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Discrete space, bfield and the lower "mass" bound 11^{th} April 2014
Let us first discuss some basic concepts and how they apply to the discrete measurement space, jspace. The finite values of divergence and curl represent the presence of a source and a boundary for a given field, let us call it F. The divergence and curl of F, being measured as continuous and irrotational are described by the following equations as,
div F = 0_{j },
curl F = 0_{j }.
In above equations 0_{j} represents the measured “null” value by the macroscopic observer Obs_{M} in discrete jspace. However 0_{j} will be measured as finite by Obs_{i}.Further the limits of an integral i.e. the interval [0, ∞] measured as an infinite interval by the discrete space observer, will be measured as a finite interval [0_{j}, ∞_{j}] by Obs_{i}. The lower and upper bounds which are usually determined based on the experimental data. Similarly the infinitesimal period used for differentiation in jspace may contain multiple PE1_{j} events measured by Obs_{i}, which means that there will be stable structures below the measurement threshold of discrete space observer Obs_{M}. We also note that two observables A and B, measured as commuting with each other i.e. [A, B] = 0 by the observer Obs_{M }in discrete jspace, will be measured as [A, B] = 0_{j}, by Obs_{i} and hence noncommuting with each other. Therefore the symmetry property exists for the observers of identical but finite capabilities in discrete jspace, but not for Obs_{i} who has infinite capability. In other words the assumption of a reference observer with identical capability to define an origin, leads to the concept of (VTS) symmetry in discrete jspace. This assumption is equivalent to defining a lower limit to an interval per measurements made in discrete jspace. We also note that symmetry defined by an observer of lower capability will not necessarily exist for an observer of higher capability, but the opposite must hold. Above conditions are rather strict as they bind the measurements in discrete jspace to very specific properties. However for the observers to make measurements in discrete space the application of the measurement force is required and hence the existence of Obs_{i} is implicit which leads to the restrictions. The arguments presented above are independent of the dimensions of the representation or the number of variables being measured in discrete jspace and they can be generalized to any representation in discrete jspace. We can now consider the condition for bfield given as, The macroscopic observer Obs_{M} with its limited capability, will measure the R.H.S. of the above equation as "0" whereas Obs_{i }will measure R.H.S. as "0_{j}" or finite. The bfield for Obs_{M} will appear to be sourcefree and follow commutation rules, however per Obs_{i} criterion the bfield exists due to a source which can not be physically measured by Obs_{M}. The commutation property observed by Obs_{M} will actually be anticommutative and hence it will give rise to the Lie Bracket in the description provided by Obs_{M}. The lower mass bound is due to the limitation on the observer's (Obs_{M}) capability. We would also like to point out that the timeaxis and spaceaxes will not converge to "origin" simultaneously without the assumption of the symmetry i.e. the time is not necessarily zero when the space is. We can rephrase it by saying that time and space for a finite capability observer Obs_{M}, do not start simultaneously unless a symmetry is assumed. We will discuss this rather important point in detail a little later on. Information on
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