19th Issue (Oct-2012) - Department of Physics - The Chinese ...

C. N. Yang
Int. J. Mod. Phys. A 2012.27. Downloaded from www.worldscientific.com
by THE CHINESE UNIVERSITY OF HONG KONG on 10/09/12. For personal use only.
Institute, J. M. Luttinger and I happened to talk about the Ising model.
Luttinger said that Bruria Kaufman had simplified Onsager’s method
so that the solution could be understood in terms of the representation
of a system of 2n anticommuting Hermitian matrices. I knew such representations
well and understood quite readily the main points of the
Onsager–Kaufman method. After arriving at the Institute, I worked out
the essential steps of this approach and was very happy at finally understanding
Onsager’s solution ...
I kept thinking about it, and realized that Onsager and Kaufman had
obtained much more information than just the partition function ,...
I was thus led to a long calculation, the longest in my career. Full of
local, tactical tricks, the calculation proceeded by twists and turns. There
were many obstrctions. But always, after a few days, a new trick was somehow
found that pointed to a new path. The trouble was that I soon felt
I was in a maze and was not sure whether in fact, after so many turns, I
was anywhere nearer the goal than when I began. This kind of strategic
overview was very depressing, and several times I almost gave up. But each
time something drew me back, usually a new tactical trick that brightened
the scene, even though only locally.
Finally, after about six months of work off and on, all the pieces suddenly
fitted together, producing miraculous cancellation, and I was staring
at the amazingly simple final result.
Why was I able to “understood quite readily the main points of the Onsager–
Kaufman method”? One: I had thoroughly studied the representation of 2n anticommuting
Hermitian matrices when I studied Dirac’s equation in Kunming. Two:
I had spent several weeks in 1947 in Chicago trying to understand Onsager’s 1944
paper, without success. But that experience had thoroughly familiarized me with
Onsager’s paper. Combining these two earlier efforts I was able to rapidly understand
what Luttinger had said about the real strategy of Onsager’s solution.
What I had learned from Professor J. S. Wang in Kunming made me deeply
interested in statistical mechanics, leading to my preparatory efforts in Chicago
on Onsager’s work. The final breakthrough came from a short conversation with
Luttinger. This process:
Interest → Preparation → Breakthrough, (5)
I think, is the necessary three-step process for any research work. In the experience
described above the breakthrough came from an external stimulation (conversation
with Luttinger). But in most cases the breakthrough is a sudden inspiration without
external stimulation: After long preparation, when one is not thinking about
the problem, apparently new conceptual combinations were still being formed subconsciously
in one’s brain. When the right combination occurs a sudden inspiration
would result. A century earlier Poincaré 8 had given a very interesting analysis of
this process.
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