It is shown that two quantum theories dealing, respectively, in the Hilbert spaces of state vectors ℌ1 and ℌ2 are physically equivalent whenever we have a faithful representation of the same abstract algebra of observables in both spaces, no matter whether the representations are unitarily equivalent or not. This allows a purely algebraic formulation of the theory. The framework of an algebraic version of quantum field theory is discussed and compared to the customary operator approach. It is pointed out that one reason (and possibly the only one) for the existence of unitarily inequivalent faithful, irreducible representations in quantum field theory is the (physically irrelevant) behavior of the states with respect to observations made infinitely far away. The separation between such ``global'' features and the local ones is studied. An application of this point of view to superselection rules shows that, for example, in electrodynamics the Hilbert space of states with charge zero carries already all the relevant physical information.

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);
R.
 
Haag
 and
B.
 
Schroer
,
J. Math. Phys.
 
3
,
248
(
1962
);
Google Scholar
Crossref
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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in die Axiomatische Quantenfeldtheorie,” Lecture notes at the Eidgenössischen Technischen Hochschule, Zürich, 1961/62, unpublished.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
);
Google Scholar
Crossref
Search ADS
 
D.
 
Ruelle
,
Helv. Phys. Acta
 
35
,
147
(
1962
);
Google Scholar
 
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
);
Google Scholar
 
K. O. Friedrichs, Mathematical Aspects of the Quantum Theory of Fields (Interscience Publishers, Inc., New York, 1953).
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
);
Google Scholar
Crossref
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A. S.
 
Wightman
 and
S. S.
 
Schweber
,
Phys. Rev.
 
98
,
812
(
1955
);
Google Scholar
Crossref
Search ADS
 
R.
 
Haag
,
Kgl. Danske Videnskab. Selskab Mat.‐Fiz. Medd.
 
29
, No.
12
(
1955
);
Google Scholar
 
I. E.
 
Segal
,
Trans. Am. Math. Soc.
 
88
,
12
(
1958
);
Google Scholar
Crossref
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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
).
Google Scholar
Crossref
Search ADS
 
The following papers discuss the existence and uniqueness of a vacuum state for specific models.
H.
 
Araki
,
J. Math. Phys.
 
1
,
492
(
1960
);
Google Scholar
Crossref
Search ADS
 
D. Shale, Ph.D. Thesis, Department of Mathematics, University of Chicago, 1961, unpublished;
I. E.
 
Segal
,
Illinois J. Math.
 
6
,
500
(
1962
);
Google Scholar
Crossref
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H. J.
Borchers
,
R.
Haag
, and
B.
Schroer
,
Nuovo Cimento
29
,
148
(
1963
).
Google Scholar
Crossref
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7.
J. M. G.
Fell
,
Trans. Am. Math. Soc.
94
,
365
(
1960
).
Google Scholar
 
8.
H. J. Borchers, R. Haag, and B. Schroer, see Ref. 6.
9.
I. E.
Segal
,
Ann. Math.
48
,
930
(
1947
).
Google Scholar
Crossref
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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 A(B) 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, A(B) containing it due to the uniqueness of the C*‐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);
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 |Φ(A)|/‖A|.
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,
|Φ(I)−1|<ε0
.
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 B1 and B2 is finite but only that it becomes complete in the limit of infinite spacelike separation.
30.
Let A1 and A2 be two subalgebras of 𝔄 and A1⊥, A2⊥ those subspaces of A* which are composed of the linear forms vanishing respectively on u1 and u2. If the “partial states” over A1 determine those over A2 we have
A2⊥⊃A1⊥
. Thus
A2⊥⊥⊂A1⊥⊥
. But Ai⊥⊥, considered as a subset of 𝔄 coincides with the uniform closure of Ai, i.e., with Ai 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 Bφ 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 Φ0 via the GNS construction. See Appendix I, Ref. 59. It is irreducible if Φ0 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 Φ0 is Lorentz‐invariant. Hence there exists a 1‐parameter group of unitary operators U(τ) 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
)
Google Scholar
 
and Proceedings of the Conference on Analysis in Function Spaces, Massachusetts Institute of Technology, Cambridge, Massachusetts, 1963.
42.
For general sources of information on C*‐algebras see M. A. Neumark, Normierte Algebren (VEB Deutscher Verlag der Wissenschaften, Berlin, 1959), Chap. V, Sec. 24;
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
).
Google Scholar
 
47.
Here we depart from Rickart’s terminology who calls B*‐algebra the abstract C*‐algebra and reserves the term C*‐algebra for a concrete norm‐closed algebra of operators on a Hilbert space (we speak in this case of a concrete C*‐algebra). Our C*‐algebras (Rickart’s B*‐algebras) are called by Neumark vollreguläre vollständige Algebren. Note that the condition ‖A*A‖ = ‖A‖2 is evidently fulfilled in an operator algebra and that the distinction between abstract and concrete C*‐algebras is important because different concrete C*‐algebras can define the same abstract C*‐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
).
Google Scholar
Crossref
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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 ξ∈H if the set of vectors R(A)ξ is dense in 𝔄.
59.
This construction is due to
I. M.
 
Gelfand
 and
M. A.
 
Neumark
,
Isvertija Ser. Mat.
 
12
,
445
(
1948
);
Google Scholar
 
and
I. E.
 
Segal
,
Bull. Am. Math. Soc.
 
53
,
73
(
1947
).
Google Scholar
Crossref
Search ADS
 
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
). ω(R)⊆ω(R′) means that R is unitarily equivalent to some subrepresentation of R′.
Google Scholar
Crossref
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61.
For this characterization of quasi‐equivalence, see
Z.
Takeda
,
Tôhoku Mat. J.
6
,
212
(
1954
).
Google Scholar
 
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 E⊕Ē with respect to the bilinear scalar product
g(x⊕ȳ,x′⊕ȳ′) = {12(x,y′)+(y,x)}xεE,yεE
. Here Ē is the “complex conjugate” of 𝔈 i.e., it is isomorphic to 𝔈 as an additive group:
xεE⇊x̄εĒ
, but its scalar multiplication reverses the sign of i:
((α+iβ)x)¯ = (α−iβ)x̄
. Since (x,y) is Hermitian symmetric the form g is bilinear. ℭ can be defined as the quotient of the tensor algebra over E⊕Ē by an ideal ℑ which is generated by the tensors ξ×ξ−g(ξ,ξ) with ξεE⊕I. One has a *(x) = x⊕0 mod I and a(y) = 0⊕ȳ mod I. By extension of the adjoint operation
x⊕ȳ⇊y⊕x̄
one defines on ℭ the structure of a *‐algebra. Equation (23) defines a faithful realization of ℭ by the linear operators on the space H(E) 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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