an autonomous, emergent framework for fundamental physics

An Autonomous, Emergent Framework for Fundamental Physics:Particles as Waves Trapped by Gravity

Randall C. O’Reilly∗

University of Colorado Boulder345 UCB

Boulder, CO [email protected]

Copyright 2003-4 by Randall C. O’Reilly,initially published online at URL listed below on 9/10/03

(Dated: 26th September 2004)

A computational framework for simulating fundamental physics is presented, wherein a number ofthe most central phenomenological features of physics emerge naturally from a simple autonomoussystem. As in a cellular automaton, space and time are discretized, with space tessellated in uniformcubic cells that are synchronously and locally updated in uniform time steps. This produces a fixedupper limit on the rate of propagation (i.e., the fixed speed of light). Each cell has two real-valuedstate variables: a quantum wave field, and a gravitational field that also propagates with wave dy-namics. These fields are coupled, such that the local energy in the wave field drives the gravitationalfield, and the gravitational field determines the effective distance between cells and the effective“mass” of the wave fluid, thereby warping space and time locally as in general relativity. Sufficientlyhigh-energy waves drive a gravitational well that traps the wave oscillations (i.e., a miniature “blackhole”), producing a stable dynamic state with particle-like properties. These particles exhibit chaotic,random brownian motion, and are appropriately accelerated by a constant gravitational gradient. Theparticle oscillations may correspond to quantum spin, and particles simultaneously exhibit wave andparticle-like properties, while also being fluidly transformable into and out of raw energy.

I. INTRODUCTION

The elementary particles of nature (electrons, quarks, etc) can be created out of raw energy, and can beconverted back into energy (i.e., E = mc2). Particles simultaneously exhibit both wave-like and particle-like properties, and have a somewhat mysterious quantum spin property. The speed of light in a vacuumhas a fixed constant value. These and other fundamental properties of physics emerge naturally from thesimple computational framework described in this paper. They are not assumptions of the theory, but ratheremergent consequences of a simple set of assumptions regarding the propagation of two coupled fields(a quantum wave field and a gravitational field). Particles in this framework emerge as stable patterns ofgravitationally trapped wave energy (i.e., miniature black holes), in contrast to the hard corpuscles envisagedby traditional particle theories. As emergent patterns, it is easy to see how such particles can be created fromraw energy, and destroyed back into energy. The close fit between the emergent properties of this frameworkand some of the most salient qualitative aspects of our physical universe suggests that this framework isworthy of further investigation.

As will be clear in its full explication below, the theory developed in this paper has several precedentsin the literature. It is a form of cellular automata (CA) [1–8] in that it relies on autonomous, recursiveequations operating over state variables in cubic cells distributed uniformly throughout space. However,it uses continuous, real-valued state variables instead of the discrete ones typically used in CA models.

∗URL: http://psych.colorado.edu/˜oreilly/realt.html

2 Real-valued Emergent Autonomous Liquid Theory

The emergence of elementary particles as miniature black holes (energy trapped by a sufficiently stronggravitational field of its own making) has been informally suggested by some theorists (e.g., [9, 10]). Asimilar notion of particles as emergent structures within a uniform field (as vorticies instead of black holes)goes back to J.J. Thompson in the late 1800’s (as discussed in [11]).

The present theory is known by the acronym REALT (Real-valued, Emergent, Autonomous LiquidTheory). As an automaton theory, its natural expression is in the form of a computer program, available fordownload from http://psych.colorado.edu/ ˜oreilly/realt.html. In the next section,the central assumptions of the theory are described, followed by a discussion of the emergent properties ofthis system. Then, the limitations of the framework are presented, followed by a general discussion.

II. ASSUMPTIONS OF THE THEORY

The central assumptions of the theory are as follows. Space is composed of a discrete, three-dimensionalmatrix of cells, in the form of a perfect cubic tessellation of 3D space that extends for an infinite distancein all directions. Each cubic cell, indexed by i, contains two coupled scalar state variables, wave (Wi)and gravity (Gi). The wave field represents the wave dynamics underlying quantum physics, but not asan abstract probability function as in the standard interpretations. Instead, it is a physical entity in thisframework, similar to the de Broglie-Bohm pilot-wave/quantum theory of motion interpretations of quantummechanics [12–15]. The gravitational field warps space and time in a way that conforms to general relativity,at least in the simplified Schwarzchild metric. The precise quantitative relationship between the wave fieldand quantum mechanics has yet to be established. Each cell may be on the order of 2.0x10−17cm on a side,but this remains uncertain (see Table I).

The state variables are updated in discrete steps synchronously across all cells, with an update rate thatproduces the speed of light as the maximum rate of wave propagation within the system (e.g., for cell length= 2.0x10−17cm, an update occurs every 6.6713x10−28s). The wave (and gravitational) fields generallyobey the equations for a perfect wave medium with no friction or other loss:

∂2Wi

∂t2=

c2

mi

∇2Wi (1)

where c2 is the speed-of-light squared and mi is a mass term that will be specified later. The laplacian∇2Wi

function is computed in discrete time and space, as a function of the immediate 26 neighbors (denoted N )of a given cell. The following approximation provides the starting point for the calculation:

c2∇2Wi ≈ γw3

13

∑

j∈N

kij(Wj −Wi) (2)

where the sum over the neighborhood N is weighted by coupling factors kij that reflect the squared dis-tances of the neighbors from the central element i:

kij =1

d2ij

faces: d2ij = 1

edges: d2ij = 2

corners: d2ij = 3 (3)

and 0 < γw ≤ 1 determines the effective speed of light (wave propagation speed). This discretization willpropagate a wave at the maximum speed of light of one cubic element per unit time with γw = 1. The useof all 26 neighbors as opposed to smaller subsets (e.g., only the 9 faces, as the laplacian is more typically

O’Reilly 3

discretized), is essential for stable and isotropic behavior (i.e., minimizing grid effects), especially in thecontext of large amplitude oscillating patterns at the cellular scale updating with γw > .1. It is clear fromthe particle motion reported below, and other observations of wave propagation within the system, that theunderlying grid would be invisible to any observer within the system.

The gravitational field enters into the wave equation by increasing the distances between neighboringpoints, in effect warping space. To fit with general relativity as explained below, G must make a linearcontribution to the coupling factors. Interestingly, other forms (e.g., squared distance as in equation 3) alsodo not produce stable particle states, providing a tight coupling between the emergent phenomena of thissystem and its underlying physical basis in general relativity. Therefore, the wave laplacian is defined as:

∂2Wi

∂t2=

c2

mi

∇2Wi ≡3

13

γw

mi

∑

j∈N

kgij(Wj −Wi) (4)

where the coupling factors in the neighborhood N are:

kgij =

1

d2ij + 2|Gj |

(5)

The absolute value |Gj | is required because although G is typically positive, wave propagation can causeit to go briefly negative, and this might lead to a singularity. It is also possible to simply prevent G fromever going below 0 — this affects some wave propagation but may be more “plausible” in some sense. Thefactor of 2 in equation 5 is required for the fit with general relativity, as explained below.

Gravitation also warps time, by determining the effective “mass” of the wave field m i:

mi = 1 + 2|Gi| (6)

Thus, larger gravitational values effectively slow down the rate of change of the wave field, achieving thetime warping of general relativity. The complete wave update equation in discrete time is thus:

Wn+1

i =Wni + (W

ni −Wn−1

i ) +3γw

13(1 + 2|Gi|)

∑

j∈N

Wnj −Wn

i

d2ij + 2|Gj |

(7)

where the superscripts indicate time step; the kgij coupling values are computed based on the G values attime n.

The gravitational field also propagates according to a discretized wave equation, and is driven at eachcell by the energy of the wave field (kinetic plus potential energy, Ej) averaged over the local neighborhood:

Gn+1

i = Gni + (G

ni −Gn−1

i ) +3γg13

∑

j∈N

Gnj −Gn

i

d2ij

+γg

∑

j∈N1

d2

ij

∑

j∈N

En+1

j

d2ij

(8)

Note that this gravitational field is not affected by itself: the coupling factors are the standard squareddistance terms without the extra gravitational distances, and the effective “mass” of each point in the grav-itational field is 1. Nevertheless, the gravitational field indirectly couples with itself via its effects on thewave field. The wave energy of a cell is:

En+1

i =1

2(1 + 2|Gi|)

(

Wn+1

i −Wni

)2+1

2

3γw13

∑

j∈N

(Wni −Wn

j )2

d2ij + 2|Gj |

(9)

where the first term is the kinetic energy (12mv2), and the second is the potential energy (gravitationally

weighted) represented by the displacement of the current cell relative to its neighbors.


The emergence of stable particles in the system depends critically on this form of coupling betweenwave energy and gravitation, including the use of En+1 instead of En, and the integration of En+1 overthe local neighborhood. Intuitively, these properties enable the gravitational field to “stay in front” of thewave field: the neighborhood integration of En+1 effectively doubles the rate of propagation of informationabout wave energy. Likely as a result of this doubled rate, the system is not stable with both γw = 1 andγg = 1, but it is stable with γw = 1 and γg = .5, which is the default configuration. Values below theselevels slow down the propagation of information in the system, but do not otherwise change the essentialcharacter of its emergent properties.

Finally, it is possible to introduce a coupling parameter that scales the impact of wave energy E on thegravitational values G. However, this does not affect the essential dynamics of the particle attractors, and istherefore assumed to be 1 for the time being.

A. Edges

Computational limitations require simulations of the system to be performed in a relatively small matrixof cells (e.g., around 2503 cells on the author’s 28 node “beowulf” cluster), meaning that edges pose animportant problem. One can approximate the Sommerfield boundary conditions to minimize echos off ofthe edges. The update equations for edge elements (indexed e) thus differ from those for internal cells byexcluding the first temporal derivative term, and by only interacting with internal (non-edge) cells (denotedby the neighborhood I instead of N ):

Wn+1e = κWn

e +γw

(1 + 2|Ge|)∑

j∈I kgej

∑

j∈I

kgej(W

nj −Wn

e ) (10)

Gn+1e = κGn

e +γν

∑

j∈I kej

∑

j∈N

kej(Gnj −Gn

e ) (11)

Note that the gravitational field is also not driven by wave energy at the edges, and a decay term 0 < κ ≤ 1(.99 typically) can be used to capture the dissipation of the field values that would occur in a larger space.It is also possible to simulate an anechoic chamber with velocity-damping spikes around the edges, but thisis not essential.

B. Relationship To General Relativity

The space-time curvature imposed by the gravitational field in the present system is identical to that ofgeneral relativity, at least within the simplified Schwarzchild metric. This metric represents an idealizeduniverse with a single static spherical object of mass M (e.g., an idealized star), using a radial coordinatesystem. Space is stretched along the radial direction r as follows:

grr =1

1− 2Mr

(12)

and time is stretched by a similar factor. In the present system, the neighborhood coupling factors kg warpspace according to their inverse (i.e., weaker coupling = greater effective curvature), so that the effectivestretching can be conveniently computed in terms of the value of 1

kg as a function of the distance r′ along agrid axis from a single fixed point mass:

1

kgij

= 1 + 2Gj = 1 +2M

r′(13)

O’Reilly 5

where the gravitational field Gj in the absence of any driving energy falls off as Mr′

(and is thus proportionalto the Newtonian gravitational potential field). These two curvature values are equivalent if we use an offsetof r′ = r − 2M (i.e., measuring the radial distance from the Schwarzchild radius 2M instead of fromthe center of the mass). The same math goes through for time dilation. Although the present frameworksupports wave propagation of the gravitational field, as in general relativity, this and other more complexrelationships have yet to be firmly established.

It is also important to emphasize that although the present framework clearly has a privileged referenceframe in the form of the underlying matrix of cells and the rate of synchronous updating, this does notcontradict the central postulate of relativity banishing the notion of such a privileged reference frame. Thisis because the presence of the matrix is completely invisible to any observer located within the systemitself, due to the relativistic stretching of space and time. In short, these relativistic transformations of thegeometry of space-time, which can be used to convert from one reference frame to any other, can also beused to convert a fixed underlying “privileged” reference frame into a relative one. In other words, despitebeing functionally consistent with general relativity, the present framework embodies a very classical senseof space, completely opposed to the famous statement of Einstein to the effect that if you removed all matterfrom the universe, space would disappear too; instead, space is primary, and matter merely state variablesfluctuating within it.

The flat underlying reference frame in this system may also have important explanatory benefits, forexample in providing an explanation for the otherwise somewhat coincidental flatness of empty space (forwhich the controversial cosmological constant was invoked in the framework of general relativity) — in theabsence of all matter, space in the present system reflects the cartesian flatness built into the matrix.

III. EMERGENT PHENOMENA AND GENERAL PROPERTIES OF THE SYSTEM

At the lower wave energies associated for example with a photon moving through open space, the waveand gravitational fields can be considered to be essentially uncoupled — gravitational distortions (e.g.,driven by large concentrations of mass) affect the propagation of waves by warping space and time, butthe waves do not sufficiently distort local gravity to significantly alter the way they propagate. However,at sufficiently high energies, the coupled dynamics of the wave and gravity fields can exhibit complexemergent phenomena, including stable particles as described next. Other emergent and general propertiesof this system are also described.

A. Particles as Stable Attractors

When the local energy of the wave field exceeds a critical value (approximately 6 in the natural unitsof the system), the resulting gravitational distortions are sufficient to trap the propagation of the wavefield, producing a stable oscillatory pattern in the wave. This can be induced for example by perturbingan otherwise flat wave field with a wave packet of wavelength 4 cells, gaussian width of 4, and amplitude1.5. The energy of the trapped wave sustains the gravitational distortion necessary to prevent the wave fromdissipating. These patterns correspond to the elementary particles of nature in this system. In essence, thisstable pattern acts like a miniature black hole, except that there is no strong singularity at the Schwarzchildradius of this system, and some wave energy continues to radiate out from the particle. As noted earlier, thisnotion of particles as stable attractors in a field medium goes back to J.J. Thompson (as discussed in [11]),and the general idea of particles as trapped light or miniature black holes has been discussed informally bysome theorists [9, 10].

Several central properties of particles emerge naturally from this framework. First, the energy in parti-cles can be released when attractors are disrupted, and energy can be absorbed into new attractors to formnew particles. Indeed, the primary means of constructing a particle in the system is to impose a sufficiently


1 2 3

4 5 6

Figure 1: Sequence of 6 states for a stable particle. The left of each panel shows the wave state, and the right thegravitational state. The display shows 15 x 15 x 5 (horiz, depth, vert) cells, with the state value represented both bythe height of the plane at each point, and by color & transparency (solid yellow = highly positive, transparent grey= zero, solid light blue = highly negative). The wave state exhibits strong disparities one cell apart, producing highenergy levels that drive a gravitational well (actually a bump in the G field, as shown) that traps these waves. Thegravitational well also exhibits important dynamics; a static well will not produce a stable particle.

energetic wave pattern and this automatically induces collapse. This corresponds well with the highly fluidnature of particle destruction and construction as observed in high energy physics experiments. Further-more, there is no separate construct of “rest mass” anywhere in the system — the rest mass of a particlecorresponds to the amount of energy trapped in its attractor. In other words, the E = mc2 relationshipemerges naturally and necessarily from this system (note that c = 1 in the natural units of the system).

A sequence of images of a stable particle are shown in Figure 1, showing how the wave field developsstanding oscillations within a gravitational well. The fact that the particle has strong oscillating patternsat the single-cell scale (i.e., one cell having a strong positive wave state while its neighbor has a strongnegative one) is likely necessary for generating enough wave energy to sustain a particle. Thus, the discretenature of space in the system likely plays a critical role in its ability to support emergent particles. Itis also important to appreciate that the emergence of stable particles depends critically on the equationsas currently configured — numerous other formulations failed to produce such structures. Furthermore,particle stability depends on complex interactions between the wave and gravitational fields — freezing thegravitational field at any point results in the dissipation of the particle wave field energy. Understanding theprecise basis of particle stability is an important open question at this point; the difficulty of doing so is anunfortunate property of complex emergent phenomena.

B. Particle Rest Mass

The system supports a stable particle configuration with a peak gravitational or energy value that fluctu-ates around the range of 3-5 in the natural units of the system, with an overall average of around 3.6. Ideally,the system would support a range of stable particle masses, which should correspond with those of known

O’Reilly 7

ToFrom Realt cm s Mev g

Realt 1 2x10−17 6.6713x10−28 2.6463x10−7 4.7172x10−34

cm 5.0x1016 1 3.3356x10−11 1.3231x1010 2.3586x10−17

s 1.499x1027 2.9979x1010 1 3.9667x1020 7.0710x10−7

Mev 3.7789x106 7.5577x10−11 2.521x10−21 1 5.6098x1026

g 2.1199x1033 4.2398x1016 1.4142x106 1.7826x10−27 1

Table I: Conversion factors for REALT units into standard physical units, under the assumption that the rest mass ofthe basic particle (taken as 3.7799 Realt units for convenience; the actual value fluctuates between 3-5 with an averageof 3.6) equals 1 eV, or the upper limit guess on the rest mass of a neutrino. The units of length, time, energy and massare all equivalent in Realt.

elementary particles. However, this is not (yet) the case. To perform a comparison with actual particle restmasses, we need to transform them into the natural units of the system, which requires (and indeed fixes)the length of a cell (from which all other quantities can be derived).

If we consider this particle to be equivalent to an electron (rest mass .511 MeV), this would require thecell length to be around 1.022x10−11cm, which is too large by several orders of magnitude given otherknown sizes (e.g., this is larger than the diameter of the nucleus). If we instead consider it to be equal toan upper-limit guess on the mass of the lightest particle thought to have a rest mass, an electron neutrino(1eV), this gives the “default” cell length of 2.0x10 −17cm, which is at least plausible (Table I). At thislength scale, the electron mass would be roughly 1.93x106 in the natural units of the system. It is plausiblethat the considerable additional rest mass results from the energy associated with the electric charge andcorrespondingly strong electromagnetic field of the electron, which is not included in the framework as yet(as discussed later).

A cell length equal to the planck length (√

~Gc3= 1.616x10−33cm) would require the rest mass of even

the upper limit on an electron neutrino (1eV) to be very large (4.67x1016) relative to that of the extantemergent particles. Therefore, although the planck length is widely thought to be the natural length scalefor quantum gravity, it does not seem be the natural scale for this system.

C. Particle Motion

In otherwise empty space, a particle exhibits random patterns of motion, suggestive of complex emergentdynamics in the interaction between the surrounding field and the particle (Figure 2). To characterize thismotion, the mean and variability of the particle location displacements∆ were computed for different timeincrements m in different directions ~d:

∆n(m, ~d) = (~xn+m − ~xn) · ~d (14)

where n indexes time and ~x is the particle location (maximum of the gravitational field). Figure 3a showsthat the mean displacement is essentially zero, while the variance increases linearly with time increment m.This is indicative of brownian random motion. It is also noteworthy that the variance is roughly equal in alldifferent directions ~d, including those along grid axes and off-axis diagonals, indicating that the underlyingsquare grid is not evident in overall particle motion. At small time increments (m < 100), oscillation of theparticle itself produces oscillatory variance in these displacements (Figure 3b).

As might be expected from the random brownian character of particle motion, it is also highly sensitiveto initial conditions. For example, entirely different trajectories of motion are produced by starting a particleat (25,25,25) in an otherwise empty 503 universe, compared to one started at (24,25,25) (Figure 2). In thissense, one can characterize the particle motion as chaotic.


Figure 2: Trajectories of a particle on 11 different runs in a 50x50x50 universe with minor differences in initial startingconditions, demonstrating the chaotic and random nature of particle motion. Display is 41x41x33, and trajectories arecolor coded for time: early-late goes from light blue, dark blue, grey, dark red, to yellow.

To measure the particle’s motion under a net force, gravitational gradients along a given direction wereimposed. Figure 4a shows that this gradient produces a constant acceleration of the particle’s motion “down”the gradient, consistent with Newtonian dynamics. Furthermore, the particle acceleration is proportional tothe gradient (Figure 4b). Because of the brownian random motion, which persists in the presence of thegradient, there is considerable variability in the measured accelerations of different runs. Each run differedonly in the starting location of the particle by one cell, again demonstrating the chaotic nature of particlemotion.

Finally, when two particles are placed in close proximity (i.e., within 10 or less cells), their mutualgravitational attraction draws them toward each other, at which point they merge into a single particle andproduce high amplitude waves that carry off the excess energy.

D. Wave-Particle Duality and Quantum Mechanics

A form of wave-particle duality, one of the central puzzles of quantum mechanics, emerges naturallyfrom this framework. Particles exhibit wave-like properties because they are fundamentally oscillatingwaves within a wave propagating medium. They exhibit particle-like properties because the gravitationalwell localizes them in spacetime and discretizes their energy state (rest mass). As noted earlier, the wavefield in this framework is conceived of in terms of the de Broglie-Bohm pilot-wave/quantum theory of mo-tion interpretation of quantum mechanics [12–15], as contrasted with the standard purely probabilistic in-terpretation. Specifically, the wave field is physically real, and deterministically (but chaotically) influencesparticle behavior. Nevertheless, the uncertainty principle holds because it is impossible to measure the stateof such a fluid system without irreparably altering its state under the influence of another unknown state,

O’Reilly 9

a) b)

0 2500 5000 7500 10000Time Increment

0

2

4

6

8

10D

isp

lace

men

t M

ean

/Var X mean

Y meanZ meanXY meanYZ meanXZ meanXYZ meanX varY varZ varXY varYZ varXZ varXYZ var

Average Movement, No Gradient

0 10 20 30 40 50Time Increment

0.00

0.05

0.10

0.15

0.20

Dis

pla

cem

ent

Mea

n/V

ar

Average Movement, No Gradient

Figure 3: a) Mean and variance of particle motion for a single particle in empty space (50x50x50 universe, computedover 11 runs with different starting locations), showing displacements in different directions for different time incre-ments. The essentially zero mean and linearly increasing variance with time increment are consistent with brownianrandom particle motion. Also, motion is essentially equivalent along grid axes and off-axis diagonals, indicatingthat grid effects are not a problem. b) Oscillations present in average particle displacement variability for small timeincrements, likely resulting from the fundamental oscillation of the particle itself.

a) b)

0 25000 50000Time

0

5

10

15

20

25

30

X P

osi

tio

n

Trajectory for G Grad = .03166

a = 1.92e-8

0.00 0.01 0.02 0.03Grav Gradient

0.0

5.0×10-9

1.0×10-8

1.5×10-8

2.0×10-8

Net

Acc

el A

lon

g G

rad

ien

t Particle Acceleration from Grav Gradient

Figure 4: a) Particle location in the X axis over time in a gravitational field gradient G along this axis of .03166 Gper cell. This trajectory is an average over 5 runs varying only by 1 cell in the Y,Z axes from the starting point of(10,25,25) in a (50,50,50) universe. Averaging reduces some of the variability associated with random patterns ofmotion along the other axes, which persisted throughout the trajectory. b) Summary results for different gravitationalfield gradients, showing a linear effect on particle acceleration. The two straight lines reflect linear regression fits forthe entire set of points (dashed line) and just for the fastest points for each gradient value (dotted line), while the solidline reflects the average. The slopes of the two regression lines are similar (5.873e-7 and 6.878e-7), and it is likelythat the intercept will pass through the origin for a direct trajectory without the slowing effects of brownian motion.

such that accurate measurements and predictions about specific events on the quantum scale are impossible.This is particularly true given the highly nonlinear properties of particle motion in this system. Considerablework remains to be done, however, to explore the quantitative fit of this system to the well-established dataof quantum physics.


E. Spin, Superposition, and Pauli Exclusion

The somewhat strange quantum mechanical quantity of spin may correspond to the regular oscillations ofthe wave field that are produced by the stable particle configuration. Although these oscillations are clearlynot a simple spinning as in a classical fluid vortex, they can potentially have an axis of oscillation in space(e.g., an oscillation with a non-uniform spatial pattern over the neighboring cells, such that cells along oneaxis are anti-coupled whereas those along other axes are coupled). In quantum physics, massive particles(fermions) such as electrons and quarks are all spin 1

2, while massless particles (bosons, e.g., photons) are all

spin 1. This distinction may also emerge from the framework, in the sense that the gravitationally trappedwave oscillation that produces particle rest mass is clearly different from a wave oscillation uncoupledwith any gravitational formation that moves at the speed of light (which is the model for a photon withinthe system). Quantum physics also says that bosons can superpose, whereas fermions cannot (i.e., thePauli exclusion principle). Correspondingly, waves in the wave medium traveling at the speed of light cansuperpose, whereas two particle attractor states cannot.

IV. LIMITATIONS AND OPEN QUESTIONS

Although the present system has a number of encouraging emergent properties, it also suffers from anumber of limitations. Some of these may be more apparent than real, but some are certainly very reallimitations that hopefully reflect an incomplete, rather than entirely incorrect, theory. In addition, there area number of open questions that need to be resolved.

A. Quantum States and Quantum Entanglement

The quantum mechanical wave function is in general 3N dimensional and defined over complex num-bers, where N is the number of relevant particles. In contrast, the wave field in this system is a strictly 3dimensional real scalar field evolving over time. It is possible that the complexities of the coupled wave andgravitational fields capture the interactions between particles that are otherwise represented by the quan-tum wave function, but this remains to be seen. The other obvious problem is the apparent non-locality ofquantum entanglement (e.g., as demonstrated by tests of Bell’s inequality in the EPR thought experimentframework, [16–19]), given the local nature of interactions in this system.

Mitigating against the locality concern are the following points. First, the wave field is distributedthroughout space and constantly interacting with itself, whereas the typical EPR framework assumes hid-den variables local to each of the moving particles (indeed recent analyses show that Bell’s inequalitydepends critically on perhaps overly strong localist assumptions; [20–22]). Thus, it seems at least possiblethat the observed correlations are communicated throughout the distributed wave field over the duration ofthe entangled particle’s trajectories. Given that this system exhibits chaotic behavior, it would be highlysensitive to small changes in the wave field associated with the entangled particle’s mutual interaction.Finally, entanglement always results from a superposition of states, and the wave field similarly supportssuperpositions of traveling waves — such superpositions are difficult to reason about intuitively, and thusit would be prudent to reserve judgment about the entanglement behavior of this system until appropriatesimulations (or accurate analyses) can be conducted.

B. Energy Conservation and Gravitational Self-Energy

The total wave energy in the system (computed by summing equation 9 over all cells) is constant un-der some conditions but not others. In a gravitationally flat space (e.g., achieved by artificially fixing the

O’Reilly 11

gravitational field to a constant value), total energy is always constant when averaged over time, but it canvary across individual time steps in some cases. For example, when a wave packet is propagating by itselfthrough empty space (without bouncing off the edges or passing through other waves), the total energy isexactly constant (to machine precision) at each time step. However, when a wave packet passes throughanother wave, the total energy oscillates around a central, conserved value. This is presumably due to theinaccuracies associated with computing local energy values in discrete space and time. Nevertheless, localenergy must be computed to drive the gravitational field, so such inaccuracies may be inevitable. It mayalso be that some interesting physically-measurable effects derive from such equations, but this has yet toexplored.

With the gravitational field in effect, the conservation picture is less clear. For example, the stableparticles maintain the same time-averaged energy value, even with dissipative Sommerfield edge conditionswith decay. Thus, they are in effect creating energy that is being carried off the edges. However, this seemsinevitable in order for stable emergent particles to not simply diffuse away into a flat uniform sea.

A related issue concerns the computation of gravitational self-energy. According to general relativity,the propagation of gravitational waves contributes to the energy that drives gravitation. When this notionwas implemented in the system by computing gravitational energy in the same way as wave energy, thesystem inevitably became unstable, producing infinite gravity field values (even with very small scalingfactors for the gravitational self-energy contribution). Instead of directly adding a self-energy term in thesystem, the propagation of time and space warping via gravity indirectly affects the energy computed in thewave function, and this might be sufficient to account for the predicted (and observed) self-energy factorsof general relativity. This hypothesis remains to be tested, and in general more work needs to be done tounderstand the global energy dynamics of the system.

C. Missing Forces

Despite exploring a large number of ways of perturbing the state and the particles within it, it appearsthat the fundamental forces of nature beyond gravitation (electromagnetic, weak and strong) do not emergenaturally from within the system as presently constituted. In further support of this conclusion, it seemsthat the current emergent particles are too fluid to support something like a force — bombarding theseparticles with waves of all sorts has very little predictable effect on their otherwise random motions. Thewave medium is essentially a linear medium, and thus the incoming waves simply superpose their way onthrough the particles, despite the gravitational nonlinearities (while nevertheless contributing in some com-plex manner to the random-like motions observed). In short, there seems to be not enough of a nonlinearityfor a force to “catch” on in this purely liquid system. The only reason the “gravitational force” is exhib-ited is because these liquid particles ooze “down” the gravitationally warped space. Some stronger form ofnonlinearity would appear to be required to capture the effects of the other forces.

Several facts of nature may place important constraints on the nature of the additional mechanismsrequired to support the electromagnetic force. For example, the electric charge is highly discrete, coming inonly two magnitudes (1, 1

3) with two polarities (and possibly just one magnitude according to quark charge-

sharing theories; [23–25]). This is not particularly consistent with the fluid and graded nature of energyand particle rest masses as captured in the present system. Furthermore, although everything has energy(and thus impacts the gravitational field), some particles have charge and others simply do not. Therefore,it seems likely that a discrete charge state variable will need to be introduced, which then moves from cell-to-cell under the influence of the existing wave and gravitational fields, together with an electromagneticfield that is driven by the presence of charges. The propagation of such an electromagnetic field shouldbe achievable using the same kinds of wave equations currently in use. This should for example producethe characteristic inverse-square force behavior, and photons would find natural expression as propagatingwaves in both the electromagnetic and wave fields. The fact that the quantum wave field reflects the energy


of a given particle suggests that the energy of the electromagnetic field should then drive the wave field,which can then drive the gravitational field as in the present system.

Beyond implementing a plausible electromagnetic force field and charges, it remains to be seen whetherthe weak and strong forces would then naturally emerge through the complex interactions of the resultingsystem, or perhaps they would require additional mechanisms as well. The peculiar short-range natureof these forces raises the appealing prospect that they would indeed emerge as some kind of collectiveinteraction of closely-located massive particle attractors, allowing the two long-range forces of gravityand electromagnetism, together with the quantum wave field, to be the only fundamental forces supporteddirectly by the system.

V. DISCUSSION

Perhaps the most compelling feature of this framework is the relatively large number of physical phe-nomena that emerge “for free” from the interaction between two uniform fields (wave and gravity) oper-ating largely according to the simplest form of wave dynamics. The range of phenomena captured withinthe present framework includes: stable particles that exhibit both random, chaotic motion and Newtonianacceleration in a gravitational field; particle rest mass that reflects the raw energy trapped within a gravita-tional well, and conversion of particles into raw energy and vice-versa (i.e., E = mc2), wave-like propertiesco-existing with localized particle-like properties; a spin-like oscillation associated with massive particlesthat differs from those of massless particles; and the fixed speed of light.

Furthermore, these emergent phenomena are strongly dependent on the specific form of the underlyingwave equations and their coupling, and the gravitational field’s properties are largely constrained by theproperties of general relativity (and provide a particularly simple way of implementing this complex theory).This is indicative of a highly constrained, parsimonious theoretical framework: only a very narrow rangeof equations and parameters produce anything approximating the known physical reality. This stands incontrast to other frameworks such as string theory, which seem to suffer from excess degrees of freedom.Of course, it remains to be seen whether the present highly constrained theory has enough power to captureall of the fundamental forces of physics; it may just be elegant but wrong.

At this point, it is clear that the framework is at least incomplete. It does not capture the electromagnetic,weak, and strong forces. As discussed above, there are possible ways forward to add these forces, but itseems that they will not emerge quite as naturally and elegantly as particles do in the present system. Inaddition to resolving this incompleteness, the precise quantitative relationship of the framework to quantummechanics remains to be explored. It is clear that the wave field can influence particle motion in a way thatseems to qualitatively capture the wave-like aspect of particle dynamics in quantum mechanics. However, aprecise quantitative fit will likely require the presence of electromagnetic forces that can more reliably anddynamically move particles around, compared to the relatively weak and difficult to manipulate gravitationalforce. Furthermore, additional work remains to be done in quantitatively relating the gravitational field inthis framework with the fully general mathematical expression of general relativity.

In conclusion, the present framework raises many more questions than it answers, and considerable workremains to be done even to just understand the emergent behaviors of the current set of simple equations.Furthermore, it is clear that additional elements need to be added to simulate any actual physical phenomena,and thus the theory may be essentially untestable until it has been suitably extended. Despite all theselimitations, hopefully this approach has enough potential merit that people with relevant expertise willinvest the effort necessary to evaluate and develop it further.

O’Reilly 13

Acknowledgments

Jeffrey M. Beck ([email protected]) contributed extensively to the author’s un-derstanding and presentation of the equations presented herein, and to a number of other issues discussedin this paper.

[1] S. Ulam, Proceedings of the International Congress on Mathematics, 1950 2, 264 (1952).[2] J. von Neumann, Theory of Self-Reproducing Automata, edited and completed by A. Burks (University of Illinois

Press, Champaign, IL, 1966).[3] E. Fredkin and T. Toffoli, International Journal of Theoretical Physics 21, 219 (1982).[4] S. Wolfram, Review of Modern Physics 55, 601 (1983).[5] S. Wolfram, Physica D 10, 1 (1984).[6] S. Wolfram, Communications in Mathematical Physics 96, 15 (1984).[7] E. Fredkin, Physica D 45, 254 (1990).[8] T. Toffoli and N. Margolus, Physica D 45, 229 (1990).[9] B. Greene, The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory

(Vintage Books, 2000).[10] G. t’Hooft, In Search of the Ultimate Building Blocks (Cambridge University Press, Cambridge, 1996).[11] T. Y. Cao, Conceptual Developments of 20th Century Field Theories (Cambridge University Press, Cambridge,

1997).[12] D. Bohm, Physical Review 85, 166 (1952).[13] D. Bohm, Physical Review 89, 458 (1953).[14] D. Bohm and B. J. Hiley, The Unidivided Universe: An Ontological Interpretation of Quantum Theory (Rout-

ledge, London, 1993).[15] P. R. Holland, The Quantum Theory of Motion: An Account of the de Broglie-Bohm Causal Interpretation of

Quantum Mechanics (Cambridge University Press, Cambridge, 1993).[16] A. Einstein, B. Podolsky, and N. Rosen, Physical Review 47, 777 (1935).[17] J. S. Bell, Reviews of Modern Physics 38, 447 (1966).[18] J. S. Bell, Speakable and unspeakable in quantum mechanics (collected papers in quantum philosophy) (Cam-

bridge University Press, Cambridge, 1987).[19] A. Aspect, Nature 398, 189 (1999).[20] K. Hess and W. Philipp, Proceedings of the National Academy of Sciences 98, 14224 (2001).[21] K. Hess and W. Philipp, Proceedings of the National Academy of Sciences 98, 14228 (2001).[22] R. D. Gill, G. Weihs, A. Zeilinger, and M. Zukowski, Proceedings of the National Academy of Sciences 99,

14632 (2002).[23] P. M. Ferreira, hep-ph/0209156v4 (2002).[24] M. Han and Y. Nambu, Physical Review 139, B1006 (1965).[25] J. C. Pati and A. Salam, Physical Review D 8, 1240 (1973).

an autonomous, emergent framework for fundamental physics

Documents