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Hence, the family of coherent states is to be viewed as the orbit of the Fock vacuum under the action of operators D (z), z C Let us show how we derive from (353) the coherent states in the position representation From |z = D (z)|0 , and from D ( z) x ( y ) = e 2 q p e i pQ e iqP x ( y ) = e 2 q p e i pQ x ( y + q) = e 2 q p e i px x q ( y ) , one nds the following expression for the coherent states:



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structured in the bottom, making it difficult to find the information. In the top the information has been ordered into meaningful categories with blank spacing between them, making it easier to select the necessary information. Perception refers to how information is acquired from the environment, via the different sense organs (e.g., eyes, ears, fingers) and transformed into experiences of objects, events, sounds, and tastes (Roth, 1986). It is a complex process, involving other cognitive processes such as memory, attention, and language. Vision is the

z (x)





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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 ,

115 116 117

(183)

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 =

493 333 172 133 126 102 60 59 53 45 43 37 36 24 18

f (x)|x x| E (dx) ,

expression1 IS NOT NULL THEN expression1 expression2 IS NOT NULL THEN expression2 expression3 IS NOT NULL THEN expression3 NULL

(184)

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

Source: United Nations: 2007 Status of Contributions to the Regular Budget, International Tribunals, Peacekeeping Operations and Capital Master Plan

4096 4096 0 0 0

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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