It would be interesting to see the foundations of mathematics necessary for physics reduced to groups, groupoids, monoids and categories such as motives in a Grothendiecke type of system. I have I think found a possible route towards this using discrete systems.

There are various entanglement schemes, and the GHZ entanglement is 1/8 supersymmetric. There are bipartite and tripartite entanglements that are ½ and ¼ supersymmetric. This means that on BPS black holes these entanglements of states associated with the BPS charges have this number of supersymmetric generators that are unbroken. The algebraic geometry of these entanglements involves a quotient homology of projective varieties. These are systems between the Hilbert space and a projective Hilbert space with the geometric phase as the fibration. This system is categorically the same as a quotient homology on the moduli space for quaternionic bundles, such as with SO(4), or SO(3,1) in the hyperbolic case. The isometries for the this system is SO(4,2) and the moduli is AdS_5 ~ SO(4,2)/SO(4,1).

The two quotient systems are given by discrete groups. In the case of the AdS_5 the quotients are Kleinian groups, which are quotients with a discrete group, such as an elementary Z_n ~ Z/nZ, or a more complex polytope group. For the case of projective varieties on the Hilbert space these are a system of discrete orbits that have a discrete geometric phase ~ e^{nEt}. The two orbit spaces are I think categorically equivalent.

The various quotient groups correspond to cobordants of one dimension lower.

For instance, with AdS_5 there is a boundary spacetime, and for the quotient group on the moduli AdS_5 defines two boundaries with different spacetimes that may have different topologies. The equivalency between quantum projective varieties and the Kleinian orbit space of different spacetime topologies connects topology changes with different quantum states or sets of quantum states.

I think these correspondence goes beyond one particular type of entanglement. There is a whole algebraic category of entanglements by Micheal Duff and his research partners. This algebraic system of entanglements is connected to this structure of quotient homologies and algebraic varieties. The categorical equivalency with the AdS_5 is a surprising aspect that I have suspect exists with respect to the mathematics of four manifolds as found by Atiyah, Donaldson, Freed, Singer, Uhlenbeck and others. The moduli space when reduced to a finite group is equivalent to the orbit spaces of quantum states with a discrete structure. This structure is given by the Kirwan polyhedra of holomorphic coadjoint orbits. This is categorically equivalent to the quotient of the moduli space or AdS_5 with a discrete or Kleinian group.

The theory does of course connect with Raamsdonk's observation that entanglement can be converted to geometric content. In particular the entanglement of a quantum system with states associated with gravitation is equivalent to the entanglement of that system with the stretched horizon of a black hole or similar system. The "large N limit" means the set of states entangled with the gravitational system become entangled with a system that has a coarse grained structure, such as how an event horizon has lots of Planck area units that states can be shuffled around in.

By doing this I think we can reduce physics to certain topological invariants, and all of physics can be reduced to a homotopy theory of logic gates. I expect in time to see physics rely upon mathematics that is less motivated by concerns with infinite and infinitesimal elements or sets, and more motivated by discrete structures. For now the more traditional geometric interpretation of things is a necessary aspect to how these are derived, but in time these things may be of less importance.

LC

Philip,

A very nice essay. I am also intrigued by the multiplicity of landscape solutions, and the thought that a priori we can only assume that all consistent vacuum solutions have equal probability of being correct. Such indetermination may not be an obstacle for progress though. We can certainly define the maths that use indeterminates and at some level, that might be enough. Curious as to what you think of just letting go of the idea that we will ever know the exact vacuum state.

Best,

Harlan

    Dear Sir,

    You have raised some very important questions that can be answered only if we think out of the box. The problem is that we collect lots of data and without proper examination, reject most (like at LHC) that could have given us equally plausible ideas about the natural laws. Secondly, we follow the beaten path without reviewing it and reconciling the apparent contradictions that are being increasingly observed. However, your essay provoked us to expand a few thoughts.

    Even though the viability of the loop quantum gravity is questionable, one of its predicted scenarios is the big bounce. If we replace the big bang with the big bounce, add to it the laws of thermodynamics and some ancient ideas about time, we get a totally different picture.

    Let us start from the last. Time is the ordered interval of events, which are measurement of observables at various coordinates. There is a fixed pattern of all events. These are: being (situation leading to its creation), becoming (its creation itself), (growth due to addition of other particles/events), transformation (as a result), transmutation (due to the same effect - incompatible/excess addition), destruction (change of form as a consequence) to start a new chain. Since galactic blue-shift has been observed putting a question mark to dark energy concepts, let us assume a steady state universe, where everything follows this pattern. Everything is measured/perceived through the radiation it emits - thus, through thermodynamic processes. Condition of maximum entropy is the final stage of the cycle. Then, in the Universal scale, big bounce will be the beginning of a cycle. At that stage, it will be only creation through redistribution. There is the universal space and universal energy, but no one to perceive or measure. The one energy is all pervasive. The emergent energies can be different, local or unknown. Structure formation being an event, must have followed the beginning of the cycle. Since space is the base and interval of structures, space as we know it, must be an emergent property after time. But how did it all start?

    If you look at motion and action, you will find that action is momentary, but it creates a pair of equal and oppositely directed inertia that create local disturbances to create composite and differential inertia that tends to restore equilibrium in a multiple reaction mode. On the other hand, motion is mechanical - it perpetually responds to density fluctuation in all sorts of manners: energy, material density, air density, charge density, etc, created by all sorts of manners including heat (electric), cold (magnetic), etc. Anything subject to strong interaction has the capacity of confining motion. It generates inertia that also acts mechanically till local equilibrium is restored (weak interaction). This is followed by redistribution (electromagnetic). But action is different. It is induced by a conscious agent that breaks the stability or equilibrium. Thus, at the creation event, inherent instability of the conscious system of the universal observer starts the process by creating a perturbation. Some may question this as religious belief. But can quantum theory survive without observer?

    We have a fully developed theory that explains many things. In our essay, we have discussed the Wigner's view of unreasonable effectiveness of mathematics and Gödel's incompleteness theorems as well as Einstein's formulations.

    Regards,

    basudeba

    Philip,

    Thanks, for sharing your ideas. It is nice to see so many different points of views regarding this topic.

    Best Regards,

    -D.C.Adams

    Professor Gibbs,

    The new paradigm shift (from one Universe to multiple Universes) is hard even on Western thought. At first I was reticent but then... . It is actually amazing when one thinks about it compared to the concept of the one singular Universe our egocentric minds have somewhat logically been led to believe in (hard to let go). There are hints to the Meta-Laws. Frank Wilczsek stated something to the order that why are the gauge coupling forces unifying (GUT except gravity) at one point in our Universe (I am assuming SUSY makes the unification more exact)? Thinking about this, is it probable that most successful Universes has this form of unification of gauge forces more or less in some tolerant area near a unifying energy (Planck energy type) with gravity? This cannot be attributed to coincidence. You stated in your essay, "It is known that the combination of quantum theory and general relativity imposes tough constraints on the possible range of consistent space-time models." Also, the GUT point being (how much?)variable with other Universe formations this does rule out 'fine tuning" and maybe one should not look at just one number (say the Higgs boson mass) and say it has some 'unnaturalness' because it looks random and especially 'fine tuned'. It is best just to look at the GUT points in any formed successful Universe that has a 4D space-time. It may be hard to trust just a few numbers whereas the GUT points are more of a gestalt of what is going on. Why 4D? It is well know that 4D manifolds are the most interesting manifolds/topologies in mathematics. Even more so than higher dimension topologies. Recently, the Triangulation Conjecture was disproven. Coverings of simplices (triangles or tetrahedrons) cannot completely cover higher dimension topologies (past the 4D) based on some simple rules. This leaves higher dimension coverings 'foamy' or full of holes. Makes one wonder whether this means that higher dimensional Universes can exist in a 'physical' sense. I am not sure that 'all solutions exists'. It is bound to be super variegated (though with 57 varieties ;)). I am thinking that one should not even consider 'fine tuning' anymore but to look at the tolerance or range of solutions in the hierarchal space as a mathematical structure that can eventually be computed in a Scientific framework. All other Universe that have the 4D space-times would somehow have a computational relation to each other (a dictionary?). The GUT point would also be a measure of how fast nuclear rates (and chemistries) and gravity combine to create the Universe. In some Universes (even if 4D) that stars would burn too fast or gravitational collapse only produces black holes where life could not have enough time to form or exist. Or the GUT point is such that stars cannot ignite and that Universe is a cold dead world with not much going on. Where there is no GUT point for a Universe it perhaps collapses to nothing. I think a lot of people misunderstand the Multiverse especially when it is hyped to the point that suggests 'all outcomes are possible' somewhere in a Universe we will never know. And there is no parallel person that is me somewhere else with a different set of beliefs or lifestyle as that is hogwash. I think that the other Universes have some sort of 4D path integral sensibility which produces some similar outcomes but not the same or nearly the same outcomes found in the other Universes. Finally, here is a direct quote from a white paper "Higher-Order Intersections in Low-Dimensional Topology" by Conant, Schneiderman and Teichner, "The Whitney move, sometimes also called the Whitney trick, remains a primary tool for turning algebraic information (counting double points)into geometric information (existence of embeddings). It was successfully used in classification of manifolds of dimension > 4 specifically in Smale's celebrated h-cobordism theorem (implying the Poincare conjecture) and the surgery theory of Kervaire-Milnor-Browder-Novikov-Wall. The failure of the Whitney move in dimension 4 is the main cause that, even today , there is no classification of 4-dimensional manifolds in sight." I quoted this to make the point that there is a lot of work to be done and that perhaps the unreasonable effectiveness of math in physics is really related to the 4D space-time issue of physical-ness.

    Philip,

    Your mathematical description of doing frontier theoretical physics is noted. However I believe that there is a need to change our emphasis from the mathematical development to the physical model development.

    In the past 100 years theoretical physics and cosmology developments have been conducted almost exclusively on a mathematical basis, leading to non-physical objects or processes such as fields, space-time, curvature in space-time, time dilation, length contraction, virtual particles, action at a distance, curled-up dimensions, Entanglement, Dark Energy, Dark Matter....etc. I believe that these abstract mathematical objects are different aspects of one physical model of our universe. Therefore I urge that we devote more efforts on the physical model development.

    Regards,

    Ken Seto

    Sorry for not responding to comments for a while. I will get back to it at some point soon.

    I am what you call an amateur scientist. In 2001 I discovered what I thought something significant. I submitted a paper to the "American Physical Society, Physical Review D", put up a web site, published a book: nothing. I discovered FQXi and thought it great. I have submitted contest entries with the simple hope that someone would review my material. Now, I have found viXra. The purpose of this note is to express extreme thanks for your involvement in that device. Now, I will continue to read your essay.

    Thank you for your well written essay.

    I like your idea showing the symmetry between the complexity of mathematics and physics, going hand by hand. I am not sure they are attracted towards a point of universality. It is an interesting subject.

      Philip,

      In my previous response, I said that I am sceptical about the universality.

      The reason is not found in arguments, but in our position. We humans are close from our evolutionary relatives (for example chimpanzee). How can we pretend that our differences with them make us more susceptible of reaching "universality" more than they are?

      In my view, understanding can be compared to a sense. From an evolutionary perspective, we humans may have developed this ability to perceive nature/our environment through another way. At first, our understandings were weak, fuzzy, black or white, but as this sense developed, our ability to perceive our environment became richer, varied and even colorful though it is still a long way to the deepness of visual perception.

      Hi Philip,

      It is interesting that we have partly similar opinions on the links between math and physics, that is, a mathematical Platonism carrying all possibilities, and the idea that the physical universe comes as a particular case of deep mathematical theories. Here are my remarks:

      "It has been observed for years that the nature of physical laws appears fine-tuned for the convenience of life (...) Almost every natural occurring element of the periodic table plays some essential role in the making of multicellular life form."

      Sorry, while a case for fine-tuning can be made indeed, I don't see it well expressed in this specific way. If I saw well, only few of the physical constants really matter in all crucial processes of the evolution of stars and nucleosynthesis, and they already need to be tuned to fulfill the vital requirements. This does not let many available degrees of freedom to also fine-tune the details of chemical composition, which is actually not so fine-tuned but more guided by its own rigid necessities (mathematically necessary list of nuclear orbitals).

      To make water, hydrogen came first and did not need to be produced; oxygen is much needed but its abundance is no mystery, as 8 protons is a magic number for nucleus stability, not much sensitive to physical constants.

      After oxygen, the next 3 most abundant elements in the Earth's crust are silicon, aluminum and iron. Silicon is only 2ﾃ--10^-5 of mass in the human body, and "very few organisms have a use for it" (wikipedia). Aluminum "has no known function in biology". Iron is useful but in very small proportion only. Phosphorus is needed as 1% of mass of human body but it is only 0.1% of continental crust and 6ﾃ--10^-8 the mass of sea water (as it better stays in rocks).

      "Some physicists have speculated that there is an eternal process of inflation with vacua decaying to different solutions so that our own universe is just one bubble inside a larger arena. "

      Hum, if I understand well it requires 2 inflation periods, one before and one after decay ; inflation before decay is natural, since, by definition, the void had higher energy, but then the problem is that for different possible lower levels the inflation needs to still go on for some time and then stop in a synchronized manner (but what would happen otherwise ?). Hard stuff.

      "I think it is more parsimonious to accept that all solutions exist in some higher sense, whether inside or outside our universe (...) I take it as self-evident that logical possibilities exist even if only in some metaphorical sense that we don't understand. It is just a way of saying that some things are possible"

      I agree that, according to the nature of mathematics, all logical possibilities exist. However I see this as very clearly, formally defined and not mysterious at all, only subject to the well-understood limit of undecidability. As you should know, existence is a mathematical concept, expressed by a specific symbol, that only needs an axiomatic theory to describe the shape of a universe where it is interpreted. The typically suitable framework to state the existence of all mathematical possibilities, is set theory, which admits itself many possible variants to specify the details (due to the undecidability of existence of many kinds of infinite systems we cannot construct).

      "What then would happen if we treat the whole of mathematics as a statistical physics system or as a path integral over the moduli space of all possible theories ? Would some universal behavior emerge that could describe the meta-laws of physics?"

      A big problem to define an integral over a range of "all possibilities", is that it requires some kind of measure to compare their weights. Such a measure usually requires, at least, a kind of fixed size of a local part of body whose variations are considered at a time. However no such comparison is possible between infinite systems that cannot be precisely described by a common specific theoretical framework.

      I think that before flying to such highly speculative, ill-defined generalizations (that I would consider not really mathematical anymore, since I see mathematics as the science of definiteness), you need to consider the obvious first step and particular case of application of such ideas of "admitting the coexistence of all possibilities" that combines the advantages of being very well-known and well-defined, a natural direct consequence of the known laws of physics, with a well-defined measure of the weight of the many possibilities that is very directly and massively verified by observations, and for which, at the same time, this idea of coexistence of all possibilities has the amazing advantage of being still highly controversial. You see what I mean ? Answer below.

      You wrote "I was going to write about what might happen if there were only mathematicians and no physicists. How many ideas from physics would they invent without any input from the real world. You can imagine that they even have no direct contact with the physical world. They could just be brains in a vat left to ponder on logical problems. It may even be possible one day to see this happen using artificial intelligence. To be more specific we might program an AI system using Sparse Acataleptic Bayesian Inference algorithms to solve integer diophantine equations.(...) Diophantine equations are very rich in terms of the kind of mathematical tools are required to solve even simple cases."

      This is quite interesting as I have a similar project of rebuilding mathematics from scratch, starting with purely logical concepts before reaching physics. I use my own intelligence instead of an AI system, and of course I do know much physics at the start but I care to put things into an optimal logical order so as to make every step appear sufficiently motivated by purely logical concerns without any feeling of arbitrariness nor external (physical) source of motivation or inspiration.

      Though the properties, and proofs of properties, of integer diophantine equations may potentially involve lots of mathematics indeed, I'm afraid they would be a quite inefficient way of rebuilding maths from scratch. In fact, I guess the power of development of high mathematics is much less a matter of what problem is supposed to be tackled, than a matter of what kind of intelligence or algorithm is tackling it. Indeed, this AI needs to know from the start what is a proof and what is a definition that can be used to make shorter proofs... finally you need to give it a huge a priori knowledge of mathematics beyond diophantine equations, before it starts searching. But the worst point in my opinion is that, even considering deep mathematics as an intrinsic necessary reality to be discovered, I guess the act of discovering them may require a conscious mind (not AI) to be efficiently done. The concepts of elegance and universality in mathematics may be themselves diversely interpreted and not always in mathematically well-defined manners. Insofar as they would be mathematically definite, I would compare discovering deep maths to a problem of breaking a cryptographic key, for which the solution may indeed be mathematically unique and well-verifiable, but purely mathematical systems could not efficiently discover the solution themselves if it is not revealed from the outside.

      Of course I was referring above to the many-worlds interpretation of quantum physics. The most famous difficulty with this interpretation is how to make sense of probabilities. You wrote "The Mathematical Universe Hypotheses tells us that all logical possibilities are equal". Consider the simple example of a polarized photon that is measured in another direction with an arbitrary angle. Both possible observed states are well-defined elementary states, not any variably large numbers of possible states, and have no intrinsic difference of quality having anything to do with the entropy of the measurement apparatus or whatever. Still the theory says one is more likely than the other, with a probability that can take any value depending on the angle of measurement. How to make sense of their difference of likeliness if they are equally real ? See my longer analysis of the paradoxes in this interpretation. Finally I invite you to read my essay where I defend another interpretation.

        Thanks Lawrence, I hope to find some more time now. I see you have an essay up, that's great. In fact quite a lot of essays already.

        Sylvian, thank you for your detailed points. These are all interesting things and I will give my responses one at a time. Overall I would say that these are things where different people have very different opinions and have been the subject of interesting debates. I recognise that my opinion is not likely to be the right one on everything but for the purposes of argument I will put my best case in defense of how I see them.

        On fine-tuning in chemistry: First let me correct a few misunderstandings. When I said that "Almost every natural occurring element of the periodic table plays some essential role in the making of multicellular life form." I did not only mean that every element is incorporated into biochemistry. Silicon has only minor roles to play in biology itself but it forms rocky planets without which life as we know it would not exist. I also did not mean to imply that every element is used in proportion to its abundance. Some elements are used in only very trace amounts but the role they play is still very important.

        Now it is true that there are not enough free parameters in the standard model to fine-tune every element to a specific role. That is not how it works. The coupling constants are however fine tuned to control the richness of chemistry. Small differences would mean different numbers of elements with different properties and we can expect chemistry and abundances to vary quite dramatically and perhaps even chaotically as the parameters change. It is very difficult to work out what chemostry would really be like with different values of constants and even harder to try to work out what forms of biochemistry may be possible based on different sets of chemical elements. Perhaps science and computation will make that possible one day but another thing that will happen (hopefully) is that we will get an idea of what other lifeforms exist in the universe. If the fine-ytuning idea is right then there should be exactly one major form of biochemistry on which complex life can be based. If we find that there are two different types of biochemistry that lead to sophisticated lifeforms then the fine tuning argument is wrong. I don't think we will.

        Christophe, the attraction towards universality is a feature of complexity theory and self organisation. Think about the theory of strange attractors in chaos theory for example. I dont think we fully understand why it works that way but it does.

        I don't think a chimpanzee is qualitatively different from us. he is just a little less intelligent. It is not us that reach universailty, it is a feature of mathematics that arises independently of us. The only role we play is in establishing a selection criteria in that the solution to the equations of universality which is actually realised has to be able to support us. This is the anthropic principle and it applies to some extent to chimpanzees too.

        Of course the details of modifications of chemistry would be very hard to find out but the main principles of dependence with respect to the fundamental constants are clear.

        As for nucleosynthesis, we have this:

        A fine-tuning of constants is needed for the Triple-alpha process: " 8Be + 4He has almost exactly the energy of an excited state of 12C ".

        The ratio of nuclear to electrostatic strength of interaction between protons (the latter being essentially given by the fine structure constant), gives the approximate weight of the most stable element (iron)

        As for chemistry with given elements, only 2 physical constants seem involved:

        The fine structure constant gives the average speed of electrons compared to the speed of light, which may result in relativistic effects but as far as I know the consequences on chemistry are quite small. One of the main effects I heard of is that it gives the color of gold, due to the properties in the excitation of innermost orbitals, that of electrons having higher speed, closer to the speed of light because they come close to the nucleus. Generally, the fine structure constant determines the intensity of the photon emission/absorption processes, and also the wavelengths of photons, in case that matters.

        More importantly, the electron-to-proton mass ratio determines the width of the Heisenberg uncertainty on the distance between atoms with a given bond in its ground state. Namely, this distance uncertainty is proportional to (k.m)-1/4 where k is the rigidity of the bond and m is the ratio of the mass of the atom to that of the electron.

        In the case of covalent bonds (k close to 1) this uncertainty is quite small anyway (such as 0.1 邃ｫ), since m is so big, despite being put to the power (-1/4).

        The sensitivity, then, may come for weaker bonds (small k), especially the inter-molecular bonds (including the lateral degrees of freedom) packing small molecules into solids or liquids, however I'm not sure how much it stands as compared to the role of temperature, which should be the main factor in many cases (letting the ground state of the bond unlikely and thus irrelevant). This latter uncertainty on position is proportional to sqr(T/k). Where temperature happens to produce a significantly bigger position uncertainty of a given bond than the Heisenberg uncertainty of the ground state (even twice bigger may suffice), the sensitivity to the mass ratio becomes insignificant.

        For details and explanations, I gathered in my site some relations of dimensional analysis that give the orders of magnitude of a number of phenomena out of the fundamental constants of physics.

        But I do not see there a point to consider fine-tuning done for a specific biochemistry that would exclude other forms of biochemistry. Instead, I see the possibility of biochemistry as a very general property of chemistry, that is its ability to develop complex molecules with complex reactions. As soon as complex chemistry is possible in general, I do not see a point why the specific efficient combinations should be unique. Just take an example : without leaving this Earth, Arsenic in significant amounts is toxic for most organisms, however a few species of bacteria have a different biochemistry that tolerates it, and even uses it, to thrive where it is abundant.

        Dear Phillip,

        I thought your essay was well done and very interesting. I am not sure that I understood it all, but I agree that the concept of universality is critically important (no pun intended) to understanding the underlying *process*, which I think we perceive as dualistic aspects of reality. I emphasize process because my life experience (my "lazy process" that includes graduate education (physics, math, electrical and nuclear engineering, medical physics, and national security/ strategic studies) and as a nuclear submariner and clinical medical physicist) has given me a perspective that is more focused on process (especially the unity of space and time as opposed to the differences). I don't recall learning about universality in my statistical mechanics class, so I have to look it up, but from what I just read on line, it seems to be an excellent direction for continued research.

        Scientific writing has never been one of my strong points, and I've struggled with putting my ideas in a format acceptable to scientific journals, so allow me to express my sincere gratitude to you for vixra. If I never succeed in getting it published in a journal, at least I now have a chance to share my philosophy about the unity of space and time, especially my space-time-motion model (see http://vixra.org/abs/1402.0045) with people who are much smarter and knowledgeable than I. My only hope is that it will be useful in the quest for understanding the importance of unity (the metaphorical center of the ring) as a foundational concept. I believe that the entire world (not just physics) is in crisis because science has proven the utility and power of reductionism yet failed to recognize the importance of concepts such as unity and universality (except physicists like David Bohm and Fritjof Capra).

        I took a very different approach to presenting space-time-motion unity in this essay contest, because the guidelines emphasized "Original and Creative" ways of pushing forward understanding "in a fresh way or with new perspective". So I invite you to read and comment on "Doctors of the Ring - The Power of Merlin the Mathematician to Transform Chaos into Consciousness."

        Best regards,

        Ted St. John

          Thank you for your comments. I am glad you like the central idea of universality. I look forward to reading your essay

          Sylvian, thank you for these detailed and interesting points. The only thing I would add at this point is that the fine structure constant does not just affect the chemical bonds. It also affects nuclear stability because the electrostatic repulsion is balanced against the strong force. A small change would have a profound affect on which elements are stable.

          I think it would be interesting nut hard exercise to work out the chemistry and nuclear properties of elements as constants vary. Until someone does that I am not sure what the real situation is.