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1.Throughout this paper, “Lorentz group” always refers to to the six‐parameter homogeneous group and “Poincaré group” to the ten‐parameter inhomogeneous group. “Conformal group” refers to the fifteen‐parameter group of transformations which preserve the local conformal structure of Minkowski space‐time. If in any particular context it is important to exclude the space reversing and time reversing transformations, then this is made explicit, for example, by the use of the term “restricted”.
2.The use of homogeneous coordinates in space‐time would afford a simple example of such a formalism (but apparently one of limited physical interest). More significant, of course, is the representation of physical quantities in terms of Hilbert space, which has the advantage that the (infinite demensional) analogs of are unitary [cf., particularly E. P. Wigner, Ann. Math. 40, 149 (1939)
2.V. Bargmann and E. P. Wigner, Proc. Natl. Acad. Sci. U.S. 34, 211 (1948)]. However, it will be essential here to preserve the finite dimensionality of the operations at this stage, so that geometrical questions can be kept in the forefront.
3.B. L. van der Waerden, Nachr. Ges. Wiss. Gottingen 100, 1 (1929).
4.F. Klein, Gesammelte Mathem. Abhandlungen (J. Springer, Berlin, 1921);
4.cf. also H. Weyl, The Classical Groups (Princeton University Press, Princeton, New Jersey, 1939);
4.R. Brauer and H. Weyl, Am. J. Math. 57, 425 (1935).
5.Quantities which are essentially twistors have been described by W. A. Hepner, Nuovo Cimento 26, 351 (1962).
5.See also Y. Murai, Nucl. Phys. 6, 489 (1958);
5.Y. Murai, Progr. Theoret. Phys. (Kyoto) 9, 147 (1953);
5.Y. Murai, 11, 441 (1954). , Prog. Theor. Phys.
5.Similar quantities have also recently gained prominence in, for example, the work of A. Salam, R. Delbourgo, and J. Strathdee, Proc. Roy. Soc. (London) A284, 147 (1965).
6.E. Cunningham, Proc. London Math. Soc. 8, 77 (1910);
6.H. Bateman, Proc. London Math. Soc. 8, 223 (1910); , Proc. London Math. Soc.
6.P. A. M. Dirac, Ann. Math. 37, 429 (1936);
6.J. A. McLennan, Jr., Nuovo Cimento 10, 1360 (1956);
6.H. A. Buchdahl, Nuovo Cimento 11, 496 (1959)., Nuovo Cimento
7.R. Penrose, Proc. Roy. Soc. (London) A284, 159 (1965).
8.H. A. Kastrupp, Phys. Letters (Amsterdam) 3, 78 (1962);
8.H. A. Kastrupp, Phys. Rev. 142, 1060 (1966).
8.A recent additional suggestion is that the mass splittings of strong interaction physics may be derivable from conformal group symmetry: D. Bohm, M. Flato, D. Sternheimer, and J. P. Vigier, Nuovo Cimento 38, 1941 (1965).
8.For a discussion of the relevance of the conformal group in physics, see T. Fulton, F. Rohrlich, and L. Witten, Rev. Mod. Phys. 34, 442 (1962).
9.L. Infeld and B. L. van der Waerden, S. B. Preuss. Akad. 9, 380 (1933).
10.See any standard work on quantum field theory. The essential matters referred to here can be found in R. F. Streater and A. S. Wightman, PCT, Spin and Statistics, and All That (W. A. Benjamin, Inc., New York, 1964).
11.I. Robinson, J. Math. Phys. 2, 290 (1961).
12.World vectors and tensors are labeled by lower case Latin indices running over 0,1,2,3, the Minkowski metric being given by Capital Latin index letters, primed or unprimed, denote spinor indices and run over 0,1 or Greek indices are twistor indices and run over 0,1,2,3. The summation convention applies to each of these four types of index separately. Thus, in particular, no summation takes place between primed and unprimed spinor indices even when the same letter is used, e.g., J and are regarded as distinct letters in This allows us to write the tensor‐spinor correspondence in a definite way by simply using the two capital versions of a tensor index as its spinor translation: etc. (Rindler’s convention).
13.For more complete accounts see Refs. 3 and 9 and, for example, W. L. Bade and H. Jehle, Rev. Mod. Phys. 25, 714 (1953);
13.E, M. Corson, An Introduction to Tensors, Spinors and Relativistic Wave Equations (Blackie & Son Ltd., London, 1953);
13.F. A. E. Pirani, in Brandeis Summer Institute in Theoretical Physics, 1964, Lectures on General Relativity (Prentice‐Hall Inc., Englewood Cliffs, New Jersey, 1965), Vol. 1.
14.E. T. Whittaker, Proc. Roy. Soc. (London) A158, 38 (1937);
14.W. T. Payne, Am. J. Phys. 20, 253 (1952);
14.R. Penrose, “Null Hypersurface Initial Data”, in P. G. Bergmann’s Aeronautical Research Lab. Tech. Documentary Rept. 63‐56 (Office of Aerospace Research, U.S. Air Force, 1963).
15.To give a rigorous definition of a spinor which takes into account its sign, it is usual to appeal to the theory of fibre bundles. This is not essential, however, and an elementary (nonlocal) geometrical description will be given in an appendix to a forthcoming book by R. Penrose and W. Rindler on the applications of spinors in relativity.
16.Any temptation to identify the twistor (2.13) with a Dirac spinor should be rejected here, since their transformation properties are quite different [cf., for example, (7.13), (7.17)].
17.The operation of twistor (or spinor) complex conjugation is denoted by a bar extending only over the base symbol involved and not over the indices. If the bar extends also over the indices, this denotes simply the complex conjugate of the complex number that the symbol represents.
18.N. H. Kuiper, Ann. Math. 50, 916 (1949);
18.H. Rudberg, dissertation, University of Uppsala, Uppsala, Sweden (1958);
18.R. Penrose, in Proceedings of the 1962 Conference on Relativistic Theories of Gravitation, Warsaw (Polish Academy of Science, Warsaw, 1965)
18.A. Uhlmann, Acta Phys. Polon. 24, 293 (1963). Rudberg also mentions the four‐valuedness of spinors.
19.The term “congruence” is used to denote a system of curves (or surfaces, etc.) for which there is just one member of the system (or at most a discrete number) through a general point in the space.
20.First investigated by I. Robinson (private communication).
21.The screw sense arising here depends, of course, on the “handedness” of the choice of matrices in (2.4).
22.I am grateful to J. Terryl and J. E. Reeve for this observation concerning the Hopf fibring.
23.See Ref. 7, Eqs. (10.1), (10.7), and (10.8) for the relevant formulas.
24.P. Jordan, J. Ehlers, and R. Sachs, Akad. Wiss. Lit. Mainz, no. 1 (Mainz 2) (1961);
24.E. T. Newman and R. Penrose, J. Math. Phys. 3, 566 (1962).
25.A symbol indicates r dimensionality in the real sense.
26.See any standard work on classical algebraic geometry, for example, J. A. Todd, Projective and Analytical Geometry (I. Pitman, London, 1947);
26.J. G. Semple and L. Roth, Algebraic Geometry (Clarendon Press, Oxford, England, 1949).
27.Strictly, we should also show that the concept of a Robinson congruence is “geometrical” in M. An explicit construction in terms of incidence of null lines is given in Sec.IX, but in any case, the geometric (and conformally invariant) nature of a Robinson congruence is already implied by Sec.IV and V.
28.In fact, the transformations of C would be the same if we specified only that be proportional to the inverse of since the are projective coordinates for C. However, the stronger requirement (7.2) is adopted here since the factor of proportionality of a twistor is required when the more complete representation in accordance with Sec. V is used.
29.The representation of twistors in terms of components given in this paper is perhaps the most convenient; but is by no means the only one possible. It amounts to insisting that the coordinate basis twistors are null and satisfy with all the other scalar products vanishing. Another possible choice would be to require (b not summed!) in which case twistor complex conjugation would take the form instead of (3.5), and the fixed matrix in (7.5) would become diag (1, 1, ). (The simple connection with spinors (2.13) would be lost, however.) The only essential restriction on the way in which the twistors are represented is that the signature of the form X must be (+ + − −) [cf., (6.1)].
30.The staggering of the spinor indices is to indicate that, in each case, the left‐hand index labels rows and right‐hand index labels columns. Also, for notational consistency, is used here instead of and instead of although they are all numerically equall.
31.I. Robinson and A. Trautman, Proc. Roy. Soc. (London) A265, 463 (1962).
32.R. P. Kerr, private communication;
32.cf. also R. P. Kerr, Phys. Rev. Letters 11, 238 (1963);
32.R. P. Kerr and A. Schild, in Proceedings of the American Mathematical Society Symposium, April 1964. According to Kerr’s original construction, a general shear‐free null geodetic congruence is defined by an analytic relation where
33.The rotation‐free, shear‐free null congruences which are not analytic in the real sense emerge here simply as the system of null lines meeting a nonanalytic curve in M.
34.Explicit shear‐free null congruences of this type have been used to generate explicit solutions of Einstein’s equations. For example, in Kerr’s construction of the field of a rotating body, is quadratic. For details, see Ref. 32.
35.When is timelike we may represent P as a parallelism on M (with torsion; left‐handed if ) which is closely related to Clifford parallelism on S8. The sets of null directions which are to be regarded as parallel are those of the Robinson congruences represented by the points of the line P in C (i.e., by planes through P̄). A transitive four‐parameter group of (conformal) motions of M preserves this parallelism, namely that given by twistor transformations (7.1) for which the line P is left pointwise invariant. This group is readily seen to be the group of unitary matices and leads to Uhlmann’s representation (see Ref. 18) of the points of M in terms of such matrices.
36.Since a real null line in M defines both a point in C and a plane in C through this point, the configuration of Fig. 6 is represented in the C picture as a pair of mutually inscribed and circumscribed tetrahedra—a configuration familiar to geometers. We may note that the full complexification of a null line in M leads, in the C picture, strictly to a point in C together with a plane through it. This gives a five‐complex‐dimensional system as we would expect.
37.It only fails if there is a circle through O meeting U, V, and W which lies on a null cone through U.
38.In practice, the and of the decomposition (9.3) (which may be specilized if desired) often turn out to be more convenient coordinates than This is of value in connection with physical fields and will be discussed elsewhere.
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