“Let’s assume for a moment, that you are a dishonest man.”
“Assume away.”
– Leo Bloom and Max Bialystok, from The Producers
Introduction
The author stumbles into a dim room where a tangled game of thought is in progress. Players at a strangely-curved table hunch over fistfuls of colorful cards blooming with indecipherable tokens, friends whispering and gesturing close by. Over the middle of the table a cloudy image rides unsteadily, its soft glow a shifting patina of ghostly light on the players’ faces. A wobbly bass hum from nowhere fills the room. No ceiling can be seen in the darkness, and even the walls and floor fade in shadow.[1]
A deep, disembodied baritone voice, soothing but clear, is speaking.
[LEONARDO]: “This is a mere snapshot, a flicker of entirety: our familiar realm of physics, embedded in a greater realm. Our physics may manifest unseen and powerful symmetries and resonances as we advance into that greater realm.”
The author mutters to an onlooker.
[AUTHOR]: “Who is that?”
The onlooker ignores him. The voice continues:
[LEONARDO]: “Our cosmos may be considered a global singularity of very-high dimension, in which our familiar spacetime, with all its perceived quantum and gravitational properties, is a mere projection from a richer dimensional regime. Our spacetime worldlines, in this model, appear to us as apparently-discontinuous trajectories which, in the embedding and projecting space, are in fact smooth transitions.”
A player slaps cards down on the margin of the uneven table, their faces showing tangles of characters from many languages.
[TIBERIO]: “No! What you are saying is nothing more than the tale of Plato’s Cave! Our world as a shadow of reality! That idea is thousands of years old. Here are the cards that prove it!”
A flourish of new cards, littered with blooms of multicolored symbols, sprouts directly beneath the suspended image.
[LEONARDO]: “You limit your thinking. Information – that which bestows form and pattern – is fundamental to our universe. From where we are situated, we see information as spatiotemporal organization: structure in space and dynamics in time. But that view limits the meaning of the term ‘information’ to the measures we can make of it where we are. It is all we know. But given the idea of an embedding space, it may not be bounded in this way. My turn.”
The image begins to shift and mutate, gradually becoming a stretched torus, 
its motes of light moving as in a smoke-ring. Words appear.
[LEONARDO]: “See that little shaded ring at the bottom level? That’s our home, our Minkowski spacetime. The arrows suggest its emergence from an unknown origin.”
[ALLEGRA]: “Information flow in this model is symmetrical, perhaps? Black holes? From our thin spacetime ring at the bottom of the image it moves into the overall embedding space shown above it, via singularities at the stellar and galactic scales. And I see that information moves from the embedding space into our spacetime via singularities… but what kinds of singularities are those? I’d think they’d be at the quark scale and below.”
After a lengthy pause:
[LEONARDO]: “Yes… let’s take some time to play with that. Perhaps, metaphorically speaking, we exist entirely on the rind of a very deep, very rich fruit, and the meat of that fruit is apparent to us only indirectly, through our awareness of mass and gravitation – and the information they bring us.”
He grins.
“Let’s have some dishonest fun.”
A hubbub of voices, and then:.
[LODOVICO (rumbles)]: “Now wait a minute. You can’t go flying off like this. All our evidence points to our universe’s emergence from a singularity. What you’re showing us looks like the old ‘steady-state’ models.”
Heads nod vigorously.
[LEONARDO, nodding also]: “Yes, and we need to clear away that apprehension. The problem is time. We live in its passing, fish in its current, and it encompasses all of our physics. To reach the fringes of thought where time is just another measure, contained in a greater realm of being beyond being, takes us out of our language and experience altogether. Hence my term ‘dishonest’ – I don’t really know what I’m talking about, but it’s fun. Others have done it. Stephen Hawking suggested that time itself might be a complex variable, not just an ordinary real-valued measure.[2]
“Let me tell you a story. Cosmologists do it, so why can’t I?”
Laughter. In a teasing voice, Giovanna beckons to Leonardo.
[GIOVANNA]: “Tell us a story, Uncle Leo! Pleeease?”
More laughter.
[LEONARDO]: “Far beyond all time and care, there was and is all that we do not know. Some of what we do not know flows together and blows apart, and time and space are born.”
The image changed.
“We could call this event a ‘startup’ or a ‘big bang’, but all our terms have implications of time moving steadily through the event, as time does for us. But this was no event in time – it is the ‘super-event’ that created what we call time itself. It also created space itself, woven with time Minkowski-fashion.”
“In the blowing-apart, expansion began, and what started as terribly dense quickly grew and cooled. This is pretty much what our current models tell us from the evidence.”
The image changed again. No one spoke for a few moments.
[LODOVICO]: “Go on, please. I want to see where this is taking us.”
[LEONARDO]: “The expansion has been traced out in existing models quite well, and we can see the earliest stages by looking back to a few beats following the initial super-event. At first the expansion ruled over gravitation. Mass-energy was balanced far towards energy. Expansion, in cooling everything, provided for increasing coalescences of mass from energy.”
“But as time developed onward, the coalescences of mass began to interact with the spacetime itself. At the larger scales, those coalescences grew quickly to the level where their warping of spacetime affected vast sweeps of space.”
[LODOVICO]: “I see that we are still within current thinking with all this, except for those arrows showing flows continuing into our universe. You’ll offer some thoughts on those, I hope. Go on.”
[LEONARDO]: “Yes, but let’s just see how the model and current thinking relate, first, all right? We can image the effects of mass on space here.”
The image continued in its shifts as Leonardo spoke.
[LEONARDO]: “Here we see the formation of large gravitational wells, some of stellar size and some much larger. We show them here as a distinct part of spacetime, simply to show how they evolve. In our experience gravitational wells embrace us at many levels from the planetary to the supragalactic. But as we scale up to see them, the potential for collapse of the wells increases.”
“We study those collapses from far outside them, and they present us with paradoxes over which we still debate with vigor.”
“Here’s the next stage in our lookbacks that we are examining now. As we peer back into time, we see uncountable galaxies reeling here and there, many of them with apparent gravitational singularities at their hearts. We also catch the signals of inconceivably-violent gamma-ray bursts that signal total gravitational collapses of great mass.”
[LODOVICO]: “So far, so good. All fitting with the standard model. But it looks from this picture as if you’re building up to something else.”
[LEONARDO]: “Of course I am! Think of it as some infinite cosmic jest if you will. Deeper and deeper go the gravitational wells, and at collapse we see only the explosive signals of the bursts – if we’re lucky – along with the flashes of the exploding outer stellar shells and the shuddering of gravitational waves. If we’re lucky, looking in the right spot at the right time. And that doesn’t happen much, because there is just too much to look at, out there, all over the universe.”
He pauses.
[GIOVANNA]: “The story isn’t finished there, is it?”
[ALLEGRA]: “You showed us the closing of the flow of information earlier. That’s where you’re going now, isn’t it?”
[LEONARDO]: “That’s where Lodovico raised his objection, but, yes – and first we should see to the limits of what we can surmise in the existing models. We’re nearly coming into existence at this stage, in which spacetime has grown to where planets and molecules and other goodies can form, and as in the earlier images you can see the hints of a convective flow into and out of our universe. Those flows are not in the existing models.
“They are not flows in time, in any sense. They represent a structure of sustenance of time and space, one that transcends what we can know in any direct way. What we are looking for here is a symmetry of sorts, some kind of closure that situates us firmly. So why not have a symmetry of these flows, those entering and those leaving our universe? Like this.”
The image becomes a torus, and the completion of the convective cycle becomes plain.
[LUCREZIA]: “What’s that stuff at the top of the image? We can’t access any of that, can we?”
[LEONARDO]: “No. That’s why it’s omitted from other images – to remind us that we can’t know anything directly at that level. But what we can do is to nibble at the fringes of our spacetime to reveal increasing hints of what lies out there beyond those fringes. One bit of our nibbling is the idea of the Hawking radiation at the edges of the event horizon of a gravitational singularity. It’s one case where quantum effects and gravitational collapse appear in the same realm.
“But having said all that, we may find that directing our nibblings, using ideas of symmetry like those we have here, will give us better results, even bring us to testability for specific consequences of those ideas. So here is one of those symmetries to look at. That’s the punch line at this level.
Nods.
[LODOVICO]: Where does this take us now?
Upcoming Considerations
[LEONARDO]: “So. To get deeper into this, let’s assume for a moment that our fuller realm is what the model shows. Not our four-dimensional continuum, but a space of much-greater dimension. What we call spacetime may then be assumed to be embedded in this larger realm. [3]”
Several at the table roll their eyes. A snicker, and then a softer, insistent voice penetrates:
[MINERVA]: “Why stop there? Why restrict dimensionality to integral values? Forget all that. The universe doesn’t count on its fingers just because we do. Fingers are simply our primate convenience for our computation and conceptualization.” [4]
Groans.
[LEONARDO]: “You’re ahead of me! But assume that gravitation, at all scales, corresponds to curvature of spacetime, again consistent with current thinking. But now also assume further that particles having mass represent distortions of the same general class as macroscopic gravitational singularities.[5] Combine these two assumptions. First, Minerva’s: a space of high and nonintegral dimensionality; and second, Allegra’s: the applicability of gravitational effects at the particle level. Then we face the possibility that space itself is fractal in character[6].”
Our Realm and Its Context
A plaintive complaint, almost a whine, penetrates the narrative, and a few cards hit the table.
[VALERIA]: “We don’t have the mathematical instruments for all that! How can you expect us to test or validate these wild ideas if we haven’t got the tools to grasp them?”
Heads nod vigorously.
[LEONARDO]: “Too true. But in everyday life we do it all the time, and our minds and bodies seem to grasp clearly what embarrasses our intellects to define. We can catch a thrown ball without any equations that predict its course. We can fly a stunt place through corkscrews with no step-by-step proof every time that we will not fall from the sky in a specific maneuver.”
[RAFFAELLO]: “Yes. And we can refer to our mother’s mother, but the computer programs that manage knowledge struggle to make sense of this simple idea. Our brains do all this effortlessly, with no help from our pencils and computers.”
[GIOVANNA]: “But these mathematical tools DO exist. It’s just that not enough of us are familiar with the ones we might need.”
[MINERVA]: “We can only perceive, measure, and consider details within our spacetime. But I like this, to think of where we live as a projection from a larger realm of higher-dimensional space.”
She raises a finger to the floating image, and it drifts and coalesces.
[MINERVA]: “Since we can’t draw higher-dimensional pictures, we’ll have to depict the situation in a more abstract form. Let’s make the number (or level) of dimension into a dimension itself. Then we can draw a line and make scale marks on it indicating growth in dimensional complexity, as you see here ( ). Note that it’s a continuous line, not a set of integer marks.
“Now we use that scale in a cosmological context. For the most part, we experience the universe as a seemingly-flat, four-dimensional, uniform space. The figure shows a circle around the ‘4’ to signify the importance of that four-dimensional appearance of things to us. It’s where we live. Only in the last hundred years have we learned that this appearance is only local to our perceived scale of space and time, though there seem to be some good physical reasons for this[7]. The reality of things may go much deeper.”
Now the cloudy shape over the table gathers itself into a vertical line anchored over the circled number 4, and from the middle of the line a deep horizontal thrust shows a sharp spike along the axis toward greater values. The faint and mysterious words “quark width” mark the beginnings of the spike at the vertical line. Another voice from the table’s surroundings.
[LUCREZIA (reading)]: “‘Quark width’? What does that mean?”
The response is lighthearted.
[MINERVA]: “You mentioned Plato’s Cave? The shape you see in the figure shows at right a vertical line, representing our spacetime continuum, with an opening leading to the left, representing a connection from our projected spacetime to dimensions beyond our physical reach. Notice the smooth curve – the dimensionality is not jumping, but flowing. The connection to these added dimensions is proposed here to be the gravitational distortion of spacetime. It may be macroscopic or microscopic, but the result is the same.”
“Now for that ‘quark width’ idea. Assume – dishonestly, of course, Leonardo – that the opening represents a subatomic particle, say, a proton. We experience the proton mostly in the vertical line – our spacetime. But probing the proton has shown us the existence of quarks within it. The strong nuclear force, not apparent in our macroscopic world, binds the quarks. For all practical purposes, quarks occupy a space with properties different from those of the space of our senses that we inhabit.
“The accepted definitions of the interactions in our world rest on algebraic structures called Lie groups that specify and categorize those interactions, or ‘forces’. The Lie groups[8] defining electromagnetism reflect our spacetime. The Lie groups defining the strong nuclear force reflect a more-complex realm[9]. Treating that realm as an extension of our spacetime, or treating our spacetime as an embedded component of the larger realm, seems potentially useful.”
Now a hubbub of voices, from which one emerges more clearly:
[VALERIA]: “You’re just munging together a bunch of unrelated things here! These other spaces, like the spaces defining spin for different kinds of particles, are just abstractions – they’re nothing like the space or spacetime we live in!”
A soft sound, a sigh, perhaps, and a reply.
[MINERVA]: “Yes, a lot of people share your view. But reality has already shown us that we need to open ourselves to more possibilities. After all, time itself wasn’t even a dimension until relativity left us with no other choice. Isn’t our science simply the process of matching our abstractions better and better to realities?”
Silence hung briefly in the mist around the diagram’s lines.
[LEONARDO]: “Gravitation is partner to mass. Theoretical physicist John Archibald Wheeler said, “Spacetime tells matter how to move; matter tells spacetime how to curve.”[10] Gravitation’s manifestation arises from mass interacting with spacetime. The degree of curvature of spacetime partners with the strength of gravitational attraction.”
A lighter voice enters the conversation.
[GIOVANNA]: “But in contemporary physics, curvature is intrinsic to the curving medium – there is no external embedding space from which one measures the curvature of a medium embedded in that external space. This approach provides a clear picture of highly-complex curvatures purely in the four dimensions of spacetime. Why do we need more dimensionality?”
Murmurs, as this new voice continues.
[GIOVANNA]: “We see spacetime as a smooth ‘manifold’ – a kind of formal ‘space’ in which any local region can be mapped as if it were a nice, plain, flat hunk of easy Euclidean reality. Computation is straightforward in such a region. Many such regions can be assembled into a ‘map’ of a broader expanse having curvature or connection properties not intrinsic to any single region. Isn’t all that enough for our science?”
The softer voice responds.
[MINERVA]: “Some of us think not. Once we encounter a sufficiently-intense concentration of matter, as in the case of a gravitational singularity, we come to the precipice, metaphorically speaking, of the usefulness of curvature in the contemporary sense. A singularity implies a discontinuity or break in smoothness, which forces us from the realm of smooth manifolds into something more challenging. And we haven’t even viewed the quantum implications.
“So… an embedding space. Consider gravitation’s relationship with the larger realm as a correspondence between gravitational intensity, or curvature, in a region, and the actual dimensionality of that region. The stronger the gravitational force, the greater the curvature, and the more dimensionality the local region possesses. In the voids, far from great masses, no connection with greater dimension can be made except in the infinitesimal ‘tucks’ of far-scattered particles.
“But when galactic, galactic-cluster, and supercluster masses aggregate, the result is curvature on a grand scale. A few particles here and there can have negligible effect on dimensionality, per our assumption. But large masses, concentrated, may affect subtly the dimensionality of the space they inhabit. In effect, mass in this evolving model gives access to more of the global realm in which spacetime is assumed to be embedded.
“What we see locally, from our vantage point within our galaxy in the Local Group in the Virgo Supercluster, corresponds to a space in which mass presence affects the properties of the space itself, and therefore possibly the degrees of freedom of its contents, as may be reflected by the Lie groups of physics we’ve found so far.
“This proposed model suggests a rationale for the apparent weakness of gravitation in our limited spacetime. We may live on the four-dimensional ‘rind’ of a realm in which gravitation’s influence must be diffused, unlike that of electromagnetism. Only a tiny fraction of gravitation may be apparent to us[11]. With a few rash assumptions relating gravitation to electric charge, say, we might be able to estimate the dimensionality of the greater cosmos in all its measure.”
Singularity
Another interruption.
[RAFFAELLO]: “Back up a bit. What do you mean by ‘tucks’ of particles? What are you getting at here?”
And another interrupter jumps in.
[ALLEGRA]: “Wait! Are you saying that there can be some significant relationship between a quark, for example, and gravitation?”
Minerva sketches rapidly in the air, and the figure morphs again.
[MINERVA]: “Yes! Look here, where we see gravitational singularities for two particles. But here we have a pair of dotted vertical lines that mark off different regions of dimension. On the far right is our spacetime, at four dimensions. To its immediate left is the region in which the strong nuclear force operates. To the left of that, and on across the diagram to its left edge, is the realm of successively-enriched dimension culminating in singularity. We could consider the diagram’s horizontal axis as a real-valued measure of dimension, increasing from right to left.”
[LUCREZIA]: “What is this ‘singularity’?”
[LEONARDO]: “There is no way to know exactly, but assuming its presence informs the proposed model here. One might consider it to be a totality, with our spacetime a projection of some of its aspects into the limited range of dimensions and properties we perceive.”
[TIBERIO, a bit sarcastically]: “So you’re adding a turtle to the usual stack of explanatory turtles.”
[LEONARDO]: “Not really. Extending such a consideration allows us to relate the singularity we trace at our cosmic beginnings to the singularities of collapsed stars and galaxies, and even to the proposed singularities of the particles at their own small scale. In effect, all the singularities become in this approach simply our views of one singularity: the only one.”
[TIBERIO]: “Turtle soup, then? With… fruit rinds?”
Snickers.
[PELLEGRINO]: “What might superstrings[12] mean in this model? There’s a distinction between fermionic and bosonic strings, the former open-ended, the latter closed. Could our view of fermionic strings as open-ended reflect their properties only as seen from spacetime? Maybe we could consider them as originating in this one original singularity itself that you’ve proposed, issuing through the particles we see, and looping: returning to the one singularity through the large-scale gravitational singularities of stars and galaxies. Then the open-appearing strings would be closed. The difference between fermionic and bosonic strings disappears in this model because all are closed.”
[RAFFAELLO (excitedly)]: “The symmetry of such a view implies a conserved quantity[13]! Maybe we’re just seeing fermionic strings from within the strings themselves, like a person standing inside a pipe and looking along its interior. But here, when we consider strings, we can look at them with theoretical eyes that can see at the proper scale.”
Over the table, the vaporous figure shifts again.
[MINERVA]: “Yes, this is very good. Now we see the gravitational wells – or maybe ‘wellsprings’ is a better term – of two quarks. Here we see the quarks in close proximity, so that their interaction via the strong nuclear force can be considered as having raised the dimensionality of the region between them, in our model.
“Again, on the far right is our spacetime. The region in which the strong nuclear force operates is shown as broader here, and we are modeling the difficulty of separating the quarks by considering such separation attempts as trying to ‘compress’ the space generated by the quark masses into a lower level of dimension, like trying to flatten a three-dimensional ball into a piece of paper. The energy devoted to such an attempt merely turns into mass, squeezing out more quarks without confining them to our 4-dimensional spacetime.”
Lucrezia has been frowning at the image.
[LUCREZIA]: “This distortion of dimension we’re talking about – where’s the evidence that we see anything but four-dimensional Minkowski space? Our number appears to be exactly four, not 4.27 or 3.98. How would a physical space of nonintegral dimension be detectible as such?”
[MINERVA]: “Good question, and it suggests that we may be able to find and define tests and formulations that make such conditions apparent to us – along the lines of current attempts to find ‘curled-up’ dimensions at small scales.
“We haven’t really intersected the quantum world with the fractal world that well, not in this model yet. At the smallest scales, that is an absolute requirement, but we’re just getting started.”
The Strong Nuclear Force
Minerva speaks slowly, choosing her words with pauses here and there.
[MINERVA]: “In most physics where such things as point masses are considered, discontinuities appear. When we embraced quantum dualities – wave and particle – and the uncertainties inherent in the model, we tiptoed away from the issue as well as we could… but we still ended up with infinites and renormalization – right back at the same discontinuity enigma.
“This is why I like treating the space of the problem as embedded in a space of higher dimension[14]. Treat the embedded space for such problems as a subspace. Discontinuities appear, but in the embedding space, the discontinuities are merely artifacts of the projection of the embedding space into the embedded space – it engages the Plato’s Cave metaphor all over again.”
Giovanna gestures excitedly.
[GIOVANNA]: “So the embedding space may be a smooth manifold (‘smooth’ meaning infinitely differentiable), while its projections contain abrupt transitions!”
Tiberio breaks in.
[TIBERIO]: “How about this? Couldn’t the embedding space alternatively be defined in discrete algebraic terms, with its projections as various choices of its subalgebras?”
Groans again, and the image wavers, flickering between coils of vapor and flashing motes of light.
Algebraic Aside
[LEONARDO]: “Why not? Maybe the Planck-scale elements of the universe are simply the elements of simplicial complexes… complexes of fractal and variable dimensionality?”
[MINERVA]: “Does it matter? We’re learning to maneuver our mathematics between the continuous and the discrete, and with fractal scaling we are getting better at it.”
[TIBERIO again]: “We already have the SU(3) group as a subgroup of SU(5), where we find the SU(3) × SU(2) × U(1) subgroup embedded. How dishonest can we be? How do we bridge between SU(5) and SU(3)? What a sharp and narrow bridge that would be![15] Can we consider some kind of transitional, fractional, maybe even fractal pattern, say, SU(4.7)?”
Several at the table throw cards at Tiberio, who ducks, laughing. Minerva gestures at him impolitely, and the image stabilizes.
[MINERVA]: “We still count SOME things on our fingers! We mortals have a long way to go, and this entertainment has its limits!”
She leans back, and continues.
Looking in the Well… spring?
[MINERVA]: “After all, the Plato’s Cave metaphor works either way. The gesturing real person is, from a casual perspective, smooth, but her projected shadow displays edges. She could just as easily be a higher-dimensional set of pixels, points, simplicial elements, what have you, in some discrete algebra. Her shadow is just a projection, with one or more dimensions removed in formal terms.
“Algebras have some advantages in their kinds of structure relationships and rigor. That may be why some researchers lean to their use.
“Either way, in the human image, the edge of her shadow may collide and overlap with the edge of a shadow of her friend nearby, but the apparent ‘collision’ represents no real collision except when the two friends actually collide themselves. Likewise, using our model here, a point-mass particle is not a point but is instead a view into the higher dimensions of the overall realm we are modeling. Its apparent interactions with other such particles may then be considered in a more-complex perspective than 4-dimensional spacetime.”
Leonardo clears his throat.
[LEONARDO]: “If a mass is represented this way what do we see? Is it a well, or a wellspring?”
[MINERVA]: “Let’s see. We look into the well – yes… maybe wellspring is a better word – of the mass at the quantum scale and see… the wave function of the mass itself. The aperture of the wellspring corresponds to the value of the wave function, in this model.”
[RAFFAELLO]: “But what about energy? Since our spacetime supports the simplest of the Lie groups, the quanta of energy are supported by our spacetime alone.”
He slows to pick out thoughts.
[RAFFAELLO]: “It is as if to see mass… we look inward (down the well, into the wellspring, whatever you call it) to higher dimensions, but to see energy… we look… across our own spacetime.”
Tiberio laughs again.
[TIBERIO]: “All right. Just how deep does the rabbit hole of this “mass wellspring” go? In this model image, the well narrows and lengthens ( ), to the left — the horizontal scale of the drawing is hugely shortened at the left end) all the way down to the original singularity itself you’re proposing. So… if I get the idea here, this model proposes that every mass, however small or large, is a view into some ‘source of everything’. The two distinct points shown at the left end of the drawing actually converge onto one single source point, if the term ‘point’ is appropriate for the high-dimensional ‘container’ of all things. This is getting almost theistic.”
Fractal Spacetime
A silence descends over the table, the scatters and clumps of played cards forming some hints of a pattern. Everyone stares up at the image, which is now softening and reshaping itself to accommodate Tiberio’s words. Minerva takes up the theme, and the image enriches.
[MINERVA]: “So now we can put four particle masses very close together, see? But at such proximity, are the masses sufficient to bring about a gravitational distortion of the near-infinitesimal region they inhabit? The closely-dotted line shows the near-flat aspect of spacetime as in the previous illustration, but the solid line shows how the densest concentration of mass raises the dimensionality of the local region in the present model. Again, the left-hand ends of the mass profiles here would extend to reach the original singularity’s high dimension, as Tiberio puts it. Is the dimension bounded?”
A tenor voice breaks the silence that follows the question.
[MALATESTA]: “Fractional dimensions are a current research topic. Their use has simplified some calculations, but they are difficult to visualize at all. They take the entire idea of dimension, which we have rigidly cast into an integer framework, into the realm of set theory, but we wrestle with such ideas as Hausdorff dimension and fractal spacetime.
“What groups define the classes of structure and interaction among entities of physics, but in a fractal regime? Suppose that in the diagrams here, the distance from the vertical line on the right represents the fractal dimension of the local space. How do different Lie groups come into play at different ranges along the solid line at the bottm? Is there some continuous or refined progression from, say, SU(3) to SU(5)?”
A whine from the opposite side of the table. [16]
[ALLEGRA]: “That gives me a headache.”
Laughs and nods.
[MINERVA]: “We’ll get there, step by step. Maybe. Let’s take a few steps, one at a time. One way to model such transitions is to treat group behavior as a form of ‘resonance’ with the characteristics of the dimensional substrate accommodating the behavior of the group, the way that fuzzy nonrational frequency ratios resolve into the neat loops of oscilloscope Lissajous figures with integral numbers of points along their edges.
“But four dimensions seem to be stable for us. That property may connect with a mathematical truth about four-dimensional manifolds: that they allow for stable complexity in ways not possible in higher- or lower-dimensioned settings. Go to five or more dimensions, and complexities of manifolds resolve or simplify easily. Go to three or fewer, and there aren’t enough complexities to make the realm of much interest.
“Sooo… electromagnetism can be represented by a U(1) Lie group ‘resonance’ farthest to the right along the dimensionality line ( ), the weak nuclear force by an SU(2) ‘resonance’ at some interval to the left, and the strong nuclear force by an SU(3) ‘resonance’ further to the left, each unfolding as dimensionality expands, as shown along the horizontal axes here. The deeper we go into the complexity of cosmic structure, the denser become the group structures possible at each deeper level. But how do the Lie groups map to the dimensional line, and does some form of structure lie between them? Good questions.”
[LEONARDO]: “Concerning the theme of further unification of interactions beyond current theories, physicist Edward Witten said[17]:
‘Having at least partly unified the weak and electromagnetic interactions in a U(2) or SU(2) × U(1) gauge theory, one naturally wonders if it is possible to do better… The most obvious simple Lie group that contains SU(3) × SU(2) × U(1) – which describes the strong, weak, and electromagnetic interactions – is SU(5).’
“He continued by suggesting a supersymmetric extension of SU(5) that gives a successful prediction for the experimentally-measured weak mixing angle[18]. And then he added:
‘It may well be that [the SU(5)] model is part of a more complete description of nature.’
“He concluded with the express wish of nearly every theoretical physicist since Einstein:
‘Beyond unifying the usual elementary particle forces, one would like to also include gravity.’”
Minerva nods.
[MINERVA]: “We’re just exploring here, and raising more questions than we can answer. The implications of higher-dimensional space and the constructs that can be derived within it are baffling and beyond intuition. What can we make of Calabi-Yau manifolds[19] and more? All we can do in this game is to try to find ways to connect and harmonize our bewildering array of outlooks into this awesome realm. We’re just sitting on a bit of its crust.”
[LEONARDO]: “Just a moment. That figure with the four wellsprings? We could turn each wellspring into a well if we change the label phrase from ‘particle cluster’ to ‘Abell cluster’, yes?”
Nods and shakes of heads. Silence.
Time, Determinism, and Completeness
[LODOVICO (grumbling)]: “Time and determinism cause us endless trouble, don’t they? In the present model, time is an artifact of our limited realm — it carries no particular power in this larger picture. Determinism here seems complete. Is our desire for, our belief in free will no more than another artifact of our existence as projections from the larger realm? Such an idea can be most challenging – even frightening, or worse, utterly depressing. This pushes us straight into philosophy and religion.”
The figure morphs once more.
MINERVA]: “We can take a step anyway. Here’s a larger perspective ( ): four gravitational wells, leading as before in this model back to the ‘center’ of the realm, whatever that may be. But our spacetime at the right is now shown as curved, embedded in the complete space shown. The wells might represent singularities of quadrillions of solar masses each, monstrous collapse events feeding some indescribable, undefinable ‘center’ of spacetime.”
A long silence ensues as the image drifts over the middle of the table. Now and then it flickers, and a brief glimpse of a three-dimensional enrichment of its projection appears ( ), tantalizing the players, and then evaporates back to the linear view.
One More Thing…
Giovanna and Tiberio both break the silence, stop speaking, and then Giovanna bursts out.
[GIOVANNA]: “One possible ramification of the ideas sketched here is the symmetry of information ‘flow’ in the model! A singularity of a given size contains a limited amount of information, right?[20] If we model our universe as one huge singularity with our spacetime embedded in it as a kind of ‘hypersurface’, we turn the whole problem inside out, and all information is contained in the singularity!”
[TIBERIO (nodding vigorously)]: “Then our spacetime hypersurface becomes a recipient of projections of information from the singularity of which it is a part.
“So we can then treat the quark-scale singularities of the present model as information outflows from the overall singularity, so that the information outflow at the micro level balances the inflow, via stellar and galactic collapses, at the macro level. As with fermionic strings, the notion of symmetry implies a conservation of some quantity.”
The dotted circle of the image grows to show a sample of the inward and outward information flows ( ). Giovanna continues,
[GIOVANNA]: “The outward arrows here present the experience of mass to us in the form of particles. The inward arrow signifies our experience (or lack of it) for a large-scale gravitational singularity.
“But how does gravitation harmonize with the other three forces we know? Its requirements for symmetry groups don’t match those required by the strong force. But consider the inflow/outflow symmetry idea just described. Gravitation and the strong force seem to have qualitatively-opposite characteristics. The former has universal range, attenuates slowly, and registers weakly in our spacetime; the latter has extremely short range, rises very sharply, and registers very strongly in our spacetime. Might it be possible to find a transformation, a mapping that harmonizes or symmetrizes the two forces in one framework?”
Turning Things Inside Out
Tiberio tosses a flourish of cards on the table and flips one over. Again the image turns on itself.
[TIBERIO]: “But there is no reason to assume that we are on some ‘skin’ of an enclosure – we can just as easily visualize our universe as an interior surface, seeing from inside a ‘bubble’ outward. The same openings and flows apply, but in opposite directions.
“Some have suggested that our universe itself might be inside a gravitational singularity, but that specific proposal is at best debatable. Our universe appears on the best evidence to be expanding, which would imply that we are not on the inside of anything. Yet if we are to consider ideas, it seems better to open the field to them and let investigation reshape and select among them. We have had to do this throughout the advance of science in every field.
“Honestly, dishonestly, with humor and insight, we need to keep playing, to work nature’s puzzles, find better solutions, and repeat the process. Endlessly.”
He gathers in all the cards. The table is clean.
Visualizing All Things
[VALERIA]: “If we could see it all with eyes capable of taking in the whole — and of course, we can’t – the complete realm in which our existence is embedded might look like that image we thought of earlier: some huge hyperspherical fruit, with every particle, mass, and aggregate of our existence having a root leading to its innermost infinite core point. And then we could see our lives as a far-richer form of Plato’s shadow play, with all the mysteries and breaks and ambiguities no more than the overlays of brighter shadows from regions of greater richness, depth, smoothness, and completeness.”
[MINERVA]: “In that vision, some might feel terror, while others would gain a sense of deep assurance. Our conversation in the playful mists of thought is essentially an entertainment, a trip into realms that echo fragments of harmony from the greater place in which those realms of thought find themselves embedded. We long for everyone to explore it, to find it stimulating.”
“Many wonderful works[21], [22], [23], [24] offer us deeply-considered (and constantly-evolving) overviews of our physical existence, and this presentation might contribute one more piquant flavor to the banquet of such ideas. By the time our conversation here reaches the awareness of others, the banquet will have moved on, with new dishes, new recipes, new chefs.”
The Conversation Leaves Us
[MALATESTA]: “How would we ever prove or disprove any of the kinds of assertions made here, or even in the current theoretical realms, since the Planck scale is not accessible to us? What’s the point?”
[LEONARDO]: “Who knows? Exciting mathematics and physics lie ahead. Time and again, mathematical theory has entered realms so abstract that no one except the mathematicians could understand them, much less see any application for them. Yet quite often, apparently-impractical and unusable components of mathematics have suddenly surged into prominence in physics and other scientific fields as applications arose. We kept thinking that ideas were untestable, and then tests appeared, and we learned.
“The twentieth century examples are all over the place. Hermann Weyl’s book The Classical Groups is a prime example; Einstein’s use of tensors and Riemannian geometry is another. The notion of nonintegral dimension is pure mathematics now – what might it become in the hands of gifted physicists? Isn’t the unexpected application of information theory to singularities another step along the great road of exploration?
[MINERVA]: “And here emerges our central theme, more mysterious than that question mark in our images at the place where all the singularities meet. We can see all that we accomplish not as some finality at the edge of some comfortably-bounded realm within human reach, but as an endless journey. An infinite range of conceptual search and application lies ahead. Even with the best tools and learning, the process will extend out for hundreds, thousands of years: unending time. We will never know all of it. That’s what makes any conventional idea of a creator of this endless, infinitely-complex realm completely pointless. We can’t have a limited creator of an unlimited creation.”
Coda
Into the silence, a woman who has not spoken throughout the whole conversation brings a soft alto:
[MARIA]: “Those convection images we saw earlier bring to mind an essential theme from the works of faith: the idea of the arcs of descent and ascent. Concerning our progression in the universe, consider this:
‘…those who have thoroughly investigated the questions of divinity know of a certainty that the material worlds terminate at the end of the arc of descent; that the station of man lies at the end of the arc of descent and the beginning of the arc of ascent, which is opposite the Supreme Centre; and that from the beginning to the end of the arc of ascent the degrees of progress are of a spiritual nature. The arc of descent is called that of “bringing forth” and the arc of ascent that of “creating anew”. The arc of descent ends in material realities and the arc of ascent in spiritual realities.’[25]
“For me, the similarity of patterns of this model of ours and that pair of arcs is striking.”
Minerva smiles.
[MINERVA]: “I love this summary, from the Writings of Bahá’u’lláh:
‘A drop of the billowing ocean of His endless mercy hath adorned all creation with the ornament of existence, and a breath wafted from His peerless Paradise hath invested all beings with the robe of His sanctity and glory. A sprinkling from the unfathomed deep of His sovereign and all-pervasive Will hath, out of utter nothingness, called into being a creation which is infinite in its range and deathless in its duration. The wonders of His bounty can never cease, and the stream of His merciful grace can never be arrested. The process of His creation hath had no beginning, and can have no end.’[26] ”
The gathering draws to a close. The table is swept clean of cards, the air is cleared of images, and everyone heads out on the next stage of the journey.
NOTES
[1] Astrophysicist Brian Greene moderated a presentation titled “Quantum Reality: Space, Time, and Entanglement”, featuring four distinguished colleagues as they explored for non-specialists the principles of quantum physics and relativity, from the problem of quantum nonlocality to the holographic principle. The discussion leaves one in awe of such intelligence and discipline, set atop the rich, deep foundations of centuries of work leading up to their insights into these incomparably-cryptic realms of extremity.
How can this author even begin to write coherently about such things? One feels like the doorman at some infinite Golden Hilbert Convention Center, welcoming the passing heroes and heroines of understanding as they stream into the gigantic lobby, strains of music wafting from inner doorways to numberless ballrooms where musicians play from scores on their music stands – scores with million-octave staves, instruments from gravitational tympani to swelling voces angelicae of light, while servers bear trays of cosmic-neutrino dainties everywhere – and all one can do, greeting the guests at the door, is to inhale the aromas, feel the urgent pulses and beats, hear a few sweet warbling notes, and be content that one can welcome gladly the great who pass within.
[2] Stephen Hawking, The Universe in a Nutshell, p. 59.
[3] Higher-dimensional space is consistent with current extensions of spacetime now in research: Lisa Randall, Warped Passages
[4] Nonintegral dimensionality relies on fractal mathematics, for its basis, to recast our metrics to ‘crowd’ its points closer together, smoothly, letting the degree of crowding represent the dimensionality of the set of points. But care is needed to keep clear the distinction between ‘fractal’ and ‘fractional’. See.Serge Dubuc, Models of Irregular Curves, in Fractals: Non-integral Dimensions and Applications, G. Cherbit, ed., Wiley 1991; Bruce J. West, Mauro Bologna, Paolo Grigolini, Physics of Fractal Operators.
[5] This ‘wormhole’ assumption was tried out by J. A. Wheeler, but led nowhere at the time. Edwin F. Taylor and John A. Wheeler, Spacetime Physics, W. H. Freeman & Co. 1971
[6] Some explorations of this idea are already being done: K Svozil, Quantum field theory on fractal spacetime: a new regularisation method, 1987 J. Phys. A: Math. Gen. 20; A. M. Selvam and SuvarnaFadnavis, Cantorian Fractal Spacetime, Quantum-like Chaos and Scale Relativity in Atmospheric Flows, arXiv:chao-dyn/9808015 v1 13 Aug 1998; Laurent Nottale, Fractal Space-Time and Microphysics: Towards a Theory of Scale Relativity, World Scientific 1993.
[7] Hermann Weyl, Why is the World Four-Dimensional? in his book of essays Levels of Infinity, Dover 2012.
[8] Robert Gilmore, Lie Groups, Lie Algebras, and Some of Their Applications, Dover 2005.
[9] See Bruce Schumm, Deep Down Things: The Breathtaking Beauty of Particle Physics, for a useful grasp of this.
[10] From John Archibald Wheeler, Geons, Black Holes, and Quantum Foam, p. 235.
[11] Lisa Randall, in Warped Passages, treats added dimensions as “curled up” below the scale of testability. More work is continuing in this area, but the “curling-up” may require classes of tests that the apparatus of physics is not yet prepared to apply.
[12] Michio Kaku, An Introduction to Superstrings and M-Theory
[13] Anthony Zee, Quantum Field Theory, Princeton University Press 2003
[14] Catastrophe theory uses such a strategy to study and classify different kinds of abrupt transitions. See Tim Poston and Ian Stewart, Catastrophe Theory and its Applications
[15] Tiberio is recalling mischievously from the Thousand and One Nights: ‘What is the length and breadth of the bridge Es Sirat [the bridge between this world and the next] ?’… ‘Its length is three thousand years’ journey, a thousand in descent, a thousand level, and a thousand in ascent: it is sharper than a sword and finer than a hair.’ Quoted from the John Payne translation in One Thousand and One Nights – Complete Arabian Nights Collection (Delphi Classics)
[16] Hermann Weyl, The Classical Groups, Princeton, 1939
[17] Edward Witten, From Superconductors and 4-Manifolds to Weak Interactions, Bulletin of the American Mathematical Society, Vol. 44, No. 3, July 2007 (Witten 2007)
[18] The weak mixing angle is a parameter of known experimental value specifying a relationship between electromagnetism and the electroweak force, and the particles interacting via those forces.
[19] See Brian Greene, The Fabric of the Cosmos.
[20] The Bekenstein bound.
[21] Roger Penrose, The Road to Reality: A Complete Guide to the Laws of the Universe
[22] Craig Callender and Nick Huggett, eds., Physics Meets Philosophy at the Planck Scale: Contemporary Theories in Quantum Gravity, Cambridge University Press 2001
[23] Lee Smolin, Three Roads to Quantum Gravity, Perseus 2002 (Smolin 2002)
[24] Brian Greene, The Elegant Universe
[25] ‘Abdu’l-Bahá, Some Answered Questions, from No. 81, Reincarnation.
[26] Bahá’u’lláh, Gleanings from the Writings of Bahá’u’lláh, from XXVI.