1/N Page: 7 of 72
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precpding diagrams in the larRe_s litnlr.
This is tlie first step in construct Inc Mu' 1 /-•" expansion. We
r.,wst decide what parameters to held fixed as « becomes large. If
we make the wrong choice, we can obtain either a trivial theory
(only the Born term si. vivos) or one without a 1/N expansion (there
are graphs proportional to positive powers of N). Of course, we
have not yet shown that the second possibility docs not occur in
the theory at hand. However, there are clearly an infinite number
of graphs proportional to l/N, times various powers of gQ; two of
them t,ro shown in Fig. 2. (To keep the graphs from being hopeless
jumbles, 1 have left out the index labels; 1 hope you can figure
out where they go.)
To keep all these diagrams straight, and to show that there
are no diagrams proportional to positive powers of N, Is a combina-
toric challenge. We van simplify life considerably b*» Introducing
an auxiliary field, a, and altcr'ng the I^igrangv density:
^ * ■2’'1 ’ e“(° " 5 T? ,°*1)2 ' <2'3>
This added c«. rm has it-’ effects on the dynamics of the theory.
This is easy to see fro.n the viewpoint cf Tunclional integration.
The functional integral over j is a trivial Ccusgian integral; its
only effect if Co multiply the generating functional of the theory
by an Irrelevant constant. Tt is also easy to see from the view-
point of canonical quantization. The Euler-Lagrange equation for
1 •,0 sttui
Here’s what’s next.
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Coleman, S. 1/N, report, March 1, 1980; California. (https://digital.library.unt.edu/ark:/67531/metadc1070576/m1/7/: accessed April 23, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.