Richard,

Thanks for the response and the question. This is something other people have wondered about. There is a small probability the photon will loop around the slit! This is predicted by the Feynman path integral. Feynman said the purpose of the path integral is to derive quantum properties from all possible paths corresponding to amplitdes. A particle can leave a source and reach a target with some probability that it looped around Mars. Of course the amplitdue for this looping is very small, but it is there. Again Feynman once said that most or all of quantum mechanics can be studied with the two-slit experiment. This means that with the actual experiment there should be some very small effect due to the looping of a particle around the topological obstruction that is the slit. This is something that at first seems utterly impossible until you think more closely about it.

Cheers LC

Dear Lawrence,

Your essay taught me certain things that I didn't know. I liked the historical way the essay was written starting slowly and rising to a crescendo.

I would have wanted included in the essay concise definitions for what geometrically you mean by 'continuous' and 'discrete', in the light of Euclid's definitions of a what a line is.

In contrast to the assertion, "Geometry is then not fundamental", which I disagree with, I leave you a quote from Galileo to ponder:

"He who attempts natural philosophy without geometry is lost"

- Galileo Galilei, Dialogo, Opere 7 299 (Edizione nazionale, Florence, 1890-1909).

A very informative essay worth keeping and reading more than once.

Regards,

Akinbo

    Geometry is important, but I think it is built up from quantum entanglement. Of course there are a number of papers in this contest which purport that Bell's theorem, entanglement and the quantum physics of nonlocal correlations is wrong. One is wise to not bite on this. Quantum mechanics is one of the most experimentally tested areas of physics. So far all of the strange consequences of quantum physics hold up, this is even if they appear to be so utterly bizarre and upside down.

    Quantum states entangle with a black hole. The observer on the outside easily loses information on the exact black hole horizon state her part of the EPR pair is entangled with. This is a form of entropy; the entropy illustrates the lack of knowledge. The horizon area of the black hole is then a direct measure of entropy, where any quantum bit in our outside world entangled with the black hole is entangled with a Planck unit of area on the stretched horizon. It is difficult to localize that of course. If one tries to find which region of the horizon your EPR part of the pair is entangled with the Heisenberg microscope argument tells us the other part of the EPR pair will be sent into a huge uncertainty in position. Hence one is not able to recover this information.

    The horizon of a black hole is then built up from entanglement. Further, the null boundaries of spacetime contain the holographic information of the entire region. We then have the physics of space or spacetime really being built up from entanglement of quantum states. Geometry, at least geometry used to model physics, is then an emergent property.

    LC

    Lawrence,

    Much to ponder here, Lawrence. The Penrose triangle suggests the circularity of the widespread view that math arises from the mind, the mind arises out of matter, and that matter can be explained in terms of math. My connections I don't feel have a Platonic flavor, as you mention, only an attempt to mimic the mind in mathematical models for understanding natural connections between the quantum and the classical worlds. I see a functional relationship between mind, physics and math, making possible giant strides in physics and other sciences.Quantum entanglement has been found to have a role in classical phenomena such as navigation of birds, turning new pages in quantum biology.

    Great essay.

    Jim

      As I remember Penrose and his triangle in "Road to Reality" had consciousness or mind with physical reality and mathematics as a triality. The I recall that he had mathematics as the foundation of physics, and physics the foundation of mind, and to complete the cycle mind gave conscious recognition to mathematics.

      Platonism is not something I take that strongly. I will put on the "hat of Platonism" when it suits me, and at other times I will not. I discuss largely the aspect of mathematics that I think has "meat," while a lot of mathematics involving infinity and infinitesimals is what I call "soul." I am not out to deny the "soul," but I do think the "meat" has a more direct connection with physical reality.

      LC

      Lawrence, this is possibly your most readable essay yet in these contests, but you have still managed to maintain a high level of novel mathematical ideas. I liked the historical introduction that puts the relationship between Mathematics and physics in perspective.

      The HoTT ideas are very interesting and they mesh well with my own ideas of higher category theory as a system of multiple quantisation so it is good to see this presented.

      You should do well.

        Thanks for the positive word. I wrote the bulk of this essay up in a single day. I spent another day correcting it. I tried to keep the mathematics somewhat "physical," in that the discussion on incompleteness and numbers is oriented towards what one might actually encounter in computing these things.

        I read your essay early on, and as with most essays I have read I have yet to assign a score to it. So far you are running at the top. I will probably have to re-read yours. I have been a bit busy and unable to attend to this contest that much.

        Cheers LC

        Lawrence

        Nice work. I see you mentioned entanglement of quantum black holes. In light of twistor-scattering amplitudes, I suspect there is an operad structure, to be defined for all such quantum black holes. In the complex case, the operad is already defined (by Loday), and can be viewed as a chain of punctured Riemann spheres. This has an interpretation in terms of associahedra and binary tree diagrams. In essence, once can tile a moduli space with associahedra. The vertices of the associahedra correspond to interactions that contribute to the relevant n-object scattering amplitudes.

          The connection with the Bott periodicity of large N, or SU(N), entanglements with associahedra is with the some sort of projection of this thing. The Stasheff polytope K_5 has 14 vertices, and this might be seen as composed of two copies of 7 elements mapped to the Fano plane, with each of these "sevens" associated with a projective point "в€ћ", or as associated with eight elements by the Hopf fibration. So the associahedra might be some sort of "double cover" with the Bott periodicity on A&D type Lie groups.

          I think to construct such operads one has to work more from the ground up. Spacetime as a monoid, groupoid or so called magma might lead to the sort of spacetime algebra that leads to the sort of category theory. The set of possible operations would then form these "trees" and are associated with certain homotopies and categories. My essay discusses this in an informal sense, and homotopy as a route to category theory with monad/monoid structure.

          I can discuss this more tomorrow or later. I see that you have an essay as well, which I will try to get to in the coming days. These are coming in faster than I can read them.

          Cheers LC

          Dear Lawrence,

          very good essay, I have to go more deeply into the details to post a substantial comment. Also thanks for the emails (I will answer soon).

          In the meantime I also wrote an essay which appears now. Her is the link to

          my essay.

          Best Torsten

          The construction of spacetime, from say, category theory would be elegant. Through the lens of noncommutative geometry, one comes very close to the spirit of category theory, as classic geometrical points are replaced by pure states of a C*-algebra. If one takes the lessons learned from D-branes seriously, gauge symmetries arise from configurations of branes, and natural brane coordinates are noncommutative. In fact, the brane coordinates arise from a noncommutative C*-algebra. Let's take the case of N coincident branes, which have a U(N) gauge symmetry as long as the branes are coincident. The N D-brane positions in spacetime are packaged in a matrix, say X, in the adjoint representation of the unbroken U(N) group. Upon diagonalizing X, the N eigenvalues give the classical spacetime positions of the N D-branes, corresponding to the ground state of the system.

          In category theory language, the noncommutative C*-algebra of NxN complex matrices Mat(N,C) can be interpreted as the algebra of noncommutative functions over a finite point space, with N objects. The elements of the noncommutative C*-algebra serve as morphisms over these N objects. Therefore, we can define a category C with N objects in Obj(C) and Hom(C) consisting of Mat(N,C) morphisms.

          Michael,

          I have been a bit tied up with a number of things. Your discussion about D-branes and the NxN matrix of their symmetry in U(N) (SU(N)) or SO(N) is close to what I have been working on. The Bott periodicity of these matrix systems gives an 8-fold structure. This 8-fold system has a connection of E8. I am interested in 4-qubit entanglements of 8-qubit systems that are E8. The structure of four manifolds involves a construction with Plucker coordinates and the E8 Cartan matrix. This seems to imply, though I have not seen it in the literature, that for 8 qubits there is not the same SLOCC system based on the Kostant=Sekiguchi theorem. However, I suspect that the structure of 4-spaces might hold the key for something analogous to KS theorem and the structure of 2-3 (GHZ) entanglements that are constructed from G_{abcd}. If the universe has this sort of discrete structure via computation, then it makes some sense to say the universe is in some ways a "machine" that functions by mathematics.

          Cheers LC

          Hi LC,

          Once again, you made an excellent work with a very profound Essay. Here are my comments:

          1) This should have been an excellent Essay also in previous contests "It From Bit or Bit From It?" and "Is Reality Digital or Analog? ".

          2) My recent work on black holes, that you know, goes in the same direction that "topology and the computation of topological charges and indices, quantum numbers, and connection to logical switching theory are likely to supplant concerns of geometry, metrics and infinitesimal structure of manifolds."

          3) On the other hand, I am not sure that such a statement goes in one specific direction. I try to clarify: you claim that

          i) "This means that the fundamental description of reality is not with space, spacetime or anything geometric. Geometry or metric space is something which is a measure of entanglement of quantum bits with black holes and the inability to follow the entanglement phase. Geometry is then not fundamental."

          But you also claim that:

          ii) "Spacetime is built up from entanglements".

          Thus, in my opinion, statements can be inverted. One could claim that "Entanglement of quantum bits is something which is a measure of geometry or metric space". In other words, this could be a sort of duality and/or complementarity of the fundamental description of reality. Geometry and entanglement could be two different aspects of the fundamental description of reality, but we could be unable to decide which one is the most fundamental.

          In any case, I stress again that you wrote an intriguing Essay deserving the top score that I am going to give you.

          I wish you best luck in the Contest.

          Cheers,

          Ch.

            Dear Christian,

            Event horizons are a way that a quantum state or EPR pair of qubits can be entangled with a black hole. The entanglement is coarse grained, for one does not know which of the qubits on the horizon one's EPR state in the pair you are entangled with. Any attempt to find out runs into limitations of the Heisenberg microscope argument. So horizons are an ensemble or Bayesian set of priors for quantum states. This is sort of the meaning of how geometry is built up from entanglements. The converse has some element to it, in that entanglements are coset structures of geometric elements.

            I will look at your new entry and score in the next few days. I have been on travel lately and time is a bit tight.

            Cheers LC

            Hello Lawrence,

            As a layperson I am not going to pretend to understand these new approaches and the maths being utilised but in your conclusions you state interestingly,

            "Chaitan has advanced ideas that mathematics is not something that exists in any sort of coherent whole-

            ness. It is more a sort of archipelago of logically consistent systems that sit in an ocean of chaos [21].

            This chaos is a set of statements that are purely self-referential and have truth or falsehood by no logical

            reason.

            Possibly the quantum vacuum is similar. It may be a tangle of self-referential quantum bits, where

            some sets of these exist in logical coherent forms. These zones of logical coherence might form a type of

            universe. These logical coherent forms are then accidents similar to Chaitan0 s philosophy of mathematics.

            It is very dicult to understand how this could be scienti cally demonstratedｬ yet maybe regularities inｼｯpｾｼpｾphysics described by mathematics exist for no reason at allｮ｢ｼｯpｾｼpｾﾂasicallyｬ the mystery of the quantum vacuum is really expressed as the ｧhierarchy problemｧｮ ﾉ think if you stare at the so called hierarchal gap then that is where the state of physics is at todayｮ ﾉ like the idea of entanglement defining geometry somehow and this may be inherent in the blank space ｨvoidｩ where the particle desert existsｮﾔhe ﾂig ﾇap ｨor whatever you may call itｩ maybe the potential pool from which the multiverses are generaed or at least some indication that the multiverse do indeed existsｮ ﾍaybe ﾇﾕﾔ convergences are different for different ﾕniversesｮ ﾁlsoｬ maybe it is a matter of physics becoming more aligned with mathematics toward the quantum gravity realmｮ ﾉn that we live in a ｴﾄ world our physics ｨﾆrom ﾇreek physikaｩ is generally found to work ｨempiricallyｩ from the platform of the sensory ｨsensationｩ because of the physical nature of things at our low energyｮ ﾓensory as apart from intelligenceｮ ﾁt some point the physics may blur into the mathematics and the senseｭdata ｨempiricalｩ will be left behindｮ ﾔhis implies that intelligence will discover things by ｧexplanatory powerｧ and that the empirical apparatus ｨmachinesｩ no longer yield anything useful in upward and onward explorationsｮ ﾍaybe ｴﾄ is unique in that it implies a physical world in which physics is a workable contextｮ ﾉf so then there might be an ｢ end of the machineｨsｩｧ such as the ﾌﾈﾃ where it just cannot yield the pure mathematical result as a physical ｨsensationｭobservationｩｮ ﾔhen we are at the ｢end of physics｢ in a different sense and a brave new world of mathematics ｨexplaining thinsｩ is awaitingｮ ｨﾍaybe the meaning of life is not ｴｲ but ｴﾄ ｻｭｩ ｩ ﾉf you look at the hierarchal gap as a ｧzen koanｧ then there is more there than meets the eyeｮｼｯpｾｼpｾmark

              Mark,

              There are some indications that this might be happening. With high energy physics the LHC and tests of the standard model, supersymmetry and maybe some hints of exotic physics are probably the last of the sort of direct tests physics has enjoyed or wanted. The future may see theories tested in increasingly oblique or indirect ways. I hope that progress can be had this way through the 21st century.

              I see the prospect that physics becomes purely a mental game a bit disturbing. If we end up in the future where we are unable to make any test, no matter how indirect, of our physical theories we will be in a problem. In some ways it will be the end of physics as a science. Science does depend upon experimentation.

              Theoretically the heirarchy problem is solved with supersymmetry. It is just an open question of whether the LHC with 13TeV beam can find evidence of supersymmetry.

              Cheers LC

              Dear Lawrence,

              Now I had time to read yiour essay. I agree with a comment above: it is one of your best. Here are my own comments:

              1. You spoke about Bott periodicity but SU(N) has a 2-periodicity. It is the SO(N) group which admits an eight-fold periodicity (with integer coefficients, it is 4-periodic like the symplectic group for rational coefficients).

              2. Your double slit experiment is very interesting. You view it from the topological point of view. Maybe one should remark that this approach wa already done by Berry and others using geometric phases.

              3. you discussed it that HOTT will overcome the continuum approach. But homotopy needs a continuous family of maps (the deformation). It is central point in the approach and many results using implicitely the continuum (like Cerf theory, Whiteheads theorem etc.)

              4. I also don't understand why you want to change from continuum to discrete. I showed in a previous essay that a smooth manifold contains only finitely many information (from topology). Furthermore, the dynamics in quantm mechanics (or field theory) is smooth (and continuous). Only the spectrum of the operators is discrete.

              5. HOTT is a good approach but this proposal don't change the logic. Therefore Gödel works. Fro your approach, you need model theory (including forcing) to go over it. I remembered on a approach of Landsman to quantum mechanics using this approach. But my friend (and co-worker) Jerzy is the real expert.

              I like your part explaining the Turing machine (and the relation to the Entscheidungsproblem)

              Very good work

              Torsten

                In the end there is a bit of a duality here, or a dialectic of sorts. I think that what is measured in physics is discrete. We measure certain observables that have finite values, and quantum physics in particular bears this out pretty seriously. The continuum aspects to physics is pretty much a mathematical issue. Experimental data does not have any reference to infinitesimals or infinities. The calculus is based on the limit where the difference between two points becomes infinitesimally small. Physical experiments have not direct bearing on this.

                It is the case that homotopy does involve curves that are smoothly deformed into each other, but this is used to get the value of the homotopy group that is usually Z_2 or Z, where Z could be interpreted as just unbounded and infinity is avoided. The homotopies are then more directly related to the actual measured aspects of physics.

                Spacetime is a bit odd with regards to this. The Planck scale does indicate that one can't isolate a qubit in a region smaller than sqrt{G徴/c^3}. The Heisenberg microscope argument indicates that if one tries to isolate the Planck unit of area a quantum state is contained that it will scatter violently. This illustrates that using a large value of momentum to isolate particle demonstrates that spacetime has a discrete structure. This has an interpretation in the generalized uncertainty in string theory. On the other hand the FERMI and Integral spacecraft measurements of distant burstars found no dispersion of photons predicted by loop quantum gravity. This is a discrete form of quantum gravity, and it appears to be in trouble. In this experiment a very large ruler (measurements out to a billion light years) found that spacetime appears very continuous. This suggests a more general form of the uncertainty principle, where at one limit spacetime is continuous, and on the other limit discrete.

                The problem is that physics is not completely discrete or continuous. One of these FQXI essay contests went into this. The main thrust of my essay though is that the physical observables we measure, and physics is an experimental science, are discrete. Mathematics has what I might call a "body" and a "soul." The body is what is computed, and can be computed on a computer. The soul is all of the continuum stuff, calculus, infinitesimals etc, which have a weaker connection to experiments. I am not committed to any metaphysics about whether the soul exists or not. That is to say I have no belief or lack thereof with respect to what some might call Platonism.

                LC

                Lawrence,

                Ok I see the point. Of course the outcome of experiments is not a real number but as you also point out, one has problems to confirm the discrete structure of spacetime.

                I see one reason in the underlying topological nature of physics. You also discussed it in your essay. I will illustrate it in a an example:

                If two curves intersect then we measure the number of intersections (a discrete number, gauge or diffeomorphism invariant) but in most cases we are not interested in the coordinates of the intersection. Even sometimes we have problem to determine the coordinate system.

                I see the measurement values in physics in this fashion. But then one has a dichotomy between discrete (number of intersections) and continuous. The measured values are in principle discrete but you need the continuum to express the probabilites of quantum mechanics.

                I don't see any contradiction in this picture. Of course you will never measure that spacetime has a continuum structure but you can measure a discrete structure. And as you correctly point out: every experiment failed up to now.

                In principle I agree with you very much. In particular I like your body-and-soul picture

                Best

                Torsten

                In the subject of gauge theory a central aspect is the interaction form. These types of continuous homotopy or homotopy-like constructions involve curves that can be adjusted in certain ways so that an index is invariant or constant.

                I am back home, but of course I have a lot of things to attend to here. I will try to expand on things in the future. The whole subject involves orbit spaces, or quotents of groups or spaces. The subject of four-manifolds is centered around the moduli, a 5-dim space that in a hyperbolic setting can be the AdS_5. Of course the hyperbolic setting is not Hausdorff and there are other problems. However, this is a form of orbit space that is mapped to the quantum SLOCC types of theory.

                Cheers LC