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first published 26. Jun 2002 revised upload |
| Bargmann Wigner and Irving Segal | A ℝ¹×𝕊³-approximation ( perhaps even a 𝕊¹×𝕊³-curved time one instead ) approximates a space-time with gravitation better ( there is no point in space-time without gravitation ) than flat Minkowski space ( this first was seen by Irving Segal [ Seg ] ) - like the curvature circle with radius ፫ approximates a curve better than the tangent ( if the curve is not a line ). These approximations not necessarily describe the large scale structure of the universe. Here there is a huge ,but': Loos gives a description of local symmetric spaces [ Loo chapter II ]. Applying this to space-time - whatever the large scale structure of the universe really is, it is a local symmetric space and its structure is described better by a principal fibre bundle, with base space-time and fibres symmetric spaces, than one with Minkowski space fibres. | ℝ are the real numbers, 𝕊n(፫) is the sphere of radius ፫ in n dimensions and 𝕊n(1) = 𝕊n is the unit sphere | ||||||||||||||||||||||||||||
| Space-Time Approximation and - Invariance Group | Generate a transformation group on this variant of flat Minkowski space by the 4-dimensional commutative Lie group of left multiplications ( corresponding to translations in Minkowski space ) and the full automorphism group ( inner automorphisms correspond to rotations in 3-dimensional space, outer to special Galileitransformations ) to get a non-linear variant of the Poincaré group ( the dimension of which may be greater than 10 ). This space-time invariance group contracts to the flat space Poincaré group if the (curvature) radius of 𝕊³ increases. | describe space-time directly - not by its invariance group | ||||||||||||||||||||||||||||
| Ray Representations in Hilbert Space | Derive by a Bargmann-Wigner classification of this variant of the Poincaré group a discrete ( because 𝕊³ is compact ) mass spectrum of elementary particles. Bring this mass spectrum in coincidence with the actual Pomeranchuk spectrum to get the unknown curvature radius of our position in space-time. The rest masses of elementary particles depend on this curvature radius, which for 𝕊³ is a multiple of the scalar curvature [ Ti l ] – and thus are not universal constants - they depend on position and gravitation there. The mass of an hydrogen atom thus depends on the position in curved space-time, which leads to a new version of the blue / red-shift formula and to the possibility [ Seg ] to escape big bang theories and / or a flat large scale structure of the universe. | must be irreducible but not unitary | ||||||||||||||||||||||||||||
| A Variant of Special Relativity | Formulate an entire variant of special relativity [ Irving Segal ] on ℝ×𝕊³ - especially electromagnetism should have interesting effects in regions of strong gravity [ Pos ]. Singularities in space-time ( not only black holes ) should be defined as 0-, 1- ,2- or even 3-dimensional sub( not necessarily vector ) spaces where the concept of curvature circle breaks down. | are there global effects in electrodynamics and optics? | ||||||||||||||||||||||||||||
| Cosmology Gravitation Space-Time Group | Instead of classifying unitary ray representations of the above generalization of the Poincaré group, there are two other possibilities for a group theoretical Bargmann-Wigner approach: (i) The cotangent bundle of ℝ×𝕊³ , itself an 8-dimensional Lie group, leads to a phase-space approach: The Lie group-input for the Bargmann-Wigner classification is a transformation group on this phase-space, containing inner automorhisms of the 𝕊𝕌(2)-sphere as a substitution and generalization of translations in the flat Minkowski space approach ( there may be more transformations, like outer automorphisms of ℝ×𝕊³ ).
Or better, instead of blowing up dimensions, reduce it to 4 :
(ii) Classify the irreducible ( not unitary - see below ) ray representations of the space-time symmetric space ℝ×𝕊³ itself ( instead of those of a transformation group thereon ). This is a cosmological, direct approach to elementary particles, which thus can be seen as quantized universes. 𝕊³ is besides the one-dimensional case ( up to a radius-dependent diffeomorphism ) the only non-trivial sphere which is a group – the compact Lie group 𝕊𝕌(2). Particles thus are parametrized by a discrete mass spectrum. The parameters - the rest masses - depend on the curvature radius of the position. Near a black hole rest masses differ from those at our far outside position in the galaxy, near the spiral-arm Sagittarius. | what is the physical meaning of additon in flat Minkowski space? | ||||||||||||||||||||||||||||
| How Many Alternatives to 𝕊³ Do Exist? | Variants of this variant of special relativity are given by substituting 𝕊³ by other 3-dimensional differentiable manifolds. There are exactly seven more orientable manifolds which are mutually non-homotopic ( hence neither homeomorphic nor diffeomorphic ) or to 𝕊³ [ Col p 93 ]. Among these eight manifolds not only 𝕊³ has the special property, being a Lie group, also
A Lie group is generated by exponentiation of its (tangent) Lie algebra elements, which makes it handable. The two non-Lie groups carry a symmetric space multiplication in the sense of O. Loos [ Loo ]. In this case too its elements are generated by exponentiation of elements of its (tangent) Lie triple and things are simpler. Note that every group is a symmetric space [Loo], but the converse is not true: The n-sphere ( even more general -hyperboloid ) is a symmetric space for all n, but a Lie group only for n=1 and n=3. Any of those 7 manifolds could serve as a handable variant, with a Bargmann-Wigner quantization in the above sense. However, our choice is the most ,symmetric', hence the most natural one, besides the - non-performing - ℝ³ . 𝕊³(፫) has the special property, that its main parameter, the radius ፫, is given in the most simple way by second derivatives. If the remaining 8th 3-dimensional oriented manifold is not a symmetric space - is there a more general algebraic multiplication, generalizing Loos' symmetric one, like this one generalizes group multiplication? How does its large scale structure look like ( avoiding the word ,topology' )? Another question: Are there more non-orientable alternatives to 𝕊³(፫) besides Klein's bottles? Is there an algebraic process ( this always makes mathematics easier ) which derives every such non-orientable manifold from an oriented one? For symmetric spaces and hence for Lie goups for instance in the form of a short exact sequence with a discrete kernel ( model covering )? Any such alternative may even better approximate the universe than this 𝕊³(፫)-model, at least in regions with extremely strong gravitation. Or conversely, orientable or not, may be approximated themselves by a sphere 𝕊³(፫). | If there is any mathematical meaning behind strings, here is one. But this would be bad luck, since an explicite description of this number 8, a global or local structure or an explicite construction in terms of exponentiation, is totally unclear – it just exists. | ||||||||||||||||||||||||||||
| Disadvantage of the Group Approach to 𝕊³(፫) | We forget for the moment the time component and concentrate on the sphere-part 𝕊³(፫) of our cosmological model / approximation. We start with the well-known group structure thereon: It is clear that the non-trivial sphere with radius ፫ has the structure of a (compact) Lie group. In fact to multiply two of its elements to get a third one, we have to go ,down' to the sphere by dividing twice by ፫, then multiply both elements in the group 𝕊𝕌(2) we have arrived at, and finally go back ,up' again by multiplying with ፫. Since these operations are diffeomorphisms, it is clear that 𝕊³(፫) thus can be given a Lie group structure. But there are severe short-comings of this procedure. 1stly the 3-dimensional unit-sphere has the very elegant formulation by the three (complex) Pauli-matrices together with the 2-dimensional identity matrix, which even, with respect to the canonical trace form, is a 4-dimensional (real) Minkowski space. This elegant formulation, loved by physicists since the early times of Wolfgang Pauli, is totally lost by blowing up or squeezing down the sphere with radius 1 to one with arbitrary radius. There seems to be no process to do this entirely in matrix form, at least not over the real or complex numbers. Undoubtedly numerous physicists have tried this in vain. And there is a 2nd severe handicap, which cannot be overcome in the traditional framework. Spheres are not diffeomorphic to planes, that is, they don't have a global chart. In fact there must be at least two charts, but even two do not suffice to get a neat description of the sphere, as any cartographer knows ( in general there must be an atlas of even more - this being the case for the most symmetric non-flat manifold was the starting point of modern differential geometry ). But beware: Making a (black) hole into an atlas does not mean that the underlying manifold has a singularity - only the atlas is blown apart - hey Schwarzschild❗ The elegant representation of the group structure in one and three dimensions in terms of matrices out of sin and cos, or sinh and cosh in the indefinite case, is one in local charts! This leads directly to the 3rd severe disadvantage: In invariant form, i.e. without using charts, there is no closed and basis-free representation of the product of two group elements of the group, even if its Lie algebra has elegant basis-free commutation relations in the space of antisymmetrized 2nd power elements of the Clifford algebra. We do not know any procedure to include the mathematical elegance of which into the product of two group elements. | The standard approach to black holes does not convince | ||||||||||||||||||||||||||||
| The Elegance of the Symmetric Space Approach to 𝕊³(፫) | We compare the two algebraic structures on 𝕊³(፫) , again dropping for the moment the time-component and concentrate on the sphere-part of our space-time approximation. Therefore we use symmetric spaces in the elegant formulation of O. Loos' [ Loo ] two books: We denote (t)his multiplication, which is not commutative, by ₪ in the following. One of his main examples is given by the invertible elements of a Jordan algebra, thus going back to the early days of quantum mechanics when Pascual Jordan started the operator structure of quantum mechanics, long before any C*-algebraic approach. The invertible elements of the natural Jordan algebra on any pseudo-orthogonal vector space has as invertible elements exactly those outside the null-cone, with a canonical symmetric space multiplication ₪. Spheres ( or hyperboloids, depending on the signature of the defining bilinear form ) are symmetric subspaces with respect to this ₪-multiplication, with an invariant, i.e. basis-free closed form for the ₪-product of two elements. There even is a Lie group embedding for any symmetric space ( making it a homogeneous one ), but this leaves four dimensions for the physical case. So it may be forgotten for the first approach, like the Jordan algebraic one, which only automatically arises in a quantization process. Denoting the n+1-dimensional symmetric non-degenerate bilinear form this time by ( , ), the Loos multiplication ₪ is such simple that we can write it down here
(ο) ⸻⸻⸻( x ₪ y , x ₪ y ) = ፫² .
This repairs the main handicap of the group approach, besides the simple closed invariant form of the right hand side of ₪ :
(σ)⸻⸻⸻ ( 𝕊n(፫) , ₪ )⸻⸻is a symmetric space .
Mathematically this is due to the asymmetric weight of the two entries on the left hand side of (o₪), also seen from the fact that Loos' multiplication
(g₪) ⸻⸻x ₪ y = x y−1 x .
makes every group a symmetric space.Note dimensions - we started from a pseudo-orthogonal vector space of dimension n+1 to arrive at a n-sphere, in physics from a 4-dimensional orthogonal vector space to arrive at the 3-sphere of radius ፫. The additional dimension is not time and has to be interpreted in a Jordan algebraic quantization process. In addition, although we started from a n+1-dimensional flat space, the manifold in question ( sphere or hyperboloid ) carries intrinsic curvature. Also remarkable – the ( symmetric space- ) isomorphy of multiplications of any element with a positive real scalar. From (ο), blowing up a unit sphere to one of arbitrary radius is a very natural operation in the symmetric space approach. | The Loos product x ₪ y of two elements x,y on the same n-sphere ( or n-hyperboloid ) in dimension n and for a positive real radius ፫ brings in the elegance of Artin's school of mathematics | ||||||||||||||||||||||||||||
| Quantization in a New Old Form | Collecting the facts on symmetric spaces and Jordan algebras of Hilbert spaces, found inon this webpage, we now can turn to the quantization of space-time: In non-quantized physics, space-time in greek philosophy and modern general relativistic physics plays the role of non-observable states, mathematical parameters, as C. F. von Weizsäcker has pointed out in his Hamburg lectures in the early 60th. Non-quantized observables are real-valued functions on space-time, not necessarily polynomials only. Generators are real-valued functions of space-time as well, thus their action on space-time being given. They act via the left action of the Poisson bracket on observables. Although mathematically observables and generators are the same, conceptually they belong to different categories. In classical physics
In quantum physics, with the same details,
Quantization of a classical dynamic means to classify, i.e. to find and reduce them to irreducible ones, all (ray) representations of a given classical dynamic into a quantum one on Hilbert space, i.e. find all injective (true) inequivalent symmetric space-morphisms of the given classical dynamic, defined by (σ), into a quantum dynamic on Hilbert space, given by (u₪). If the configuration space is compact, the Hilbert space is finite-dimensional, otherwise infinite-dimensional ( for the proof take traces or determinants somewhere in the Jordan algebraic world ). Thus unobservable space-time points, so-called events, are mapped into unobservable states. In addition the tangent bundle of space-time will be unobservable as well, since it involves only first derivatives of space-time objects. Only multiplyNing with masses makes the cotangent bundle of momentum observables. Thus the whole quantization process of Bargmann-Wigner can be restarted, but from different mathematical categories: Classify ( not unitary but invertible and symmetric, we have to leapfrog a third time, in order not to cross the divide between observables and invariance ) ray representations of this symmetric space. Since the tangent functor for symmetric spaces arrives in the category of Lie triples, in our Hilbert space example we proved that it even arrives in the category of Jordan algebras, which can easier be started locally and then be integrated. It is unlikely, that we loose representations that way. Switching to local, i.e. tangent formulations, a quantum dynamic leads to an ordinary differential equation as described in Loos' book, in which there is no unmotivated complex unit, because we are localising from the category of symmetric spaces to that of Jordan algebras and not from that one of unitary groups to their Lie algebras. | symmetric irreps of symmetric spaces | ||||||||||||||||||||||||||||
| Possible Generalizations and a Gedankenexperiment | Tangent spaces are defined by first derivatives, curvature radius by second order ones. There might be differential geometric spaces, defined by higher order derivatives, not being used in differential geometry so far. Such higher order spaces approximate actual curved space-time even better than the second order spaces of constant curvature spheres. A Gedankenexperiment sheds some light onto our approach of quantization of space-time to get a mass spectrum of elementary particles: Today experiments give the rest masses of particles, for instance the rest masses of neutrinos as approximately 2.3 eV ( in Carlsbad, California ). Now we know that the time of one rotation of our galaxy is some 250 million years, and that some 450 millions years ago there has been a catastrophy, which wiped out every higher live on earth, restarting the evolution of species from the simplest creatures. Most likely it originated in a nearly touch of our solar system with a neighboring one such that the two Oorts clouds got in touch, showering the two solar systems by solid objects. To avoid such a second catastrophy, which will take place within 50 million years, mankind will decide to move our solar system from its current position near the inner border of the spiral arm Sagitarius to some isolated position between this spiral arm and the main body of our galaxy. Thats easily done by installing a battery of large rockets on Uranus, being ignated in certain intervals. This will move Uranus from its position, but by gravitation it will move our whole solar system as well. Having succeeded in moving our solar system at a more secure position and assuming, than we still use our current theories of relativity, experiments in Carlsbad will give different rest masses of neutrinos. Before having derived the exact relationship of rest masses of particles in terms of the curvature radius, we have to guess this relationship. Assume that the relationship is such that rest masses decrease with increasing gravity ( the most likely case ), i.e. with increasing curvature radius. This means that now experiments in Carlsbad will give the rest masses of neutrinos to, say 0.00023 eV. With nowadays relativity physicists will conclude, that relativity totally is wrong, because that little measured mass of elementary particles never can account for the make up of all observed galaxies in space-time. Any danger now being removed, politicians will decide to spend money instead here elsewhere ( remember NASA ), and within some generations the whole procedure totally will be forgotten. However, there is inearta! This means that our solar system moves on, and after some (million) years we enter the main body of our galaxy, where gravitation increases. From Carlsbad there will come alarming news: The rest masses of elementary particles increase. Assume now two facts: The masses do not converge only to a certain limit but increase unbounded. And secondly that we are not wiped out by too narrow solar systems and such a catastrophy that mankind tried to avoid many (million) generations before. After some time physicists will conclude that there must be a concept of negative mass to account for the make up of our universe. In the end our solar system will be shattered because of too many stars near by or - if not - we will enter the event horizon of the huge black hole in the center of our galaxy - and that was it. | a Gedankenexperiment is not exactly science fiction | ||||||||||||||||||||||||||||
| Expectation grav. waves | As a Consequence, only e ≝: ⓔ, h ≝: ⓗ, c ≝: ⓒ and the Gravitational Constant G ≝: ⓖ will Remain as Universal Constants. Even Sommerfeld's Fine-Structure Constant may Turn out to Depend on Curvature. If Rest Masses Increase with Gravity, there is no Need of Dark Matter or - Energy, and there Results a Different, Radius-Dependent Redshift Formula. |