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functions f (x) (think of the combination ( p 2 + q 2 )/2 in the example of one-dimensional motion) lose their status of observable in the previous sense: the spectrum is not the same as the set of all the values assumed by the function It is then necessary to build a formalism able to account for this new evidence: the customary way is to view f (x) as a nondiagonal matrix, a more general operator, acting on some linear space, with spectrum the set of observed values in conformity with E The question becomes nding the right mathematical framework in which f (x) is realized as an operator Clearly, the adoption of the new protocol E , possibly together with the new measure space (X , E ), results in the emergence of a generically noncommutative algebra of operators We have observed that this noncommutativity occurs along with a reduction of the Hilbert space L2 (X , E ) to a closed subspace, say, K This reduction should be thought of as the action of a projector PK on that Hilbert space: K = PK L2 (X , E ) (182)
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Note that the restriction of PK to the subspace K is the identity operator I K on the latter It is precisely at this point that the existence of a family of normalized vectors |x resolving, as elements of L2 (X , E ), the projector PK , or, as elements of K, resolving the unity I K , |x x| E (dx) = PK or I K ,
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(183)
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allows us to implement the construction of the operator corresponding to an observable f Indeed, such a function, which was viewed as a multiplication operator M f according to the former protocol E, becomes, under the projection PK , the operator A f = P K M f PK =
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f (x)|x x| E (dx) ,
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(184)
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the quantized version of f under the quantization provided by the family {|x , x X } Actually, we have followed a mathematical procedure known in speci c situations as the Toeplitz quantization of the set X Of course, we could be faced with ambiguities or con icts of the type: within the same protocol, two different observables could lead to different projections, and so to incompatible physics or to different interpretations of the original set X A change of protocol is then necessary These possibilities raise deep questions, beyond the scope of this book
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Appendix A The Basic Formalism of Probability Theory
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Many excellent textbooks exist on the subject I recommend one of them, which is based on a course by Sinai [220]
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A1 Sigma-Algebra
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Let X be a set A family F of subsets of X is a -algebra if and only if it has the following properties: (i) The empty set is in F (ii) If A is in F , then so is the complement of A (iii) If A1 , A2 , A3 , is a sequence in F , then their (countable) union is also in F From (i) and (ii) it follows that X is in F ; from (ii) and (iii) it follows that the -algebra is also closed under countable intersections -algebras are mainly used to de ne measures on X, as are de ned in the next section An ordered pair (X , F ), where X is a set and F is a -algebra over X, is called a measurable space
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1 The family consisting only of the empty set and X is a -algebra over X, the so-called trivial -algebra Another -algebra over X is given by the power set of X, that is, the set P(X ) of all subsets of X 2 If {Fa } is a family of -algebras over X, then the intersection of all Fa is also a -algebra over X 3 If U is an arbitrary family of subsets of X, then we can form a special -algebra from U , called the -algebra generated by U We denote it by (U ) and de ne it as follows First note that there is a -algebra over X that contains U , namely, the power set of X Let be the family of all -algebras over X that contain U (ie, a -algebra F over X is in if and only if U is a subset
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Coherent States in Quantum Physics Jean-Pierre Gazeau Copyright 2009 WILEY-VCH Verlag GmbH & Co KGaA, Weinheim ISBN: 978-3-527-40709-5
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