|Discrete space, b-field and the lower "mass" bound
11th April 2014
Let us first discuss some basic concepts and how they apply to the discrete measurement space, j-space. 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 = 0j ,
curl F = 0j .In above equations 0j represents the measured “null” value by the macroscopic observer ObsM in discrete j-space. However 0j will be measured as finite by Obsi.
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 [0j, ∞j] by Obsi. The lower and upper bounds which are usually determined based on the experimental data. Similarly the infinitesimal period used for differentiation in j-space may contain multiple PE1j events measured by Obsi, which means that there will be stable structures below the measurement threshold of discrete space observer ObsM.
We also note that two observables A and B, measured as commuting with each other i.e. [A, B] = 0 by the observer ObsM in discrete j-space, will be measured as [A, B] = 0j, by Obsi and hence non-commuting with each other. Therefore the symmetry property exists for the observers of identical but finite capabilities in discrete j-space, but not for Obsi 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 j-space. This assumption is equivalent to defining a lower limit to an interval per measurements made in discrete j-space. 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 j-space 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 Obsi 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 j-space and they can be generalized to any representation in discrete j-space.
We can now consider the condition for b-field given as,
The macroscopic observer ObsM with its limited capability, will measure the R.H.S. of the above equation as "0" whereas Obsi will measure R.H.S. as "0j" or finite. The b-field for ObsM will appear to be source-free and follow commutation rules, however per Obsi criterion the b-field exists due to a source which can not be physically measured by ObsM. The commutation property observed by ObsM will actually be anti-commutative and hence it will give rise to the Lie Bracket in the description provided by ObsM. The lower mass bound is due to the limitation on the observer's (ObsM) capability. We would also like to point out that the time-axis and space-axes 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 ObsM, do not start simultaneously unless a symmetry is assumed. We will discuss this rather important point in detail a little later on.
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