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Energy Conservation & Criteria for Replacement Rules

Posted 4 years ago

Mr. Wolfram describes energy as "energy: flux of edges in the multiway causal graph through spacelike (or branchlike) hypersurfaces". Take this idea of flux seriously and assign energy to be directly proportional to the quantity of edge changes on a particular vertices. For the purposes of conservation, we may set the proportionally constant to one and simply allow the energy to be the number of edges changed per update. Now, say that the unit energy will be the planck energy and that we may set that equal to one change per update. Now, let us look at any two updates A, and B. Count the total number of edges changed during A and again for B. The total for A will be the total energy content of the universe at that point in 'time'. The same will be true for B.

What energy conservation tells us is that we can not destroy or create energy, which in this model translates to the number of changes of edges each update will not change. Because if A and B were to have different numbers of edge changes or energy contents, then the universe would have changed its energy content. So, if we are to keep energy conservation within this model, then it seems clear that we must do one of two things:

  1. Assume that every update contains the same number of edge changes. Which would put a hard criteria upon the replacement rules since we have seen through the registry of notable universes that a conservation of edges changes seems a rare creature.

  2. That the hypergraph splits in such a way to causally begin with some net energy, but end with a tilted scale in each of the splits without violating conservation on the whole. Which also seems like an incredible claim to force upon whatever replacement rule that generates the whole structure.

Both of these paths of criteria for replacement rules provides very strict rules directly from energy conservation and the concept that energy may be a flux of edges. My question here is whether this very brief analysis seems valid and if so, does it in fact provide a strict limitation for replacement rules?

Furthermore, if such a criteria be accepted, there is one interesting application that emerges quite immediately: Take the concept of energy as edge flux or edges changing upon update. Then say that a particle is some persistent and stable subset of the hypergraph that seems relatively invariant under hypergraph transformation up to isomorphism. Then, imagine the situation of a particle with some rest mass m which is really some energy content E. By forcing said particle to be stationary, one then sees that the there still exists an energy content of the particle despite a lack of motion. The only way in which this particle may contain energy, i.e. have edge changes without moving and changing its vertices, is some form of internal change of edges between the vertices that construct the particle. This internal change of edges may, potentially, be thought of as intrinsic spin or perhaps some other intrinsic property of the particle which is directly related to the fact that there is no situation where this particle may be without an intrinsic energy content.

POSTED BY: Phoenix Smith
4 Replies
Posted 4 years ago

As far as I knew the conservation of energy, even at cosmological scales, is still held. I know of theories that use violations to explain the origins of dark energy (https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.118.021102), but am unaware of any experimental evidence of violations. So I was coming from the point of view that energy conservation holds for the entire universe. If you have any evidence to convince me that energy conservation is not held at cosmological levels, I'd gladly see it.

What I had written above was under the assumption that energy is conserved at all scales of the universe. Holding that assumption, I proposed a very simple model for edge flux as simply the changes in the number of edges on vertices and explains that if one vertice lost an edge, then another vertice gained one and this was the interpretation of energy conservation as an edge conservation through updating events. If energy conservation was indeed synonymous with a conservation of edges, then if energy was to be conserved globally by the updating events, the number of edges did as well. (If, at this point we how the universe could expand without changes the quantity of edges, we simply give it a very large number of vertices with a comparably small number of edges as the initial conditions for the universe.)

If however, we are assuming that energy is not to be conversed globally, then it still must be nearly conserved globally because the measured value of the cosmological constant is still very small. Models, like the one I already gave the link to above, show how early violation and a leveling off violations could occur and explain dark energy. Therefore, whether we decide to keep energy conservation globally or not, we must still construct some replacement rule and initial condition that either maintains a constant quantity of edges or, after some amount of time, produces some logistic feature for the population of edges, leaving one with a nearly constant number of edges. Either will give global criteria for replacement rules as far as their long-term structures global structures are concerned.

Furthermore, even if we have assumed that energy conservation does not hold globally, then it is held extremely well at small scales because we have, as far as I'm aware, never found experimental evidence of such a violation. Therefore, the arguments I gave for rest mass, the internal rotations of edges, interpretations of quantized spin, and the difference of fermions and bosons are still up for grabs.

POSTED BY: Phoenix Smith

Energy conservation (which, as mentioned above, is violated cosmologically) only places constraints upon the flux of causal edges associated with updating events which maintain baryonic matter in the hypergraph, which in turn only places constraints on the classes of topological obstructions in the hypergraph that can be associated with elementary particles. As far as I know, this does not place any direct restrictions on the structure of the rules. Have I missed something?

POSTED BY: Jonathan Gorard
Posted 4 years ago

I'm not sure I agree with your assessment of the criteria of a fixed number of edges. Say we keep the total number of fixed, but allow the number of vertices to change. These rules for vertices and edges would correspond to the global average of number of edges per vertice to decrease, which means that the energy density of the universe was actually decreasing. A decreasing energy density, seems to me, to potentially indicate an inflationary behavior.

Additionally, past the criteria for replacement rules to globally increase vertices while conserving edges (which may hint toward families of initial conditions), the simple model of energy also provides an avenue for some kind of quantized spin. Since we can imagine a particle as a discrete structure, we will then have a quantized 'motion' in that changing edges, through replacement rules, must occur in a discrete manner. So, if we are to take the case for rest matter seriously and purpose that edges undergo some kind of 'rotation' of edges, then this rotation much in tern be discrete. This would be a very intuitive model for spin since the edges that construct a particle are not, due to their stable nature, a component of space. Therefore, the spin of the particle is not only discrete, resemble an actual angular system, but also is intrinsic due to the inaccessibility of the interior edges of from exterior edges or 'space'.

Another point I'd like to bring up is that we have modeled a particle as a kind of structure which undergoes some periodic dynamics. If particles can be modeled as stable periodic localities of the hypergraph, then we may, potentially, be able to model force carriers (the bosons) as a kind of 'extension' of this periodic behavior into the exterior edges that surround the particle(to be considered fermion obviously). In this way, fermions modeled as a kind of discrete waveform would be 'driving' periodic patterns in the surrounding 'space' which, themselves, would be discrete waveforms and could be considered bosons. The behavior of fermions and bosons would then be defined by wave-like phenomenon in a discrete structure.

POSTED BY: Phoenix Smith
Posted 4 years ago

I find you analysis very clear and useful but there is one confounding factor that we need to take into account, energy is not conservatived insofar as new space is constantly being generated in the real universe. There are constantly vertices being created as the universe expands so conditioning on a fixed amount of causal edges would not work. Particularly if inflation is real then we need a rule that can vary in its creation of new verticies. You conditioning proposal would work however if we had a way to delineate changes in causal edge flux due to spatial expansion versus the causal edge flux associated with particles, a kind self sustaining edge configurations.

This makes me think that we need a rule that has a fixed quantity of self sustaining energy(particle energy) from the beginning that gets diluted as more space is created, this would mimic the high energy at the time of the big bang. Perhaps the rule would in the beginning have a structure that creates a large number of self sustaining constructs in the graph but as the number if vertices increase this behavior ends. One possibility could be whatever process creates virtual particle pairs that anhiliate in our current vacume would create non-anhilating particles in the early universe.

POSTED BY: Alex Jorjorian
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