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Certain Bootstraps and Related Equations
1.M. M. Broido, J. Math. Phys. 6, 1702 (1965).
2.M. M. Broido and J. G. Taylor, Phys. Rev. 147, 993 (1966).
3.G. F. Chew and S. C. Frautschi, Phys. Rev. Letters 7, 349 (1961). A large bibliography appears in Ref. 2 above.
4.B. L. van der Waerden, Modern Algebra (Frederick Ungar Publishing Company, New York, 1950).
5.M. A. Krasnosel’skii, Topological Methods in the Theory of Nonlinear Integral Equations (The Macmillan Company, New York, 1964).
6.J. G. Taylor, Proc. Roy. Soc. (London) (to be published).
7.(a) J. G. Taylor, J. Math. Phys. 6, 1148 (1965).
7.(b) J. G. Taylor, Rutgers University preprints: “Topological degree for Non‐Compact Mappings”; “On the Existence of Field Theory, I and II”, cf. J. Math. Phys. 7, 1720 (1966).
8.J. Schauder, Studia Math. 2, (1930).
9.E. A. Michael, Am. Math. Soc. Memoirs. No. 11 (1952).
10.J. G. Taylor, Nuovo Cimento Suppl. Pt. 1, 857 (1963).
11.H. Lehmann, K. Symanzik, and W. Zimmermann, Nuovo Cimento 25, 425 (1955);
11.K. Symanzik, CERN preprint (1961).
12.We sketch the argument involved, since it has not appeared in print. We have to construct the involution and to show that M is invariant under it. Since the algebra is generated by M, it is sufficient to do this for polynomials in M. The ring anti‐isomorphism is constructed simply by reversing the legs in the s channel. We are, of course, working with the complex field K, and the required anti‐linearity in K will be the natural one. We are still free to fix a factor in the transformation and we do this precisely so as to have This is possible since regarded as a numerical function of four‐momenta, in any case invariant under the ring anti‐isomorphism (say, by field‐theoretic crossing, see Ref. 10, p. 991) (even off‐the‐mass‐shell, compare the continuation equation, the τ‐function analog of Eq. 23) (p. 881, Ref. 10). Thus we also maintain as we must. We regard this involution as fixed before we embed the algebra of polynomials into a suitable complete locally m‐convex algebra, (i.e., by taking an appropriate uniform structure on the polynomial algebra), since the completion of the algebra of polynomials will depend a priori on the involution. The connection with TCP is now logically redundant, but can be seen easily enough by writing down the expression for M as the Fourier transform of the vacuum expectation value of the time‐ordered product of fields. (We are indebted to Dr. J. G. Taylor for pointing out the difficulties which arise when one tries to establish a direct connection between TCP and the involution.) Finally, we note that this is not quite the same involution as that used in Appendix 1 of Ref. 1.
13.A. Salam, Nuovo Cimento 25, 224 (1962).
14.R. Jost, General Field Theory (American Mathematical Society, Providence, Rhode Island, 1965).
15.M. M. Broido, Proc. Cambridge Phil. Soc. 62, 209 (1966);
15.M. M. Broido, ibid. (to be published).
16.M. M. Broido, Ph.D. thesis, Cambridge University (1965).
17.R. Brout and F. Englert, Bull. Am. Phys. Soc. 11, 21 (1966).
18.M. M. Broido (to be published).
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