Dear S': I note with distress
The length of your yardstick is less
And please wind your clock
To make it tick-tock
More briskly.
Your faithful friend, S.


- Relatively Good Advice
by Edward H. Green.



















Two photons,
Close-coupled at start,
Flew several parsecs apart.
Said one, in distress,
"What you're forced to express
Removes any choice on my part.

- Einstein, Podolsky and Rosen
by David Halliday.




















The chairman of AT&T
Said, "Your graduate physics degree
Is not worth a - penny,
Of your kind we've too many.
Perhaps you can program in C?


- See You at Work
by Steve Langer



















There was a young fellow named Cole
Who ventured too near a black hole.
His dv by dt
Was quite wondrous to see
But now all that's left is his soul.

- Cole's Lost Soul
by A. P. French

***

And it is characterized by the time-independent j-space constant j_ML defined as:
 

where m_o is the rest mass for elementary particles, and  λ_Compton  is Compton wavelength.  However Aku lives in a dynamic world, dynamism which is the consequence of the entropy inherent in the nature of the discrete measurement space.  Then how do we visualize the j-space and the resulting universe for Aku? 

    What we would like to do in next few blogs, is to develop a multifaceted fabric in discrete measurement space, a fabric in which physical structures can be introduced.  The physical structures here, could be elemental or cosmic.  Usually the measurements are defined for a point in space-time.  When we speak of stitching together a discrete measurement space, we are discussing how the measurements corresponding to different points in space-time, are to be combined together.
 

1. Resources are needed even for the Simplest measurement:

    The simplest measurement, is defined as the zero-entropy measurement.  In this case all the relevant information, is extracted in one measurement (S = k × log_e1 = 0).  However we require a minimal amount of resources to perform this single measurement.  Essentially we are describing "Action" here.  The action associated with a system is defined as the minimum amount of resources provided to the system, so that it can travel the path between initial and final states, unassisted. 

    We must clarify that we do not require the action to be time independent in j-space.  For example we could have strung together a sequence of zero-entropy measurements.  The time-axis τ corresponding to this measurement string however, would have been very different than the time-axis t followed by Aku.  In fact, we can say that Aku (S-reference frame) will measure the clocks corresponding to τ (S'-reference frame), as stopped frozen. 

    The time-independence of action is assumed only when we write wave-equations in Aku's measurement space using t, for a conservative system.  However any conservative system is a very small subset of information contained in a discrete measurement space.  The concept of a conservative system is extremely useful for developing a mathematical structure for an observed physical phenomenon, but it also has fairly limited prediction capability.

2. The simplest and the desired measurement is "COUNTING":

    If we think carefully, in essence the process of visual measurement is based on "counting".  We first count different types of objects in our vision and then fill in details for each object later.  For example a simple sequence of measurements leading to a deduction may be:

1 (object) containing 4 (objects) and 6 (objects), sitting on one (object), followed by
1 (basket) containing 4 (apples) and 6 (oranges), sitting on one (table), followed  by
1  (wicker basket) containing 4 (red apples) and 6 (orange oranges) sitting on one (wooden table), followed by
..(more and more details)..and so on. 

    In real life the process of visualization is invariably extremely fast and constantly changing, as the visual information continues to stream in.  Therefore for an observer the sequence of events are almost simultaneous.  This is a truly remarkable capability hidden in human brain.  It combines what an observer knows and what an observer measures at every instant into tangible information, the observer can use to make decisions. 

    We can not simply reverse the process described above, by lumping all the details together and then start sorting out the different types of objects from a big gigantic blob of information¹.  We will be stuck in a quagmire of sheer madness.  No decision making will be possible, which is actually all nature wants to do with minimum or no hassles, with the process inherent in nature we can call "continuation".  The "continuation" is defined as the progression through successive zero-entropy measurements. Going around in circles, does not count as continuation.  Nature does not repeat itself, an observer does.  We will discuss "continuation" in j-space in further details later on. 

   
    In discrete measurement space the process of counting, is equivalent to completing the circuit.  Completing the circuit in turn, leads to forming the Surface of Least Disorder.  For each object the nature of the Surface of Least Disorder will be different, though they will be correlated to each other by Lorentz Invariance and hence the condition of Locality (i.e. the definition of origin 0_j and VT-Symmetry).  Similarly in tensor algebra, we look for the capability to count, when we require a tensor to reduce to a scalar by performing successive contraction operations.

    An important concept in this context, is the Axioms of Countability.  Since we already know that ∞_j has a finite value in a discrete measurement space, the second axiom of countability is satisfied. And since the second axiom of countability is satisfied, the first axiom of countability follows automatically.

    The Axioms of Countability allow us to develop the description in j-space with simply-connected, two-sided oriented covering surfaces i.e. we can think of manifolds in j-space.  This is a completely different discussion, which we will take up some other time. (A zero-entropy measurement is essentially a point with a neighbourhood, on a manifold.  In discrete measurement space we call it a j-pixel.)


3. The presence of a field is required for performing measurements:
 
     A measurement can be performed only by the application of a measurement-force.  A force is equivalent to an interaction and an interaction requires the presence of a field.  Therefore to perform a measurement a field must be established.  In other words, in non-classical cases once a circuit is completed i.e. a particle or a system is identified, the presence of field is required to perform measurements.  The existence of a field means that for a given system, at every point in space-time a unique tangential vector can be associated with the system. 

    For measurement fields in j-space we also place an additional requirement, that for every point in space-time a unique normal vector is associated with the system.  The normal vector represents the memory of the initial state, <t = 0_j>², thus the uniqueness part.  We must not confuse this normal vector, with the conventional normal vector in Euclidean surface.

    The memory of the initial state, is actually an important property for entities in j-space.  Consider a system with no memory whatsoever of the initial state.  In this case the system will be completely-random, similar to gas inn statistical thermodynamics.  However the lack of information about the initial state means that as we lower the temperature, the expected phase transitions to liquid and solid, will not take place.  The system will remain completely-random even down to 0^oK.  There will be absolutely no presence of information-structures of any sort, which actually is the definition of a true black-hole.  When a system hits a true black-hole in j-space, the information about the initial-state is irrevocably lost. 

    As a matter of fact, a black-hole in j-space is much more catastrophic than the maximum-entropy state.  A system in the state of maximum-entropy, will extract no further information from its measurements, while maintaining its structural integrity.  However inside a j-space black-hole, a system with no memory of the initial state will be devoid of any structure, with no chance of recovery.  A j-space black-hole feasting on an information-system, is essentially a one-trick pony with one huge-bang (~Δm_oc²; Δm_o = equivalent rest mass of the information system; an extremely high value) and nothing afterwards. 

    It is highly unlikely that a universe that supports life, would have originated from such a barren structure.  Hence all structures in j-space, are required to have a unique normal vector at every point in space-time, corresponding to the initial state <t = 0_j>.  We keep in mind that we are in Aku's reference frame S.  The description looks like as shown below:

 

Notice that the direction of the tangential component t , remains the same, however the direction of the normal component n, changes across locations in the space-time grid in j-space.  The unit vectors n and t , are similar to the fundamental dyad of a metric tensor g_μν .  We will discuss the geometrical significance of having a normal vector in j-space, and why we need tensors in j-space when we go surfing in (t, x, y, z) phase-space a little later.

Let us sum up:

•  We need resources even for a zero-entropy measurement.
•  Simplest method "COUNTING". (Axioms of Countability)
•  We build a picture by completing various circuits, add them up, and then fill in details based on the a priori knowledge (a priori knowledge ≡  initial state <t = 0_j>).
•  Measurements require the presence of a field. 
• In j-space for each point in space-time field, a system must have both tangential and normal components.

To be continued..

___________________

1. Actually we sort of can, by assuming various Symmetries and forming Groups etc. (aka matrix formulation), then describe measurements in corresponding fields by Differential Equations and Boundary Conditions (aka wave formulation).  But then nobody understands what is really happening inside M-elephant.  Clearly in this case, the measurement-circuits can not be completed in j-space and hence the precision in the description will be missing.  For example, one may end up with a wooden basket with 6.2 red colored oranges and 3.8 orange colored apples with one leg each, sitting on a wicker table with no legs.  That will still be okay perhaps, if the number of dimensions turns out to be 26.

2. Imagine a phase space with a single cell at t = 0_j.  This single cell will contain all the information about δ_i.  And the phase space consisting of only this single cell, is represented as the state <t = 0_j>.  As the entropy of the system increases, we can think of this single phase space cell, dividing and multiplying as time t progresses, according to observer's or system's capability (v << c), while combined phase space still has the same information content, i.e. δ_i.  The combined phase space has its own clock τ (reference  frame S'), which stays frozen for Aku's reference frame S, hence the assertion that the universe is equivalent to a single zero-entropy measurement.

***