# Some Aspects of Functional Integrals and Many Body Theory

• David Sherrington
Part of the NATO Advanced Study Institutes Series book series (NSSB, volume 34)

## Abstract

Conventionally the quantum many body problem has been formulated in terms of a Hamiltonian language. In second quantization this involves the use of field operators which obey commutation (for bosons) or anticommutation (for fermions) relations for equal times;
$$\psi \left( {\underline x ,t} \right)\,\,{\psi ^ + }\left( {\underline x ',t} \right) - {\psi ^ + }\left( {\underline x ',t} \right)\,\,\psi \left( {\underline x ,t} \right)\, = \delta \left( {\underline x - \underline x '} \right)$$
(1a)
$$\psi \left( {\underline x ,t} \right)\,\,{\psi ^ + }\left( {\underline x ',t} \right) + {\psi ^ + }\left( {\underline {x'} ,t} \right)\,\,\psi \left( {\underline x ,t} \right)\, = \delta \left( {\underline x - \underline x '} \right)$$
(1b)
Here, and below, we use x to denote symbolically all the relevant coordinates other than time; i.e. space, spin, etc. The 6-function is to be interpreted as Kronecker or Dirac according to whether the corresponding coordinate element is discrete or continuous.

## Keywords

Partition Function Green Function Spin Glass Functional Integral Perturbation Expansion
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 1.
J.S. Bell, 1962, ‘Lectures on the Many Body Problem (Naples Spring School)’ Ed. E.R. Caianiello (New York, London: Academic Press), pp 81–9Google Scholar
2. 2.
S.F. Edwards and D. Sherrington, 1967, Proc. Phys. Soc. 90, 3–22
3. 3.
D. Sherrington, 1967, Proc. Phys. Soc. 90, 583–4
4. 4.
D. Sherrington, 1971, J. Phys. C4, 401–416
5. 5.
L.P. Kadanoff and G. Baym, 1962, ‘Quantum Statistical Mechanics’ (New York; Benjamin)
6. 6.
D. Sherrington, 1966, Ph. D. thesis (University of Manchester) unpublishedGoogle Scholar
7. 7.
B. Mühlschlegel, 1977, these lecturesGoogle Scholar
8. 8.
S.F. Edwards, 1955, Proc. Roy. Soc. A 232, 371–6
9. 9.
I.M. Gel’fand and A.M. Yaglom, 1960, J. Hath. Phys. 1, 48–69
10. 10.
R.L. Stratonovich, 1957, Dokl. Akad. Nauk SSSR 115, 1097–1100 (Sov. Phys. Dokl. 2, 416–9)
11. 11.
J. Hubbard, 1959, Phys. Rev. Lett. 3, 77–8
12. 12.
B. Mühlschlegel, unpublished notes University of Pennsylvania, referenced by Wang et al, 1969, Phys. Rev. Lett. 23, 92–5
13. 13.
See for example S.K. Ha, 1976, ‘Modern Theory of Critical Phenomena’ (New York: Benjamin) or C. Domb and M.S. Green (ed), 1976, Phase Transitions and Critical Phenomena (New York: Academic Press)Google Scholar
14. 14.
S.F. Edwards, 1970, in ‘4th Int. Conf. on Amorphous Materials’ (ed. R.W. Douglas and W. Ellis, New York: Wiley)Google Scholar
15. 15.
S.F. Edwards and P.W. Anderson, 1975, J. Phys. F5, 965–74
16. 16.
D. Sherrington and K. Mihill, 1974, Proc. Int. Conf. Mag. (Moscow 1973) Vol. 1 (1), 283–87;Google Scholar
17. 16a.
D. Sherrington and K. Mihill, 1974, J. de Phys. 35, C4, 199–201.Google Scholar
18. 17.
P.W. Anderson, 1958, Phys. Rev. 109, 1492–1505
19. 18.
See for example the review by K. Fisher, Int. Conf. on Magnetism (Amsterdam 1975)Google Scholar
20. 19.
D. Sherrington, 1975, AIP Conf. Proc. 29, 224–228
21. 20.
G. Toulouse, 1977, Comm. Phys. 2, 115–119Google Scholar
22. 21.
B.W. Southern, 1976, J. Phys. C9, 4011–4020
23. 22.
B.R. Coles, A. Tari and H.C. Jamieson, 1974, Proc. L.T. XIII, 414Google Scholar
24. 23.
D. Sherrington and S. Kirkpatrick, 1975, Phys. Rev. Lett. 35, 1792–96