Dirac quantum fields on a manifold

Dimock, J.

doi:10.1090/S0002-9947-1982-0637032-8

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ISSN 1088-6850(online) ISSN 0002-9947(print)

Dirac quantum fields on a manifold

Author: J. Dimock
Journal: Trans. Amer. Math. Soc. 269 (1982), 133-147
MSC: Primary 81E20; Secondary 46L60, 81E05
MathSciNet review: 637032
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Abstract: On globally hyperbolic Lorentzian manifolds we construct field operators which satisfy the Dirac equation and have a causal anticommutator. Ambiguities in the construction are removed by formulating the theory in terms of ${C^{\ast}}$ algebras of local observables. A generalized form of the Haag-Kastler axioms is verified.

[1] S. Avis and C. Isham, Quantum field theory and fibre bundles in a general space-time, Recent Developments in Gravitation (Levy and Deser, eds.), Plenum Press, New York, 1979.
[2] P. J. M. Bongaarts, The electron-positron field, coupled to external electromagnetic potentials, as an elementary 𝐶* algebra theory, Ann. Physics 56 (1970), 108–139. MR 0260291 (41 #4919)
[3] Y. Choquet-Bruhat, Hyperbolic partial differential equations on a manifold, Battelle Rencontres. 1967 Lectures in Mathematics and Physics, Benjamin, New York, 1968, pp. 84–106. MR 0239299 (39 #656)
[4] Yvonne Choquet-Bruhat, Cecile DeWitt-Morette, and Margaret Dillard-Bleick, Analysis, manifolds and physics, North-Holland Publishing Co., Amsterdam, 1977. MR 0467779 (57 #7631)
[5] Bryce S. DeWitt, Charles F. Hart, and Christopher J. Isham, Topology and quantum field theory, Phys. A 96 (1979), no. 1-2, 197–211. MR 534588 (80e:81056), http://dx.doi.org/10.1016/0378-4371(79)90207-3
[6] J. Dimock, Algebras of local observables on a manifold, Comm. Math. Phys. 77 (1980), no. 3, 219–228. MR 594301 (82i:81071)
[7] Sergio Doplicher, Rudolf Haag, and John E. Roberts, Fields, observables and gauge transformations. I, Comm. Math. Phys. 13 (1969), 1–23. MR 0258394 (41 #3041)
[8] Robert Geroch, Spinor structure of space-times in general relativity. I, J. Mathematical Phys. 9 (1968), 1739–1744. MR 0234703 (38 #3019)
[9] Robert Geroch, Domain of dependence, J. Mathematical Phys. 11 (1970), 437–449. MR 0270697 (42 #5585)
[10] Rudolf Haag and Daniel Kastler, An algebraic approach to quantum field theory, J. Mathematical Phys. 5 (1964), 848–861. MR 0165864 (29 #3144)
[11] S. W. Hawking and G. F. R. Ellis, The large scale structure of space-time, Cambridge University Press, London, 1973. Cambridge Monographs on Mathematical Physics, No. 1. MR 0424186 (54 #12154)
[12] Peter Hajicek, Observables for quantum fields on curved background, Differential geometrical methods in mathematical physics, II (Proc. Conf., Univ. Bonn, Bonn, 1977), Lecture Notes in Math., vol. 676, Springer, Berlin, 1978, pp. 535–566. MR 519628 (81j:81079)
[13] C. J. Isham, Quantum field theory in curved space-times, a general mathematical framework, Differential geometrical methods in mathematical physics, II (Proc. Conf., Univ. Bonn, Bonn, 1977), Lecture Notes in Math., vol. 676, Springer, Berlin, 1978, pp. 459–512. MR 519626 (81d:81037)
[14] C. J. Isham, Twisted quantum fields in a curved space-time, Proc. Roy. Soc. London Ser. A 362 (1978), no. 1710, 383–404. MR 0503525 (58 #20257)
[15] B. Kay, Linear spin-zero quantum fields in external gravitational and scalar fields. I, II, Comm. Math. Phys. 62 (1978), 55; 71 (1980), 29.
[16] André Lichnerowicz, Champs spinoriels et propagateurs en relativité générale, Bull. Soc. Math. France 92 (1964), 11–100 (French). MR 0169667 (29 #6913)
[17] -, Topics on space-times, Battelle Recontres (DeWitt and Wheeler, eds.), Benjamin, New York, 1968.
[18] Jean Leray, Hyperbolic differential equations, The Institute for Advanced Study, Princeton, N. J., 1953. MR 0063548 (16,139a)
[19] W. Pauli, Contributions mathématiques à la théorie des matrices de Dirac, Ann. Inst. H. Poincaré VI (1963), 8.
[20] Herbert-Rainer Petry, Exotic spinors in superconductivity, J. Math. Phys. 20 (1979), no. 2, 231–240. MR 519205 (80c:55011), http://dx.doi.org/10.1063/1.524069
[21] I. E. Segal, Foundations of the theory of dynamical systems of infinitely many degrees of freedom. I, Mat.-Fys. Medd. Danske Vid. Selsk. 31 (1959), no. 12, 39 pp. (1959). MR 0112626 (22 #3477)
[22] Joseph Slawny, On factor representations and the 𝐶*-algebra of canonical commutation relations, Comm. Math. Phys. 24 (1972), 151–170. MR 0293942 (45 #3017)
[23] A. Wightman, The Dirac equation, Aspects of Quantum Theory (Salam and Wigner, eds.), Cambridge Univ. Press, Cambridge, 1972, pp. 109.
[24] A. S. Wightman, Relativistic wave equations as singular hyperbolic systems, Partial differential equations (Proc. Sympos. Pure Math., Vol. XXIII, Univ. California, Berkeley, Calif., 1971), Amer. Math. Soc., Providence, R.I., 1973, pp. 441–447. MR 0342075 (49 #6821)

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