What is the point of so‐called “axiomatic field theory”?

SEP 01, 1969

This method is incapable at present of calculating cross sections and is often thought to be obsessively mathematical. Neverthless it offers a conceptual clarity indispensible for understanding the quantum mechanics of systems with an infinite number of degrees of freedom.

DOI: 10.1063/1.3035782

Arthur S. Wightman

QUANTUM FIELD THEORY is well described by an adjective that Robert Oppenheimer once applied to the Fermi theory of beta decay; it is robust. It has had its ups and downs, but it somehow seems to survive all vicissitudes and continues to provide the best available language for the description of elementary‐particle phenomena. Here I shall try to describe the origins and some of the achievements and objectives of the particular brand of field theory customarily referred to as “axiomatic.” I will thereby try to answer the question posed by the title.

References

1. N. Kemmer, Proc. Cambridge Phil. Soc. 34, 354 (1938).https://doi.org/PCPSA4
2. W. Pauli, Meson Theory of Nuclear Forces, Interscience, New York (2nd edition 1948).
3. L. Van Hove, Phys. Rev. 75, 1519 (1949).https://doi.org/PHRVAO
4. M. Levy, Phys. Rev. 88, 72 and (1952).https://doi.org/PHRVAO
5. M. L. Goldberger, in proceedings of the Midwest Conference on Theoretical Physics, Purdue University, April 1960.
Quoted in Ann. Rev. Nucl. Sci. 11, 95 (1961).https://doi.org/ARNUA8
6. A. M. Jaffe, Phys. Rev. 158, 1454 (1967).https://doi.org/PHRVAO
7. R. F. Streater, A. S. Wightman, PCT, Spin and Statistics and All That, W. A. Benjamin, New York (1964).
8. R. Jost, The General Theory of Quantized Fields, American Mathematical Society, Providence, R. I. (1965).
9. J. von Neumann, Math. Ann. 104, 570 (1931).https://doi.org/MAANA3
10. Bibliography of reference 8.
11. H. J. Borchers, Nuovo Cimento 15, 184 (1960).https://doi.org/NUCIAD
12. Proceedings of the 45th Varenna course, 12–24 August 1968, Academic Press, New York.
13. K. Symanzik, “Many‐particle Structure of Green’s Functions” in Symposium on Theoretical Physics, vol. 3, p. 121, Plenum Press, New York (1967).
14. R. Kadison, “Operator Algebras” in Cargèse Lectures in Theoretical Physics 1965 (F. Lurcat, ed.) Gordon and Breach, New York (1967).
15. S. Doplicher, D. Kastler, Comments in Math Phys. 7, 1 (1968) and references cited therein.

More about the Authors

Arthur S. Wightman. Princeton University.