Complex numbers employed in physics have consequences. One is unitarity that is important in quantum physics, another is holomorphy that is important in gauge theory. Quaternions, which are hypercomplex numbers, can be used to derive Maxwell's equations. So these things are useful in making calculations.
LC
Lawrence,
Apologies for taking so long to respond. I likewise have returned the favor and rated your essay with a 10.
I right honorable essay.
Absolutely fascinating discussion about the numbers we can never count. The idea that we can define spaces we can never explore does make one feel small in a seemingly much larger universe. Is it ultimately the human condition that we must accept that there are things we can never know? What are the true limits of knowledge? When do we know we cannot go further?
On the other side, it is somewhat refreshing to know that there is always places we can go that have not been explored. The question I have is how far will we get?
Cheers!
Harlan
Garrison Keillor has his "Guy Noir," who "On the tenth floor of the Atlas building still seeks answers to life's persistent questions." If you have ever listened to his "Prairie Home Companion" you know this well. There are persistent questions, such as "Does God exist," that will probably never be conclusively answered.
The problem is that we transition from physics to metaphysics when we start pondering what the relationship is between mathematics and physics. Smolin has written an essay that I think rather favorably of, but his stance is naturalism, which is not something one can prove. Naturalism is a conjecture about things that is really metaphysics. Platonism, which in some ways is a bit too mystical for my tastes, is also a metaphysics.
What I outline is a mathematics that has what I tend to think of as meat or body. The pure mathematics taught and studied in mathematics departments often involves what might be called "soul." I am not out to disprove or even disavow this soul, but I do tend to think that it has a questionable applicability to modern physics. This is particularly the case with matters of infinity or the continuum. Mathematicians are mostly objectivists who consider their work to involve the discovery of "something," which is not physical. This is an appeal to the existence of this soul, or in its extreme form a Platonic reality. That is fine with me, and I may put on the cap of Platonism when it suits me, and take it off at other times. In my essay I tend to keep the cap off. My essay is largely concerned with what sort of practical aspects of mathematics are likely to impact physics in the next few decades.
I think these more metaphysical questions are not going to be answered, or answered very easily. There is Tegmark's conjecture of the MUH (mathematical universe hypothesis), where in view of Goedel's theorem he replace M with C for computation. This may be the case in some ultimate sense. However, I don't see how this can ever be empirically supported. This may amount to trying to "prove too much." This attempt to get away from dualism between mathematics and physics is a sort of monism. The debate between monism and dualism may never be resolved. The two are sort of the two sides of a Buddhist satori.
LC
Your conscious effort to subjectively narrate the developmental science is well-written.
Sincerely,
Miss. Sujatha Jagannathan
Thanks for the words of encouragement. I tried to write the description of Goedel's theorem and related matters in a physical sense, and I wonder if I fell far from the mark on that.
I see that you are in the contest as well. I will try to get to your essay as soon as possible. I have been a bit unable to read many of these the last couple of weeks.
Cheers LC
Hello Laurence,
I enjoyed your essay which covered many mathematical topics of great interest. I was intrigued by your diagram: 'Topological winding numbers in the two slit experiment'.
Taking the viewpoint that a photon is a real physical wave that passes through both slits of the interference apparatus it is hard to imagine that a path looping back through the slits is a real possibility.
Regards
Richard
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
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