### 15. NOTES AND REFERENCES

(1) Perhaps this seems unduly harsh. An idea - Hamiltonian theory, for example -
may have immense utility and lead to new insights without, in this sense, having any new
physical content. Thus in this sense it was, I suppose, the advent of quantum theory
which saved the Hamiltonian viewpoint from the dustbin!

(2) It is not appropriate for me to dwell here at length on all the numerous other
motivations, some vague and some fairly clear-cut, which influenced the direction of the
development of twistors. Among these was a desire for a formalism tailored to the
four-dimensional (+---) structure of our space-time, rather than for something not so
specific. The holomorphic nature of the space of null directions, in our particular
dimension and signature, seemed to he a highly suggestive clue. Other motivations were
provided by the experimental facts of left-right asymmetry and non-locality (cf. Lee &
Yang 1956; Bohm 1951, Aharonov & Bohm 1959). Twistors have emerged as very
compatible with these objectives.

(3) Perhaps in the present climate of eleven-dimensional generalized Kaluza-Kiein
theories this objection would carry little weight with most people. However, to me it was,
and still is, a fundamental drawback.

(4) Apparently on 1, December 1963 - for which date I thank Zsuzsi Ozsvath.

(5) He had not quite succeeded in satisfying this final requirement, but a slight
modification of his solution (found a few months later by twistor-type methods, cf. Penrose
1965, provided what he had been seeking. We now refer to such solutions as elementary
states (see Penrose 1975a) and they have importance in twistor theory.

(6) He was probably also explaining to me about self-dual null bivectors, but their
relationship to spinors seems to have been something I learnt later!

REFERENCES

AHARONOV, Y. & BOHM, D. (1959), Phys. Rev. 115, 485.

ATIYAH, M. F., HITCHIN, N. J., DRINFELD, V. G. & MANIN, YU. 1. (1978), Phys. Lett.,
65A, 185.

ATIYAH, M. F. & WARD, R. S. (1977), Commun. Math. Phys., 55, 111.

BATEMAN, H. (1904), Proc. Lond. Math. Soc. (2) 1, 451.

BATEMAN, H. (1910), Proc. Lond. Math. Soc., (2) 8, 223.

BATEMAN, H. (1944), Partial Differential Equations of Mathematical Physics (Dover, New
York).

BOCHER, M. (1914), Bull. Amer. Math. Soc., 20, 185.

BOHM, D. (1951), Quantum Theory (Prentice-Hall, Englewood Cliffs).

BONDI, H., PIRANI, F. A. E. & ROBINSON, 1. (1959), Proc. Roy. Soc. (Lord.), A251, 519.

BRINKMAN, H. W. (1923), Proc. Nad. Acad. Sci. (U.S.), 9, 1.

CARTAN, A. (1914), Ann. Acole Norm. Super., 31, 263.

CAYLEY, A. (1860), Quart. J. of Pure and Appl. Math., 3, 225 (see Collected Mathematical
Papers, 4, p. 447).

CAYLEY, A. (1869), Trans. Camb. Phd. Soc., 11 (2), 290 (see Collected Mathematical
Papers, 7, p. 66).

COXETER, H. S. M. (1936), Ann. of Math., 37, 416.

CUNNINGHAM, E. (1910), Proc. Lond. Math. Soc., (2) 8, 77.

DARBOUX, G. (1914), Lecons sur la Theorie Generale des Surfaces, Pt. 1, 2nd edn.
(Gauthier-Villars, Paris).

DIRAC, P. A. M. (1936a), Proc. Roy. Soc. (Lond.), A155, 447.

DIRAC, P. A. M. (1936b), Ann. of Math., 37, 429.

GEL'FAND, 1. M., GRAEV, M. 1. & VILENKIN, N. Ya. (1966), Generalized Functions, vol.
5: Integral Geometry and Representation Theory (Academic Press, New York).

GINDIKIN, S. G. (1983), Math. Intelligencer, 5, no. 1, 27.

HEPNER, W. A. (1962), Nuovo Cim., 26, 351.

HITCHIN, N. J. (1979), Math. Proc. Camb. Phd. Soc., 85, 465.

HODGES, A. P. (1983), Proc. Roy. Soc. (Lond.) A385, 207; A386, 185.

HODGES, A. P. (1985), Proc. Roy. Soc. (Lond), A 317, 41, 375.

HODGES, A. P. & HUGGETT, S. A.; HUGHSTON, L. P.; PENROSE, R.; TOD, K. P.; WARD,
R. S. (1980), Surv. High Eenergy Physics, 1, 333, 313, 267, 299, 289.

HUGHSTON, L. P. (1979), Twistors and Particles; Lecture Notes in Physics 97 (Springer,
Berlin).

KLEIN, F. (1870), Math. Annalen, 2, 198.

KLEIN, F. (1926), Vorlesungen über höhere Geometrie (Springer, Berlin), pp. 80, 262.

KUIPER, N. H. (1949), Ann. of Math., 50, 916.

LEE, T. D. & YANG, C. N. (1956), Phys. Rev., 104, 254.

LIE, S. & SCHEFFERS (1896), Berührungstrsfn. 453.

McLENNAN, Jr., J. A. (1956), Nuovo Cim., 10, 1360.

MOUSSOURIS, J. (1983), Quantum Models of Space-Time Based on Recoupling Theory (D.
Phil. thesis, Oxford).

MURAI, Y. (1953/4), Progr. Theoret. Phys., 9, 147; 11, 441.

MURAI, Y. (1958), Nucl. Phys., 6, 489.

NEWMAN, E. T. (1976), Gen. Rel. Grav., 7, 107.

NEWMAN, E. T. & PENROSE, R. (1968), Proc. Roy. Soc. (Lond.), A305, 175.

PENROSE, R. (1959), Proc. Camb. Phil. Soc., 55, 137.

PENROSE, R. (1965a), in Proceedings of the 1962 Conference on Relativistic Theories of
Gravitation, Warsaw (Polish Acad. Sci. Warsaw).

PENROSE, R. (1965b), Proc. Roy. Soc. (Lond.), A284, 159.

PENROSE, R. (1967), J. Math. Phys., 8, 345.

PENROSE, R. (1968), Int. J. Theor. Phys., 1, 61.

PENROSE, R. (1969a), J. Math. Phys., 10, 38.

PENROSE, R. (1969b), in Quantum Theory and Beyond (ed. E. Bastin; Cambridge Univ.
Press, Cambridge).

PENROSE, R. (1972a), in General Relativity (ed. L. O'Raifeartaigh; Clarendon Press,
Oxford).

PENROSF, R. (1972b), in Magic without Magic (ed. J. R. Klauder; Freeman, San Francis
co).

PENROSE, R. (1975a), in Quantum Gravity, and Oxford Symposium (eds. C. J. Isham, R.
Penrose & D. W. Sciama; Clarendon Press, Oxford).

PENROSE, R. (1975b), in Quantum Theory and the Structures of Time and Space (eds. L.
Castell, M. Dreischner & C. F. von Weizsacker; Carl Hanser Verlag, Munich).

PENROSE, R. (1976), Gen. Rel. Grav., 7, 31, 17 1.

PENROSE, R. (1979), in Advances in Twistor Theory, Research Notes in Mathematics 37
(eds. L. P. Hughston & R. S. Ward; Pitman, San Francisco).

PENROSE, R. (1980), Gen. Rel. Grav., 12, 225.

PENROSE, R. (1982a), Bull. (New Set.) Amer. Math. Soc., 8, 427.

PENROSE, R. (1982b), Proc. Roy. Soc. (Lond.), A381, 53.

PENROSE, R. & MACCALLUM, M. A. H. (1972), Phys. Repts., 6C, 241.

PENROSE, R. & RINDLER, W. (1984), Spinors and Space-Time; vol. 1, Two-Spinor Calculus
and Relativitic Fields (Cambridge Univ. Press, Cambridge).

PENROSE, R. & RINDLER, W. (1985), Spinors and Space-Time; vol. 2, Spinor and Twistor
Methods in Space-Time Geometry (Cambridge Univ. Press, Cambridge).

PENROSE, R. & WARD, R. S. (1980), in General Relativity, One Hundred Years after the
Birth of Albert Einstein (ed. A. Held; Plenum, New York).

PERJÉS, Z. (1975), Phys. Rev., 111), 2031.

PERJÉS, Z. (1979), Phys. Rev., 201), 1857.

PERJÉS, Z. & SPARLING, G. A. J. (1979), in Advances in Twistor Theory, Research Notes
in Mathematics 37 (eds. L. P. Hughston & R. S. Ward; Pitman, San Francisco).

PLÜCKER, J. (1865), Phil. Trans. t., 155, 725.

PLÜCKER, J. (1868/9), Neue Geometrie des Raumes gegründet auf die Betrachtung der
geraden Linic als Raumdement (ed. F. Klein, Leipzig).

RADON, J. (1917), Ber. Verb. Sächs. Akad., 69, 262.

REGGE, T. (1961), Nuovo Cim., 19, 558.

ROBINSON, I. (1956), Report to the Eddington Group, Cambridge.

ROBINSON, I. (1961), J. Math. Phys., 2, 290.

ROBINSON, I. & TRAUTMAN, A. (1962), Proc. Roy. Soc. (Lond.), A265, 463.

RUDBERG, H. (1958), dissertation, Univ. of Uppsala, Uppsala, Sweden.

RZEWUSKI, J. (1958), Nuovo Cim., 9, 942.

SCHRÖDINGER, E. (1952), Science and Humanism (Cambridge Univ. Press, Cambridge).

SHAW, W. T. (1983), Proc. Roy. Soc. (Lond.), A390, 191.

SPARLING, G. A. J. (1975), in Quantum Gravity, an Oxford Symposium (eds. C. J. Isham,
R. Penrose & D. W. Sciama; Clarendon Press, Oxford.)

TOD, K. P. (1983), Proc. Roy. Soc. (Lond.), A388, 457.

WALKER, M. & PENROSE, R. (1970), Commun. Math. Phys., 18, 265.

WARD, R. S. (1977), Phys. Lett., 61A, 81.

WEIERSTRASS, K. (1866), Monatsberichte der K. P. Akademie, 612 (collected works,
vol. 3).

WHITTAKER, E. T. (1903), Math. Annalen, 57, 333.

WHEELER, J. A. (1962), Geometrodynamics (Acad. Press, New York).