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be determinate In consequence, there exist irreducible limitations, namely, N (Q) and N (Q), in the exploration of small and large distances, and both limitations have the correlation N (Q) N (Q) u 2 l 2 c Suppose that there exists, for theoretical reasons, a fundamental or universal minimal length, say, l m , which could be the Planck length l Pl = G/c 3 W 10 35 m or something else, depending on the experimental or observational context, or, equivalently, a universal ratio u = l c /l m v 1 Then, from N (Q) v l m , we infer that there exists a universal maximal length l M given by l M W 2 u l c (1239)
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Of course, if we choose l m = l c , then the size of the universe is l M W 2 l m , a fact that leaves no room for the observer and observed things! Now, if we choose a characteristic length appropriate for atomic physics, such as the Bohr radius, l c W 10 10 m, and for the minimal length the Planck length, l m = l Pl W 10 35 m, we nd for the maximal size the astronomical quantity l M W 1016 m W 1 light year, which is also of the order of one parsec On the other hand, if we consider the (controversial) estimated size of our present universe Lu = cT u , with T u W 13 ~ 109 years, we get from l p Lu W 2 l 2 a characteristic length l c W 10 5 m, that is, a wavelength in the c infrared region of the electromagnetic spectrum Another interesting outcome of the monotonic increase of the product l m l M 2 l 2 c
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is that the reasoning leading to (1239) can be reversed Suppose that there exists an absolute con nement, of size l M , for the system considered Then, at large N, there exists as well an impassable core of size l m W 2 l2 c lM (1240)
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Then, the allowed range of values of the characteristic length l c is l m u l c u l M / since, for the upper limit l c = l M / , we have l m W l M , whereas at the lowest limit we recover l M W 2 l m Let us turn to another example, which might be viewed as more concrete, namely, the quantum Hall effect in its matrix model version [169] The planar coordinates X 1 and X 2 of quantum particles in the lowest Landau level of a constant magnetic eld do not commute: [X 1 , X 2 ] = i , (1241)
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where represents a minimal area We recall that the average density of N electrons is related to by o = 1/2 and the lling fraction is = 2 0 /B The quantity l m = can be considered as a minimal length The model deals with a nite number N of electrons: [X 1,N , X 2,N ] = i (1 N |N 1 N 1|) (1242)
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In this context, our inequalities read as N (X i ) N (X i ) u 2 l 2 , c i = 1, 2 , (1243)
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where l c corresponds to a choice of experimental unit Since l m = affords an irreducible lower limit in this problem, we can assert that the maximal linear size LM of the sample should satisfy lc l M u 2 l c (1244)
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for any nite N The experimental interpretation of such a result certainly deserves a deeper investigation As a nal comment concerning the inequalities (1237), we would like to insist on the fact they are not just an outcome of nite approximations Q N and P N (or X 1,N and X 2,N ) to the canonical position and momentum operators (or to X 1 and X 2 ) in in nite-dimensional Hilbert space of quantum states They hold however large the dimension N is, as long as it is nite Furthermore, let us advocate the idea that a quantization of the classical phase space results from the choice of a speci c (reproducing) Hilbert subspace H in L2 (C, (dz d z )) in which coherent states provide a frame resolving the identity This frame corresponds to a certain point of view in dealing with the classical phase space, and this point of view yields the quantum versions Q N and P N (or X 1,N and X 2,N ) of the classical coordinates q and p (or x 1 and x 2 )
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