I had the pleasure earlier this month of giving a plenary lecture at a meeting of the American Astronomical Society. Unfortunately, as far as I know they don’t record the lectures on video. So here, at least, are the slides I showed during my talk. I’ve been a little hesitant to put them up, since some subtleties are lost if you only have the slides and not the words that went with them, but perhaps it’s better than nothing.
My assigned topic was “What We Don’t Know About the Beginning of the Universe,” and I focused on the question of whether there could have been space and time even before the Big Bang. Short answer: sure there could have been, but we don’t actually know.
So what I did to fill my time was two things. First, I talked about different ways the universe could have existed before the Big Bang, classifying models into four possibilities (see Slide 7):
- Bouncing (the universe collapses to a Big Crunch, then re-expands with a Big Bang)
- Cyclic (a series of bounces and crunches, extending forever)
- Hibernating (a universe that sits quiescently for a long time, before the Bang begins)
- Reproducing (a background empty universe that spits off babies, each of which begins with a Bang)
I don’t claim this is a logically exhaustive set of possibilities, but most semi-popular models I know fit into one of the above categories. Given my own way of thinking about the problem, I emphasized that any decent cosmological model should try to explain why the early universe had a low entropy, and suggested that the Reproducing models did the best job.
My other goal was to talk about how thinking quantum-mechanically affects the problem. There are two questions to ask: is time emergent or fundamental, and is Hilbert space finite- or infinite-dimensional. If time is fundamental, the universe lasts forever; it doesn’t have a beginning. But if time is emergent, there may very well be a first moment. If Hilbert space is finite-dimensional it’s necessary (there are only a finite number of moments of time that can possibly emerge), while if it’s infinite-dimensional the problem is open.
Despite all that we don’t know, I remain optimistic that we are actually making progress here. I’m pretty hopeful that within my lifetime we’ll have settled on a leading theory for what happened at the very beginning of the universe.
Bill S., thanks for the redress.
Your question and my response points out the key issue in a structural context. It’s exactly where we want to be in the contrast between a ‘bottom up’ and ‘top down’ perspective, especially in the context about the meaning and state of entropy.
In the current state of our universe (the ‘top down’ view), temperature and densities are bound by existing physical conditions. We have an ‘absolute’ zero that experimentally can almost be reached but never in total. We have a maximum of temperature that is set by the limitations on the density that can be approached under compression. These limits are topological boundaries. ‘Infinitely hot’ and ‘infinitely cold’ cannot be properly defined within this context, those concepts set outside conventional boundaries. That boundary is set by the conditions found between the observable universe and ‘whatever’ happen during the ‘Big Bang.’ That boundary is conventionally reached by reversing time under the assumption that ‘gravity’ should dominate in the distribution of forces with polarization within the electromagnetic cancelling out. Under this scenario, a point of thermal equilibrium is almost reached, hence a choke point of maximum entropy. Consequently, it is also a point with a certain degree of topological complexity.
In the ‘bottom up view,’ all boundaries are part of a process of ‘emergence’ which can be approximated in a process of dimensional evolution, so it must start from a pre-dimensional state. The simplest expression of that state is an unbound state, a singularity with zero-entropy. That state has ‘infinite’ degrees of freedom, there are no boundaries or preferred directions. By the same token, it must be ‘infinitely’ unstable. It is also geometrically flat and of ‘infinite’ extent, it cannot terminate in any direction. To change, it must become more complex. To become more complex, it must evolve. These are causal changes of state driven by its inherent instabilities. As a singularity defined by unity, it also must be a self-referential process relating to it dimensionality. Since there are no existing parameters (or bias) for its evolution, it must be expressed as a geometric process, related to fractalization. Here, however, the difference is that this is not an abstract mathematical construct which would yield infinite dimensions. In the process of self-referencing, another topological limit is reached despite further extensions into complexity. Dimensional evolution comes to a point of saturation, a point defined by topological closure which terminates this process of inflation. It is also a point, with a certain degree of topological complexity. It is a local limit on associative processes. Further evolution, is no longer dimensional, but recursive. All fundamental physical constants are defined by this process.
Unity entails infinite possibilities for numeric values. In the context of an initial state, the notion of thermal equilibrium must be expressed without bounds, hence the range of plus and minus infinity. In that context of a gauge, thermal equilibrium sets at zero.
I think, as with particle/wave duality, the solution is a mixture of the two possible outcomes.
It is neither an infinite dimensional Hilbert space to expand into, or a finite space for our universe to exist in… it is as observed, an expanding universe.
As our universe is observed to be expanding, we can conclude that there is enough states for it to expand into, without reoccurrence. Thus we cannot conclude it is cyclical, hibernation prone or infinitely dimensional.
Bill S., thanks for the redress. (Sorry for the duplicated copy.)
Your question and my response points out the key issue in a structural context. It’s exactly where we want to be in the contrast between a ‘bottom up’ and ‘top down’ perspective, especially in the context about the meaning and state of entropy.
In the current state of our universe (the ‘top down’ view), temperature and densities are bound by existing physical conditions. We have an ‘absolute’ zero that experimentally can almost be reached but never in total. We have a maximum of temperature that is set by the limitations on the density that can be approached under compression. These limits are topological boundaries. ‘Infinitely hot’ and ‘infinitely cold’ cannot be properly defined within this context, those concepts set outside conventional boundaries. That boundary is set by the conditions found between the observable universe and ‘whatever’ happen during the ‘Big Bang.’ That boundary is conventionally reached by reversing time under the assumption that ‘gravity’ should dominate in the distribution of forces with polarization within the electromagnetic cancelling out. Under this scenario, a point of thermal equilibrium is almost reached, hence a choke point of maximum entropy. Consequently, it is also a point with a certain degree of topological complexity.
In the ‘bottom up view,’ all boundaries are part of a process of ‘emergence’ which can be approximated in a process of dimensional evolution, so it must start from a pre-dimensional state. The simplest expression of that state is an unbound state, a singularity with zero-entropy. That state has ‘infinite’ degrees of freedom, there are no boundaries or preferred directions. By the same token, it must be ‘infinitely’ unstable. It is also geometrically flat and of ‘infinite’ extent, it cannot terminate in any direction. To change, it must become more complex. To become more complex, it must evolve. These are causal changes of state driven by inherent instabilities. As a singularity defined by unity, it also must be a self-referential process relating to its dimensionality. Since there are no existing parameters (or bias) for its evolution, it must be expressed as a geometric process, related to fractalization. Here, however, the difference is that this is not an abstract mathematical construct which would potentially yield an infinite number of dimensions. In the process of self-referencing, another topological limit is reached despite further extension into complexity. Dimensional evolution comes to a point of saturation, a point defined by topological closure which terminates this process of inflation. It is also a point, with a certain degree of topological complexity. It is a local limit on associative processes. Further evolution, is no longer dimensional, but recursive. All fundamental physical constants are defined by this process.
Unity entails infinite possibilities for numeric values. In the context of an initial state, the notion of thermal equilibrium must be expressed without bounds, hence the range of plus and minus infinity. In that context of a gauge, thermal equilibrium sets at zero.
Bill S., You have missed a key point about topology, especially one described as a topological singularity. There is no other phase space. What comes out of the ‘Big Bang’ is all you get. Sean’s notion of reproducing cosmologies come close to a correct idea but fails to note that the process of reproduction is locally constrained by quantum processes. The notion of time moving in two directions is defeated by structural consequences related to the distribution of matter and antimatter within a unitary singularity. What comes out of the ‘Big Bang’ is the emergence of ‘matter’ and ‘energy’ (including its ‘dark’ aspects) it’s a single shot that moves complexity from zero out towards infinity. Everything that has the potential to exist must emerge within the singularity as a manifestation of continuity. The local expression of structure is an equilibrium of matter and energy, usually expressed as E=mc2. The fact that it comes into existence close to thermal equilibrium is because E=mc2 is an approximation and not quite precise. It is missing a factor which relates to the mechanisms of structure [(think in structural terms of single particles, not universes) and (to be precise, particles only exist as patterns of structure which are constantly maintained by persistent processes of dimensionality, wave forms)], specifically m=(kE/c2). The value of k equates to the difference between the electromagnetic field and what is often referred to as the Higgs field. There is a constant of displacement related to each field. Due to the unitary nature of topology, the consequence of spatial expansions and related contractions also hold a fundamental key to the dynamic characteristics of our universe and the role of gravity.
Fifty years ago, I might have enjoyed bandying a salmagundi of sesquipedalia, just for its own sake, but age and shortage of available time have changed that.
Unless we are willing to look seriously at the terminology we might slip into using, assuming that it means something to others, there will always be scope for misunderstanding. Infinity provides a prime example of such lax thinking.
Any (almost?) scientist or mathematician will agree that infinity is not a very large number, yet that same person may use infinity as though that is what it is. As a non-mathematician, I am not qualified to comment on the mathematical use of infinity, but I take issue with its use as though it were the “end” of a sequence, or some sort of point that could be reached by increasing/decreasing a finite sequence.
You use, for example, “infinitely cold”. Surely you are not suggesting that anything could become infinitely cold. Do you equate it with 0K? If so, why not say that? If not, what do you mean? Do you really understand the term you are using, or does it just sound impressive?
Bill S., We are not talking nonsense, but perhaps you are. I am sorry you are ignoring what I have already said. Like you, I have no desire to beat a dead horse. So, I will not retrace what I have already said. If you truly believe, that scientists and mathematicians believe that infinity is not a very large number. You are wrong on two counts. First, the concept of infinity is not a ‘cardinal’ number. Second, it mathematically reflects the problem of computability whenever you try to divide any number by zero. Likewise, it reflects the direction of magnitudes unreached by compounding the results of any repeated process that does not terminate. The sum of a series of computations can, in some cases, approach a finite limit, which implies a convergence that might be reached by a definable point. Cases to the contrary, will diverge. Simple ratios will express numbers with different properties dependent on how they are expressed, both finite and transfinite. We could even dive into the realm of number and set theory, but neither of us have time for that. The study of ‘infinity,’ is well represented by Cantor sets. You might consider the work of the mathematicians, Henry John Smith or Georg Cantor. So, yes, I do understand the term I am using; and, no, I am not using the term just to sound impressive. I respect the term and understand the context of its history.
“We are not talking nonsense”.
There’s something I didn’t say.
“….but perhaps you are”
Possibly. If asking for a straight answer to an honest question is talking nonsense – yes, definitely.
“So, I will not retrace what I have already said”
Please, don’t.
“If you truly believe, that scientists and mathematicians believe that infinity is not a very large number. You are wrong……”
It’s a question I have asked several scientists and mathematicians and, so far, the answer has always been that it is not a number. Apparently, you know some who think differently. If so, I would be glad to know of them, and their reasoning. I prefer to be aware of as wide a spread of opinions as possible.
“We could even dive into the realm of number and set theory, but neither of us have time for that.”
I have absolutely no quarrel with the use of infinity in set theory.
“The study of ‘infinity,’ is well represented by Cantor sets.”
I have some familiarity with Cantor’s work and admire it, but remember that even he made a distinction between mathematical infinities that are amenable to manipulation, and “absolute infinity”, which is not.
Bill S., I was going to move on to the question of Hilbert space, but your comments here are very important to the understanding of continuity in conjunction with quantized spaces and that might affect the understanding of Hilbert spaces. But putting that aside, your comments here have identified the source of your aggravation.
Let’s take this one first, you said, “I have some familiarity with Cantor’s work and admire it, but remember that even he made a distinction between mathematical infinities that are amenable to manipulation, and “absolute infinity”, which is not.”
That is entirely true, and I fully agree with all of those who have made that observation.
You also said, “It’s a question I have asked several scientists and mathematicians and, so far, the answer has always been that it is not a number. Apparently, you know some who think differently. If so, I would be glad to know of them, and their reasoning. I prefer to be aware of as wide a spread of opinions as possible.”
I am glad to know that there is a consensus about ‘infinity’ not being a number. I would also worry if it were otherwise. And frankly, I know of none who would think differently.
But to address the basis of your misdirected assumption, I would point out that there is nothing within a pure continuity that is countable, except in its entirety as a singularity. However, as Cantor and others noted, most processes that generate a non-terminating numeric series can be mapped one-to-one onto the series of counting numbers. As you noted, these mathematical infinities are amenable to manipulation. All quantized fields fit into that category no matter how complex. A pure continuity, enters the realm of countability under the geometrics of folding and mixing processes. The real question you are asking is how do you distinguish between one field of infinite extent from another when it appears to have the same degree of complexity as far as the count of domains is concerned—and moreover, dimensionally, how is one field generated or associated to all possible fields? I would say there are many scientists and mathematics who would at least partly know the answer to this question if properly asked in the usual context of combinatorics. But, I am the only one (as far as I know) to apply it in the context of what happened during the ‘Big Bang.’ It is that bit of esoteric knowledge you inferred that I must have earlier comments. At the time, I admitted to having knowledge that was ‘esoteric.’ But I don’t recall you asking what that knowledge was.
You said:
“If you truly believe, that scientists and mathematicians believe that infinity is not a very large number. You are wrong on two counts”
“I am glad to know that there is a consensus about ‘infinity’ not being a number. I would also worry if it were otherwise. And frankly, I know of none who would think differently.”
Interesting contrast!
Also:
“The real question you are asking is how do you distinguish between one field of infinite extent from another when it appears to have the same degree of complexity as far as the count of domains is concerned”
Absolutely not. Perhaps this explains your reluctance to answer questions directly. You assume I don’t know what I’m asking.
Actually, it was never my intention to talk about infinity. I came here in the hope of clarifying a few points with Sean, from his excellent book. Possibly he is too busy to field trivial questions from “hitch-hikers”. Perhaps your role is to field such questions on his behalf. In either case, I might do well to use my limited time elsewhere.
“At the time, I admitted to having knowledge that was ‘esoteric.’ But I don’t recall you asking what that knowledge was.”
Your recollection is commendable; I didn’t ask.
Sean, Bill S. is correct in pointing out that anyone that is posting on your site is a “hitch-hiker” and that there is indeed an ‘interesting contrast’ in my response when I referred to “. . . scientists and mathematicians believe that infinity is not a very large number.” I should have been more specific by saying, “If you truly believe, that scientists and mathematicians believe that infinity is a bound numeric concept, like a finite number.” I apologize to him and others for being less precise than I should have been. And to Bill S. for being so loose in my language. Imprecisions can be problematic in technical communications. But, I have only asked one question, “How do you extract the finite from the infinite?” Which has not been answered, at least not yet. As you well know, the question is far from trivial. I assume Bill’s obtuse drill was to determine if I really understood the concepts of ‘infinity.” Of course, he has not announced the intent of his conversation with me. I would even agree it is still possible that his time could be spent elsewhere. But that is for him to decide.
Sean, based upon the extent of your balanced presentation, I am sure your book reflects the same excellence. Like others, I would have liked to hear how you talked about each slide. So, though we might or might not agree, I believe I can safely recommend it without having read it. You obviously do your homework and monitor what goes on here. In that context, I can honestly say I am not your surrogate here and wouldn’t be that presumptive. But having said that, I also do believe we are seeking the same ends.
That neither you or Bill has yet directly challenged any of my assertions tells me that you fully understand the significance and consequences of what I have asserted up to now. Providing of course that my application of principle is correct and that it acceptably fits into the context of your views on cosmology. My work is not yet vetted and sets outside the mainstream of practice and certifications. So, it should be approached with extreme caution. However, I will also add that the esoteric principle I am applying to this question about entropy and Big Bang inflation has already been proven in a different context. Moreover, the predictive power that it demonstrates honestly surprises me, despite my original insights into this unresolved problem. As I stated earlier, I am still sorting out and getting use to the predictive details. Starting from where I began, it could have easily failed to demonstrate any kind of productive veracity. But,since I have not yet fully aligned my case with your presentation, I will try one more post on your site before I bow out. I will only reengage if you have relevant questions for me.
Charles, I recognise that your last post was addressed, mainly, to Sean, but there are one or two points on which I must comment.
You say: “I have only asked one question, “How do you extract the finite from the infinite?””
If you asked me that question, it must have been masked by the exuberance of your verbosity, such that I missed it. I try to make it a strict rule that I do my best to answer questions.
Such is the English language that the “you” in your question could be interpreted in two ways; I’ll try to cover both.
If “how do you” means how do people in general, then my answer is that I am far from qualified to speculate, in spite of having spent a lot of time reading and thinking about infinity, and discussing it with others.
If “how do you” is directly asking me for my personal view, then the simplest answer is that mathematical “infinities” are convenient approximations which have their value in the appropriate context, but probably have little relevance to your specific question.
Beyond mathematics, infinity is not a number and eternity is not an expanse of time. Time and quantity may be unbounded, but this relates only to our finite perception. Infinite is an entirely different concept.
So, how would I “extract the finite from the infinite?” (To keep things in a physical perspective, let’s consider an infinite cosmos). This cosmos is eternal, infinite and changeless. Our perception of a finite Universe existing in linear time is a “shadow” of that underlying physical reality. It arises from our 3+1D perspective, and is essential to our ability to form any understanding of our world.
Having said that, I remind you of two things: 1) This is just my person, definitely non-expert opinion, and is wide open to modification. 2) Becoming involved in a discussion about infinity was not my purpose in coming here.
Possibly I stepped out of line by introducing a question about Sean’s book to a discussion about his presentation. If that is the case, I apologise to Sean, and offer as explanation the fact that my IT knowledge/skills are limited, so I took what looked like an opportunity.
Damn, I thought it just “Blowed up”
Thanks, great post.
Recently I’ve read something about time-symmetric crystals. Is it crazy to think of the universe as something like that time-symmetric crystal, so that it would go on forever without expending energy?
Initial response:
Action without energy – perpetual motion – run away. 🙂
Perhaps you could say a bit more about time-symmetric crystals for those of us who have not met them, and seriously lack “research” time.
My apologies, it should have been time-asymmetric crystals, predicted by Frank Wilczek and now confirmed by scientists at the University of California, the University of Maryland, and Harvard. Wilczek seems to be of the mind that studies of such time crystals would shed light on some deep questions about the origins of the universe, how it evolved, and deepen our understanding of the spacetime continuum. Any thoughts on these…?
Thank you.
Sean, I was going to refer to a specific slide in your presentation to discuss Hilbert spaces. But, you may have pulled it and replaced it with another slide. Not a problem, I can still discuss the comparison of views.
Let’s start by creating a mathematical shorthand. Infinite-dimensional Hilbert space equals H1. Finite-dimensional Hilbert space equals H2. The bottom up view holds that H2 is a subset of H1.
The relationship is revealed by some additional shorthand. For this demonstration, I will use the idea of phased field evolutions, starting with the initial singularity (the concept of unity) as P(0). P(0) must equal 1. If you wish, you can think of it as Schrodinger’s conception of certainty based on distributions of probability. As you know, H1 also has potential as an unknown volumetric, which can be expressed by H1^0, such that P(0)<H1^0 as a subset of H1.
To maintain unity in generic evolutions of complexity [P(0) to P(x)], where x is a process that goes from zero(information) within the unity of P(0) towards infinity. In that process, P(x) must reflect two basic characteristics of fundamental association. First, a ‘quantitative’ function basically related to the complexity of compounded forms; let’s simply label it as N. In that sense, N has an equivalency to H1 as a subset. And second, a ‘qualitative’ function related to fundamental limits on association, let’s simply label it as Q. In that sense, Q = H2. Q is not a subset of H2 for this definition.
Following the context for that point of view, H1 is functionally populated by a process which runs from zero information towards a potential for unlimited information.
We can now state a phased function for P(x). P(x) = NQ. Since P(0) must equal 1, then specifically, as a phased process within the context of P(0) (the singularity of unity), P(x) must also always equal 1 to retain that meaning of a profound limit. Consequently, N represents a simple count of ‘reproductive’ complexity in terms of Q. To ensure that N always equals 1, N must be written in terms related to x, but in a process of inversions. Hence, N = n1/n2, n2=n1. Or to be more precise, N takes the form that counts functional reiterations (reproductions) of Q. As a simple count related to Q, the value of n1 runs from 1 towards infinity. However, since n1 is also a byproduct in a compounded series of associations means n1 is also entails an exponential function, say a function(x), where f(x) precisely reflects the limited structures and processes entailed in Q. During inflation, these are recursive processes. So, H1 and N are related in restricted terms of process (specifically, closure on the dimension evolutions of ‘local’ space now expressed in H2). So broadly, in contrast, H1 is infinitely dimensional in a mathematical sense; which is mathematically consistent, but not necessarily reflective of the physical processes defined by the subset of P(x). To properly express physicality, the variables of those mathematical functions must reflect dimensional limitations globally within the bulk of P(x) due to entanglements, again specifically, by the finite processes entailed in H2=Q. Therefore, there is an important substantive conceptual difference between the mathematic potential of dimensions in H1 and the physical dimensions in P(x) which are formulated by N in the limited evolution of physical dimensions entailed in Q.
H2=Q determines the limitations which define the emergent properties which are functionally related to our conventional notions of space and time. In that context, N determines the extent of those physical distributions. That P(0) in some way equates to the infinite reach of Euclidian fields postulated by states in quantum field theory (QM), approaches the correct point of view for physical interpretations of P(0) in H1, especially as N expresses the global extent for physical modes of inflation.
Consequently, the reduced formulation of H2=Q is the primary focus for this discussion. It is responsible for how inflation locally terminates at a point of decoupling. The function and operations of Q also enable the interactive interactions (coupling and decoupling) seen in the observable universe in conjunction with the associated dynamics of a universal ‘now’, a phase of state indicated by P(5).
P(5) is a failure of a fifth spatial dimensional to emerge and is directly related to the emergent functions of dynamic time within four-dimensional spatial displacements. These interactions are arbitrated through displacements related to three-dimension space. This formulation of interactive dynamics is captured by Q = {(q1+q2)/(q2+q1)} = 1. In this case, the components of Q represent an average between two oscillating states. For the purpose of demonstration, phase(1)=q1/q2 and phase(2)=q2/q1. Phase(1) times phase(2) = 1. The fields represented by q1 and q2 are similar, but not quite equal. The two persist as instabilities within Q as temporal oscillations relating to the structure and emergence of time—specifically as clock states. Consequently, Q also synchronizes events within P(5), defined and detailed just below.
In looking at the basic components of any two-dimensional quantum field, a field cannot be bound by limits on complexity. It can only bound by the number of unique associations which are formed between domains expressed within the plane, despite the potential for unlimited extents. That degree of ‘uniqueness’ is numerically defined by the type of association. Types of association form a unique set, let’s label it U (it is conceptually related to symmetry groups). In this context, U expresses the degree of saturation within the field and can only run from zero to four. This fact holds true whether the space is opened as in a Euclidian plane, or closed as in the surface of a sphere (caveat, the expression of a sphere also requires a volumetric space in which to exist). The property of ‘unique’ associations underlies the solution and proof in the ‘Four Color Theorem.’ [Not liking the method of the original ‘proof,’ I did another proof using different methods.] However, what is important here is that all simple two-dimensional fields are limited in terms of uniqueness, U(1), U(2), U(3), and U(4), an extension of association to U(5) does not exist with in the field, except perhaps in modified terms of U(0). Then, U(5) = U(0) only in the sense that each reflects a potential association which does not properly exist as a domain or boundary within the field.
The corollaries to the physical dimensions of space follow this sequence: U(0) equates to pre-dimensional space; U(1) equates to non-dimensional space; U(2) equates to one-dimensional space; U(3) equates to two-dimensional space; U(4) equates to three-dimensional space; and U(5) equates to the dynamics of four-dimensional space. However, in the phasing of P(x) the relationship is staggered due to limitations on topology that are not a conditional limitation on fields. P(0) implies U(1); P(1) implies U(2); P(2) implies U(3); P(3) implies U(4); P(4) implies U(5); but, by the rules of unique association, P(5) cannot exist in terms of U(6). U(6) cannot be a field related association in terms of unique associations. Or in strict terms of U(5) or U(0) for existing as a proper associations within a two-dimension field. In evolutionary processes of progression, this means that U(5) and U(0) only exist as a genetic part of the self-associated processes which enable the dimensional emergence of fields in QM. The operative connection between U(5) and U(0) forms a looped circuit which expresses the gravitational effects of ‘causality (spatial expansions)’ and ‘gravity (spatial contractions).’ Consequently, the process underpins and sets up the emergence of GR couplings.
To reiterate, if the topology required for existence fails, then the expressed case for no topology or zero topology means physical oblivion. So, in the evolutions of H(2), P(0) must equal a state defined by U(1). In terms of quantum fields, the meaning of U(5) and U(0) expresses an allowable process for the creation and destruction of local fields. Essentially, in field evolutions, U(0) creates a potential for a field that is then extended by processes of U(5). It is in the dynamic sense of field relationships that U(0)=U(5). The topology expressed by P(0), which through intrinsic instability, fundamentally relates to the phases of global folding and mixing. Overall, as the foundational state, it refers to physical continuity and the phased processes of a singularity. Again, that process comes to dimensional closure at P(5). Closure on dimensional evolution does not terminate systemic instabilities. It only defines a point of dimensional saturation which can easily be viewed as a form of thermal equilibrium.
A U(5) association can only be defined in the interactive relationship between two saturated fields. Geometrically, those relationships are orthogonal. The saturation point of all fields is U(4). The minimum of associations which permits a Hamiltonian walk within a field or between fields is defined by U(2). Both are fundamental to folding and mixing processes. U(1) and U(3) potentially define emplacements [combined form (neutron)], [decoupled form (proton and electron)] and as specific processes [consequences of coupling and decoupling (neutrino)] within fields, but are not strictly a byproduct of global or local folding and mixing processes. They arise due to polarization within H2 and between the coupled fields of H1 and H2. In the collection of couplings expressed by H1 and H2, are five fermionic inflection points (think ‘quarks’) and two sets of eight gluon-like bosonic processes (think gluons, photons) which are dynamic reflections of internal and external processes (exchanges of energy)—all patterns of perpetual recursion within a saturated field. The energy comes from the potential energy and instabilities of the initial continuity through continuous inflationary processes.
In saturation, after closure on dimensional inflation, U(1), U(2), U(3), U(4), and U(5) are no longer variables but constants of association. At that point, folding and mixing processes can be expressed as a mostly fixed functional series in terms of U(2) and U(4). Topologically U(2) relates to electromagnet processes and U(4) relates to what might be called the Higgs field, both are related to spatial expansions and, physically, are properly ‘dimensional’. The dynamic of U(5) relates to gravitational processes in terms of expansions and contractions of and within fields. Consequently, U(5) also represents a failure for a fifth dimensional of uniqueness to be manifested within any two-dimensional field. However, {U(5), U(0)} still forms a circuit of displacements which produces and binds all fields together, it links ‘causality’ (a process of expansion) to ‘gravity’ (a process of contraction). The functional series, as an expression of spatial saturation, illustrates the relationship between ‘expansion’ and ‘contraction’. Expansion (the sum of the series) is related to inflation. Contraction (the product of the series) can be correlated to the formation of mini black-hole singularities by being related to unity. The sum and product of the series illustrate the ‘cycle’ and ‘bounce’ of between the maximum and minimum limits of these operations of dimensional compression and dimensional inflation.
As constant terms, these relationships can be easily stated. The function, f{U(2)}, is defined by the set of terms {2^8, 2^4, 2^2, [2^1, 2^-1], 2^-2, 2^-4, 2^-8}—a two-fold process. And the function, f{U(4)}, is defined by the set of terms {4^4, 4^2, 4^1, [4^0.5, 4^-0.5], 4^-1, 4^-2, 4^-4}—a four-fold process. In orders of uniqueness, f{U(2)} precedes f{U(4)}. Note that f{U(2)} = f{U(4)} = 1 as products of each series. To jointly gauge and scale these function, the two function need to be related, but also considered in the inflationary process which leads up to the point of saturated displacements. Zero Point Unity (ZPU) allows the functional series in terms of U(2) and U(4) to be dimensionally equated and aligned according to centers of displacement within Q and points of coupling and decoupling between N and Q. The centers of differential displacement within f{U(2)} and f{U(4)} are offset by square brackets. The offset value in f{U(2)} has a combined expectation value of 2.5, a sum of the bracketed set, {2, 0.5}. The offset value in f{U(4)} has a combined expectation value of zero, a sum of the bracketed set, {+2, +0.5, -2, -0.5}. This condition within the square brackets, sets up the potential for functional constants of displacement. The sums of f{U(2)} and f{U(4)} minus the expectation values for each spatial field are essentially equal. Let’s set that value to a constant and arbitrarily label it as, D. The expectation values can be combined as constants of differential displacement, C1. Note that the expectation value for C1 cannot be zero. A value of zero, would defeat the potential of the function under the resulting product of compression.
To include the evolutionary processes between N and Q and to place them in the analysis of the functions which define Q. Let’s define the results of the folding and mixing processes from P(0) to P(5). This number is another constant of displacement, C2. A final statement of analysis produces an inflated set of terms which can be written as, {[C2], D+[C1], [C2]} with C2 setting and the external points of processes which set at the limits in expression of Q, but which also must be included in the gauging defined by ZPU. The process which defines the differential constant of displacement between N and Q shows up in two places in the analysis. The first is directly reflected global processes of folding and mixing during inflation. What happens precludes Sean’s notion of a reproducing cosmology on universal scales.
Basically, half of the gross field defined by P(0) to P(5) goes to zero while the other half doubles in number. The past and future share the same topological space in the local and global conjunctions of now. The amount of positive matter and energy is doubled, while the temporal displacement of antimatter cease to exist in persistent forms, except ephemerally in the durable dynamic vacuum relate to space. In the expression of Q, this doubling appears in consolidation as a masked factor, 2/2. Within ZPU, it shows up as a phased oscillation related to the dimensional synchronizations of time between N and Q that persists as a differential constant of offset as a percentage of those active relationships that locally span the coupling and decoupling within P(5).
The combinations give you a precise statement about the functions of energy, q1; as well as a precise statement about matter, q2. Those relationships lead to the common equation generally expressed by E=mc^2. However, there is a hidden factor in that relationship which express the structural ratio between q1 an q2.
In short, q1 and q2 specifically defines the structural density of P(5) in terms of dimensional constants as abstracted from the principles of unique associations. Those parameters are inherent to the genetic processes of dimensional evolution. The geometric structure of P(5) exhibits all the structural properties of our universe and entails all fundamental constants as already indicated. Those constants can be mathematically extracted from the saturated field density as defined.
The book I wrote is an informal demonstration of this application of principle. The title, “The Topology of Quantum Timespace: A Theory of Everything,” is an accurate statement. However, it is not ‘The Theory of Everything.” To finish the project, it must be turned into a formal statement. To do that, I would have to invent a new field of study that might be titled as, “Quantum Vector Mechanics.” Mathematically, it would be a subset of vector analysis. The project is within the reach my capabilities. Fortunately, since I am short of time and inclination, I do not have to take on that project by myself. I can share what I have done or pass it on to others who, as a professional community, already have the necessary qualifications and techniques as scientists and mathematicians. Any volunteers?
Another possibility: 5. A Mirror Universe.
* In this fifth alternative, the Second Law of Thermodynamics runs in the opposite direction prior to the Big Bang, and the part of the universe on the opposite side of the time dimension from us with respect to the Big Bang is made up predominantly of antimatter. The universe extends both forward and backward in time, in each of the two cases away from the Big Bang and towards lower entropy. The Mirror Universe on the other side of the Big Bang would be basically like our own.
* This would solve the matter-antimatter asymmetry problem, would eventually allow the universe to expand indefinitely in both directions, and would also make it possible to think about the explosion of the Big Bang itself as being associated with the brief time period in which matter and antimatter are not well sorted and annihilating like crazy.
Bill S., communications are central to any of our understandings. Unfortunately, comprehensions are seldom immediate and take time to communicate. Still, the dynamic arch of our communications reflects who and what we are as individuals—our philosophies about our lives and how we interpret all the events within the reach of our perceptions. [All important points of personal and cultural context entailed in your last statement, to which I should respond to be fair to the points of those arguments. Despite telling Sean of my intent to “bow out.”]
In that sense, live in two worlds. An objective one, which expresses all the ‘lawful’ physical limitations in which we exist and subsist; and, a subjective one, in which we compound all our personal understandings and interpretations about the dynamics of our lives.
Our perceptions of an objective, physical reality is tested daily through personal experience in our interactions with everything around us. It can also be more empirically tested and scientifically examined through careful measurements in repeatedly demonstrable and controlled circumstances. Within the limits of objective laws, we can even manipulate those lawful circumstances to our advantage. But, we cannot change the fundamentals of those laws due to the lawful nature of the mechanism which enable our existence. Those are dimensionally wrapped up in a series of ‘quantum’ entanglements which mostly defy any of our technical abilities to fully isolate.
The subjective world is a highly-biased construct of assimilated and recollected experiences which is retained and constantly modified through our limited capacities for reason and memory. As individuals, that process helps us navigate the world we discover around us. It enables each of us to place ourselves in individual and communal contexts. We are self-aware.
Due to this context, I will agree and disagree on certain points of your statement.
1. “. . . mathematical “infinities” are convenient approximations which have their value in the appropriate context, but probably have little relevance to your specific question.”
[I tend to agree with the first part of the statement; but disagree about the relevance of “infinities” to my specific question.]
2. “Beyond mathematics, infinity is not a number and eternity is not an expanse of time.” Time and quantity may be unbounded, but this relates only to our finite perception. Infinite is an entirely different concept.”
[I would argue that “infinity” in not a number, both inside and outside the context of mathematics. I would argue in the same sense, that “eternity” is not a number so cannot specifically define a duration. However, it can refer to an “unbounded” duration. Both relate to our ‘finite’ perceptions.]
3. “This cosmos is eternal, infinite and changeless.”
[True, for “eternal” and “infinite.” But only in the very limited sense of “singularity” and “continuity.” However, not true for “changeless.” Anyone would be hard-pressed to explain all the processes of “change” relating to events before and after the “Big Bang” and the aggregations of matter and energy which complement spatial expansion.]
4. “Our perception of a finite Universe existing in linear time is a “shadow” of that underlying physical reality. It arises from our 3+1D perspective, and is essential to our ability to form any understanding of our world.”
[For the reasons cited above in an explanation of ‘objective’ and ‘subjective’ perceptions, I would agree. However, having that philosophical understanding, does nothing to enlighten our physical understanding about how that circumstance comes into ‘being’ and ‘becoming.’]
Sean, while you are thinking and taking care of more immediate business. I will add a footnote to my comments above to more tightly focus on the cosmological consequences of singularity as absolute unity within the context of continuity.
The Topology of Quantum Timespace (TQM) equals P(5) in terms of N and Q (functions for the GR formulation for mass and energy [in terms of QM] as well as the QM formulation for the notion of wave-particle duality). Ohwilleke’s post (A Mirror Universe) is essentially very much like your observations about the potential for a for ‘reproducing’ cosmology, albeit on a slightly smaller scale. Returning to the context for Schrödinger’s functions in terms of probabilities within Hilbert space (specifically your discussion of Eigenstates in slides 21 to 24 in reference to time, as well as the physical notion of clocked states (slide 25) and dimensional compactification (slide 15), ‘you’ should note that the conjunction of space-time and matter—anti-matter is well served. Or, in general, Schrödinger’ perspective, such that in the conjunctions for ‘now’, the components of TQM oscillate topologically. Where TQM^t defines the dimensional limits of those oscillations, such that the exponent of ‘t’ oscillates between the values of +1 and -1 in the systemic progressions of conjunctions that generate differential exchanges (bosonic attributes) between the physical states of structure (fermionic attributes)—nature’s means for handling information. If you like, ‘you’ can think in terms of ‘qubits.’
Ohwilleke,
I have always liked the mirror universe option. It could (obviously) explain any and all asymmetries that we see, or will ever see, in our universe.
However, there are very massive questions that remain, and they are largely overlapping with the same ones as in several of the other options.
Just as an example…
How is this at all different from other universe options? The only hint is that every particle in ‘this half of the universe’ has to have be somehow balanced by an antiparticle in the other half. But… except within the tiniest bit of ‘time’ when it arose, there was no interaction at all. Why the balance at all? What if the big bang was just slightly asymmetric? Etc.
Overall, when you factor in what a person thinks of when they think they say ‘time’, there is little difference.
A ‘symmetrical universe’ in your #5 iimagination is identical to an anti-universe that collapses and then explodes as our positive universe at the big bang in someone elses’ who is thinking of time as common people do.
The general nature of symmetry in everything else is a strong indication, but not really proof.
Hi Sean,
A very interesting talk. Do you have references for the theories you mentioned (the string gas cosmology, primordial degravitation, baby universes etc).
Thank you