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An Algebraic Approach to Quantum Field Theory
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1.At the present stage this claim is an overstatement, but it is a reasonable extrapolation of results described in Ref. 3.
2.R. Haag, Colloque Internationale sur les Problèmes Mathématiques de la Théorie Quantigue des Champs, Lille, 1957 (Centre National de la Recherche Scientifique, Paris, 1958);
2.R. Haag and B. Schroer, J. Math. Phys. 3, 248 (1962);
2.H. Araki, “Einführung in die Axiomatische Quantenfeldtheorie,” Lecture notes at the Eidgenössischen Technischen Hochschule, Zürich, 1961/62, unpublished.
3.R. Haag, Phys. Rev. 112, 669 (1958);
3.D. Ruelle, Helv. Phys. Acta 35, 147 (1962);
3.H. Araki, see Ref. 2.
4.In a heuristic manner the commutation relations and field equations of a conventional quantum field theory provide such an abstract characterization.
5.It was first noticed in the example of various algebras associated with infinitely many creation and destruction operators. See J. von Neumann, Comp. Math. 6, 1 (1938);
5.K. O. Friedrichs, Mathematical Aspects of the Quantum Theory of Fields (Interscience Publishers, Inc., New York, 1953).
5.For further discussions of this phenomenon in its relation to various models in quantum field theory see, for instance, L. Van Hove, Physica 18, 145 (1952);
5.A. S. Wightman and S. S. Schweber, Phys. Rev. 98, 812 (1955);
5.R. Haag, Kgl. Danske Videnskab. Selskab Mat.‐Fiz. Medd. 29, No. 12 (1955);
5.I. E. Segal, Trans. Am. Math. Soc. 88, 12 (1958);
5.J. Lew, Ph.D. thesis, Princeton Univ., 1960, unpublished; and the papers cited in Ref. 6.
6.In Wightman’s approach the existence of a vacuum state and the relevant properties of this state are postulated on physical grounds. See, e.g., A. S. Wightman, Phys. Rev. 101, 860 (1956).
6.The following papers discuss the existence and uniqueness of a vacuum state for specific models. H. Araki, J. Math. Phys. 1, 492 (1960);
6.D. Shale, Ph.D. Thesis, Department of Mathematics, University of Chicago, 1961, unpublished;
6.I. E. Segal, Illinois J. Math. 6, 500 (1962);
6.H. J. Borchers, R. Haag, and B. Schroer, Nuovo Cimento 29, 148 (1963).
7.J. M. G. Fell, Trans. Am. Math. Soc. 94, 365 (1960).
8.H. J. Borchers, R. Haag, and B. Schroer, see Ref. 6.
9.I. E. Segal, Ann. Math. 48, 930 (1947).
10.For definitions and relevant theorems see Appendix 1.
11.I. E. Segal, Collogue Internationale sur les Problèmes de la Théorie Quantique des Champs, Lille, 1957 (Centre National de la Recherche Scientifique, Paris, 1958).
12.Physically speaking: 4‐dimensional regions with finite extension.
13.The union of all has an obvious *‐algebra structure due to the isotony assumption. Furthermore, the norm of one of its elements is the same in all local algebras, containing it due to the uniqueness of the ‐norm (see Appendix 1).
14.A is the collection of the uniform limits of all (bounded) observables describing measurements performable in finite regions of space‐time. By taking uniform limits we do not essentially change the local character of the observables (hence the name quasilocal).
15.We adopt Segal’s terminology in which the word state is used for any statistical ensemble. If the ensemble cannot be decomposed into purer ones it is called a “pure state,” otherwise an “impure state” (“mixture” in von Neumann’s terminology).
16.We find it preferable to base our discussion on the notion of “operations” as defined above instead of “observables” as used by Dirac and von Neumann. An “observable” in the technical sense is an idealization, which in general implies suitably defined limits of an infinite number of operations. It is thus a far less simple concept.
17.We hope to discuss this question in another paper.
18.J. von Neumann, Mathematical Foundations of Quantum Mechanics (Princeton University Press, Princeton, 1955);
18.I. E. Segal, see Ref. 9.
19.It is not crucial to assume that the algebra contains the unit element; see Appendix 1. The norm is then defined as sup
20.It has been emphasized by H. Ekstein that, in general, one algebraic element will correspond to many different laboratory procedures which are equivalent insofar as they produce the same transformation of the states. For simplicity we shall, however, always speak of an “operation” instead of an “equivalence class of operations.”
21.J. von Neumann, see Ref. 18.
22.I. E. Segal, see Ref. 9.
23.This will be discussed in a separate paper. See also Ref. 3.
24.We chose (arbitrarily) the normalization of the state. For the sake of symmetry with Eq. (7) we could write, instead of (6), equally well, .
25.See Ref. 7 and the last paragraph of Appendix I.
26.The kernel of a representation is the collection of all elements of A which are represented by zero.
27.Compare B. Misra, “On the algebra of quasi‐local operators of Quantum Field Theory,” to be published. See, however, Appendix II for an example of a nonsimple algebra of physical interest.
28.See, e.g., M. A. Neumark, Ref. 42.
29.From the physical point of view it would not be necessary that the uncoupling is complete if the separation distance between and is finite but only that it becomes complete in the limit of infinite spacelike separation.
30.Let and be two subalgebras of A and those subspaces of which are composed of the linear forms vanishing respectively on and If the “partial states” over determine those over we have . Thus . But considered as a subset of A coincides with the uniform closure of i.e., with itself.
31.Compare Appendix I, Kadison’s Theorem, Ref. 53.
32.“Far away” means the limit of an infinite spacelike separation.
33.An automorphism of the type (12) is called an inner automorphism of the algebra. Our argument here shows that the Lorentz transformations are outer automorphisms.
34.The symbol is used here to denote the partial state in B resulting from the restriction of φ.
35.For the sake of simplicity we pretend that the electric charge is the only superselected quantity and thus use the word “charge” in lieu of “superselected quantities.”
36.The theorem tells us that one sector is equally as faithful as the collection of all sectors taken together.
37.In the sense of Dixmier: Les Alghèbres d’operateurs dans l’espace Hilbertien (Gauthier‐Villars, Paris, 1952), Chap. I, Sec. 2.2.
38.The representation is determined by via the GNS construction. See Appendix I, Ref. 59. It is irreducible if is pure. The separability would follow from the irreducibility if the algebra were separable in the norm topology.
39.It follows from (α) that the representation obtained by the GNS construction from is Lorentz‐invariant. Hence there exists a 1‐parameter group of unitary operators representing the time translations in H. The Hamiltonian is the infinitesimal generator of this group.
40.This question was studied in Ref. 8 but the argument given there is inconclusive in some respects.
41.For some conjectures in this direction see R. Haag, Ann. Physik 11, 29 (1963)
41.and Proceedings of the Conference on Analysis in Function Spaces, Massachusetts Institute of Technology, Cambridge, Massachusetts, 1963.
42.For general sources of information on ‐algebras see M. A. Neumark, Normierte Algebren (VEB Deutscher Verlag der Wissenschaften, Berlin, 1959), Chap. V, Sec. 24;
42.and C. E. Rickart, General Theory of Banach Algebras (D. Van Nostrand, Inc., New York, 1960), Chap. IV, Sec. 8. In general we use Rickart’s terminology.
43.Called symmetrische Algebra by Neumark.
44.In Rickart’s terminology. Neumark’s is: normierte symmetrische Algebra.
45.In Rickart’s terminology. Neumark’s is: vollständige normierte symmetrische Algebra.
46.I. M. Gelfand and M. A. Neumark, Mat. Sb. 12, 197 (1943).
47.Here we depart from Rickart’s terminology who calls algebra the abstract ‐algebra and reserves the term algebra for a concrete norm‐closed algebra of operators on a Hilbert space (we speak in this case of a concrete algebra). Our ‐algebras (Rickart’s ‐algebras) are called by Neumark vollreguläre vollständige Algebren. Note that the condition is evidently fulfilled in an operator algebra and that the distinction between abstract and concrete algebras is important because different concrete ‐algebras can define the same abstract algebra.
48.We always use the word irreducible to mean topologically irreducible.
49.Whether or not A contains an identity. See Rickart (Ref. 42), Theorem (4.1.20).
50.Neumark uses reduziert for *‐semisimple and reduzierendes Ideal for *‐radical (in the case of Banach *‐algebras).
51.M. A. Neumark, Ref. 42, Chap. V, Sec. 24, Theorems 6 and 3.
52.M. A. Neumark, Ref. 42, Chap. V, Sec. 24, Theorem 4.
53.R. V. Kadison, Proc. Natl. Acad. Sci. U.S. 43, 273 (1957).
54.See G. W. Mackey, “The Theory of Group Representations,” University of Chicago, mimeographed lecture notes.
55.See Mackey, Ref. 54.
56.For the notion of quasi‐equivalence, see Mackey, Ref. 54, Chap. I.
57.See C. E. Rickart, Ref. 42, Theorems (4.5.14), (4.5.11), and (4.8.14).
58.A representation R in the space H is called cyclic with cyclic vector if the set of vectors is dense in A.
59.This construction is due to I. M. Gelfand and M. A. Neumark, Isvertija Ser. Mat. 12, 445 (1948);
59.and I. E. Segal, Bull. Am. Math. Soc. 53, 73 (1947).
59.We call it the GNS construction. See M. A. Neumark, Ref. 42, Chap. IV, Sec. 17.3 or, for the case of an algebra without unit, C. E. Rickart, Ref. 42, Chap. IV, Sec. 5.
60.These results can be inferred from R. V. Kadison, Trans. Am. Math. Soc. 103, 304 (1962). means that R is unitarily equivalent to some subrepresentation of
61.For this characterization of quasi‐equivalence, see Z. Takeda, Tôhoku Mat. J. 6, 212 (1954).
62.We are indebted to Professor H. Araki for pointing out to us the main facts described in this appendix.
63.E is the Clifford algebra over the space with respect to the bilinear scalar product . Here is the “complex conjugate” of E i.e., it is isomorphic to E as an additive group: , but its scalar multiplication reverses the sign of i: . Since is Hermitian symmetric the form g is bilinear. C can be defined as the quotient of the tensor algebra over by an ideal I which is generated by the tensors with One has a and By extension of the adjoint operation one defines on C the structure of a *‐algebra. Equation (23) defines a faithful realization of C by the linear operators on the space which latter coincides with the Grassmann algebra over E.
64.Since it is known to have many inequivalent irreducible representations it is an NGCR‐algebra.

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Scitation: An Algebraic Approach to Quantum Field Theory