Transcript Why asymptotic freedom?

From Yang-Mills to Asymptotic Freedom
to Quantum Chromodynamics
Jiří Chýla, Institute of Physics, Prague
From Yang-Mills to Asymptotic Freedom
to Quantum Chromodynamics
Jiří 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
From Yang-Mills to Asymptotic Freedom
to Quantum Chromodynamics
Jiří 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.
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 Wigner’s supermultiplet theory to QCD
G. ‘t Hooft:
When was the asymptotic freedom discovered?
A. de Rujula: Fifty years of Yang-Mills theories: a phenomenological point of view
D. Gross:
Oscar Klein and gauge theory
Premature burial
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.
From Nambu’s book Quarks
In middle 1950’s Landau and Pomeranchuk attempted to give the
renormalization procedure in QED good physical meaning and
mathematical sense. They put a finite “bare” electric charge
e0=e(r0) on a sphere of radius r0 , placed it in the QED vacuum
and calculated how it appears at a finite distance r>r0.
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.
Landau pole in QED ...
Turning the argument around, they
could have asked how would the
bare charge e0=e(r0) or rather α(r0)
have to depend on r0 to yield a
finite effective electric coupling α(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 procedure in QED is not the fact that bare electric charge
diverges, but that it does so at a finite (though very
small) distance!
... 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 β0 for the bare
as well as effective charges to be
well-defined, and actually vanish,
at small distances
Modern, inherently nonperturbative, approach to the renormalization, which lies at the heart of lattice gauge theory, is just
to construct the dependence α0=α(r0) in such a way to yield
finite values of physical quantities in the limit of vanishing r0.
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!
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!
The beginning of all
starting point: isotopic dublet of nucleons:
 
p
n

In the present paper we wish to exlore the
possibility of requiring all interactions to be
invariant under independent rotations of
'S
the isotopic spin at all space-time points,..
We then propose that all physical processes (not involving
electromagnetic field) be invariant under the isotopic gauge transformation  '  S


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 isospin 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.
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 however subject to the criticism that, like all field theories, the b-field
is beset with divergences and dimensional arguments are not
satisfactory.
mass of the gauge boson to be
determined by its full propagator
b
b
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 existence of the b-field is consistent with experimental information.
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 interaction 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.
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
p
 
3  3 matrix
 
   n   1  iH  
they got the octet of selfinteracting
   infinitesimal gauge gauge vector mesons
  transformation
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,
As an alternative to the Sakata model based on the relation
_
3  3  3  15  6  3  3
Y. Neeman and M. Gell-Mann proposed in early 1961 the
Eightfold Way
which starts with the product of three SU(3) triplets
_
3  3  3  10  8  8 1
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.
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.
Discovery of vector mesons
proceeding as quasi two-body process
followed by ρ in May, Φ in July and ω in August 1961
Who was afraid of gauge theory?
MGM’s preprint
is truly fantastic for the straightforwardness with which the
idea is presented.
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 therefore contribute to the current which is their source. The problem 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.
leptons played the role of quarks:
gauge transformations on all particles involved:
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!!
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!
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 …
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.
Zweig:
Quark model according to 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 inteinteractions. Extensive spacetime and group theoretic
structure is then predicted for
both mesons and baryons …
An experimental search for
the aces is suggested.
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
Λ 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
MGM’s 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 sometimes employed in French cuisine: a piece of pheasant meat is cooked between two slices of veal,
which are then discarded.
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.
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.
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.
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.
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!
Bardeen, Fritzsch, Gell–Mann 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, ….
Why asymptotic freedom?
Because only asymptotically free QFT could explain
surprisingly good scaling behavior of nucleon structure
functions observed since 1967 in deep inelastic electronnucleon 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!
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 non–singlets
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 don’t want to load ourselves with so much
spurious information.
Their hope that
We might eventually abstract from the quark vector–gluon
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.
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 non–singlets 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 Yang–Mills equations.
they, however, ignored this option:
In the next three Sections we shall usually treat the vector
gluon, for convenience, as a color singlet.
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.
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.
Conversion of Saul of Tars to St. Paul
Asymptotic freedom had been discovered in these two papers
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.
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.
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
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 polarization 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.
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
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 won’t, somebody else will,” he
said. I did not follow his advice.
‘t Hooft now likely regrets his decision.
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.
One typical DIS event
from H1 experiment
Nice H1 event with 3 clearly separate and different jets
Potvrzení asymptotické volnosti QCD
Výsledky měření z
různých experimentů
Z přednášky F. Wilczeka

Data
LEP
≈1/r→

Z přednášky
F. Wilczeka v
Karolinu 2003
dilepton
dilepton
+foton
Jets at LEP
dva jety
tři jety
ALEPH
μ-
μ+
jet
jet
jet
jet
jet
jet
jet
jet
jet
Experimental evidence for the basic feature of
nonabelian gauge theories
Triple gluon coupling (1990)
measured by the angular distribution of four jet events at LEP:
taking into account that
as well as ZWW and WW vertices (1998)
Thanks to LEP2 we now
see the effects of triple
gauge boson coupling.
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 vectors to be compared not only at the same point,
but at any arbitrary separated points.
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.
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.
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
2
g'e g
Though mathematically beautiful, Weyl’s 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!
Weyl’s 1929 classic: Electron and gravitation
The Dirac field equations for ψ 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
i
 e 
and

fp  fp  p
x
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 accompanying phenomenon not of gravitation, but of material wave field
represented by ψ. Since gauge invariance involves an arbitrary function λ it has the character of “general relativity” and
can naturally be understood in this context.
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:
Klein’s 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.
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 equations were written down, but an intrinsic overall understanding
of them was later supplied by mathematics.
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 Maxwell’s theory. We knew of no geometrical
meaning of Maxwell’s 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.
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 mathematics, 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
Yang: This is right.
are more important to learn?
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 mathematics 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 ....
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 scattering can be explained by assuming that leading singularities 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.
END
Color as a dynamical variable
Nambu’s model has been resurrected and cast into modern language by Lipkin shortly after the discovery of asymptotic freedom.
The color part of the interaction between pairs of quarks is assumed 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 Mq=cv/2,
we finally get
i.e. only color singlet states have zero
(small) mass, whereas all color nonsinglet ones have masses of the order
of Mq and cannot thus be observed!
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.
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
red
s
Plausible idea, but
• mixes strong and electromagnetic interactions
• excluded by data
blue
yellow