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0II. 1973-1974: From Asymptotic Freedom to Lattice Gauge Theory (1)
May 1973: Asymptotic Freedom made known
(Gross-Wilczek and Politzer )
I face problem: I have not learned details of renormalization of continuum gauge theories
I need to catch up - fast. What do I do?
Answer: try putting theory on a lattice
with hope that it will be easier to understand than the continuum version
Confinement?
How could I know in advance - that confinement would result?
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II. 1973-1974: From Asymptotic Freedom to Lattice Gauge Theory (2)
By summer 1973, the formalism is clear:
the gauge variables must live on nearest neighbor links and must be unitary matrices.
Fermion variables are anticommuting objects living at lattice sites. The gauge theory is formulated on a Euclidean lattice.
There are nearest neighbor couplings denoted K and a gauge coupling g, the latter appearing as 1 / (g*g).
I talk about this at Orsay (summer, 1973).
It is also easy to see that there is a strong coupling expansion for the limit:
g large and K small:
The expansion resembles high temperature expansions in statistical mechanics.
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II. 1973-1974: Origins (3)
I run into a problem:
formulating physical interpretation of the strong coupling expansion
I spent many hours going back and forth between the approximate formulae for the lattice theory and the concepts of continuum string theory
B. I struggle for many months to arrive at the obvious:
The nearest neighbor couplings in the Euclidean time-like direction define large quark masses, of order ln(K),
Gauge plaquettes in one space, time direction result in large string bit masses on space-like links, of order 2 ln g.
I found it hard to separate the two roles of K producing large unperturbed masses from time-like nearest neighbor terms but much smaller, momentum dependent corrections from space-like nearest neighbor terms.
Getting straight that masses of strings were additive over each link of space-like lattice strings was even more difficult
C. It was a complete surprise to me that:
The lattice theory requires no gauge fixing;
and that color confinement results from the strong coupling limit of g large and K small.
As is customary, my article (accepted in May of 1974) made no mention of my struggles with these concepts, but was limited to announcing and justifying the results.
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.II. 1973-1974: From Asymptotic Freedom to Lattice Gauge Theory (4)
When and Why did I learn about high temperature expansions in statistical mechanics?
Two leading researchers on critical phenomena: Michael Fisher and Ben Widom are at Cornell in Chemistry
After 1965, I learn from them about the challenges of critical phenomena,
and the uses of high temperature expansions, e.g. for the Ising model, to determine behavior at the critical point.
B. It becomes apparent to me that the Ising Model can serve as a field theorist s laboratory.
It exhibits renormalizability with anomalous dimensions.
I gain experience with the transfer matrix formalism and high temperature expansions, both invaluable for my initial research on lattice gauge theory.
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RII. 1973-1974: From Asymptotic Freedom to Lattice Gauge Theory (5)
Retrospective
The lattice gauge theory was a discovery waiting to happen,
once asymptotic freedom was established.
There were several physicists that pursued the concept of lattice gauge theory independent of my work,
Franz Wegner: a theory with plaquettes of Ising-like pins, published in 1971
J. Smit: work performed at UCLA in 1972-1973 while a graduate student
M. Polyakov: unpublished work
[See K. Wilson in the Proceedings of the 1983 International Symposium on Lepton and Photon Interactions at High Energies, published by Newman Laboratory, page 827.]
Before my 1974 article was completed, both Kogut and Susskind and Balian, Drouffe, and Itzykson had started their research on lattice gauge theories
I benefited from conversations with both these groups, as I acknowledged in the 1974 article
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>III. 1958 1971: A Thesis Project and its Aftermath (1)
Like many second-rate graduate students, it took me a long time (fifteen years) to disengage from my graduate thesis topic and from research directions based on that topic
1958: Gell-Mann gives me a problem to work on: applying the Low equation (one meson approximation) to K-p scattering.
Low equation: a one-dimensional integral equation for the pi-p (or K-p) scattering amplitude, if one neglects two-meson states (such as pi-pi)
I disregard Gell-Mann s goal; instead I become fascinated by the high-energy behavior of an equation that, at best, is a low energy approximation to the physics of scattering of pi s and K s.
By becoming irrelevant to physics for almost ten years I manage to find something useful that might not have been found otherwise.
In 1959, Gell-Mann goes off to Paris; I (by choice) stay behind, working on my own
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III. 1958 1971: A Thesis Project and its Aftermath (2)
Jon Mathews introduces me to Cal Tech s 1970 s computer, programmed in assembly language. I also used early IBM machines: the 704 and 709
B. Using the computer, I learn that the pi-p amplitude f(E) has an expansion in powers of ln(E) for E large;
the coefficients are integers a(n), and (to my surprise) I find an analytic form for a(n) versus n
C. I also learn that the Gell-Man and Low version of renormalization group theory applies to the Low equation
D. Feynman anecdotes
Presenting thesis
The Wilson theorem seminar
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vIII. 1958 1971: A Thesis Project and its Aftermath (3)
I spend from 1963-1970 in a world of my own
Eager to learn how to solve strongly coupled field theories
Even though all field theories I study are guaranteed to be wrong for the strong interactions (QCD is not known yet).
I work in a number of theorist s laboratories, the Ising model being but one example
I think about how to solve theories numerically, even though computers of the 1960 s are grotesquely inadequate for this task
I try to master as many approximation methods as I can find
I have been fascinated by techniques for approximation since my years as an undergraduate
I struggle to come up with something useful from my research
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&III. 1958 1971: A Thesis Project and its Aftermath (4)
A laboratory experiment:
The field theory underlying the Low equation, namely a free meson field coupled to a fixed source at single point (representing a proton) becomes a theme of my research.
The Interaction: g j(0) t
Since the theory, at high energies, has no practical interest, I am free to butcher it as I please, and do so
The momentum slice model
A free meson field phi coupled to a fixed source through its value at one point: j(0)
In momentum slices, k is restricted to separated ranges, eg, for mass m of order 1:
1<|k|<2; 1000<|k|<2000; 1000000<|k|<2000000, etc.
In Feynman diagrams, this replaces integrations over ALL three-dimensional k space by integrations limited to k vectors within momentum slices.
Different momentum slices have different energy scales 1<E<2; 1000 <E < 2000, 1000000<E < 2000000;
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III. 1958 1971: A Thesis Project and its Aftermath (5)
Solving momentum slice models is accomplished by treating low energy slices as perturbations on the highest energy slice
leads to a renormalization group process starting at the cutoff and working down, slice by slice
See, e.g, K. G. Wilson, Phys. Rev. 140, B445 (1965)
Momentum slice models lead to effective Hamiltonians with
an infinite number of couplings
I write a long and impenetrable article showing that the infinity of couplings does not cause a problem in the model.
K. G. Wilson, Phys. Rev. D2, 1438 (1970)
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III. 1958 1971: A Thesis Project and its Aftermath (6)
Two Payoffs!
First payoff: a one-dimensional integral equation for critical phenomena- a very rough approximation
The integral equation turns out to be soluble by an expansion about four dimensions, (the e expansion) easily applied to three dimensions!
The expansion can be computed by field theory perturbative methods not requiring any further approximations!
For critical phenomena, the momentum slice approach resulted in the discovery of a new perturbative expansion that could be used even without the momentum slices
Second Payoff (1971-1975): The Kondo problem: a problem in condensed matter physics, in which an electron field is coupled to a fixed source (representing an impurity).
It is soluble to considerable accuracy with momentum slices that are adjacent: 1 < k < 2; 2 < k <4; etc.
I use the CDC 7600 at Berkeley for the serious computation
But my reason for undertaking it is partly to enjoy using my newly purchased HP pocket calculator
The Kondo solution was not a discovery waiting to happen;
However, later an exact analytic solution appeared!
See A. C. Hewson, The Kondo Problem to Heavy Fermions, Cambridge University Press, 1997.
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III. 1958 1971: A Thesis Project and its Aftermath (7)
A Blunder: A discussion of possible high-energy behaviors for the renormalization group in strong interactions, proposing every conceivable alternative except for asymptotic freedom
I did not know of any theories exhibiting asymptotic freedom, and was focused more on possible non-perturbative behaviors.
Renormalization Group and Strong Interactions, K. G. Wilson, Phys. Rev. D3, 1818 (1971).
I also ruled out low mass scalar fields, claiming that a low mass was highly unnatural, an argument that may also prove to be a blunder, if dark energy is associated with such a field.
A Bizarre Episode
A call from the blue: a reader of my impenetrable paper on the renormalization group with momentum slices and an infinite number of couplings
We agree to meet on a Saturday morning at Newman Lab.
The person in question is not a physicist; but he explains to me his view of the world:
The world we experience is actually only a computer simulation being run in a higher-level world.
The phenomena of quantum mechanics are caused by BUGS in the simulation code.
The higher-level world is again a computer simulation being run at an even higher level, and this continues for many levels hence his interest in the repeated iterations of my renormalization group paper.
Eventually one encounters a master intelligence: a supreme being at the highest level.
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IV. 2004 New Research Questions for Lattice Gauge Theory
Are there colored partners for the gluon--with integer spin, not predictable by current models?
no model predicted the muon, and quarks have partners too: why not partners for gluons?
Is there a Renormalization Group Infrared Limit Cycle in the theoretical domain spanned by QCD?
namely for quark masses tuned to produce both a deuteron and a di-neutron precisely at threshold?
See the seminal article by Eric Braaten and Hans Hammer: PRL 91, 102002 (2003)
For a simple example of a renormalization group limit cycle: See S. Glazek and K. Wilson, PRL 89, 230401 (2002)
See these references to learn what is meant by a renormalization group limit cycle.
Are there novel expansions still awaiting discovery for solving lattice gauge theory or continuum QCD?
Perhaps a variant of weak coupling expansions (exemplified by the e expansion for critical phenomena)?
Perhaps a variant on strong coupling expansions,
Perhaps discoverable by seeking out new ways to butcher quantum field theory (exemplified by the momentum slice approach)?
Or from study of renormalized light front QCD?
See e.g., Wilson et al., Phys. Rev. D49, 6720 (1994) and more recent publications by Stan Glazek
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