Abstract
One obvious question to ask if one has a family of honest field theories obtained from Lagrangian models is the relation of the theory to the Feynman perturbation series. One is interested in this question for two reasons. First, one would like to understand why perturbation theory is such a good guide; put differently, one would like to show that perturbation theory “determines” the theory in some way. Secondly, one hopes to prove rigorously that some (or all) of the theories are non-trivial. If one could show a Feynman series is asymptotic for a truncated four-point Green’s function, one would know some theories are nontrivial in that their S-matrix is nonzero (modulo the proof of a mass gap and the existence of one particle states).
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References
F. Dyson, Phys. Rev. 85, 631 (1952).
C. Hurst, Proc. Camb. Phil. Soc. 48, 625 (1952), W. Thirring, Helv. Phys. Acta 26, 33 (1953), A. Peterman, Helv. Phys. Acta 26, 291 (1953).
A. Jaffe, Commun. Math. Phys. 1, 127–149 (1965).
W. Frank, Ann. Phys. 29, 175 (1964).
C. Bender and T.T. Wu, Phys. Rev. 184, 1231 (1969). This paper is quite basic to understanding what is going on in the anharmonic oscillator. While it is nonrigorous, its computations provide useful intuition.
E. Caianiello, Nuovo Cimento 3, 223 (1956).
D. Yennie and S. Gartenhaus, Nuovo Cimento 9, 59 (1958); A. Buccafurri and E. Caianiello, Nuovo Cimento 8, 170 (1958).
B. Simon, Nuovo Cimento 59A, 199 (1969).
F. Guerra and M. Mariano, Nuovo Cimento 42A, 285 (1966).
Carleman’s theorem is proven in T. Carleman, Les Fonctions Quasianalytiques, Gauthier-Villars, Paris, 1926. The proof of the simpler theorem is discussed in G. Hardy, Divergent Series, Oxford Univ. Press 1949 and in M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. II, Academic Press, 1973.
G. Watson, Phil. Trans. Roy. Soc. 211, 279–313 (1912); see also Hardy, ref. 10.
The general method is described in G. Baker, Adv. Teho Phys. 1, 1 (1966). The application to Rayleigh-Schrödinger series is J.J. Loeffel, A. Martin, B. Simon, and A.S. Wightman, Phys. Lett. 30B, 656 (1969).
The general method is discussed in G. Hardy, ref. 10. The application to perturbation series is in J. Gunson and D. Ng. (Univ. of Birmingham preprint).
S. Graffi, V. Grecchi and G. Turchetti, Nuovo Cimento 4B, 313 (1971).
This is an exactly soluable model solved in J. Schwinger, Phys. Rev. 82, 664 (1951). The relation to summability is found in V. Ogieveski, S.A.N.S.S.R. 109, 919 (1956) and S. Graffi, J. Math. Phys., to appear.
This is the behavior of the main term in a Rayleigh Schrödinger series, i.e. \(\begin{gathered} {{\left( { - 1} \right)}^{n}} \sum \hfill \\ {{i}_{l}} = 0 \ldots {{i}_{{n - 1}}} \ne 0 \hfill \\ \left\langle {0\left| V \right|{{i}_{l}}} \right\rangle \ldots \left\langle {{{i}_{{n - 1}}}\left| V \right|0} \right\rangle {{\left( {{{E}_{{il}}} - {{E}_{0}}} \right)}^{{ - 1}}} \ldots {{\left( {{{E}_{i}}_{{_{n}}} - {{E}_{0}}} \right)}^{{ - 1}}} \hfill \\ \end{gathered} \) For the anharmonic oscillator there is a numerical analysis by Bender and Wu (ref. 5) and a set of reasonable arguments by C. Bender and T.T. Wu, Phys. Rev. Lett. 27., 461 (1971) which predict [n(m-1)]! behavior.
B. Simon, Ann. Phys. 58, 79 (1970); S. Graffi, V. Grecchi, and B. Simon, Phys. Lett. 32B, 631 (1970).
The Padé method was proven to work for one dimensional oscillators by Loeffel et al. (ref. 12) and the Borel method by Graffi et al. (ref. 17).
It is our feeling that the localization techniques that Jaffe will discuss will eventually produce enough control on the infinite volume limit to prove strong asymptotic conditions in that case.
Similarly, we think the ideas in E. Nelson, “Time-ordered operator products of sharp-time quadratic forms” Princeton Preprint, Dec. 1971 should lead to control over the Green’s functions.
The result for E(β) is from B. Simon, Phys. Rev. Lett. 25, 1583–1586 (1970). The result for W(β) is from L. Rosen and B. Simon Trans A.M.S., to appear.
The operator H(β) are “sectorial” in the sense of M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. I, pp. 281–282, Academic Press, 1972. This is a simple consequence of the linear lower bound as noted by B. Simon and R. Hoegh-Krohn, J. Func. Anal., to appear.
I know of at least four proofs of this fact: (i) The original proof in Simon, Phys. Rev. Lett. (ref. 21) used the linear lower bound (ii) A proof using the \(\left(\phi^{4}\right)_{2}\) quadratic estimate was suggested by L. Rosen and appears in B. Simon, Adv. Math., to appear. (iii) A proof using “the method of hyperconstractive semigroups appears in Rosen-Simon (ref. 21) (iv) A proof due to B. Gidas, Univ. of Michigan preprint, 1971 which relies on techniques of P. Federbush.
This and many other very useful perturbation theory results appear in T. Kato, Perturbation Theory for Linear Operators, Springer, 1966.
P (β) is the projection onto the eigenvector with eigenvalue E (β). The formula for E (β) is very useful. One can compute the higher order of the Rayleigh-Schrödinger series from it by using a geometric series on \(\left(\textup{H}_{0}+\beta \textup{H}_{\textup{I}}-\lambda \right)^{-1}\) in the formula for P(β) and one can compute the error easily using a geometric series with remainder.
By this we mean not that \(\left (\textup{H}\left (\beta \right ) -\lambda \right )\) is entire for any particular \(\lambda\), but that for any \(\beta_{0}\) there is some \(\lambda_{0}\) with \(\left (\textup{H}\left (\beta \right )- \lambda _{0} \right)^-1\) analytic for β near \(\beta_{0}\). The entirety follows from an estimate \(\textup{H}_{\textup{I,k}} {2}\leq \varepsilon \textup{H}_{0} {2}+\textup{b} {2}\) for ε arbitrarily small and with b ε-dependent.
J. Glimm, Commun. Math. Phys. 5, 343–386 (1967); 6, 61–76 (1967).
J. Dimock, Harvard Preprint, 1971; a similar but divergent bound is found in J. Glimm and A. Jaffe, Ann. Phys. 60, 321–383 (1970).
References to Postscript
K. Osterwalder, Fort. der Phys., to appear.
J. Glimm, Commun. Math. Phys. 10, 1–47 (1968).
K. Hepp in Systèmes a un nombre infini de degrés de liberté, C.N.R.S. publ. No. 181, Paris, 1970.
J. Fabray, Commun. Math. Phys. 19, 1 (1968).
K. Osterwalder and J.P. Ekmann, Helv. Phys. Acta., to appear J.P. Eckmann, Brandeis University, Preprint.
It is unfortunate that some books call field strength renormalization “wave function renormalizations”. Glimm’s “wave function renormalization” is very different from field strength renormalization.
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Simon, B. (1973). Perturbation Theory and Coupling Constant Analyticity in Two-Dimensional Field Theories. In: Iverson, G., Perlmutter, A., Mintz, S. (eds) Fundamental Interactions in Physics and Astrophysics. Studies in the Natural Sciences, vol 3. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-4586-2_4
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