friend wrote: Or is the spin just some extra property that we assume to exist so that the particle/antiparticle can avoid each other along the way? Thanks.
friend wrote:Does the spin of an electron in the Standard Model describe something actually turning such that it predicts the phase angle of some reference point that rotates and then returns to a starting point after some period of rotation? Or is it just a vector of some magnitude along the direction of spin but no description of what is spinning with a phase of some angle as it (whatever it is) turns?
friend wrote:If these particles are indeed point particles, then it seems likely that they are completely symmetrical about their axis of rotation. So there is no feature to serve as a reference with respect to an angle as they spin. In other words, the spin is completely inherent, it does not refer to some feature that is doing the spinning. Every angle would look the same.
friend wrote:Austin Fearnley wrote:Its a while since I read about your ideas. I remember that an implication seemed interestingly like wavefunction collapse of many points in a set to a single point or implication. (Though QM does not use sets.)
As a reminder: I start with individual points and describe each with individual coordinates. (I don't have the dimensions of space or a metric yet.) And like anything else, you describe them using propositions which are either true or not. But now having propositions, we can consider conjunctions and disjunctions of them and how one might imply the other. A space consists of a logical conjunction of all the points in the space. And a conjunction between any two points implies an implication one way in conjunction with the reverse implication. Implication is represented with a complex Gaussian for reasons explained on the website. An implication looks exactly like the propogator (wavefunction) of a particle; the reverse implication is the complex conjugate that then looks like the antiparticle. Both seem to be traveling from the same start and end points at the same time, and since they start from nothing they end in nothing; they cancel each other out. In other words, we have virtual particles popping in and out of existence between every two points in the space that you can imagine, a sea of virtual particles making up the quantum fields through which particles travel.Austin Fearnley wrote:Does a wavefunction exist in spacetime?
The complex Gaussians that represent implications are functions of time and space. Without a spacetime to begin with, there are no Gaussians.Austin Fearnley wrote:If it does, then do you have to have every point in the set for an electron wavefunction avoiding every point in the set for a positron? Or is it only the (final) implication that is actually in spacetime and needs its own space (as in Pauli's Exclusion principle).
You may not need the number of dimensions to specify angular momentum. But you do need a metric to specify the magnitude of the radius times the magnitude of the velocity. Is it possible to specify a metric without specifying the dimensionality?
My website is at: logictophysics.com
gill1109 wrote:And the Gaussians are just pulled out of a hat. So your approach is not based on logic, it is based on making arbitrary choices. It does not give formulas which could be used to make predictions which could in principle be wrong so it is not science, it is fantasy. [With all respect, in my personal opinion, etc. I am impressed by your grand research programme, it is a noble enterprise, but for me, it does not come off the ground]
friend wrote:But there's a third reason. The formulation is developed by using very many implications. But in logic, there can be no preference for one implication over another; they all have the same importance and relevance and weight. So whatever function represents an implication, there cannot be any size or value differences depending on which implication is used. When the exponential of the Gaussian become complex, it gives the same absolute value of 1 for every implication. The only difference is a phase.
more at: logictophysics.com
friend wrote:friend wrote:But there's a third reason. The formulation is developed by using very many implications. But in logic, there can be no preference for one implication over another; they all have the same importance and relevance and weight. So whatever function represents an implication, there cannot be any size or value differences depending on which implication is used. When the exponential of the Gaussian become complex, it gives the same absolute value of 1 for every implication. The only difference is a phase.
more at: logictophysics.com
I've shown how the path integral can be derived starting with propositional logic. But I've not shown why this particular formulation is unique or necessary. For that I'd have to show the uniqueness of every step. I was able to introduce numbers, 0 and 1, into the logic by using the Dirac measure as a substitute for implication. In retrospect this seems like an obvious step. But to be honest, I do not know if there is any other way of introducing numbers into the logic. If numbers are necessarily the cardinality of sets, then the most basic numbers would have to be 0 for the empty set, and 1 for a set with only one element. If cardinality is the only way to get numbers out of sets, then it is necessary to use the Dirac measure to express set inclusion, and it's necessary to use set inclusion to express material implication. This would make it necessary to express the conjunction of all points of space with a disjunction of a conjunction of implications that can then be expressed as a sum of a multiple of Kronecker delta functions. This at least allowed me to derive the rule of addition and multiplication and a type of path integral from that. So the question is: are numbers necessarily always the cardinality of sets?
I then migrated to the continuous case by letting the Kronecker delta change to a Dirac delta function. When I express the Dirac delta function as a complex exponential Gaussian, the Feynman Path Integral emerged, at least for a free particle with no potential energy term. I've given some reasons why the complex exponential Gaussian should be chosen for the Dirac delta function. But this is not the same as a necessity. The Feynman Path Integral has been controversial; some question whether the measure of that integral can be mathematically justified. So if I were to show the absolute necessity of using the complex exponential Gaussian in this case, would that get us closer to a mathematical justification of the Path Integral itself? The question I have is whether the complex exponential Gaussian is the only function whose infinite multiple gives a well defined measure that can be integrated. If it is, then this necessitates its use to represent the Dirac delta function.
friend wrote:The Feynman Path Integral has been controversial; some question whether the measure of that integral can be mathematically justified. So if I were to show the absolute necessity of using the complex exponential Gaussian in this case, would that get us closer to a mathematical justification of the Path Integral itself? The question I have is whether the complex exponential Gaussian is the only function whose infinite multiple gives a well defined measure that can be integrated. If it is, then this necessitates its use to represent the Dirac delta function.
gill1109 wrote:friend wrote:friend wrote:But there's a third reason. The formulation is developed by using very many implications. But in logic, there can be no preference for one implication over another; they all have the same importance and relevance and weight. So whatever function represents an implication, there cannot be any size or value differences depending on which implication is used. When the exponential of the Gaussian become complex, it gives the same absolute value of 1 for every implication. The only difference is a phase.
more at: logictophysics.com
I've shown how the path integral can be derived starting with propositional logic. But I've not shown why this particular formulation is unique or necessary. For that I'd have to show the uniqueness of every step. I was able to introduce numbers, 0 and 1, into the logic by using the Dirac measure as a substitute for implication. In retrospect this seems like an obvious step. But to be honest, I do not know if there is any other way of introducing numbers into the logic. If numbers are necessarily the cardinality of sets, then the most basic numbers would have to be 0 for the empty set, and 1 for a set with only one element. If cardinality is the only way to get numbers out of sets, then it is necessary to use the Dirac measure to express set inclusion, and it's necessary to use set inclusion to express material implication. This would make it necessary to express the conjunction of all points of space with a disjunction of a conjunction of implications that can then be expressed as a sum of a multiple of Kronecker delta functions. This at least allowed me to derive the rule of addition and multiplication and a type of path integral from that. So the question is: are numbers necessarily always the cardinality of sets?
I then migrated to the continuous case by letting the Kronecker delta change to a Dirac delta function. When I express the Dirac delta function as a complex exponential Gaussian, the Feynman Path Integral emerged, at least for a free particle with no potential energy term. I've given some reasons why the complex exponential Gaussian should be chosen for the Dirac delta function. But this is not the same as a necessity. The Feynman Path Integral has been controversial; some question whether the measure of that integral can be mathematically justified. So if I were to show the absolute necessity of using the complex exponential Gaussian in this case, would that get us closer to a mathematical justification of the Path Integral itself? The question I have is whether the complex exponential Gaussian is the only function whose infinite multiple gives a well defined measure that can be integrated. If it is, then this necessitates its use to represent the Dirac delta function.
I don’t understand Feynman path integrals. It looks to me you have replaced one complex (in two senses!) computational approach with another. If it makes you feel good, that’s great! If you can explain it to others, that’s greater still. If it makes in-principle falsifiable predictions then that’s quite exciting. Do carry on. I can’t follow you. I’m just an ordinary statistician. And a mathematician. And not young any more.
jreed wrote:The use of path integrals in quantum mechanics was discovered by Feynman. He generalized the principle of least action in classical mechanics, where the action is given as the integral of the Lagrangian. If this integral is minimized, this gives the lagrangian equation of motion, which can be solved for the classical motion of the system. Feynman, a very smart fellow, generalized this principle so that each path has an action, and took the integral over all paths for a quantum mechanical system. He found that this expression can solve quantum mechanical problems, just like Schrodinger's equation but has a different interpretation. What this has to do with Boolean logic escapes me.
friend wrote:OK, I have a few misstatements that I have to correct. I wrote " If the conjunctions (^) can be replaced with summations and the disjuctions (V) replaced by multiplications". This should be, " If the conjunctions (^) can be replaced with multiplications and the disjuctions (V) replaced by summations". Sorry about that. It seems I read what I meant to write and not what I actually wrote. I only catch these mistakes if I let enough time pass before rereading it so I can forget what I intended to write, which is difficult. Here is a corrected version.
gill1109 wrote:I think you have found a formal analogy between certain expressions in formal logic, and certain expressions in the ‘theory’ of Feynman path integrals. But I don’t find this surprising. I wonder if it is *useful*? Could you use the analogy to make new, testable, conjectures about path integrals?
friend wrote:gill1109 wrote:I think you have found a formal analogy between certain expressions in formal logic, and certain expressions in the ‘theory’ of Feynman path integrals. But I don’t find this surprising. I wonder if it is *useful*? Could you use the analogy to make new, testable, conjectures about path integrals?
You're right. I am drawing an analogy between logic and physics by using the Dirac measure. I don't know that I have a complete mathematical analysis of propositional logic. We'll have to ask someone more skilled in analysis than I am. My framework seems to be more than an analogy; it seems to be a direct mapping from one domain (logic) to the other (math). Isn't it curious how I come up with addition for disjunction and multiplication for conjunction. It works for the numbers 0 and 1. Have I included enough information to use mathematical induction for any numbers? I've not seen anywhere else where they derive addition of probabilities for alternative possibilities or multiplication of probabilities for a conjunction of possibilities - the sum and product rule of probabilities.
I used the iterations of logical expressions to find what seems to be the various particles of the Standard Model. See logictophysics.com/StandardModel.html. This is all derived from assuming that points in space are in conjunction with each other. So one thing my model predicts is if there is space, then we should always have these particles, it would seem even during inflation and even during any phase transition from the false vacuum to the true vacuum. Some would suggest that particles did not exit yet during inflation, and others suggest that a collapse of the false vacuum would possibly give us a whole new set of particles. And others would suggest that there may be other universes that have a different set of particles and laws. But my work seems to indication that if there is space with points in conjunction, then we have the set of particles of the Standard Model.
Friend wrote:
So one thing my model predicts is if there is space, then we should always have these particles, it would seem even during inflation and even during any phase transition from the false vacuum to the true vacuum. Some would suggest that particles did not exit yet during inflation, and others suggest that a collapse of the false vacuum would possibly give us a whole new set of particles.
Friend wrote:
I've been debating gill1109 as to why the complex exponential Gaussian function should be chosen for the Dirac delta function. He thinks I've pulled it out of the air because it works. I think I have more obvious reason for selecting it.
gill1109 wrote:Boole worked out probability *and* logic. Of course there may me a one-to-one correspondence between some parts of one formalism and some parts of another. If you can *prove* that it is indeed one-to-one then you can transpose theorems in the one field to theorems in the other. This is already big business in mathematics, using theorems about theorems to prove new theorems. Category theory does this.
Austin Fearnley wrote:So consider all the photons together at the start of the universe at a point....The bosons are not sufficient alone to provide a metric of space.
Friend wrote:
Matter can stand still in space. But photons are defined as moving... at the speed of light. If there is no expanse of space for photons to move within, then there can be no photons. Right?
Austin Fearnley wrote:Well, IMO as an amateur, from our viewpoint the photon is created by a fermion, then travels alone at speed c, and then is annihilated by a fermion. If the fermions are all gone then there is no means of creation of new, nor destruction of old, photons. There may be other ways of creation/destruction that I am overlooking, but I do not think so. Penrose talks of only photons surviving to the CCC nodes but I am unclear about other bosons such as gluons.
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