[Fleegello originally wrote his critique of octan physics as section 2.3 of the *Principia*. It is listed separately here, as it frankly stands apart, and interferes with the general flow of that work. Most scholars agree that Fleegello did not introduce any novel physics in this section. Rather he took ideas already advanced by contemporary physicists, and adapted them to his own philosophical framework. Editorial comments are enclosed in brackets throughout the text, to distinguish them from primary content.]

The physical (pan)universe has been identified as the Physical Consistency Subfield (PCS) of the Consistency Ideo Field (CIF) – the complete set of abstract mathematical objects that both define physical (temporospatial) relationships and are compatible with consistency logic. The PCS may itself be naturally divided into multiple, logically self-contained physical universes, characterized by distinct physical laws and conditions. Our own universe would be one of these worlds.

Mathematical objects in the PCS may manifest (in part) as states of a physical system, or as **operators** representing **observables** or other entities that act on those states. If modern physics is a guide, they incorporate a broad class of multidimensional objects known as **dimensors** [including common **scalars**, **vectors**, more general **tensors**, and **spinors**]. Many useful dimensor operators (in particular, those representing observables) define **linear** relations between other dimensors.

In standard **Shrodiik [quantum] theory** [named in honor of the pre-Dracian physicist Shrodo], a physical state (of any sufficiently isolated system) is denoted by a **ket** symbol |*ψ*>, where *ψ* is an arbitrary label. This entity is supposed to encompass all physical aspects of a system. |*ψ*> was originally interpreted in terms of the positions and motions of material particles at a time *t* in a pre-existing three-dimensional (3D) space ** x** [here bold type indicates a physical 3D vector]. Observables then include the positions, energies and momenta of these particles.

Contemporary Shrodiik physics has a most peculiar feature. For any physical state |*ψ*>, only the *probabilities* for measuring different values of a given observable can be computed. Even granted complete knowledge of a physical system at a particular moment, the future course as seen by any octan observer cannot in general be predicted with certainty. Detailed prescriptions for computing probabilities may be found in Shrodiik physics texts.

For any physical system, there is a range of possible states. Related to its probabilistic character, |*ψ*> can consist of a linear combination, or **superposition**, of these available states. The selection of a set of fundamental **basis** states is then arbitrary, to some extent; any given set of basis states can be mixed into new combinations, to form distinct sets.

In general, |*ψ*> can be viewed as a **vector** in an abstract **space** that spans all the possible states. If the components of a state vector are defined with respect to a specified set of basis vectors, then the state may be represented by a single-column dimensor array. **Linear operators** may in turn be represented by square dimensor arrays that transform any given state (by the rules of **matrix multiplication**) into another state.

Let *Â* represent a (linear) operator corresponding to an observable *A*. Here the symbol "hat" explicitly denotes operator, versus numeric parameter, status. When *Â* is applied to an arbitrary state vector, the result is typically a linear combination of other state vectors. Suppose, however, that *Â* is applied to a state vector |*ψ _{a}*> characterized by a well-defined value

*Â*|*ψ _{a}*> =

This is what it *means* for *Â* to represent an observable. Mathematically, |*ψ _{a}*> is an

In **bra-ket** notation [physicists used this archaic script during Fleegello's era], the numeric overlap between two states |*φ*> and |*ψ*> is represented by <*φ*|*ψ*>, where the states are normalized such that <*ψ*|*ψ*>=1 for all |*ψ*>. The overlap value is a **probability amplitude** for starting with a system in state |*ψ*>, but observing it in state |*φ*>. The actual probability is the absolute square of this amplitude, or |<*φ*|*ψ*>|^{2}. For example, consider a one-particle system. If |** x**> is the state with the particle at 3D position

The **expectation value** of an observable *A*, defined as the average value *A*_{avg} over repeated measurements on identical states |*ψ*>, is given by

*A*_{avg} = <*ψ*|*Â*|*ψ*> .

The observable *A* has a definite value *a* only if |*ψ*> is already an eigenstate |*ψ _{a}*> of

In classical physics, material particles were treated as localized entities, distinct from **waves** (such as light) that propogate through underlying fields or media. Only waves could undergo self-**interference**, or **diffract** around obstacles. At the dawn of the Shrodiik revolution, ostensible particles were found to have wave properties, and nominal waves were found to sometimes act like classical particles. Observables that classically had a continuous range of values – e.g., the energy of an electron in an atom – might now be **quantized**, or restricted to discrete values.

**Energy** is one of the central observables in quantum physics. It is associated with the **Hoobitean** operator *Ĥ* [named for the classical physicist Hoobitu]. For material particles, *Ĥ* is often written as the sum of a kinetic energy term *Ĥ*_{o} plus a potential energy (interaction) term *Ĥ*_{int}. After wave-particle duality was discovered, the energy *E* of a (set of) particle(s) in an eigenstate of *Ĥ* became associated with a temporal **frequency** *f*, or **angular frequency** *ω*:

*E* = ℎ *f* = ℎ *ω* / 2𝜋 = ℏ *ω* ,

where ℎ is the minuscule but nonzero **Planko constant** [named for the pioneering physicist **Planko**], and ℏ is the reduced Planko constant. Conversely, experiments showed that the energy in a traditional wave of frequency *f* was not continuously distributed over the wave, but carried by discrete quanta with individual energies given by the same equation. *Ĥ* embodies the way a state changes in time. When *Ĥ* acts on a state vector |*ψ*>, the result is the constant (*i*ℏ) multiplied by the time rate of change of |*ψ*>, where *i* is the **imaginary unit** (square root of -1). It is remarkable how imaginary (or **complex**) quantities arise naturally in the equations of Shrodiik physics!

A 3D vector quantity closely related to energy is **linear momentum**, represented by the symbol * p*. Much as energy is associated with temporal frequency, momentum

*p _{x}* = ℎ / λ

Shorter wavelength (and so either larger momentum, or a smaller value of ℎ) generally begets more particle-like behavior. When the operator *p*̂* _{x}* acts on a state vector |

For a single material particle, the relationship between time and energy is thus analogous to that between spatial position and linear momentum. For a multiparticle system, however, the situation is more nuanced. Whereas every particle may be assigned its own dynamical position and momentum operators, all particles traditionally share a common time. Time is then treated as a numeric system parameter, and not associated with a true operator.

Using **calculus**, it can be shown that the (unnormalized) time- and space-dependent wavefunction *ψ*(*t*,*x*) of a particle (or any quantum) with pure angular frequency *ω* and wavenumber *k _{x}* (energy

*ψ*(*t*,*x*) = *e*^{-iωt} *e*^{ikxx} = *cos*( *k _{x}x*-

where *e* is the **Eulero number** of mathematics, and the "cos" and "sin" terms refer to standard **trigonometric functions**. Note that the relative probability at any moment for finding the particle at a given position (equal to the absolute-square of the wavefunction) is the same for *all* *x* values.

With quantum physics, it is found that the order in which observables are measured may be significant. Consider then two observables *A* and *B*, represented by operators *Â* and *B̂*. The observables/operators **commute** if the order of measurement is irrelevant, or equivalently if the order in which *Â* and *B̂* act on an arbitrary physical state is irrelevant – i.e., if *ÂB̂* = *B̂Â*. They do not commute if *ÂB̂* ≠ *B̂Â*. In this case the very act of measuring *Â* or *B̂* introduces uncertainty into the value of the other, **complementary** observable. A system cannot simultaneously be an eigenstate of two operators that do not commute – neither operator would alter such a state, so their order could not matter. It is then impossible to simultaneously measure the values of two non-commuting observables, since such a measurement must create an eigenstate of both.

In Shrodiik mechanics, the archetypal pair of observables with non-commuting operators are the position *x* and linear momentum *p _{x}* of a material particle along a given spatial direction. Classically, these quantities commute, and every particle simultaneously has well-defined position and linear momentum. Based on their operator interpretations, the quantum

*x̂ p̂ _{x}* -

Because the operators do not commute, an eigenstate of *p*̂* _{x}* must span a range of

*σ _{x}*

Consider then a system |*ψ*> = |*x _{o}*>, in which a particle initially has a definite position

Consider now a more general system in which one member of any pair of non-commuting observables is well defined. Mathematically, the system can be considered a **superposition** of pure eigenstates with different but well-defined values of the other non-commuting quantity. The existence of non-commuting observables is contrary to classical (pre-Shrodiik) physics. The natural law that describes physical evolution applies to superpositions of pure states, rather than to states in which all classical variables have precise values.

The state |*ψ*> of a physical system can in general be written as a coherent sum

|*ψ*> = *C*_{1}|*φ*_{1}> + *C*_{2}|*φ*_{2}> + *C*_{3}|*φ*_{3}> + . . . = Σ_{j} *C _{j}*|

over a complete set of orthonormal states |*φ*_{j}>, where the *C*_{j} are (complex) constants.

The |*φ _{j}*> are orthonormal if <

The choice of the |*φ _{j}*> is arbitrary to some extent, but they must be eigenstates of a

The probability of starting with the system in state |*ψ*> but finding it in a state |*φ _{j}*> is then

*A*_{avg} = <*ψ*|*Â*|*ψ*> = Σ_{j}Σ_{k}*C _{j}*

The off-diagonal terms *j* ≠ *k* in the sum represent nonclassical **interference** between the different states in the coherent superposition comprising |*ψ*>. These terms in general vanish only if the |*φ _{j}*> have well-defined values of

The **physical interpretation** of the state vector |*ψ*> has a long and tortuous history. **Originally** it was viewed merely as a device for computing the probability of observing a given outcome in an experiment. Reality was seen to reside in the observed positions and momenta of individual particles. The physical universe was assumed to evolve in a linear manner, with a single unfolding history, which was deterministic in only a limited, probabilistic sense. The act of observation was divorced from the natural evolution of a physical system, and treated as something special, even magical.

Yet consistency logic requires that the universe be totally deterministic. Recently, the contradictions inherent in the original interpretation of Shrodiik mechanics have led the **Evette** group to develop an alternative **multi-world** view, in which reality resides in |*ψ*> itself. Observers, measuring devices and related processes are now included as integral parts of |*ψ*>. The physical panuniversal |*ψ*> is viewed as a superposition of conventional quantum worlds, each represented by a restricted state vector, which have become **decohered** and mutually orthonormal [or minimally overlapping]. These worlds evolve [almost] independently of each other, and continually split [rarely merge] into new separate, decohered worlds through time. A given observer occupies one conventional world at a given instant. As this world subsequently splits, the observer likewise branches into multiple selves, each with a distinct future experience. An observer does not see physical evolution as completely deterministic simply because no individual mind encompasses all worlds of the unfolding panuniversal state.

[Unfortunately, only the barest references to the original Evette School survive in the historical record. The writings may have been systematically destroyed by conservative, fundamentalist religious sects that flourished at the time, and found the work heretical. These traditionalist factions believed the universe progressed in a linear fashion along a single preordained path, in accordance with a divine plan for the octan race. The random, branching character of the multi-world view demanded an even greater, and to many more threatening, decentering to the octan psyche than the recognition five octujopes earlier that Jopitar was not at the physical center of the universe, but was a nonsingular ball of ordinary matter orbiting a commonplace star in a minuscule corner of a vast ocean of space and time.]

Let |*ϕ*_{0}> represent a conventional world in the Evette sense. Then the matrix elements *A*_{0j} between this world and any other coexisting conventional world |*ϕ _{j}*> must be [essentially] zero for all observables

Suppose that |*ϕ*_{0}> incorporates a subsystem consisting of a simple superposition ( |*B*_{1}> + |*B*_{2}>) of orthonormal eigenstates of an observable *B*. Then

|*ϕ*_{0}> = ( |*B*_{1}> + |*B*_{2}> ) |Ɛ>

where |Ɛ> represents the environment of the subsystem. The environment may interact with the subsystem, so as to become correlated with its eigenstates. This happens in particular when |Ɛ> includes an observer who measures the value of *B*. If *B̂* commutes with the interaction Hoobitean *Ĥ*_{int}, the eigenstates of *B̂* are not changed by the interaction, and

|*ϕ*_{0}> ➜ |*B*_{1}> |Ɛ_{1}> + |*B*_{2}> |Ɛ_{2}> = |*ϕ*_{1}> + |*ϕ*_{2}>

where |Ɛ_{1}> and |Ɛ_{2}> are themselves eigenstates of observables that commute with *Ĥ*_{int}.

Consider now the matrix elements *A*_{12} and *A*_{21} with |*ϕ*_{1}> and |*ϕ*_{2}> for an arbitrary observable *A*. If *Â* commutes with *B̂*, then *A*_{12} = *A*_{21} = 0, since the eigenstates of *B̂* are orthonormal. If *Â* does not commute with *B̂*, then it acts only on the *B* subsystem (if *Â* were a product of operators that separately act on the subsystem and its environment, then *A* would not be a valid observable). In this case *Â* does not affect |Ɛ>, and *A*_{12}=*A*_{21}=0 if |Ɛ_{1}> and |Ɛ_{2}> are orthonormal. The states |*ϕ*_{1}> and |*ϕ*_{2}> can thus be identified as two new conventional worlds, split off from the original |*ϕ*_{0}>, if only |Ɛ_{1}> and |Ɛ_{2}> are orthonormal.

[This line of reasoning, which did *not* originate with Fleegello, helped resolve a problem with the many-world interpretation, involving an apparent ambiguity in the identification of the individual worlds. Some researchers argued that the choice of states |*B*_{1}> and |*B*_{2}> in the given example was quite arbitrary. By choosing a rotated basis set, e.g.

|*b*_{1}> = (|*B*_{1}> + |*B*_{2}>)/√2 and |*b*_{2}> = (|*B*_{1}> − |*B*_{2}>)/√2 ,

the state |*ϕ*_{0}> appeared to split into a different set of conventional worlds. Eventually it was realized that the interaction between a system and its environment naturally selects a particular (compatible) basis set. If the operator *B̂* does not commute with *Ĥ*_{int}, then |*B*_{1}>|*E*> does not evolve into |*B*_{1}>|*E*_{1}>, since |*B*_{1}> is itself transformed by the interaction.]

Conventional worlds can thus be distinguished by non-interfering "memories" of prior branchings. The storage sites of these data may include, but are by no means limited to, animal brains (and recently, scientific apparatus acting as extensions of those brains). The physical structure of a brain determines its interactions with the environment, and thus the types of conventional worlds (i.e., which observables are relevant and well-defined) generated by the observation process. If a brain is so constructed that only one value of a particular observable can communicate with (affect) other elements in a conscious field, then a state including a coherent superposition of different values of that observable at the same moment must correspond to distinct unified ideo fields, or selves, in separate (conventional) worlds. The information stored in a brain does not *define* the external reality of the associated world – a person may make faulty observations – but it may still be a point of reference by which that world is distinguished from others. Two distinct conventional worlds can even merge, if their distinguishing memories are lost or corrupted so as to become identical. Observers inhabiting the worlds would experience no sense of merger, as all valid memories of a former distinct past would be absent.

What determines useful observables, other than position? The mathematician **Noethra** has linked many such quantities to **symmetries** in the equations of motion that describe the temporal evolution of |*ψ*>. Noethra's **first theorem** states that for every continuous, **differentiable** coordinate **transformation** that does not alter these equations, there is a corresponding observable whose expectation value is **conserved**, or constant over time. For sufficiently isolated (closed) systems, the equations are in fact generally unaffected by several such transformations, including time displacement, spatial displacement, and spatial rotation. Each of these symmetries is associated with an observable and conserved quantity.

Why are the dynamical equations unaffected by the given transformations? Although physical conditions clearly vary at different locations in time-space, there is nothing else to distinguish points or directions. From an ideobasic perspective, physical law for a sufficiently closed system (which incorporates all relevant causal agents) should then depend only on extant physical conditions. Although distinct physical laws may apply in different physical universes, the same law and dependencies should apply at all times, positions and orientations within a given universe. This leads to the observed symmetries.

When the equations of motion are not affected by displacements in time (i.e., they remain the same over time), then what is commonly called **energy** is conserved. This is primarily what makes energy a useful observable. Note that only the laws of motion are unchanging; physical conditions and entire systems may change dramatically over time. When the equations of motion are not affected by displacements in spatial position (i.e., physical law is the same at different spatial points), then **linear momentum** is conserved, and is a useful observable. When the equations of motion are not affected by spatial rotations (orientation in space), then **angular momentum** is conserved, and useful. It can be shown more generally that the expectation value of any operator that both commutes with the Hoobitean operator *Ĥ*, and is not explicitly a function of time, is also a **constant of motion**. Every symmetry in *Ĥ* is thus associated with a conserved quantity, and a corresponding observable.

Classical observables may have nonclassical analogs that result from a reinterpretation (typically involving commutation relations) of associated operators. In particular, the commutation relations among the three orthogonal (mutually perpendicular) angular momentum operators imply the existence of a nonclassical type of angular momentum, known as **spin**. Elementary particles are found to inherently possess this type of angular momentum. Particle spin is naturally quantized to discrete values, characterized by a **spin number** *s*, which must be an integral multiple of 1/2. Overall spin angular momentum is ℏ√*s(s+1)*, while the maximum possible component in any 3D direction is ℏ*s*.

Spin angular momentum operators can be represented by **irreducible** (2*s*+1) x (2*s* +1) arrays. The spin aspect of a spin-s particle can then be represented by a (2*s* +1)-dimensional single-column dimensor known as a pointor, designated by Š. An overall single-particle state may in turn be represented by a pointor wavefunction Š(*t*,* x*) of time

Spinless (*s*=0) particles are represented by simple scalar (zero-rank dimensor) functions, with no inherent directionality. [No elementary spin-0 particles were known in Fleegello's era, although composite spin-0 particles (e.g., **pions**) were certainly recognized.] Spin-½ particles are represented by special two-dimensional pointors known as **spinors**. Spinors do *not* transform like geometric vectors under coordinate transformations. Spin-1 particles with mass are represented by three-dimensional pointors, which do transform like geometric vectors. [Because massless spin-1 particles (in particular **photons**) have no rest frame but are constrained to move at light speed, they must be represented by two-component pointors.] Particles with even larger spin values are represented by distinct pointor classes.

Yet particles do not normally exist in isolation. How then can the state of a multiparticle system be represented? Suppose first that the particles are distinguishable, and motions are much slower than light speed. Such systems have traditionally been represented by a **direct product** of the pointor functions for the individual particles, in which time *t* is a common system parameter, but the coordinates * x_{j}* of the various particles

Š_{a}(*t*,** x_{1}**) Š

where subscripts *a* and *b* label two different single-particle states.

Suppose now that two particles in a system are **identical**. The probability of finding either cannot be affected when their labels are exchanged – they would otherwise be distinguishable. Because the probability is equal to the absolute square of the wavefunction, and the associated exchange operator must (as an observable) be linear, the state can at most acquire a complex **phase factor** (absolute value one) under particle exchange. Since two successive exchanges must leave the state unchanged, the phase factor is limited to the values ±1. A state must then be either symmetric (unchanged) or antisymmetric (phase factor -1) under identical particle exchange.

The wavefunctions of identical **bosons** (particles with integral spin) are found to be symmetric, while those of identical **fermions** (particles with half-integral spin) are antisymmetric. The appropriate symmetry can be achieved if a system is represented by a sum over the direct pointor products, in which the functional dependencies of the particles are suitably interchanged. For example, the state of two identical fermions might be represented by

Š_{a}(*t*,** x_{1}**) Š

Symmetries in the equations of motion are not limited to continuous time-space transformations, but may also include discrete operations, such as **time reversal** and **parity inversion** (mirror reversal). [Fleegello stubbornly maintained that various discrete spacetime symmetries should generally hold, despite contrary evidence. For example, experiments seemed to demonstrate that **parity is not conserved** during certain types of radioactive decay. Parity is conserved if the equations of motion are unchanged when a system is replaced by its mirror image. Fleegello believed that physics could not be affected by such a simple transformation, and felt that crucial elements had been omitted from experimental analyses. Yet physicists soon realized that, since time and space are intimately linked, and

Indeed, the **fundamental interactions** between elementary particles are thought to derive from a variety of internal **local gauge symmetries**. For example, consider the **electromagnetic interaction**. Under a local phase transformation, the single-particle wavefunction *ψ*(*t*,* x*) is multiplied by a

The physicist **Vigno** has argued that symmetries do not merely restrict the laws of physics, but further *define* much of physical reality. While fundamental forces have been related to symmetries in the equations of motion, elementary particles have themselves been associated with (irreducible) mathematical representations of abstract **symmetry groups**. Every consistent object and process must coexist with every other consistent object and process somewhere within the PCS. This may involve a natural segregation into distinct, self-contained physical universes.

Coordinate systems do not exist a priori in nature. The choice of a coordinate framework to describe a physical system should thus be arbitrary, from a strictly mathematical viewpoint (although one frame may be more convenient than another for a given purpose). It should then be possible to describe the laws of physics in a coordinate-free manner, in which observables appear only as abstract quantities, with no explicit reference to coordinate components. Expressing physical laws in such a **covariant** manner simplifies identification of symmetries and conserved quantities.

If the PCS is to respect the inherent arbitrariness in the choice of coordinate system, then fundamental **physical constants** that appear in the laws of physics should also be the same for all observers within a given physical universe, independent of the choice of reference frame. This applies in particular to **dimensionless** constants (e.g., the **fine structure constant** of atomic physics), which carry no physical units, but can be expressed as the ratios or products of **dimensional** constants that do possess units. Changes in the values of dimensional constants are generally meaningful only with respect to changes in their dimensionless combinations. So long as the values of physical constants are individually changed in a way that maintains the values of all fundamental dimensionless constants, the physical world is unaffected. Dimensionless constants stand independent of any arbitrary choice of measurement units. Indeed, no variations over time or space have thus far been detected.

[Some quantities thought to be fundamental constants in Fleegello's era have since been found to be variable. These have been reinterpreted as functions of truly fundamental constants and local physical conditions.]

Dimensionless fundamental constants need only be the same at all points within a particular physical universe. The values in distinct, non-interacting universes may be different. If there is no fundamental reason a constant should have a particular value, then the PCS *must* encompass a host of universes covering the range of acceptable values. Yet these values must be countable (either discrete/quantized, or at least represented by rational numbers). All the worlds otherwise could not have meaningful existence within the PCS.

Even fundamental dimensional constants (whose numeric values depend on the choice of physical units) should be the same for all observers in a given universe, when measured with respect to reproducible units characteristic of fundamental physical processes. In particular, the **speed of light** in a vacuum, commonly denoted by the symbol *c*, appears to constitute a universal limit to the rate at which information can propagate through space. As first proposed by the physicist **Niestu** in his **theory of inertial invariance**, the speed *c* has the same value for all observers, irrespective of their state of motion. This is contrary to classical expectations, whereby an observer moving toward (away from) a light source detects a higher (lower) relative light speed than an observer at rest with respect to the source. That *c* is finite may be expected from an ideobasic viewpoint. An infinite speed is a special, limiting case of a general value, and the PCS should opt for the most general conception.

Niestu introduced a major paradigm shift in physics when he showed that a common value for *c* implies that time (space) intervals measured by one observer may be partially seen as space (time) intervals by an observer in a relative state of motion; time and space do not exist separately, but must be combined into a unified **timespace** [scientists of Fleegello's era apparently preferred this expression to today's more common term *spacetime*]. The effect is tiny at low velocities, but becomes significant as speed approaches *c* (so-called Niestiik speeds). The associated coordinate transformation between reference frames in a relative state of motion is distinct from that of classical physics. If the equations of motion are to remain invariant under a velocity transformation, then those equations must be modified as well. A remarkable consequence of inertial invariance is that any mass *m* is associated with an energy *mc*^{2}. For a free particle, the relationship between total energy *E*, momentum *p*, and rest mass *m* becomes

*E*^{2} = *p*^{2}*c*^{2} + *m*^{2}*c*^{4} .

Niestu ultimately expanded his ideas into the **theory of general invariance**, which describes gravity in terms of distortions in the geometry of timespace.

[Fleegello overlooked a related serious inconsistency in his view of the CIF. The CIF must encompass all possible reference frames. If It experiences the same time as observers in those frames, as Fleegello envisioned, It must integrate the various time lines to maintain a single unified state of being. Yet if speed *c* is the same for all observers, events that are simultaneous in one frame may be *non*simultaneous in another. Events could then be seen by the CIF as both simultaneous and not simultaneous, a contradiction. This inconsistency is resolved only if the CIF transcends physical time, and experiences it the way corporeal creatures experience space – as **block time**. All events in the physical panuniverse then span a single, eternal moment in the mind of the CIF. Yet the CIF must still distinguish the time-like and space-like separations among physical events that define causal chains. Primacy resides in these causal chains, and not in the reference frames that observers use to describe them.]

While inertial invariance was readily incorporated into Shrodiik mechanics for single particles, problems arose for multi-particle systems. In particular, time and space coordinates were not treated coequally in the traditional equations of motion. Inertial invariance requires that time and position *both* be treated either as system parameters, or as formal operators. Currently the most widely adopted solution, based on the first approach, is to reformulate Shrodiik mechanics into a Niestiik **quantum field theory** (QFT), in which elementary particles of a given type are treated as quantum excitations of an underlying **field**. QFT covers both traditional particles with mass, like the electron, and zero-mass particles once considered pure waves, like the photon. Different particle types are represented by distinct fields, defined by a variety of attributes including rest mass, spin, and electric charge. For each field type *k*, a position **field operator** *Φ̂ _{k}(t,x*) and conjugate momentum field operator

A simple field state in QFT is characterized by the number of (identical) **quanta** occupying each of a set of allowed levels. The number of quanta is just the number of "particles" of the given type. Field quanta contain no explicit particle labels; QFT respects the exchange symmetry of identical particles in a remarkably natural way. Indeed, in QFT it can be shown that fields with half-integral spin must be antisymmetric, and those with integral spin symmetric [dictated by the distinct Niestiik equations of motion for fermions versus bosons]. Many physicists prefer not to speak of particles at all in QFT, but only quanta. A general field state can be represented by a superposition of simple states. Unlike in Shrodiik mechanics, this is not limited to states with a fixed number of particles; interactions between fields result in the routine creation/destruction of quanta. The overall state of a system is represented by the direct product of its constituent fields, or more generally by a superposition of such products.

Shortly after QFT was introduced, the mathematician Draci proposed a multi-time theory (MTT) alternative, in which both the (observer-based) times and positions (*t _{j}*,

A later version of MTT was a more radical departure from single-time theory, but more attuned to inertial invariance and the vagaries of measurement, and is adopted here. Every particle *j* in MTT has an innate **proper time** dimension, measured along the particle's **world line**. Different time lines are *not* inherently synchronized, but correlated only through interactions. To locate at time *t*_{o} the position * x_{j}* of particle

The associated wavefunction *Ψ*_{mt} must then separately incorporate the reference observer time *t*_{o} at which the (*t _{j}*,

The given *Ψ*_{mt} is a joint probability amplitude that, when the observer finds itself at time *t*_{o}, it also sees itself extended to **x**_{o}, and each particle *j* at *t _{j}* (corresponding to

Note that MTT does *not* posit that any given material particle evolves along multiple time dimensions. Each particle *j* is assumed to evolve along a single proper time line *τ _{j}*. The corresponding

If there are no particle interactions, *Ψ*_{mt} should separately satisfy the free-particle equations of motion for each particle *j*, as well as for the observer. If there are also no observer-particle interactions, then there is no way the observer can detect the particles, define observer-particle distances, or establish the relation between *t*_{o} and the *t _{j}*. Then

where

Here

With interparticle forces, interaction terms must be introduced into the equations of motion, resulting in *N*+1 coupled equations of motion for an *N*-particle plus observer system. Draci has shown that, for any MTT, the respective interaction terms must satisfy a consistency condition, or the evolution of *Ψ*_{mt} along the various *t _{j}* is not well-defined. Finding consistent MT equations with interactions is a challenge. In particular, conventional interaction potentials always lead to inconsistency. Interactions must then be implemented by other means, in particular through the use of

It is often useful to adopt composite MT timespace coordinates to replace the (*t _{j}*,

* X* = (

**center-of-mass** (COM) coordinates with *a _{j}* =

*T* = *a*_{1}*t*_{1} + *a*_{2}*t*_{2} and *ρ*_{12} = (*t*_{2} - *t*_{1}) .

The (*t*_{o},**x**_{o}; *t*_{1},**x**_{1}; *t*_{2},**x**_{2}) are now replaced by (*t*_{o},**x**_{o}; *T*,* X*;

Note that (*t*_{o},**x**_{o}) is already inherently composite, if the observer (including measurement devices) is a macroscopic entity composed of myriad material particles. If the observer reference frame is at rest with respect to the observer spatial COM, and internal time lines are sufficiently synchronized, then (*t*_{o},**x**_{o}) may be identified as composite 4D timespace COM observer coordinates (where some quantum jitter is expected around the average value of **x**_{o}). While this (*t*_{o},**x**_{o}) could in principle be included in the definition of (*T*,* X*), the observer would then become part of the system being observed, confusing its status. The observed complex would also become a three-body system, requiring a new set of 4D displacement coordinates.

The observer is nonetheless integrally connected to the observed two-particle system. Interactions between the observer and the two particles *define* the COM motion of that system. If the two observed particles are otherwise isolated, then ideally interactions between these particles directly determine the primary dependence of *Ψ*_{mt} on both *ρ*_{12} and **r**_{12}, but interactions with the observer must inevitably have some residual effect.

The (*T*,* X*) and (

Energy / linear momentum eigenfunctions for two free particles can now be written

*Ψ*_{mt}(free) ∼ *e*^{-iωoto+iko·xo} *e*^{-iΩT+iK·X} *e*^{-iωρρ12+ikr·r12}

where (*Ω*,* K*)=(

Both QFT and standard MTT assume a pre-existing, observer-based, four-dimensional (4D) timespace framework (*t*, * x*). Even if this timespace is affected (warped) by matter and energy, it is not created by them. Yet what is the origin of timespace itself, and how are its coordinates meaningfully defined at all for a multiparticle system? Neither time nor space can be measured in absolute terms. Temporal and spatial intervals are gauged only with respect to physical processes and structures, which are traditionally interpreted in terms of elementary particles and their interactions. Stripped of these vestments, timespace loses all meaning. Physical objects and dimensions of relation are inextricably linked.

As mentioned earlier, every elementary particle with mass does have an innate proper time dimension. For every external observer time *t _{j}* in standard MTT, there is a corresponding internal proper time

Interactions corresponding to the **fundamental forces** are associated with gauge symmetries. In QFT, their description can be interpreted in terms of the exchange of **phantom** elementary gauge bosons by elementary fermions. The **electromagnetic**, **weak**, and **strong** interactions involve the exchange of phantom photons, **W** and **Z bosons**, and **gluons**, respectively (all spin-1). Phantom particles have all the attributes of their real counterparts, except mass; the usual relationship between energy, momentum and rest mass is not followed, making these particles ephemeral. Many physicists consider phantom particles not real in *any* sense, but merely a bookkeeping device. Elementary fermions include **electrons**, **neutrinos**, and **quarks** (all spin-1/2). [Fleegello's archaic list omits various types of **invisible matter**, that interact with ordinary matter solely through the gravity.] Only the gravitational force, which is ostensibly associated with the exchange of phantom **gravitons** (normally massless, spin-2), has eluded incorporation into the QFT framework.

Although phantom particles were introduced in QFT, gauge symmetries in MTT should lead to expressions for interactions with an analogous interpretation. Recall that individual particles can be identified in MTT, but *not* in QFT. Because phantom bosons in either approach are superpositions spanning energies and momenta that do not respect standard mass relationships, their exchange should not be literally interpreted in terms of particle trajectories. Yet they do link interacting particles, and transfer information at light speed. Fermions may also be linked and transfer information via real gauge bosons. The exchange of either phantom or real particles can thus establish causal links (CLs) – phantom causal links (PCLs) or real causal links (RCLs), respectively. Because individual particles can be identified in MTT, but not in QFT, fully exploiting CLs requires an MT approach.

Consider then a multi-time CL of any type from fermion *j* at proper time *τ _{j}* to fermion

Define an elementary event as any point on the world line of a real elementary particle at which a CL is established with another particle. The physicists Machi and [later] Niestu have promoted the radical idea that the network of CLs among particles does not merely occur *within* timespace, but even *defines* timespace. The number of spatial dimensions is set by the number of components in * u_{jk}*. For example, consider the pair of electromagnetic links between two charged particles #1 and #2 in the diagram at right. Suppose a CL (dotted line) connects #1 (solid line) at proper time

The spatial location of particle 2 from the perspective of particle 1 (the "observer" in this case) at *τ*_{1o} = (*τ*_{1A}+*τ*_{1C})/2 is

**r**_{12B} = *r*_{12B} **u**_{1AB}

where **u**_{1AB} is a 3D unit vector pointing in the spatial direction of flow from A to B from the perspective of particle 1, and *r*_{12B} is the scalar interparticle distance

*r*_{12B} = (*τ*_{1C}-*τ*_{1A}) *c*/2 .

From the same perspective, **u**_{1BC} = -**u**_{1AB}. Note that neither *τ*_{2B} nor the relative interparticle speed *v* appear in these equations. If *τ*_{1} and *τ*_{2} are not perfectly correlated (by the probability distributions for all possible interactions, which coincidentally define *v*), a range of *τ*_{2B} values could give the same result. Timespace coordinates for event B are (*τ*_{2B},0) from the perspective of #2. Time *τ*_{1B} should be related to *τ*_{2B} by at least a probabilistic, Niestiik-like (due to *v*-dependence) transformation. For non-Niestiik motion, one can set *τ*_{1B}~*τ*_{2B} based on the single interaction. However, *τ*_{1B} must equal *τ*_{1o} *only* if *τ*_{1} and *τ*_{2} are perfectly correlated. More generally, *τ*_{1B} has a range of possible values, corresponding to the range of *τ*_{2B}, but (with even minimal synchronization) centered on *τ*_{1o}.

From the perspective of #2, the situation is more nuanced. It is *not* generally true that the two link paths are equal, or that **u**_{2BC} = -**u**_{2AB}. Interparticle distance *r*_{21B} at *τ*_{2B} now depends on *v*, with Niestiik corrections expected. While both *r*_{21B} and *v* can be related to the event times (*τ*_{2A}, *τ*_{2B}, *τ*_{2C}) from the perspective of #2 (an interesting exercise!), deriving *τ*_{2A} from *τ*_{1A}, and *τ*_{2C} from *τ*_{1C}, now depends on the correlation between *τ*_{1} and *τ*_{2}. The interaction defines a definite correspondence *τ*_{1o}=*τ*_{1B}=*τ*_{2B}, with a common interparticle separation *r*_{21B} = *r*_{12B}, *only* if time is absolute (as in classical physics), *and* the time lines are synchronized.

While multiple serial links are needed to correlate time lines and define interparticle distance, the properties of successive links must be compatible, and accord with physical principles. The directions of adjacent CLs should reflect a consistent sense in the flow of proper time, or distance is ill-defined. Relative speeds inferred from CLs must not exceed the value *c*. Changes in velocity (acceleration) should reflect the transfer of linear momentum by CLs.

Let *∩*_{12} represent the probability amplitude, from the prespective of particle 1 (the first index), for a CL connecting particles 1 and 2, with [partial] functional dependence *∩*_{12} (*τ*_{1}, *τ*_{2}, *θ*_{12}, **u**_{12}). Assume that *∩*_{12} is a complex quantity, and that the (real) connection probability per *τ*_{1}, *τ*_{2}, and solid angle in **u**_{12} is |*∩*_{12}|^{2}. There may be applications where it is useful to renormalize *∩*_{12} such that, for any (*τ*_{1},*θ*_{12}), the sum [integral] of Δ|*∩*_{12}|^{2} over (*τ*_{2}, **u**_{12}) is unity, where Δ is a (small) interval along *τ*_{1} [if proper time is discrete, Δ may be set to the minimum time interval]. For vanishingly weak interactions, *∩*_{12} is then the same, vanishingly small value for all (*τ*_{1}, *τ*_{2}), and time scales are uncorrelated.

The amplitude for link AB is *∩*_{12} (*τ*_{1A}, *τ*_{2B}, +1, **u**_{1AB}), and for BC is *∩*_{12} (*τ*_{1C}, *τ*_{2B}, -1, -**u**_{1AB}). Measured by 1 at *τ*_{1o}, distance *r*_{12} is defined by link pairs of like AB and BC, with *τ*_{1A}=*τ*_{1o}-*r*_{12}/*c* and *τ*_{1C}=*τ*_{1o}+*r*_{12}/*c*. For any *τ*_{2B} and *θ*_{AB}=+1, the average of |*∩*_{AB}|^{2} over **u**_{1AB} should peak at some *τ*^{p}_{1A}. For the same *τ*_{2B} and *θ*_{CB}=-1, the average of |*∩*_{CB}|^{2} over **u**_{1CB} peaks at some *τ*^{p}_{1C}. This defines a most probable connection between *τ*^{p}_{1o}=(*τ*^{p}_{1A}+*τ*^{p}_{1C})/2 and *τ*_{2B}. By evaluating *τ*^{p}_{1o} over a range of *τ*_{2B}, the function *τ*^{p}_{1o}(*τ*_{2}) is obtained. The width of the distribution |*∩*_{AB}*∩*_{CB}|^{2} along *τ*_{2B} is a measure of the correlation between *τ*_{1} and *τ*_{2} at *τ*^{p}_{1o}, which is *not* in general one-to-one. Single-time theory then cannot most accurately describe the system. The distribution width with respect to *r*_{12} at fixed (*τ*^{p}_{1o},*τ*_{2B}) is a measure of the uncertainty in *r*_{12}. As in standard Shrodiik physics, this may be nonzero even when *τ*_{1} and *τ*_{2} are well correlated.

Remarkably, ||*∩*_{AB}*∩*_{CB}|^{2} has units equivalent to a probability per 4D (*τ*_{2}, **r**_{12B}) timespace volume, just like the probability of an MT observer (here, particle 1) seeing particle 2 at *τ*_{2B} and **r**_{12B}. Apart from **x**_{1o} dependence, one can then identify a CL-related partial wavefunction *ɸ*_{mt}(*τ*_{1o}; *τ*_{2}, **r**_{12}), with

*ɸ*_{mt}(*τ*_{1o}; *τ*_{2}, **r**_{12}) ~ *∩*_{12} (*τ*_{1o}-*r*_{12}/*c*, *τ*_{2}, +1, **r**_{12}/*r*_{12}) *∩*_{12} (*τ*_{1o}+*r*_{12}/*c*, *τ*_{2}, -1, -**r**_{12}/r_{12}).

For a given *τ*_{1o}, the quantity |*ɸ*_{mt}|^{2} defines a joint probability distribution over (*τ*_{2B}, *r*_{12}, **u**_{1AB}). Amplitudes should be normalized such that the sum [integral] of |*ɸ*_{mt}|^{2} over (*τ*_{2}, **r**_{12}) is unity.

A student of Draci [likely inspired by QF] has investigated the idea that a physical object may even have PCLs to itself. A link connecting *τ*_{1a} to *τ*_{1b} on the world line of particle 1 effectively extends outward a distance *r*_{11}~(*τ*_{1b}-*τ*_{1a})*c*/2 at a time *τ*_{1o}=(*τ*_{1a}+*τ*_{1b})/2. For a lone particle 1, the corresponding amplitude may be represented by *∩*_{11} (*τ*_{1a}, *τ*_{1b}, *θ*_{11}, **u**_{11}). The CL-related partial wavefunction associated with self links is then

*ɸ*_{mt}(*τ*_{1o}, **r**_{1o}) ~ *∩*_{11} (*τ*_{1o}-*r*_{1o}/*c*, *τ*_{1o}+*r*_{1o}/*c*, +1, **r**_{1o}/*r*_{1o}) *∩*_{11} (*τ*_{1o}+*r*_{1o}/*c*, *τ*_{1o}-*r*_{1o}/*c*, -1, **r**_{1o}/*r*_{1o})

~ *∩*_{11}^{2} (*τ*_{1o}-*r*_{1o}/*c*, *τ*_{1o}+*r*_{1o}/*c*, +1, **r**_{1o}/*r*_{1o}) .

Self-distance is thus defined by a single link rather than a pair of links, and the two link amplitudes associated with *ɸ*_{mt}(*τ*_{1o}, **r**_{1o}) are identical. Self-links mark elementary particles as extended objects. They may even embody traditional force fields (e.g., the electric field of an isolated charged particle), and contribute to the rest-mass energies (frequencies) that dominate the time dependence of the complete multi-particle wavefunction *Ψ*_{mt} in the low-interaction limit.

In a two-particle system, a self-link may become correlated with links between the particles, such that the corresponding joint link amplitude is *not* a product of two simple amplitudes. Let ∧* _{jk}* represent the full joint link amplitude, including all self-links, for CLs involving two particles

Based on its relationship with the standard wavefunction, it is convenient to now redefine *∩ _{jk}* as a truncated amplitude only including CLs with at least one end anchored to the world line of particle

This scheme can be extended to systems with *N*>2 particles. Links between different particle pairs may themselves become correlated, such that joint link amplitudes cannot be written as products of one- or two-particle amplitudes. The *N*=3 joint amplitude for particles 1, 2 and 3 may be written ∧_{123}, and includes links 1-1, 1-2, 1-3, 2-2, 2-3, and 3-3. The truncated amplitude *∩*_{123} includes links 1-1, 1-2, and 1-3.

Because they depend on proper times with respect to distinct particles, the *∩ _{jk}* and ∧

In the traditional external-perspective MT (MTe) approach, a wavefunction is defined from the perspective of an external inertial observer, typically an organization of myriad physical objects that establishes a timespace frame (*t*_{o}, **x**_{o}), and provides a consistent perspective. To measure the position of a solitary particle 1, this observer must rely on CLs connecting its own world line at *t*_{o1} with the particle world line at *t*_{1o} (measured in the observer frame), where *t*_{1o} may be interpreted as the most likely *t*_{o} corresponding to proper time *τ*_{1o}. The full observer-based link amplitude may be represented by *Γ*_{o1} (*t*_{ooa}, *t*_{oob}, *θ*_{oo}, **u**_{oo}; *t*_{o1}, *t*_{1o}, *θ*_{o1}, **u**_{o1}; *t*_{11a}, *t*_{11b}, *θ*_{11}, **u**_{11}), and the truncated amplitude by *γ*_{o1} (*t*_{ooa}, *t*_{oob}, *θ*_{oo}, **u**_{oo}; *t*_{o1}, *t*_{1o}, *θ*_{o1}, **u**_{o1}). Function *γ*_{o1} embodies all information accessible to the observer by measurement, including self-extension and particle position as functions of observer time, and defines a partial wavefunction *ɸ*_{o1}, with

*ɸ*_{o1}(*t*_{o}, **x**_{o}; *t*_{1}, **x**_{1}) ~ *γ*_{o1} (*t*_{o}-*x*_{o}/*c*, *t*_{o}+*x*_{o}/*c*, +1, +**x**_{o}/*x*_{o}; *t*_{o}-*x*_{1}/*c*, *t*_{1}, +1, +**x**_{1}/*x*_{1})

*γ*_{o1} (*t*_{o}+*x*_{o}/*c*, *t*_{o}-*x*_{o}/*c*, -1, -**x**_{o}/*x*_{o}; *t*_{o}+*x*_{1}/*c*, *t*_{1}, -1, -**x**_{1}/*x*_{1}) .

For two particles 1 and 2, link amplitudes are *Γ*_{o12} and *γ*_{o12}, defining *ɸ*_{o12}(*t*_{o}, **x**_{o}; *t*_{1}, **x**_{1}; *t*_{2}, **x**_{2}). Links involving only particles 1 and 2 are defined in *Γ*_{o12}, but *not* *γ*_{o12}, though the two amplitudes must be compatible. *ɸ*_{o12} defines a joint probability that the observer at *t*_{o} sees a self-extension to (*t*_{o}, **x**_{o}), particle 1 at (*t*_{1}, **x**_{1}), and particle 2 at (*t*_{2}, **x**_{2}). Units of |ɸ_{mt}|^{2} are probability per 3D **x**_{o} spatial volume for the observer, and per 4D (*t _{j}*,

In the spirit of exploring how CLs might define timespace, members of the Draci group have introduced an alternative internal-perspective MT (MTi) approach, which does not rely on a pre-existing 4D timespace. Instead, the theory only assumes that every particle has its own innate proper time, and that CL (gauge) bosons are associated with mathematical objects that define common multi-D properties. Quantities are defined from the consistent but *non*-inertial perspective of a given particle, indicated by the first index in the MTi link amplitude.

For two particles 1 and 2, the full MTi link amplitude with respect to particle 1 is represented by *Γ*_{12} (*t*_{11a}, *t*_{11b}, *θ*_{11}, **u**_{11}; *t*_{12}, *t*_{21}, *θ*_{12}, **u**_{12}; *t*_{22a}, *t*_{22b}, *θ*_{22}, **u**_{22}), and the truncated link amplitude by *γ*_{12} (*t*_{11a}, *t*_{11b}, *θ*_{11}, **u**_{11}; *t*_{12}, *t*_{21}, *θ*_{12}, **u**_{12}). Here all times are measured with respect to the *proper* time of particle 1, so that *t*_{1j}=*τ*_{1j} for any *j*, and *t*_{2j} is the most probable proper time along the world line of particle 1 corresponding to the linked *τ*_{2j}. Note that for any (*τ*_{12},*τ*_{21}) there is a probability distribution over *t*_{21}, and for any (*τ*_{12},*t*_{21}) a distribution over *τ*_{21}, defined by *Γ*_{12}.

The quantity *γ*_{12} defines a CL-related partial wavefunction *ɸ*_{12}(*t*_{1}, **r**_{1}; *t*_{2}, **r**_{2}), with

*ɸ*_{12}(*t*_{1}, **r**_{1}; *t*_{2}, **r**_{2}) ~ *γ*_{12} (*t*_{1}-*r*_{1}/*c*, *t*_{1}+*r*_{1}/*c*, +1, +**r**_{1}/*r*_{1}; *t*_{1}-*r*_{2}/*c*, *t*_{2}, +1, +**r**_{2}/*r*_{2})

*γ*_{12} (*t*_{1}+*r*_{1}/*c*, *t*_{1}-*r*_{1}/*c*, -1, -**r**_{1}/*r*_{1}; *t*_{1}+*r*_{2}/*c*, *t*_{2}, -1, -**r**_{2}/*r*_{2})

This function is a joint probability amplitude that particle 1 at (proper) time *t*_{1} sees itself extended to **r**_{1}, and particle 2 at (*t*_{2}, **r**_{2}). The system may equally be represented by a wavefunction *ɸ*_{21}(*t*_{2}, **r**_{2}; *t*_{1}, **r**_{1}) from the perspective of particle 2, with link amplitudes *Γ*_{21} and *γ*_{21}.

The external- and internal-perspective approaches can both be extended to more complex systems with *N*>2 particles. If the various links become correlated, the CL amplitudes cannot be encoded using two-particle *Γ*_{ojk} and *γ*_{ojk}, or *Γ _{jk}* and

The internal-perspective approach does not rely on any pre-existing external reference frame; coordinates are defined internally by interconnections among a given set of particles. Yet there are difficulties with this approach. For any system of interacting particles, a reference frame anchored in one particular particle is inherently non-inertial. If particles are identical, exchanging indices further involves a change in coordinate prespective.

For non-Niestiik motion, these problems may be overcome by transforming coordinates to a center-of-mass reference frame through a spatial displacement vector. For two particles, the partial MTi wavefunction becomes *ɸ*_{12C}(*t*_{C1}, **x**_{C1}; *t*_{C2}, **x**_{C2}), where index "C" now indicates an inertial COM reference. If the particles are identical, *ɸ*_{12C} is easily symmetrized. Alternatively, composite COM coordinates (*T*_{C},**X**_{C}; *ρ*_{C},**r**_{C}) may be used, as decribed earlier. Both *T*_{C} and **X**_{C} are unchanged by particle exchange, while the signs of *ρ*_{C} and **r**_{C} are reversed.

If CLs transfer information between particles, their amplitudes should be functions of the associated quantities. Consider in this regard a simplified, two-particle MTi link amplitude *γ*_{12} (*t*_{12}, *t*_{21}, *θ*_{12}, **u**_{12}), where correlations with self-links and with other particles are ignored.

In addition to implicitly defining interparticle distance **r**_{12} from the perspective of particle 1, the given link may be associated with an energy and linear momentum transfer. Let *ω*_{12} be the angular frequency associated with energy transferred along **u**_{12}, and **K**_{12} the 3D **wavevector** associated with linear momentum transferred in the same direction, again from the perspective of 1. The quantity **K**_{12} should contribute to changes in relative particle velocity. The probability per *t*_{12}, *t*_{21}, solid angle in **u**_{12}, frequency, and 3D wavevector volume that a link between 1 and 2 exists is now the absolute square of *γ*_{12} (*t*_{12}, *t*_{21}, *θ*_{12}, **u**_{12}, *ω*_{12}, **K**_{12}). [Link amplitudes may also specify angular momentum transfer, and the type (phantom or real) of linking particle.]

Although information carried by a PCL effectively moves at light speed, such links do not represent real particles, so there is no fixed relationship between *ω*_{12} and *K*_{12}. Because a negative frequency *ω*_{12} moving in the positive time direction is equivalent to a positive frequency moving in the opposite *θ*_{12} direction [as well as a positive-frequency antiparticle moving in the positive time direction], then physically meaningful frequencies *ω*_{12} may be restricted to positive values.

The amplitude *γ*_{12} may in general be nonzero for **K**_{12} pointing in directions other than **u**_{12}. Whereas the *average* value of **K**_{12} is collinear with **u**_{12} for RCLs, this restriction does not apply to PCLs. Define **K**^{+}_{12} to be the vector component of **K**_{12} pointing along **u**_{12}, with **K**^{+}_{12}= *K*^{+}_{12}**u**_{12}. Forward wavenumbers *K*^{+}_{12} may be either positive, if an interaction is repulsive; or negative, if it is attractive. **K**^{+}_{12} respectively points in the same direction as **u**_{12}, or in the opposite direction.

For a PCL, it is straightforward to conserve linear momentum at both time-ordered link vertices regardless of the value *K*_{12}. However, energy is then *not* generally conserved. This is allowed, since PCLs are transient, and do not represent real particles. If frequencies closest to energy conservation are favored, then *ω*_{12}<<*K*_{12}/*c*, and the effective link mass is imaginary. The total energy of two particles may change (mainly due to momentum exchange) during a link interaction, reflecting a conversion of potential energy to or from kinetic energy.

If there is no interparticle motion, then **u**_{21}=**u**_{12}. If particle 2 moves relative to 1, then *ω*_{21} and the components of **u**_{21}, **r**_{21} and **K**_{21} fwith respect to 2 will differ from the values seen by 1. The functions relating the two perspectives should be compatible with Niestiik transformations for the given link (speed-*c* if a PCL or zero-mass RCL). Those equations maintain any collinearity between * K* and

[Fleegello failed to realize that he unnecessarily restricted each CL to the transfer of a single quantum of information. A CL may actually transfer one, two, or any number of packets. A link amplitude must then also specify the number of packets, as well as the angular frequency and linear momentum associated with each. Link amplitudes may alternatively be reformulated in a way resembling QFT, to specify the number of quanta occupying each of a set of allowed states for links between each particle pair.]

Although the functional dependence of analogous quantities in QFT is understood, the dependence of link amplitudes in MTT must be distinct. While the explicit functional forms of *γ*_{12} and MT wavefunctions depend on the MT equations of motion, it should be possible to write any link amplitude as a Foorier sum over a complete set of eigenvectors of *ω*_{12} and *K*_{12}.

The time dependence of a physical eigenvector must encode the direction of flow, in a manner consistent with *θ*_{12}. In quantum physics, a complex phase factor *e*^{-iωt} indicates flow in the positive t (time) direction, where ω is angular frequency. The amplitude *γ*_{12} for a simple link eigenvector of *ω*_{12} (again ignoring self-links) should then include a temporal phase factor

*e*^{-i ω12 θ12 (t21-t12)} = *e*^{-i ω12 δ12}

where *δ*_{12} ≡ *θ*_{12} (*t*_{21}-*t*_{12}). If *γ*_{12} is symmetric in timespace parameters, it must also include a spatial phase factor. Distance **r**_{12}=(*t*_{21}-*t*_{12}) **u**_{12}*c*, defined at *t*_{21}=*t*_{12}+*θ*_{12} *r*_{12}/*c* from the perspective of 1, is the space-like vector corresponding to *δ*_{12}. Note that **r**_{12}=**r**_{2}+**ε**_{12} and *t*_{21}=*t*_{2} for (*t*_{2},**r**_{2}) found in *ɸ*_{12}(*t*_{1},**r**_{1}; *t*_{2},**r**_{2}), where **ε**_{12}=*c*(*t*_{2}-*t*_{1})**u**_{12}. Both coordinate pairs (*δ*_{21},**r**_{12}) and (*t*_{2},**r**_{2}) have operator counterparts in MTT. The overall *γ*_{12} becomes

*γ*_{12} ∼ *e*^{-iω12δ12} *e*^{+iθ12K12·r12} .

Phase factors of this form are akin to energy-momentum eigenstates of free particles. [While these functions comprise a complete set of states, other forms – e.g., angular momentum eigenstates – are also possible, and often more convenient.]

Because *γ*_{12} and *γ*_{21} represent the same link seen from a different perspective, the exchange of labels must not alter any physical link characteristic – both link probability and direction of information flow must be preserved. Under particle exchange,

*γ*_{12} (*t*_{12}, *t*_{21}, *θ*_{12}, **u**_{12}, *ω*_{12}, **K**_{12}) => *γ*_{21} (*t*_{21}, *t*_{12}, *θ*_{21}, **u**_{21}, *ω*_{21}, **K**_{21}) .

Here the same quantities are defined from different perspectives in the respective amplitudes. Under Niestiik transforms, the amplitudes do have the same value for the given phase factor.

The eigenvector *γ*_{12} does not favor any linkage 1 to 2; its absolute value is the same for all (*t*_{12},*t*_{21}). However, a physical state of *γ*_{12} may consist of a superposition of eigenvectors. In mathematics, a summation

∑_{ω} *e*^{-i ω (t-to)}

over a range of ω peaks at *t* ∼ *t*_{o}. If *t*_{1} and *t*_{2} are correlated, then |*γ*_{12}| should peak at some *δ*_{12} ∼ *d*_{12}/*c*, where *d*_{12} is the length of a 3D interparticle distance **d**_{12}. If **r**_{12} is bounded, |*γ*_{12}| should peak at **r**_{12} ∼ **d**_{12}. In a restrictive sum over *ω*_{12} and **K**_{12}, the phase factor becomes

*γ*_{12} ∼ *e*^{-iω12(δ12-d12/c)} *e*^{+iθ12K12·(r12-d12)} .

Whereas *δ*_{12} and **r**_{12} have operator counterparts in this MT approach, **d**_{12} is strictly a parameter that captures initial conditions and particle motion, and may be a function of *t*_{21}.

How closely can such superpositions of simple eigenvectors correlate the times *t*_{1} and *t*_{2}? In the classical limit, the time lines can be perfectly synchronized, and a common, well-defined 3D distance **d**_{12}(*t*_{21}) exists at time *t*_{21}. The quantities

*η* ≡ [*θ*_{12}(*t*_{21}-*t*_{12}) - *d*_{12}/*c*] and

* ξ *≡ (

are then both zero with 100% probability. The corresponding *γ*_{12} can be approximated by a maximal summation over (*ω*_{12}, **K**_{12}) eigenvector states. In the classical extreme, the sum is over discrete values of *ω*_{12} from 0 to +∞, and each component of **K**_{12} from -∞ to +∞ (including zero), at increments Δ*ω*_{12} and Δ*K*_{12}, in the limit Δ*ω*_{12}→0 and Δ*K*_{12}→0. Each phase factor in the sum is multiplied by the four-dimensional product of the increments.

[This sum is closely related to the product *δ*(*η*) *δ*^{3}(* ξ*) of four Draci

Can such a link amplitude be realized in our physical world? RCLs and PCLs actually have distinct minimum and maximum allowed absolute values (*ω*_{min}, *ω*_{max}) of angular frequency and (*K*_{min}, *K*_{max}) of wavenumber in any superposition of simple states. For real massless bosons, an RCL travel distance *d*_{12} defines a limiting maximum wavelength *λ*_{max}=2*d*_{12}, entailing a *minimum* frequency *ω*_{min}=𝜋*c*/*d*_{12} and wavenumber *K*_{min}=𝜋/*d*_{12}. Limiting maximum values of frequency and wavenumber are defined only by the (inverse) smallest possible size of a timespace interval, and so are huge but presumably finite.

The situation is reversed for PCLs. Because a PCL represents collective phantom processes and does not comprise an independent time line, phase cannot gradually change along its length; only the net shift across a PCL is meaningful. This shift must then be limited to the range -𝜋 to +𝜋. Any outside value would be mathematically indistinguishable from a number inside the range. A PCL travel distance *d*_{12} thus defines a limiting *maximum* absolute frequency *ω*_{max}=𝜋*c*/*d*_{12} and wavenumber *K*_{max}=𝜋/*d*_{12}. Limiting minimum allowed frequency and wavenumber are both zero.

For RCLs, the *ω*_{12} and *K*_{12} must further satisfy standard mass relationships, and both *ω*_{12} and *K*^{+}_{12} are restricted to positive values. For PCLs, the *ω*_{12} must also be positive, but *K*^{+}_{12} is restricted to positive values if an interaction is repulsive, and negative values if it is attractive.

A PCL sum may not favor all frequencies and momenta equally even within the allowed ranges. Analogous sums in QFT electromagnetic calculations include factors like 1/(*ω*^{2}-*c*^{2}*K*^{2}), favoring photon-like states with effective mass near zero. As suggested earlier, sums for PCLs may be restricted to *ω*_{12}<<*K*_{12}/*c*. Applying appropriate limits, but neglecting possible weighting factors, a link summation can be converted to an **integral**, and evaluated using calculus. Probability distributions may be derived from the absolute square of the result.

For PCLs, the (normalized) probability distribution *P*_{PCL}(*η*) associated with the sum over *ω*_{12} is

2 sin^{2}(*Δ*_{ω} *η*/2)

*P*_{PCL}(*η*) ∼ ** ^{_________________}** where

𝜋

This symmetric distribution still peaks at *η*=0, now with finite value *Δ*_{ω}/2𝜋. The uncertainty *σ _{η}* in

*P*_{PCL}(*η*=0) = 0.5*c*/*d*_{12} and *σ _{η}* ~ 2

The distribution *P*_{PCL}(*ξ*) associated with the sum over **K**_{12} is more complex, but peaks at *ξ*=0, with value ~5*K*_{max}/6𝜋 if *K*_{min}=0. The uncertainty *σ _{ξ}* in

*P*_{PCL}(*ξ*=0) ~ 5/6*d*_{12} and *σ _{ξ}* ~ 8

For RCLs, there is no explicit integration over *ω*_{12} , as it is folded into the integration over *K*_{12}. However, for massless boson links, inertial invariance requires that ｜*η*｜=*ξ*/*c* with *ξ*≥0, so that *P*_{RCL}(｜*η*｜)=*P*_{RCL}(*ξ*). The distribution peaks at at *η*=0 and *ξ*=0. If *K*_{min}≪*K*_{max}, then *K*_{min}*ξ*≪1 inside the main peak of the distribution, and the peak value is again ~5*K*_{max}/6𝜋. The uncertainty *σ _{ξ}* remains ~8/

The correlation between *t*_{1} and *t*_{2} is thus in general *not* one-to-one, especially for PCLs. Single-time theory would be an approximation; a multi-time scheme should be more accurate. The classical limit is approached only through a sum of lower-energy interactive phantom processes and higher-energy real boson links. The minimum uncertainty in the correlation between *t*_{1} and *t*_{2} from phantom processes alone is ~*d*_{12}/*c* (the time light travels *d*_{12}). An uncertainty ~*d*_{12} is likewise inherent in any PCL specification of interparticle distance. Values for a macroscopic observer viewing a single fermion could be much smaller, if real particles are used as probes. Yet even for RCLs, the *ω*_{max} and *K*_{max} values associated with actual links may be generally much smaller than any upper limit imposed by timespace.

Recall that defining distance **r**_{2} at time *t*_{1} in *ɸ*_{12}(*t*_{1},**r**_{1}; *t*_{2},**r**_{2}) requires pairs of links AB (1 to 2) and BA (2 to 1), with respective amplitudes (still ignoring self-links)

*γ*_{AB} = *γ*_{12} (*t*_{1}-*r*_{2}/*c*, *t*_{2}, +1, **r**_{2}/*r*_{2}) and *γ*_{BC} = *γ*_{12} (*t*_{1}+*r*_{2}/*c*, *t*_{2}, -1, -**r**_{2}/*r*_{2}) .

Each amplitude can be separately represented by a weighted sum of pure-frequency eigenstates over a range of temporal and spatial frequencies. Define **ε**_{12}=*c*(*t*_{2}-*t*_{1}) **r**_{2}/*r*_{2}. Even if restricted to values *ω*_{12}<<*K*_{12}/*c*, a broad sum over **K**_{12} in both *γ*_{AB} and *γ*_{BC} causes the absolute value of each amplitude to peak at **r**_{2} + *c ε*

Consider now the opposite extreme, in which *γ*_{AB} and *γ*_{BC} are both represented by simple, pure-frequency eigenvectors. Dropping the irrelevant dependence on **d**_{12}, then

*ɸ*_{12}(*t*_{1}; *t*_{2}, **r**_{2}) ∼ *γ*_{AB} *γ*_{BC} ∼ *e*^{-iωAB(r2+cε12)/c} *e*^{+iKAB·(r2+cε12)} *e*^{-iωBC(r2-cε12)/c} *e*^{-iKBC·(r2-cε12)} .

Note that *ω*_{AB}, *ω*_{BC}, **K**_{AB}, and **K**_{BC} may in general be functions of *ε*_{12}. If *ω*_{BC} = *ω*_{AB} ≡ *ω* and **K**_{BC} = -**K**_{AB} ≡ -* K*/2, then

*ɸ*_{12}(*t*_{1}; *t*_{2}, **r**_{2}) ∼ *e*^{-2iωr2/c} *e*^{+K·r12} .

Imposing the realistic restriction *ω*_{12}<<*K*_{12}/*c*, then

*ɸ*_{12}(*t*_{1}; *t*_{2}, **r**_{2}) ∼ *e*^{+K·r2} ,

with an **r**_{2}-dependence similar to the traditional form of an overall wavefunction!

Might partial wavefunctions *ɸ*_{mt} generally (in both MTe and MTi) embody all information concerning time correlations and distances? For an *N*-particle (plus observer, if MTe) system, *ɸ*_{mt} then lacks only the free-particle mass-energy time dependence

∼ *e*^{-iωoto} *e*^{-iω1t1} . . . *e*^{-iωNtN}

that dominates the time evolution of the overall wavefunction *Ψ*_{mt} in the low-interaction limit. Here *ω*^{2} = *k*^{2}*c*^{2} + *m*^{2}*c*^{4}/ℏ^{2} for each object, where *m* is object rest mass. *Ψ*_{mt} must incorporate both this free-particle temporal dependence and the partial wavefunction *ɸ*_{mt}.

What are the dynamic equations that define the time evolution of MT causal link amplitudes and wavefunctions . . .

.

.

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[At this point there is a gap in the historic record, in which Fleegello purportedly tried to formulate a regorous mathematical framework for his MT ideas, and demonstrate that CLs provide a complete, novel MT framework for representing multi-particle systems. His failure to complete this project apparently caused Fleegello to doubt his own competence and worth as a natural philosopher / physicist, and to destroy all related material in a fit of despair shortly before his death. Physicists eventually did develop a complete MT quantum theory that supplanted standard QFT, in which the connections among elementary non-gauge particles (mainly fermions, plus non-gauge bosons unknown to Fleegello) inherently define spacetime, and PCLs are dictated by gauge symmetries within an MT context. For every non-gauge particle type, there is a pair of operators that create or destroy, respectively, one such particle (including its proper time line). For every gauge boson, there is a pair of operators that create or destroy, respectively, a CL between two specified points on particle world lines. Operator commutation relations ensure the appropriate symmetries of quantum states.]

Insofar as CLs specify 3D direction vectors, they also define (probability amplitudes for) the relative 3D positions of all causally connected particles in a system, regardless of their number. Yet CLs should be capable of establishing a 3D spatial framework even *without * these inherent 3D direction vectors.

Consider in this regard an isolated system of *N* distinguishable particles. If relative speeds are non-Niestiik, then interparticle distances *d _{jk}* ≅

The web of CLs and associated proper times among the world lines of elementary particles could thus determine the geometry of timespace, in particular the large-scale geometry, whether or not CLs specify spatial direction. In either case, space can unfold from the relationships among myriad connected events. To the extent this occurs, space does not have independent existence, but is defined by the connections between the mathematical objects we perceive as particles. A world without causal links would be a world without space; any particles would be independent of each other, with no meaningful positional relationships.

Yet why do inter-particle connections defined by interactions specifically generate an overall *three*-dimensional, nearly *flat* (under normal conditions) space? Classically, even without inherent 3D directions, the *N*(*N*-1)/2 interparticle distances in an *N*-particle system are sufficient to determine all relative coordinates for up to (*N*-1) dimensions!

The mathematical physicist **Wittuu** has proposed a mechanism that both restricts and defines the number of spatial dimensions. Large-scale timespace is defined mainly by the electromagnetic interaction, since it is associated with the exchange of phantom photons of unlimited range. While real photons are massless, and so restricted to two spin states, phantom photons have *three* spin states, like any spin-1 particle with mass. These correspond to three inherent, independent "directions." Because all photons are identical, exchanging their identities cannot alter a physical system, so they must share the same three directions. This causes every interparticle distance defined by CLs from photon connections to be limited to vectors in a common macroscopic 3D space.

The strong and weak interactions should establish additional spatial dimensions. Although there are eight gluon types, these are not truly independent, and together generate only three new dimensions. The three bosons of the weak force generate the same, making a total of nine spatial, or ten timespace dimensions. Yet the ranges of the strong and weak interactions are so tiny (~10^{-13} and 10^{-16} centurets, respectively), they mainly affect the small-scale geometry of timespace. Wittuu suggests that the associated dimensions are "curled up" or "attenuated," and only obvious at very small scales or high energies.

[While Wittuu's argument was sketchy, later generations of physicists demonstrated that his intuition was sound (although he missed a few small-scale dimensions). A rigorous explanation of the origin of macroscopic spatial dimensions was eventually developed. Spacetime essentially arises as an emergent property from the interconnections among primitive, abstract, timeless mathematical forms.]

What about gravity? The carrier of this interaction is ostensibly the massless graviton. Because the graviton is a spin-2 particle with unlimited range, gravity might be expected to generate its own large-scale 5-dimensional space. Yet gravity has an unusual character, related to its incompatibility with standard QFT. All other fundamental forces are carried by spin-1 bosons, and associated with unique and conserved "charges" (e.g., electric charge). But gravity couples to a system's **stress-energy tensor**, to which *every* interaction contributes. Gravity reduces all "bare" mass energies, and even couples to itself, leading to nonlinearities in the **gravitational field equations**. PCLs established by gravity are thus dependent on and flow from the other interactions, so that gravity does not add any new dimensions. It may nonetheless distort the large-scale structure of timespace, as in Niestu's theory of general invariance.

It is clear that neither standard graviton exchange nor general invariance represents the complete fundamental description of gravity, even if both are good approximations in the low-energy limit. A missing element in these theories may involve the small-scale structure of time. Physicists have traditionally considered time to be continuous. In an attempt to avoid divergent (infinite) quantities in QFT calculations [which had previously been removed for forces other than gravity by a dubious procedure known as **renormalization**], Planko has proposed that proper time is **quantized** along the world line of every elementary particle with mass. The fundamental (minimum) proper time interval, or **chronon**, is represented by the symbol *Δ*. Distance between particles is naturally quantized in integral multiples of *Δc*/2. [Spacetime volume is thus more generally quantized, and not space or time separately. By inertial invariance, this quantity is unaffected by a velocity transformation.]

Quantized proper time may be *required* by ideobasic principles. Consistency logic compels the PCS to recognize the most general conception of time. Yet continuous time is only a limiting case of quantized time. The infinity of numbers on a continuous line segment is furthermore countable only for rational values; there is no one-to-one correspondence between irrational numbers and the set of positive integers. If it is impossible to locate irrational values within the PCS field, they cannot have meaningful existence there (although the *concept* may exist). Note that time is inherent only along particle world lines; it does not meaningfully reside anywhere else.

If timespace is quantized, then the smooth **differential equations** of Shrodiik mechanics, QFT and MTT must be replaced by discrete difference equations. Observables defined in terms of derivatives must be similarly redefined. Symmetry principles and conservation laws are all affected.

[Other physicists had previously hypothesized that spacetime was quantized, but only with respect to the overall coordinate framework of a given observer, not with respect to individual particles. These **lattice approaches** were doomed to failure, as they were divorced from the very processes that define space and time.]

A minimum proper time interval *Δ* implies a maximum absolute angular frequency

*ω*_{max} = 𝜋/*Δ*

in any function of proper time, and a range of meaningful frequencies

0 ≤ *ω* ≤ +*ω*_{max}

(while this universal frequency limit applies directly to RCLs, a smaller cutoff applies to PCLs). Based on a symmetric version of the modified single-particle equation of motion, Planko has proposed replacing the linear equation relating the energy *E* of an elementary particle in its own rest frame to its proper time angular frequency *ω* by the trigonometric formula

*E* = (ℏ*ω*_{max}/𝜋) sin(𝜋*ω*/*ω*_{max}) .

This reduces to the standard equation when *ω*/*ω*_{max} << 1, and can alternatively be written

*E* = *E*_{max} sin(*E*_{o}/*E*_{max})

where *E*_{o} is a particle's uncorrected energy *E*_{o} = ℏ*ω* , and

*E*_{max} = ℏ *ω*_{max} /𝜋 = ℏ/*Δ* at the angular frequency *ω*_{max} / 2 .

Note that *E*_{o} = *m*_{o}*c*^{2} for an elementary particle with an uncorrected (bare) rest mass *m*_{o}.

[Massless particles have no rest frame, and there is no passage of time along their world lines; frequency and energy must be specified with respect to associated particles with mass.]

Proper time quantization reduces and limits a particle's effective rest mass energy, in a manner curiously similar to gravity. For every known elementary particle (characterized by a single, independent proper time line), *E*_{o} / *E*_{max} << 1. The sine function in the modified equation for energy can then be approximated by a truncated power series. Including only the first correction term in this expansion,

*E* ≈ *E*_{o} - *E*_{max}(*E*_{o}/*E*_{max})^{3}/6 .

For an elementary particle, this is

*E* ≈ *m*_{o}*c*^{2} - *m*_{o}^{3}*c*^{6}*Δ*^{2}/6ℏ^{2} .

According to standard Shrodiik theory, the **uncertainty in the position** of a particle of mass *m* cannot be smaller than ℏ/2*mc*. The rest mass *m*_{o} thus cannot be meaningfully confined to a volume with a radius *r*_{min} smaller than

*r*_{min} ≈ ℏ/4*m*_{o}*c* .

Using this relation to remove one power of *m*_{o} from the previous equation,

*E* ≈ *m*_{o}*c*^{2} - (*c*^{5}*Δ*^{2}/24ℏ)(*m*_{o}^{2}/*r*_{min}) .

The correction term is equivalent to the classical **gravitational binding energy** of a mass *m*_{o} distributed over a surface of radius *r*_{min}, if one identifies the **gravitational constant G** as

*G* ≈ *c*^{5}*Δ*^{2}/12ℏ .

Conversely, the chronon *Δ* can now be related to the gravitational constant by

*Δ* = √12ℏ*G*/*c*^{5} .

Indeed, Planko has identified the minimum distance *Δ**c*/2 with the **Planko length**

L_{P} = √ℏ*G*/*c*^{3} ≈ 10^{-33} centurets,

and the chronon *Δ* with the **Planko time**

T_{P} = √4ℏ*G*/*c*^{5} ≈ 10^{-43} nocs,

which differs from the value derived above by less than a factor of two.

[These estimates are remarkably close (within a factor of eight) to the chronon value obtained from subsequent experiments. The Planko quantities were inferred theoretically from the time/distance scale at which the quantum effects of gravity become significant.]

Planko has further proposed a **natural system** of units, in which the equations of motion are simplified. The unit of time is the chronon, and the unit of distance *Δ**c*. Interparticle separations are then half-integral multiples of the fundamental unit, and speed *c* is numerically equal to one. Units of mass and electric charge are selected so that both the Planko and the gravitational constants are numerically equal to one, and an elementary electric charge is the square root of the fine structure constant.

How does time quantization affect the energy of a multiparticle system? Consider a pair of particles 1 and 2, interacting solely by virtue of their mass, that are at rest with respect to an observer, and separated by radial distance *r*. When the particles are far apart, interactions are negligible, so proper times *τ*_{1} and *τ*_{2} should be independent and uncorrelated. The two-body wavefunction *Ψ*_{mt} is the product of free-particle wavefunctions, with bare masses *m*_{1o} and *m*_{2o} . Because an increment *Δ* along one time line does not imply an increment along the other, total system energy *E*_{far} is the sum of individual particle energies:

*E*_{far} = *ℏω*_{1}+*ℏω*_{2} ≈ *m*_{1}*c*^{2} + *m*_{2}*c*^{2}

where *m*_{1} and *m*_{2} are the respective effective rest masses of each particle, related to the bare rest masses by

*m _{j}c*

The energy limit *E*_{max} applies only to the rest mass energies of the individual particles, along their respective world lines, and not to the overall system, so the maximum possible system energy is 2*E*_{max}. In terms of Niestiik COM time coordinates (*T*,*ρ*) defined earlier, the *discrete* rate of change of *Ψ*_{mt} along *T* no longer equals the sum of the *discrete* rates of change of *Ψ*_{mt} along *τ*_{1} and *τ*_{2}. The latter instead equals the sum of the discrete rates of change along *T* for increments *a*_{1}*Δ* and *a*_{2}*Δ*, respectively. This is equivalent to the discrete rate of change along *T* for an increment *Δ _{T}*, where

*Δ*/2 (for *ω*_{1}=*ω*_{2}) ≤ *Δ _{T}* ≤

Note here that *T* is *not* a proper time measure, so the minimum increment *Δ* does not apply.

At smaller separations, *τ*_{1} and *τ*_{2} may be correlated by mass-related interactions (also required to define interparticle distance), and increments along *τ*_{1} and *τ*_{2} are no longer independent. The maximum correlation, at a minimum meaningful distance *r*_{min}, is equivalent to the particles merging into a single world line and proper time – increasing *τ*_{1} by *Δ* also increases *τ*_{2} by *Δ*, and vice versa. The time correlation parameter *ρ* is restricted to values near zero, and the individual particle wavefunctions merge into a single function characterized by a bare mass (*m*_{1o}+*m*_{2o}) and proper time *T*, now with a standard minimum increment *Δ*. Considering only bare mass and time quantization effects, the combined energy is then

*E*_{near} ≈ *E*_{max} sin[(*m*_{1o}+*m*_{2o})*c*^{2}/*E*_{max}] .

Whereas a total energy limit corresponding to a minimum system time interval *Δ*/2 applies when the particles are far apart, a lower limit corresponding to *Δ* applies when they are near.

At intermediate separations *r*, one can write

*E _{r}* = (1-

where *ε* is a function of *r*, such that *ε* → 0 as *r* → ∞, and *ε* → 1 as *r* → *r*_{min} .

This equation can in turn be rewritten

*E _{r}* =

where *E*_{int} is an effective interaction energy

*E*_{int} = -*ε* (*E*_{far} - *E*_{near}) .

If bare mass energies are much smaller than *E*_{max}, then to good approximation, correction terms of order higher than (*E*_{o}/*E*_{max})^{2} may be ignored, and

*E*_{int} ≈ -*ε* *c*^{6 }*m*_{1}*m*_{2}(*m*_{1o}+*m*_{2o})/2*E*_{max}^{2} .

The minimum separation *r*_{min} can be estimated from the smallest volume that can confine the total uncorrected rest mass energy,

*r*_{min} ≈ ℏ/4(*m*_{1o}+*m*_{2o})*c* .

[Fleegello uses semiclassical reasoning here. He contemplates in a classical manner the total energy of two motionless masses as a function of their separation; then applies a Shrodiik argument that the probability distribution of a particle's position is spread out over space, and confinement entails a degree of motion "jitter," to estimate *r*_{min}. The conclusions nonetheless have some qualitative validity.]

If the chronon is approximated by the value √8ℏ*G*/*c*^{5}, the equation for *E*_{int} can be rewritten

*E*_{int} ≈ -*m*_{1}*m*_{2}*G* (*ε*/*r*_{min}) .

This is identical to the classical (low-energy) expression for gravitational binding energy, if only the function *ε* is set to

*ε* = *r*_{min}/*r* .

This radial dependence is appropriate for a long-range interaction carried by massless gravitons. It also suggests that the degree of correlation between times *τ*_{1} and *τ*_{2} is ~ 1/*r*, consistent with a cutoff frequency *ω*_{max} ~ *c*/*r* for PCLs.

Thus, in the low-energy limit, time quantization appears to effect system energy in a way similar to the low-energy limit of standard graviton exchange, assuming only that *τ*_{1} and *τ*_{2} can be correlated by purely mass-dependent interactions. Gravity would then be similar to the other forces, in that it can be associated with the exchange of a gauge boson at low energies, but distinct in its more fundamental association with time quantization.

What if the interacting particles have electric charges *Q*_{1} and *Q*_{1}? The bare electrostatic interaction energy is then

*E*^{o}_{elec} = *Q*_{1}*Q*_{1} / *r* .

Again ignoring corrections of order greater than (*E*_{o}/*E*_{max})^{2}, and adopting the original *ε*(*r*), the residual interaction energy associated with time quantization is now

*E*_{int} ≈ -[*m*_{1}*m*_{2} + (*m*_{1}+*m*_{2})*E*^{o}_{elec}/*c*^{2} + *E*^{o}_{elec}*E*^{o}_{elec}/*c*^{4}] *G*/*r* .

The three terms in brackets can be interpreted as the gravitational interaction energies between the two masses, between the masses and the electrostatic field, and between the electrostatic field and itself. As in more traditional theories, gravity couples to all relevant sources of energy.

The PCLs generated by the electric (or any non-gravitational) force appear to only indirectly affect the correlation of the proper times of the two particles; the derived mass-to-mass interaction energy with and without an electric interaction would otherwise have a different value for a given separation. The energies associated with non-gravitational PCLs apparently engender coincident graviton PCLs, which embody the actual process by which time correlations are established. This may reflect gravity's pivotal role in defining the geometry of timespace, and the fact that gravity does not appear to add any new spatial dimensions.

The origami-like unfolding of timespace from a milieu of interwoven, correlated events may generally result in a non-Euclidean macroscopic geometry. The effective curvature of conventional timespace would then be a natural consequence of the quantization of proper time intervals and the correlation of time lines by graviton exchange.

An elementary particle's rest mass may derive from a variety of sources. Every particle is effectively surrounded by a cloud of phantom exchange bosons, corresponding to all applicable interactions. But this is unlikely the sole source of mass for common particles. For example, if the electron's effective size is the minimum volume that can contain its mass, then the electron electric field contributes less than 1% to the observed mass value (the magnitude of the negative gravitational self-energy contribution is 43 orders of magnitude smaller).

Rest mass may also originate in a particle's underlying geometric character. Wittuu has suggested that elementary particles may not be pointlike, but associated with vibrations of extended (but tiny) **geometric forms**. The size of such entities should be comparable to the radial parameter *r*_{min} computed earlier for a given rest mass.

Can quantized timespace be incorporated into a quantum field theory? Time and 3D space have traditionally been treated as continuous system parameters in QFT. Yet past attempts to include gravity in QFT have failed; the theory is not renormalizable for point particles if system timespace is continuous. Even excluding gravity, the standard renormalized version of QFT predicts an enormous **vacuum energy density**. By adopting a multi-time framework, and restricting the *proper* time intervals between events to integral multiples of a chronon, maximum energies are naturally limited, and the divergent quantities in QFT calculations may be tamed.

As discussed previously, maximum frequencies associated with phantom processes in a multi-time setting should be further restricted, and inversely proportional to the distance between interacting particles. If space is defined by and only exists with respect to real material particles, then phantom processes that are completely disconnected from real particles might also be forbidden. Any new formulation should reflect that timespace is meaningfully defined only with respect to particle world lines and interactions. It may even prove necessary to treat elementary particles as finite-sized objects.

[The approach to quantized timespace outlined by Fleegello was naive, and flawed in many respects. It does not effectively address relative particle motion, or modifications to symmetry principles and conservation laws, or how an observer can fully integrate the proper times and spatial separations of individual particles with a global timespace coordinate system, and define a total system energy and wavefunction. Fleegello did acknowledge in private correspondence that his approach to quantized timespace was simplistic and incomplete, and certainly did not comprise a testable theory. Yet by replacing QFT with a multi-time theory, quantizing proper time intervals, and identifying elementary particles with extended (though minuscule) vibrating geometric forms, physicists were at last able to integrate gravity into quantum theory, and accurately compute the rest masses of elementary particles from first principles, avoiding the infinities that had plagued previous attempts.]

Although the fundamental (microscopic) equations of motion are symmetric in time, physical processes on a macroscopic level superficially do not appear to be time-symmetric. For example, if all the air molecules in a room were clustered in a corner, they would rapidly spread out to fill the entire room; yet the reverse process is not observed to happen. Niestu has proposed that this so-called **arrow of time** is a purely statistical phenomenon.The universal state witnessed by an observer at a given moment is connected by a single time increment (chronon) to a host of other states. In Niestu's view, the number of less ordered (higher **entropy**) states corresponding to a "forward" time process is simply much larger than that for a "backward" process. If conscious experience is a random walk from one state to another, a person is much more likely to experience events along the traditional arrow of time. Reverse time steps occur, but are swamped by the sheer number of forward steps. This distinction acquires significance mainly in macroscopic systems, due to the sensitive dependence of the number of states of a given type on the number of particles in a system.

Yet this statistical feature does not in itself guarantee our experience of time. A conscious being that lacked a memory would live in an eternal present, with no sense of time's arrow. Most animal memories in a given universal state are found to be of events in connected states with lower overall entropy. This implies that the creation of memory generally involves a statistically **irreversible process**. Memories laid down in this direction are normally adaptive, and facilitate survival into an expanding realm of universal states. Memories laid down in the opposite direction could in principle also be adaptive, but only if they overtly present as precognitions, consistent with a person's walk through time.

[Shortly after Fleegello died, the natural philosopher Loh demonstrated that even the observed **expansion of the universe** could be linked to such statistical considerations, providing a critical link between cosmology, gravitation theory and Shrodiik physics.]

As discussed in section 1.18, any physical universe must have [at least] one basic (self-caused) initial state. This state can imply no previous history; the system would otherwise logically extend to an earlier time. Every physical universe must therefore evolve from an initial state characterized by infinitesimal spatial volume. Our own universe appears to have originally experienced runaway, exponential growth – the primeval **hyperburst** of modern cosmology – from a minuscule primitive state. Every newborn universe must further incorporate particles or analogous localized objects relative to which distance can be meaningfully defined. It could otherwise not expand (or contract) in any meaningful way.

[Fleegello failed to recognize that, if the experience of the CIF is timeless, a self-contained physical universe may also be cyclic, along a time-like dimension that loops back into itself. Such a system must in its entirety be the cause of itself. His basic argument has nonetheless since been extended to the **multiverse** of all possible physical worlds, whereby our own universe and its generative hyperburst may have been spawned by a pre-existing, self-caused system.]

Physics continues to evolve. Our understanding may yet be profoundly superficial. Will the physical objects and patterns identified so far prove to be unified by a single underlying entity? The multitudinous facets of one magnificent (mathematical) **jewel**? Or are they disparate, random elements, fragments tied loosely together only by the principle of consistency? Our descendants will hopefully discover the answer to this compelling question.

[During Fleegello's era, physics was rocked by conceptual revolutions every several jopes. Prominent scientists would periodically announce that a "**theory of everything**" was at hand, or that all that remained in physics was to clean up a few loose ends. These claims were invariably contradicted by new discoveries. Only after many octujopes of struggle was a viable unified theory in fact attained. Even then, physics was hardly dead. The new vision was so rich in possibilities, that its many veins continue to be mined even to this yad. Indeed, quantum physics is no longer considered the most fundamental of the physical sciences, but is viewed instead as the study of **emergent** phenomena arising from a still deeper level of mathematical reality. Other higher-level sciences (chemistry, biology, psychology, etc.) likewise continue to flourish, as an effective understanding of complex emergent reality inevitably transcends knowledge of underlying physical processes.]