Difference between revisions 601336 and 601346 on enwiki

The '''Wigner - d'Espagnat inequality''' is a basic result of [[Set theory]].
It is named for [[Eugene Wigner]] and Bertrand d'Espagnat who (as pointed out by [[John Stewart Bell|Bell]]) both employed it in their popularizations of [[quantum mechanics]].

Given a set S with three subsets, J, K, and L, the following holds:

* each member of S which is a member of J, but not of L 
:: is either a member of J, but neither of K, nor of L, 
(contracted; show full)

it is probably worth mentioning that certain non-negative ratios are readily encountered, which are appropriately labelled by similarly related indices, and which '''do''' satisfy equations corresponding to 1., 2. and 3., but which nevertheless '''don't''' satisfy the Wigner - d'Espagnat inequality. For instance:  

if three observers, A, B, and C,
 had each detected signals in one of two distinct own channels (e.g. as ''(hit A)'' vs. ''(miss A)'', ''(hit B)'' vs. ''(miss B)'', and ''(hit C)'' vs. ''(miss C)'', respectively), trial-by-trial, over a finite and nonempty set ofover several (at least pairwise defined) trials, then non-negative ratios ''N'' may be evaluated and, appropriately labelled, and found to satisfy 

# ''N<sub>(hit A) (hit B)</sub> + N<sub>(hit A) (miss B)</sub> + N<sub>(miss A) (hit B)</sub> + N<sub>(miss A) (miss B)</sub> = 1'',
# ''N<sub>(hit A) (hit C)</sub> + N<sub>(hit A) (miss C)</sub> + N<sub>(miss A) (hit C)</sub> + N<sub>(miss A) (miss C)</sub> = 1'', and
(contracted; show full)
''orientation angle( B, C ) = 1/2 arccos( N<sub>(hit B) (hit C)</sub> - N<sub>(hit B) (miss C)</sub> - N<sub>(miss B) (hit C)</sub> + N<sub>(miss B) (miss C)</sub> )'',

'''and''' if 
a set of trials isA's, B's, and C's channels are considered having been properly ''set up'' only if the constraints <BR>
''orientation angle( A, B ) = orientation angle( B, C ) = orientation angle( A, C )/2 < &pi;/4'' <BR> 
hasd been obtainfound satisfied (as one may well require, at least to good accuracyto any accuracy; where the accuracy depends on the number of trials from which the orientation angle values were obtained), then necessarily (to goodgiven sufficient accuracy)

''(cos( orientation angle( A, C ) ))<sup>2</sup> =''<BR>
: ''(N<sub>(hit A) (hit C)</sub> + N<sub>(miss A) (miss C)</sub>) = (2 (N<sub>(hit A) (hit B)</sub> + N<sub>(miss A) (miss B)</sub>) - 1)<sup>2</sup> > 0''. 

Since

''1 &ge; (N<sub>(hit A) (hit B)</sub> + N<sub>(miss A) (miss B)</sub>)'',

therefore

''1 &ge; 2 (N<sub>(hit A) (hit B)</sub> + N<sub>(miss A) (miss B)</sub>) - 1'', <BR>
''(2 (N<sub>(hit A) (hit B)</sub> + N<sub>(miss A) (miss B)</sub>) - 1) &ge; (2 (N<sub>(hit A) (hit B)</sub> + N<sub>(miss A) (miss B)</sub>) - 1) <sup>2</sup>'', <BR>
''(2 (N<sub>(hit A) (hit B)</sub> + N<sub>(miss A) (miss B)</sub>) - 1) &ge;(N<sub>(hit A) (hit C)</sub> + N<sub>(miss A) (miss C)</sub>)'', <BR>
''(1 - 2 (N<sub>(hit A) (miss B)</sub> + N<sub>(miss A) (hit B)</sub>)) &ge; (1 - (N<sub>(hit A) (miss C)</sub> + N<sub>(miss A) (hit C)</sub>))'', <BR>
''(N<sub>(hit A) (miss C)</sub> + N<sub>(miss A) (hit C)</sub>) &ge; 2 (N<sub>(hit A) (miss B)</sub> + N<sub>(miss A) (hit B)</sub>)'',

''(N<sub>(hit A) (miss C)</sub> + N<sub>(miss A) (hit C)</sub>) &ge;'' 
::: ''(N<sub>(hit A) (miss B)</sub> + N<sub>(miss A) (hit B)</sub>) + (N<sub>(hit B) (miss C)</sub> + N<sub>(miss B) (hit C)</sub>)'', 

which is in (formal) contradiction to the Wigner - d'Espagnat inequalities

''N<sub>(hit A) (miss C)</sub> &le; N<sub>(hit A) (miss B)</sub> + N<sub>(hit B) (miss C)</sub>'', or <BR>
''N<sub>(miss A) (hit C)</sub>) &le; N<sub>(miss A) (hit B)</sub>) + N<sub>(miss B) (hit C)</sub>)'', or both.

The failure of certain non-negative ratios toAccordingly, the ratios ''N'' obtained by A, B, and C, with the particular constraints on their ''setup'' in terms of values of orientation angles, '''cannot''' have been derived all at once, in one and the same set of trials together; otherwise they'd necessarily satisfy the Wigner - d'Espagnat inequalities, as shown in the example, has been characterized as constituting disproof of [[Einstein]]'s notion of ''local realism''.
Instead, they had to be derived in three distinct sets of trials, separately and pairwise by A and B, by A and C, and by B and C, respectively.

The failure of certain measurements (such as the non-negative ratios in the example) to be obtained at once, together from one and the same set of trials, and thus their failure to satisfy Wigner - d'Espagnat inequalities, has been characterized as constituting disproof of [[Einstein]]'s notion of ''local realism''.

Similar interdependencies between ''two'' particular measurements and the corresponding operators are the [[Robertson - Schrödinger relation|Uncertainty relations]] are first expressed by [[Werner Heisenberg|Heisenberg]] for the interdependence between measurements of distance and of momentum, and as generalized by [[Edward Condon]], [[Howard Percy Robertson]], and [[Erwin Schrödinger]].

====Reference====
* John S. Bell, ''Bertlmann's socks and the nature of reality'', Journal de Physique '''42''', no. 3, p. 41 (1981); and references therein.