Time, Space and Everything In Between

• Addendum: The Duality of Time
• Addendum: Wave-particle Duality

Addendum: The Duality of Time

Our model is inspired by the duality of energy as described by the holographic principle. Here we use duality in time to reveal a similar holographic structure. In our case, energy duality occurs on a quantum boundary, within a continuum.

An example of the holographic principle is the AdS/CFT correspondence from String Theory. It holds that gravity in the “bulk” is dual to a quantum field theory expressed on a lower dimensional boundary.

The analogous features of our model are a Continuum (of infinite dimension) interacting with quantum elements that form a boundary within the Continuum. The latter elements, which we termed ‘QESTs’, form finite dimensional spacetimes through quantum entanglement.

We viewed the Vacuum as a network of such entangled QESTs. We labelled this spacetime the ‘Vacuum Network’ (VN). A homogeneous spacetime metric arises as a property of network entanglement. Local excesses of energy in the VN produce gravity through feedback between elements of the network.

We extended the holographic nature of the Continuum/QEST vacuum to all forms of energy, in particular all particles and Dark Matter. By participating in this duality, such forms of energy form local spacetimes. As Hilbert spaces, these are mutually orthogonal. Energy is conserved within each, and in the VN.

To understand our version of holographic duality, we must compare the nature of time in spacetimes formed from entangled QESTs (including the VN), relative to time in the Continuum.

Our guiding concept will be the time-energy uncertainty principle. We used this extensively to describe the transmission and localization of energy in the VN.

We visualized how QESTs are restricted in holding both charge and magnetic energy over ‘tick’s of the global quantum clock. Such energy cannot be held for more than one consecutive ‘tick’. Otherwise, time passes relative to that QEST, and energy is transferred to the domain in which the QEST is entangled. Such a transfer violates conservation of energy between spacetime domains.

Using this concept, let’s see how far we can explore along the trail marked ‘Time’ in our Quantum Forest. First, we need to make our assumptions explicit. We define time in the various domains where changes in energy are expressed, including the Boundary domain. 

Assumptions:

· To begin, let’s set the stage (pun intended). Most fundamentally, some essence – some physical ‘thing’ – exists, which we define as a subset of the complement to the null set (the set where nothing exists).

· Call this subset the set S. Elements of this set are physical entities with discernable properties. The elements of S may be continuous or discrete. We exclude the null set from our model. 

· Anticipating the discussion of boundary domains, we divide S into two non-null subsets, S = B U C, where C denotes the Continuum, and B denotes all boundary domains. 

· Our focus in what follows is on point sets that are countably infinite (that is, sets of measure zero). Our universe is such a set: it is a boundary set in the Continuum. We define boundary elements as points of discontinuity in C, having measure zero. 

· S is a host for energy, where energy is defined as that which effects change in some property of S. 

· Call the set of possible properties P. For now, we do not distinguish between properties in B and C.

· The set of properties is common to all elements in S. Energy is defined separately for each property.

· Energy can influence any element in S, and property in P. Call this energy E, defined over all S x P. 

· We shall introduce two notions of time: one relative to points, the other relative to properties. Duality will show us how they are related. Duality in time reveals the holographic nature of energy in B, relative to C.

· In B, we assume time is formed from discrete instants (or ‘moments’). How ‘instants’ arise in B is a consequence of conservation of energy. In essence, quantum entanglement enables discrete clocks in subsets of elements of S. They index the ‘instants’ where time might arise. They supply the moments where energy can be either transferred or conserved.

· In P, time may form continuous intervals (or discrete moments, as a particular case). As change may be continuous, time in P is not restricted to the discrete indexing in B. In our model, we will see that time passing in P is dual to instants of time arising in B.

· ‘S-time’ is defined over points in S as a sequence of discrete instants of change in E at any point in S x P. For S-time to pass, the change for a given property must persist at a point for more than one consecutive instant. 

· To emphasize this key aspect of S-time: if the change at a point occurs for only a single instant, no S-time passes.

· If a property change persists only momentarily at a given point, and that is followed by a momentary change in a different property, then no S-time passes.

· Between instants of change at a point, no S-time passes, as time is defined as these instants of change. (In the case of P, the ‘instants’ may be continuous intervals.) 

· ‘P-time’ in C is defined as a continuous sequence of instants of simultaneous change in a given property over a subset of C. 

· P-time in B is similarly defined, but restricted to distinct points in B. For simultaneous change to occur, it must involve more than one point in B. As a boundary subset of C, points in B may experience change in a property of C. 

· If the simultaneous change in C does not persist over consecutive instants at boundary points of B, no P-time in B passes. With this property, we begin to see how the holographic expression of energy occurs in B. Continuous changes in energy (as waves) propagate in C, with no P-time passing on the boundary (where energy remains momentarily static).

Our definitions are inspired by a fundamental aspect of duality: P-time and S-time may pass independently of each other. Their joint occurrence on a boundary gives rise to the holographic nature of energy expressed on that boundary.

In summary, together P-time x S-time forms a ‘tensor-like’ field t, defined over (BU C) x P. It comprises both discrete and continuous measures of consecutive energy change.

For any property at any point or interval, the amount of change in energy will be discrete in B, and discrete or continuous in C. We shall focus on states in S x P that demonstrate ‘well structured’ evolution in both time dimensions.

By way of contrast to ‘well-structured’, let’s consider two extreme states in S x P: pure Chaos, and pure Structure, where our focus is on boundary states. How might we define “pure” for each extreme?

For pure Chaos on a boundary, a suitable definition is a state where energy changes in S x P, but with no S-time passing and no P-time passing at any boundary points of B. This characterizes the Quantum foam that we relied on in our model of the Quantum Forest. No coherent metric exists, and no net energy is expressed, in Quantum foam. Pure Chaos on the boundary B does not constrain time off the boundary in C.

Pure Structure on the boundary is defined as a non-zero, constant energy held in B over consecutive instants. Change may occur momentarily in some properties. P-time may pass off the boundary. No S-time passes, as energy is held constant throughout B.

Our definition of pure Structure is inspired by the following scenario at the first instant of the ‘Big Entanglement’: creation of a permanent boundary of discrete QESTs within the Continuum, the Quantum foam.

Drawing a parallel from thermodynamics, this required work to be performed by the Continuum. This work involved a net transfer of energy, as the discrete points constituting the foam form a state of lower entropy. No S-time passed within the foam.

This creation event required only P-time: a change in properties across a subset of C over a time interval. Some time must pass in C and B for an energy transfer to occur.

The Big Entanglement saw a further transfer of entanglement energy from the Continuum to create a permanent structure embedded in the foam: the Vacuum Network. Together, the VN and Quantum foam constitute a boundary B within the Continuum.

We will assume no excess energy is present beyond that required to form the network. This initial energy may not have been uniform, so some number if instants of change (S-time) are required for the network to reach a homogeneous equilibrium.

In arriving at this equilibrium, some S-time passes in the VN. Once it is achieved, no further S-time passes. The Vacuum Network enters a quiescent state of pure Structure. No S-time passes, but P-time may pass in C, outside of B.

Our universe is a mixed state beyond this equilibrium state of pure Structure. It is a boundary domain where S-time, but no P-time, passes. P-time passing off the boundary in C produces the holographic energy effect in B.

As a mixed state, our universe features a Vacuum Network (VN) embedded in Quantum foam that hosts additional localized, momentary concentrations of energy.

Some forms of this additional energy transmit in the VN at light speed (photons and gravitons). Others form particles and Dark Matter. These latter experience their own holographic effect. Through local entanglement, they also form part of the boundary B.

All forms of energy are sourced from the Continuum. All influence the gravitational structure of the VN. After the Big Entanglement, no net energy is transferred from C to the boundary B, so no P-time passes on B.

Let’s look at some examples of how this works.

Consider a continuous charge disturbance transmitting as a wave in C. This transient disturbance, which has zero net energy in C, interacts with elements of B. The latter is in the equilibrium state (no S-time passing). P-time may pass in C as the wave propagates.

The wave in C can continue without loss of energy if the interaction results in no net energy transferred to the boundary. The boundary experiences a local imbalance in charge. If this charge transmits out of local QESTs before the next instant, no S-time passes in B. The transmitting bundle forms a photon, supported by momentary magnetic fields formed in Quantum foam. 

The energy source is in C, but no energy is transferred to the boundary B. This is an example of the holographic principle. 

We can visualize the same process in the creation of gravitons in B. In this case, a local transient of entanglement energy propagates as a wave in C.

With particle creation in B, both entanglement and charge transients propagating in C interact with B. In this case, entanglement forms a local spacetime in B. As no energy can be transferred, this spacetime can exist only for an instant in S-time. But it can continue as a transient wave in C. Charge can transmit in the locally entangled QESTs as photons.

In the case of Dark Matter, the local transmission is a graviton. No charge is involved.

Particles and Dark Matter are only momentarily expressed on locally entangled QESTs in B. Their gravitational and charge fields can influence QESTs of the VN, but only momentarily. Otherwise, net energy would be transferred from C. We see again how energy is expressed holographically on the boundary.

The momentary interaction of the VN with all forms of energy on the boundary forms the basis of ‘projection’, which we used extensively in mapping out the Quantum Forest. We assumed projection was indexed by a perfectly precise, uniform ‘entanglement clock’.

The coordination of this clock across all elements in the boundary may be a consequence of B forming a set of measure zero in C.

The Continuum ‘sees’ the boundary, our quantum universe, as a boundary point. All Continuum/Boundary interactions are instantaneous in P-time. Energy interactions may be limited to a sequence of such instants, forming a quantum clock. With this clock, no energy is transferred, as no P-time passes.

From the perspective of wave-particle duality, the holographic interaction features waves propagating in C, producing dimensionless point particles projecting in B.

Addendum: Wave-particle Duality

We model the universe as a domain of entangled QESTs (Quantum Elements of Spacetime). These form “quantum foam” when in a state of disentanglement. When entangled, they become a coherent, discrete spacetime with a consistent metric. The global event creating our spacetime occurred with the first ‘tick’ of the discrete clock driving entanglement. We called this event the “Big Entanglement”.

We visualized this model as an extension of the AdS/CFT correspondence, which is an example of the holographic principle. With this correspondence model, a continuous spacetime arises from entanglement of a lower dimensional quantum domain. The latter is viewed as a ‘quantum boundary’ of the AdS spacetime.

Our model “extension” is more the converse of the usual holographic model. We see QESTs as discrete elements of a continuous domain. They form the “quantum boundary” (QESTs are elements of the Continuum). Our discrete, quantum spacetime arises when these QESTs become entangled.

Gravity is explicitly encoded on these boundary QESTs. Our quantum spacetime, including all particles, is a domain of measure zero. It is of lower dimension than the Continuum, which is of non-zero measure. The latter has an uncountably infinite dimension.

We extended this model of spacetime to the level of particles. They form their own individual spacetimes, also from entangled ‘boundary’ QESTs. Their local properties, including gravity (which we see as static mass), are encoded on their own local spacetime ‘boundaries’.

The dynamics of gravity in the Vacuum Network that forms our universe evolve as discrete foliations. General relativity describes the continuous time, tensor form for these dynamics. In our model, such precise dynamics could only be propagated in the Continuum.

This brings the wave-particle duality of particles into clearer focus. The wave equations that describe particle dynamics also propagate in the Continuum, with complete precision. These provide instant-by-instant state probabilities which determine the quantum properties that are ‘projected’ in discrete time. The state equations update with each ‘tick’ of the entanglement clock.

We can not directly observe the waves described by these state equations. We observe only their ‘strobed’ values, expressing the quantum properties of particles through ‘projection’. These appear as dimensionless particles. However, the waves that describe them are real: they propagate in the Continuum.

We can visualize how wave-particle duality is fundamental to our model. Duality is the expression of structure: our universe is a finite dimensional quantum domain, expressed on the ‘boundary’ of an infinite dimensional Continuum.

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