100 Years of Relativity Space-Time Structure: Einstein and Beyond ...

500 R. PenroseEinstein case) it seems to indicate some new directions of procedure whichcould open up promising lines of new development.AcknowledgmentsI am grateful to various people for illuminating conversations, most particularlyAndrew Hodges and Lionel Mason. I am grateful also to NSF forsupport under PHY00-90091.Notesa. See Penrose 1987, pp. 350, 359.b. See Einstein, Podolsky, and Rosen (1935); Bohm (1951) Ch. 22, §§15–19; Bell (1987); Baggott (2004).c. For results of this kind, see Tittel et al. (1998).d. See Károlyházy, F. (1966); Diósi (1989); Penrose (1996, 2000, 2004).e. This was around 1955, but only published later; see Penrose (1971,1975, 2004). It should be mentioned that a version of spin-networktheory is also used in the loop-variable approach to quantum gravity;see Ashtekar and Lewandowski (2004).f. However, John Moussouris (1983) has had some success in pursuingthis approach, in his (unpublished) Oxford D.Phil. thesis.g. This is a well-known correspondence; see, for example, Penrose (2004),§22.9, Fig. 22.10.h. See Terrell (1959); Penrose (1959, 2004 §18.5).i. Penrose (1976).j. See Penrose and Rindler (1986), Chapter 9; Penrose (2004) §§33.3, 5.k. Penrose and Rindler (1986) §§9.2,3; Penrose (2004).l. Penrose and Rindler (1984).m. Penrose and Rindler (1984), Chapter 2.n. Fierz and Pauli (1939); Fierz (1940); Penrose and Rindler (1984).o. See Penrose (1968, 1969); Hughston (1979); Penrose and Rindler(1986); versions of these expressions can be traced back to Whittaker(1903) and Bateman (1904, 1944).p. This type of non-singular field, termed an “elementary state”, being offinite norm and positive frequency, plays an important role in twistorscattering theory (see Hodges 1985, 1998). These fields appear to havebeen first studied by C. Lanczos.q. The more conventional term to use here, rather than “holomorphiccohomology” is “sheaf cohomology”, with a “coherent analytic sheaf”;see Gunning and Rossi (1965), Wells (1991).