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In Section 53 we drew a parallel between quantum mechanics and signal analysis: the object considered is an observation set X of data or parameters equipped with a measure de ned on a -algebra F For both theories, the natural framework of investigation is the Hilbert space L2 (X , ) (K = R or C) of square-integrable real or K complex functions f (x) on the observation set X: X | f (x)|2 (dx) < We then observed that quantum processing of X differs from signal processing since, in particular, not all square-integrable functions are eligible as quantum states We meet, precisely at this point, the quantization problem: how do we select quantum states among simple signals In other words, how do we select the true (projective) Hilbert space of quantum states, denoted by K, that is, a closed subspace of L2 (X , ), or equivalently the corresponding orthogonal projector I K K The only requirements on K, in addition to being a Hilbert space, amount to the following technical conditions: Condition 131 (Quantum states) (i) For all K and all x, (x) is well de ned (this is, of course, the case whenever X is a topological space and the elements of K are continuous functions) (ii) The linear map ( evaluation map ) x : K K (x) is continuous with respect to the topology of K, for almost all x The last condition is realized as soon as the space K is nite-dimensional since all the linear forms are continuous in this case We see below that some other examples can be found (131)



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As a consequence, using the Riesz theorem 14), there exists, for almost all x, a unique element p x K (a function) such that p x | = (x) (132)

We de ne the coherent states as the normalized vectors corresponding to p x , written in Dirac notation as |x = | px [N (x)] 2

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less energy consuming cold-water impeller machines to hotwater drum machines If a third of all machines are of the latter type, electricity usage will increase by 25 percent the equivalent of 12 large base-load power plants at 300 MW each* We can extrapolate how these household lifestyle changes like acquiring washing machines drive energy demands As the next one billion consumers begin a modern lifestyle with household appliances, one can only imagine the spike in energy demand globally throughout the twenty- rst century Public policy that could curb this demand: the introduction of metering devices to change the washing behavior of households (once they know how much energy a single wash consumes, they won t turn it on unless it s completely lled) or the introduction of energy saving washing machines Curbing wasteful behavior early in a country s life cycle and developing an energy-ef cient culture is often cited by experts as the best check on long-term energy usage

(133)

One can see at once that for any K (x) = N (x)

11/ 10

(134)

One obtains the following resolution of the identity of K that is at the basis of the whole construction: IK =

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How does the world quench this growing thirst for energy Table 32 underscores our great fossil fuel reliance Roughly 80 percent of energy comes from oil, coal, or natural gas Earth-friendly renewable sources like solar, geothermal, and wind while growing, contribute only one half of 1 percent of global energy production

(135)

Result:

(dx) 1 (x) 2 (x) = 1 | 2 ,

which hold for any 1 , 2 K Note that (x) =

Oil is the largest single energy source today, about two-thirds of it is used for transportation The world economy doesn t move much without oil: cars, trucks, boats, and planes need oil, and there is no easy alternative in sight As Table 33 shows, many countries produce the stuff, but only a few use less than they produce, and even fewer are making

(dx )

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K

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Qualitative defects can be categorised in various ways, including:

(136)

Hence, K is a reproducing Hilbert space with kernel K (x, x ) = N (x)N (x ) x |x , (137)

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