early 1965 as a dynamical
variable generating the force between quarks,
assuming furthermore that the force between
colored quarks is due to the exchange of octet
of colored gauge bosons, which induce the
effective four quark coupling of the type
and lead to (potentially infinite) gap between
colorless and colored states.
In this way his model contained all essential
elements of QCD, except that it was not Quantum
Field Theory.
34
Gell-Mann on quarks (summer 1967)
The idea that mesons and baryons are made
primarily of quarks is difficult to believe,
since we know that, in the sense of dispersion
theory, they are mostly, if not entirely, made up
out of one another. The probability that a meson
consists of a real quark pair rather than two
mesons or a baryon and antibaryon must be quite
small. Thus it seems to me that whether or not
real quarks exist, the q or q we have been
talking about are mathematical entities ......
If the mesons and baryons are made of
mathematical quarks, then the quark model may
perfectly well be compatible with bootstrap
hypothesis, that hadrons are made up out of one
another.
35
Too much scaling may be misleading
Bjorken derived scaling behavior observed at SLAC
from current algebra considerations assuming that
the nucleon structure functions stay finite in
the limit of infinite momentum transfer.

But we now know that in QCD the above assumption
does not hold and, consequently, his paper is,
indeed, empty!
36
Bardeen, Fritzsch, GellMann in 1972
(hep-ph/0211388)
One is considering the abstraction of results
that are true only formally, with canonical
manipulation of operators, and that fail, by
powers of logarithmic factors, in each order of
renormalized perturbation theory, in all barely
renormalizable models.
The reason for the recent trend is, of course,
the tendency of the deep inelastic electron
scattering experiments at SLAC to encourage
belief in Bjorken scaling, which fails to every
order of renormalized perturbation theory in
barely renormalizable models. There is also the
availability of beautiful algebraic results, with
Bjorken scaling as one of their predictions, .
37
Why asymptotic freedom?
Because only asymptotically free QFT could
explain surprisingly good scaling behavior of
nucleon structure functions observed since 1967
in deep inelastic electron-nucleon scattering at
SLAC and reconcile it with experimental fact of
quark confinement.

Because for asymptotically free quantum field
theories the renormalization procedure as
formulated by Landau Pomeranchuk can be
consistently carried through. In this sense
asymptotically free Quantum Field Theories do not
contain ultraviolet divergencies.
For these theories Dirac was thus wrong!
38
In 1972 quarks were still not taken seriously
In Summer 1972 Gell-Mann and Fritzsch presented
their view at XVI ICHEP in Chicago in a
contribution called
Current Algebra Quarks and What Else?
We assume here that quarks do not have real
counterparts that are detectable in isolation in
the laboratory they are supposed to be
permanently bound inside mesons and baryons
.........It might be a convenience
to abstract quark operators themselves, or other
nonsinglets with respect to color, , but it
is not a necessity. It may not even be much of a
convenience
since we would .... be discussing a fictitious
spectrum for each fictitious sector of Hilbert
space, and
we probably dont want to load ourselves with so
much spurious information.
39
Their hope that
We might eventually abstract from the quark
vectorgluon field theory model enough algebraic
information about the color singlet operators in
the model to describe all the degrees of freedom
that are present.
and thus
We would have a complete theory of the hadrons
and their currents, and we need never mention any
operators other than color singlets.

• has not been born out by further theoretical
  developments and experimental results, in
  particular those on
• heavy quarkonia spectra and
• jet phenomena

which require that we treat quarks and gluons in
the same way as leptons and basically forget
about confinement.
40
This paper is quoted as containing the suggestion
that gluons could form the octet of Yang-Mills
gauge bosons. In fact this option is mentioned in
the following context
Now the interesting question has been raised
lately whether we should regard the gluons as
well as the quarks as being nonsinglets with
respect to color (private communication of J.
Wess to B. Zumino). For example, they could form
a color octet of neutral vector fields obeying
the YangMills equations.
they, however, ignored this option
In the next three Sections we shall usually treat
the vector gluon, for convenience, as a color
singlet.
41
D. Gross QFT must be destroyed!
I decided, quite deliberately, to prove that
local field theory could not explain the
experimental fact of scaling and thus was not an
appropriate framework for the description of the
strong interactions. Thus, deep inelastic
scattering would finally settle the issue as to
the validity of quantum field theory. The plan of
the attack was twofold.
First, I would prove that ultraviolet
stability, the vanishing of the efective
coupling at short distances, later called
asymptotic freedom, was necessary to explain
scaling.
42
Second, I would show that there existed no
asymptotically free field theories. The latter
was to be expected. After all the paradigm of
quantum field theory QED- was infrared stable
in other words, the efective charge grew larger
at short distances and no one had ever
constructed a theory in which the opposite
occurred
Together with Frank Wilczek they succeeded in the
first step, but failed in the second because
Nonabelian gauge theories have turned out to be
(under certain circumstances) asymptotically
free!
D. Gross For me the discovery of asymptotic
freedom was totally unexpected. . Field theory
was not wrong, instead scaling must be explained
by an asymptotically free gauge theory of the
strong interactions.
43
(No Transcript)
44
Conversion of Saul of Tars to St. Paul
45
Asymptotic freedom had been discovered in these
two papers
46
shortly followed by three papers containing
complete formulation of QCD, together with
elaboration of its application to DIS.
All these papers existed as preprints by the date
of their submissions and thus months before the
submission of the paper which is often, but
incorrectly, credited with the formulation of
QCD.
47
A fourth apparent advantage of the color octet
gluon scheme has recently been demonstrated the
behavior of light cone commutators comes closer
to scaling behavior than in the color singlet
vector gluon case. However, actual Bjorken
scaling does not occur..
For us, the result that the color octet field
theory model comes closer to asymptotic scaling
than the color singlet model is interesting, but
not necessarily conclusive, since we conjecture
that there may be a modification at high
frequencies that produces true asymptotic scaling.
48
31 years after their work Gross, Wilczek and
Politzer were awarded the 2004 Nobel Prize.
Their theory provides the basic framework for
reconciling the apparently conflicting facts
that quarks do not exist as free particles but in
some situations appear to behave as almost free.
The key manifestation of their existence are
jets
49
People knowing without understanding
Perhaps the first who observed this behaviour of
a coupling constant were V. S. Vanyashin and M.
V. Terentev in 1965.
Studying the effects of vacuum polari-zation due
to loops of charged vector bosons on the
renormalized electric charge they found the
expression
and noted that these loops give the opposite
sign that those of fermion loops in standard QED!
But they attributed this result to the fact that
the theory with charged vector bosons coupled to
photons is not renormalizable.
50
G. t Hooft When was asymptotic freedom
discovered? hep-th/9808154
I knew about the beautiful scaling behaviour of
non-Abelian gauge theories. Suspecting that this
feature should be known by now by the experts on
the subject of scaling, I did not speak up
louder. Veltman warned me that no-one would
take such an idea seriously as long as it could
not be explained why quarks cannot be isolated
one from another.
By 1972, I had calculated the scaling behavior,
and I wrote it in the form
(5.3)
where Cj are Casimirs for VB, fermions and scalars
51
In June, 1972, a small meeting was organised by
Korthal Atles in Marseille. I announced at that
meeting my finding that the coefficient
determining the running of the coupling strength,
for non-Abelian gauge theories is
negative and I wrote down (5.3) on the
blackboard. Symanzik was surprised and
skeptical. If this is true, it will be very
important, and you should publish this result
quickly, and if you wont, somebody else will,
he said. I did not follow his advice.
t Hooft now likely regrets his decision.
52
Seeing quarks and gluons
D. Gross Nowadays, when you listen to
experimentalists talk about their results they
point to their lego plots and say, Here we see a
quark here a gluon. Believing is seeing,
seeing is believieng. We now believe in the
physical reality of quarks and gluons
The way in which we see quarks and gluons through
the the efects they have on our measuring
instruments is not much different from the way
we see electrons.
53
One typical DIS event from H1 experiment
54
Nice H1 event with 3 clearly separate and
different jets
55
Potvrzení asymptotické volnosti QCD
Výsledky merení z ruzných experimentu
Z prednášky F. Wilczeka
Data LEP
1/r?
56
Z prednášky F. Wilczeka v Karolinu 2003
Jets at LEP
dva jety
dilepton
dilepton foton
tri jety
57
ALEPH
jet
µ-
µ
jet
jet
jet
jet
jet
jet
jet
jet
58
Experimental evidence for the basic feature of
nonabelian gauge theories
Triple gluon coupling (1990)
measured by the angular distri-bution of four jet
events at LEP
taking into account that
59
as well as ZWW and ?WW vertices (1998)
Thanks to LEP2 we now see the effects of triple
gauge boson coupling.


60
Origins of the concept of Gauge Invariance
goes back to the attempt of Hermann Weyl in 1918
to generalize Riemannian geometry, discarding its
assumption that it makes sense to compare
magnitudes of vectors at distant points.
He observes
But Riemannian geometry described above there is
contained a last element of
geometry at a distance with no good
reason as far as I can see it is due only to
the accidental development of Riemannian
geometry from Euclidean geometry. The metric
allows the two magnitudes of two vec- tors to be
compared not only at the same point, but at any
arbitrary separated points.
61
makes his suggestion
A true infinitesimal geometry should, however,
recognize only a principle for transfering
the magnitude of a vector to an
infinitesimally close point and then, on
transfer to an arbitrary distant point the
integrability of the magnitude of a vector is no
more to be expected than the integrability of its
directions.
62
and concludes
On the removal of this inconsistency there
appears a geometry that, surprisingly, when
applied to the world, explains not only
gravitational phenomena,
but also the electrical. According to the
resultant theory both spring from the same
source, indeed in general one cannot separate
gravitation from electromagnetism in a unique
manner.
63
Weyl equipped space-time manifold with conformal
structure, i.e. with a class of conformally
equivalent Lorentz metrics g. The gauge
transformation concerned this metric
Though mathematically beautiful, Weyls theory,
did not describe reality and Weyl had to abandon
it.
In 1929, shortly after the formulation of QED by
Dirac, Weyl reformulated his theory, this time
relating electromagnetism to quantized matter
field. This time he got it right!
64
Weyls 1929 classic Electron and gravitation
The Dirac field equations for ? together with the
Maxwell equations for the four potentials fp of
the electromagnetic field have an invariance
property ....the equations remain invariant when
one makes simultaneous substitutions
and
It seems to me that this new principle of gauge
invariance, which follows not from speculation
but from experiment, tells us that the
electromagnetic field is a necessary
accompany-ing phenomenon not of gravitation, but
of material wave field represented by ?. Since
gauge invariance involves an arbit-rary function
? it has the character of general relativity
and can naturally be understood in this context.
65
Steps into extra dimensions
Kaluza, Klein and later Pauli tried to formulate
gravitational and other forces (preludes to
Theories of everything) in more-dimensional
space-time. Particularly remarkable was the
paper of Oscar Klein
On the Theory of Charged Fields
presented in 1938 at the Warsaw conference
New Theories in Physics
and discussed at length by David Gross in his
article Oscar Klein and Gauge
Theory
In his words
Kleins goal was to construct a theory of all
forces based on the U(1) gauge theory of
isospinors. He almost constructed an SU(2) gauge
theory, but not exactly.
66
The emergence of nonabelian gauge theories and
the role of mathematics in the formulation of
the concept of nonabelian gauge invariance is
discussed at length in an interview of D.Z. Zhang
with C.N. Yang in the Mathematical Intelligencer.
Some excerpts
Q How about ideas in mathematics becoming
important for physics. We may recall
Einstein was adviced to pay attention to
tensor analysis. Is that similar to your getting
help from Simmons?
Yang As to the entry of mathematics into general
theory of relativity and into gauge theory, the
processes were quite different. In the former,
Einstein could not formulate his ideas without
Riemannian geometry, while in the latter, the
equati-ons were written down, but an intrinsic
overall understanding of them was later supplied
by mathematics.
67
Q Is it true what M.E. Mayer said in 1977 A
reading of the Yang-Mills paper shows that the
geometric meaning of the gauge potentials must
have been clear to the authors since they use
the gauge invariant derivative and the curvature
for the connection
Yang Totally false. What Mills and I were doing
in 1954 was generalizing Maxwells theory. We
knew of no geometrical meaning of Maxwells
theory and were not looking in this direction.
Connection is a geometrical concept which I only
learned around 1970.
Q An interesting question is whether you
understood in 1954 the tremendous importance of
your original paper
Yang No. In 1950 we felt our work was elegant. I
realized its importance in the 1960s and its
great importance to physics in the 1970s. Its
relation to deep mathematics became clear to me
only after 1974.
68
Pair of leaves C.N. Yang and mathematics
As the concept of Yang-Mills theories has had a
profound influence on mathematics it is
interesting to know how Yang himself sees the
relation between mathematics and physics.
Q Is it important for a physicist to learn
a lot of mathematics?
Yang No, if a physicist learns too much of
mathematics, he or she is likely to be seduced by
the value judgment of ma-thematics, and may
loose his or her physical intuition. I have
likened the relation between physics and
mathematics to a pair of leaves. They share a
small common part at the base, but mostly they
are separate.
Q For a physicist, experimental results are
more important to learn?
Yang This is right.
69
The relation between mathematics and physics is
addressed also by Straumann in his essay
All major theoretical developments of the last 20
years, such as grand unification, supergravity
and supersymmetric string theory are almost
completely separated from experience. There is a
great danger that theoreticians get lost in pure
speculations. Like in the first unification
proposal of Hermann Weyl they may create
beautiful and highly relevant mathema-tics which
does, however, not describe nature. Remember what
Weyl wrote to C. Sellig in his late years
Einstein thinks that in this field the gap
between ideas and experience is so large that
only mathematical speculations ..... have the
chance to succeeed, whereas my trust in pure
speculations has declined ....
70
So, what does the Nature read?
The lesson from the preceding is, at least for
me, that there is no substitute for genuine
dynamical laws. The claim or hope of Fritzsch,
Gell-Mann Co. that
all results in deep inelastic electron and
neutrino scatter-ing can be explained by assuming
that leading singulari-ties of current products
near the light cone are determined by the light
cone algebra abstracted from the
free quark model
is simply wrong!
The nature does not read in the book of free
field theory, but it definitely prefers the
textbook on Yang-Mills theories with all the
subtleties which cannot be abstracted from free
field theory, but which are responsible for both
the confinement and asymptotic freedom.
71
END
72
(No Transcript)
73
Color as a dynamical variable
Nambus model has been resurrected and cast into
modern langu-age by Lipkin shortly after the
discovery of asymptotic freedom.
The color part of the interaction between pairs
of quarks is assu-med to have form analogous to
isospin-isospin interaction term
implying the following form of interaction energy

where C stands for quadratic Casimir operator
The total mass of a system of n colored quarks,
each of large mass Mq equals
Assuming Mqcv/2, we finally get
i.e. only color singlet states have zero (small)
mass, whereas all color non-singlet ones have
masses of the order of Mq and cannot thus be
observed!
74
Red herring integer charge colored quarks
Nambu is most often associated with the idea of
colored integer charge quarks he proposed in
April 1965 with Hahn.
75
To avoid fractional electric charges of quarks,
Hahn and Nambu made the electric charge Q, the
third component of isospin I3 and the hypercharge
Y dependent on the quark color
u d s

• Plausible idea, but
• mixes strong and electro-
• magnetic interactions
• excluded by data

red blue yellow