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Breaking of Euclidean Symmetry with an Application to the Theory of Crystallization
1.D. Kastler and D. W. Robinson, Commun. Math. Phys. 3, 151 (1966).
2.D. W. Robinson and D. Ruelle, Ann. Inst. Henri Poincaré 4, 299 (1967).
3.D. Kastler, R. Haag, and L. Michel, Seminar talk, Marseille, 1967.
4.G. G. Emch, H. J. F. Knops, and E. J. Verboven, J. Phys. Soc. Japan 26, Suppl., 301 (1969).
5.H. J. F. Knops, Ph.D. thesis, Nijmegen, 1969.
6.H. J. F. Knops and E. J. Verboven, Phys. Letters 29A, 386 (1969).
7.R. Kubo, J. Phys. Soc. Japan 12, 570 (1957).
8.P. C. Martin and J. Schwinger, Phys. Rev. 115, 1342 (1959).
9.R. Haag, N. M. Hugenholtz, and M. Winnink, Commun. Math. Phys. 5, 215 (1967).
10.N. M. Hugenholtz, Commun. Math. Phys. 6, 189 (1967).
11.D. W. Robinson, Commun. Math. Phys. 7, 337 (1968).
12.M. Winnink, thesis, Groningen, 1968.
13.H. Araki, Preprint RIMS 38, Kyoto University, 1968.
14.H. Araki and H. Miyata, Preprint RIMS 37, Kyoto University, 1968.
15.D. Kastler, J. C. T. Pool, and E. T. Poulsen, Commun. Math. Phys. 12, 175 (1969).
16.H. Araki, Preprint IHES, 1969.
17.A. Z. Jadzyk, Commun. Math. Phys. 13, 142 (1969).
18.N. M. Hugenholtz and J. D. Wieringa, Commun. Math. Phys. 11, 183 (1969).
19.M. Takesaki, Preprint, UCLA and Tohoku University, 1969 (?).
20.G. F. dell’Antonio, S. Doplicher, and D. Ruelle, Commun. Math. Phys. 2, 223 (1966).
21.G. F. dell’Antonio and S. Doplicher, J. Math. Phys. 8, 663 (1967).
22.Some important facts which we now understand as consequences of the KMS condition were already noticed by H. Araki and E. J. Woods, J. Math. Phys. 4, 637 (1963).
23.C. N. Yang and T. D. Lee, Phys. Rev. 87, 404 (1952).
24.We recall that this is the topology associated to the point‐wise convergence of the linear functionals on A; see, for instance, Dunford and Schwartz, Linear Operators (Interscience, New York, 1957), Part I.
25.See the remark following Proposition I.6.3 in J. Dixmier, Les algèbres d’opérateurs dans l’espace Hilbertien (Gauthier‐Villars, Paris, 1967).
26.J. Dixmier, Ref. 25, Theorem I.6.3.
27.J. Dixmier, Ref. 25, Corollary to Proposition I.2.2.
28.H. Araki, Ref. 13, Corollary 2.5.
29.H. Araki, Ref. 16, Theorem 10.3, for instance.
30.I.e., with denoting an invariant mean on in A and invariant states ψ on A, we actually only need here the following consequence of this assumption: for the state φ considered.
31.Throughout this paper we shall assume locality; i.e., for all local observables A and B which are spatially disjoint. As a consequence of this assumption, we have as this in turn implies all other asymptotic conditions in space which we shall use.
32.D. Ruelle, Statistical Mechanics (Benjamin, New York, 1969).
33.A previous investigation has been carried out by one of us (Ref. 5) along the lines of the present paper with, however, this condition assumed throughout.
34.S. Sakai, Trans. Am. Math. Soc. 118, 406 (1965).
35.W. Wils, Compt. Rend. 267, 810 (1968).
36.For the definition of Baire sets and their relation with Borel sets, see for instance P. R. Halmos, Measure Theory (Van Nostrand, New York, 1950), Sec. 51.
37.Defined by for all A in A.
38.We recall that a “diagonal operator” on is a mapping from into itself of the form where with denoting the identity on and λ belonging to
39.See, for instance, J. Dixmier, Les ‐algèbres et leurs représentations (Gauthier‐Villars, Paris, 1964), Sec. 8.
40.For instance, the ‐algebra of the canonical commutation relations is not separable; see 3.3.3 in J. Manuceau, Ann. Inst. Henri Poincaré, 139 (1968).
41.When A is separable, G becomes metrizable and the concepts of Baire and Borel sets coincide so that Wils’ theory (Ref. 35) contains Dixmier’s (Ref. 39) as a particular case.
42.For general reference on this theory, see R. R. Phelps, Lectures on Choquet’s theorem, Van Nostrand Mathematical Series ♯7 (Van Nostrand, Princeton, N.J. 1966).
43.There are at least five equivalent characterizations of simplices (Choquet’s theorem; see, for instance, Ref. 42, p. 66); the one taken here is the most appropriate to out purpose.
44.See the proof of Lemma 9.6 of Ref. 42.
45.Theorem I.4.1 in Dixmier, Ref. 25.
46.Example I.3.6c in Dixmier, Ref. 25.
47.C. E. Rickart, General theory of Banach Algebras (Van Nostrand, Princeton, 1960), Appendix, Sec. 1.
48.O. Lanford and D. Ruelle, J. Math. Phys. 8, 1460 (1967), Proposition 4.2; for a proof see D. Ruelle, Ref. 49.
49.D. Ruelle, Comm. Math. Phys. 3, 133 (1966); see, in particular, pp. 146–7.
50.See, for instance, p. 30 in R. R. Phelps (Ref. 42).
51.We recall that is already the unique measure satisfying the axioms of a central measure. The assertion of the present theorem is clearly of a different type. Furthermore, [see Theorem II.3.1 in Knops (Ref. 5)] an even stronger kind of uniqueness occurs, namely, that is the only (regular) measure on concentrated in the Baire sense on whenever η Abelianess in time holds.
52.Proposition 4.3(ii) in Lanford and Ruelle (Ref. 48).
53.See Lemma 3 in Kastler (Ref. 3).
54.See Corollary to Lemma 2 in Kastler, Ref. 3.
55.For details, see Knops, Ref. 5, p. 81.
56.The interchange of the mean and integral can be justified as follows. We first notice that the integrand is a function of positive type in g. The mean can hence be written as as where denotes an integral over a compact subset of The argument is then completed by a successive application of Fubini’s and Lebesgue’s theorems.
57.(iii) implies in particular that and are Borel isomorphic. In the separable case, this is always the case, irrespective of transitivity; see a remark in the beginning of Sec. 3 in Kastler.
58.See, in this spirit, G. G. Emch, H. J. F. Knops, and E. J. Verboven, Commun. Math. Phys. 8, 300 (1968).
59.We recall that
60.See Example 53.7 in Halmos, Ref. 36.
61.Corollary 9.8 in Phelps, Ref. 42.
62.Corollary 3.2 in Lanford and Ruelle, Ref. 48.
63.See pp. 104–5 in Knops, Ref. 5.
64.Prop. 5.4.11 in Dixmier, Ref. 25.
65.See Lemma 10.2 in Araki, Ref. 16 or Lemma 2.2.2 in Knops, Ref. 5.
66.See Lemma 2.2.1 in Knops, Ref. 5.
67.P. R. Halmos, Lectures on Ergodic Theory (Chelsea, New York, 1956), p. 40.
68.See Theorem 3 in S. Doplicher, and D. Kastler, Commun. Math. Phys. 7, 1 (1968).
69.G. E. Uhlenbeck, in Fundamental Problems in Statistical Mechanics II, E. G. D. Cohen, Ed. (North‐Holland, Amsterdam, 1968); see, in particular, p. 17.
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