This article is arranged as a series of learning steps, and is a 'creation sequence' for the universe (which some theoretical physicists and natural philosophers pursue).
Building 'exclusion' into physicality
Our solution for fermion events is the 'first unique' solution. This is another way of expressing the Exclusion Principle, requiring a latent phase offset to satisfy the unique solvability of more than two entities.
Basic dynamic physicality
Given that we are describing physics, there needs to be a definition of what ‘something’ is. This initial step contains a potential minefield of perspective-based definitions that could be distracting, because most of the words that we would casually choose are inappropriate in some way, loaded with assumptions we're not yet ready to make. We sidestep these issues by declaring minimal abstractions, to build upon them using simple requirements for a physical picture.
Our description of reality begins by asking how an entity may be minimally described. We then require a means for multiple entities to exist and to be uniquely reconciled among themselves.
Entity
We could begin with void, but it is not informative, and cannot be said to exist by any definition. The simplest form of something that is not void is simply zero. Mathematically, zero assumes a basis axis and a value on that axis, so we need a form of geometry from the very beginning of our construction. The value of zero is an attribute of something, so it may only apply where there is entity. Our simplest structure of something is therefore a single entity, with a single-valued attribute of zero.
To usefully approach a physical system like the one we inhabit, we need to incorporate more entities. These need to be different from each other, otherwise they would be indistinguishable as separate entities.
Caution with 'loaded terminology'
There are many 'loaded terms', and we need to be careful using them when describing the foundations, because each loaded term carries some assumptions into the mix, each of which must be validated before we can confidently include it.
Attribute is a loaded term because it assumes the basics of Noether's Theorem. For something to hold a value, it needs also to have a variable, and a variable is a term that is liable to change (assuming dynamics), or may hold a range of values (implying the requisite structure for number types). Having variables assumes that invariant (symmetric) and divergent (changing) terms are identified. But that's much more complicated than we need, so we'll rein ourselves back here, and assume no dynamics at this point, and specify no range of value. Just the notion of value, and the notion of the entity having a value of zero will suffice.
Where is 'here'?
We don't really need a "where", because the entity doesn't need to know where it is; there need be no positional context, and we haven't described it in relation to anything else, because there is not anything else yet; just an entity. We'll need more complexity in order to achieve a system that looks like our everyday physics, which has lots of other entities. We won't be jumping immediately to the notion of other entities (in the everyday sense), because that would be a short-cut that misses out a lot of important detail. We need to go via a more abstract route to better justify our approach.
Multiple static entities
With one variable to represent state, entities may not co-exist if they have the same value; for entities to be separably identifiable, there must exist some difference between them. We may quantify this by introducing the concept of spectrum: a set of possible values that a (generally quantized and observable) state may assume.
For example, if all entities have a state spectrum of two possible values, then two entities may exist at a point provided they have different state values. Such a physicality is static not dynamic; there is no need for (nor means of) change.
The number of entities that may occupy a point depends on the fundamental variables: how many variables, and the spectrum of values that the variables may assume (fig.1).
Fig.1: Spectral capacity for entities. A–C: filling a spectrum; D, E, F: options for increasing the capacity of a spectrum.
To generate complexity by introducing more entities, we must use one of the following options:
Create additional variables (bases) for each entity, so that entities may be identified by more than one variable (fig.1: D); or
Create a spectrum that has a larger set of allowed values, or a continuous (non-quantized) spectrum (fig.1: E,F).
When those options have been exhausted, any additional (identical) entities would be incompatible with the system, and an alternative means must be found to include them.
Exclusion, locality, dynamics
So far our picture is static and non-local; many states may exist at a single point, without there being any need for anything to exist outside that point. Indeed, the notion of a point is not yet important, because there is no need for any definition of space, and if there is a background space (which we have not yet invoked) it is reasonable at this stage to assume that the whole manifold (if there is one) is covered by all entities with there being nothing to uniquely identify any part of the space as being different from any other.
To increase the complexity of the system from static to dynamic, we propose a dynamic process whereby an entity’s state is regarded as a point on a sinusoidal wave, and if we ‘push’ an entity out of its initial phase, then the capacity of a single point is no longer a limitation. Incidentally the previously-occupied position in the spectrum then becomes available.
This process introduces new properties for the system: implicit locality and causality are introduced, and identical entities are now allowed, but not at the same point. We also introduce important new laws:
Concept: universal equivalence
- Phase propagation acts universally for all entities.
- Time is equivalent to phase propagation.
This implies that all matter propagates, and phase has the same meaning for all waves. Consequentially all waves propagate similarly (when related to classical or relativistic space, we know this as ‘light speed propagation’), even though the classical aggregated speed of a conserved particle may be significantly slower. Matter does not remain at a position; it immediately deconstitutes and radiates, and always at light speed. [More...]
The existence of similar entities
We distinguish between connected and similar entities (fig.2):
Fig.2: Connected and similar states.
- A: Reference;
- {A,B}: Disjoint;
- {A,C}: Connected;
- {A,D}: Similar;
- {A,E}: Similar and Connected.
Connected states would be identical (excluded) if we take phase offset into account, e.g. fig.2: pair {A, C}, and pair {A, E}.
Specifically, we take one entity A as ‘base state’ and another entity E as ‘state A + phase operator f’, where both entities would be identical if they were resolved to the same point: fA = E. This is just like identifying two spacetime coordinates as being on the same world line, having the same form as Minkowski spacetime equivalences in special relativity.
Similar states are not necessarily directly connected, but have the same states in different localities or times, e.g. fig.2: pair {A, D}, and pair {A, E}.
This illustrates that the latent phase offset allows multiple copies of a variable to exist with the same value in different places.
Background space
The phase offset implicitly creates a background space, but minimal systems will not create the 3D Euclidean space that we recognise in classical physics. Instead, each entity is regarded as having a direct phase offset from at least one other entity, so any background space is derived from these intervals, rather than the intervals being defined by the entities’ positions in the background space. 3D space is just a generalization of the simplest reduction of larger systems, and time is a generalization of phase propagation.
Unambiguous connection: unique solvability
In having two entities, their attributes should make them unique, otherwise they will look like one entity, or be undetectable. Extending the notion to three entities (while still dealing with a single positional point), we enter an important part of the model: the uniqueness of 'other entities', a feature that we call 'unique solvability'.
Assume there are three entities. Given these three entities, they must somehow relate to each other; of three entities {A, B, C}:
- A must relate to B,
- A to C, and
- B to C (three pairs).
At this point, we should make clear that we do not seek to explain how the universe came into existence, nor do we speculate whether the universe was created at a big bang event, or it always existed. We do say this: if there is an entity, then it cannot exist alone, and if there is more than one seemingly identical entity, then bosons are needed to maintain them, and this implicitly creates the causality and intermediate space.
This is where uniqueness becomes really important: if A is the same as B, then C will not be able to tell them apart, and if we require a relation between entities to return only one solution, then all must be unique to each other.
Concept: exclusion
An entity cannot receive two identical signals about other entities. Likewise, two entities sharing the same point may not share the same state values. This is another way of stating the Exclusion Principle, but on more general terms.
Summary
Our introductory definition of exclusion encompasses locality, latency, causality, dynamics, uniqueness, and background. In the context of a matter network, there are two aspects to exclusion:
- Similar entities may not occupy the same point; and
- For any given state, all of its directly-connected entities must be individually unique.
This is our origin for the Exclusion Principle: an entity must be uniquely resolvable for it to become real, otherwise it remains degenerate and virtual. In the next section we will define fermion and boson states in this context.
Fig.3: Concept diagram of 'entity'.
- Fermion
- Concept diagram for the propagation mechanism, expanding the 'quantization condition' loop on the diagram above.