Is GR effective geometry from an hypersphere? It would give many benefits.
I read from https://arxiv.org/ftp/quant-ph/papers/0604/0604210.pdf
In quantum physics, and consequently in quantum computation and information science, the Bloch sphere representation for transformations of two state systems has been traditionally used. While this representation is very useful for two state systems, it cannot be generalized to multiple states and when it is generalized, it looses its simple geometrical representation... asymmetric structures appear in N-level systems which do not have rotational invariance
How? https://arxiv.org/abs/quant-ph/0408014
We show that the maximum radius in each direction, which is due to the construction of the Bloch-vector space, is determined by the minimum eigenvalue of the corresponding observable (orthogonal generator of SU(N)). From this fact, we reveal the dual property of the structure of the Bloch-vector space; if in some direction the space reachs the large sphere (pure state), then in the opposite direction the space can only get to the small sphere, and vice versa.
and from https://pdfs.semanticscholar.org/7ef8/70e87d6589c9fb5046ead0469142c5aa85cf.pdf
...we provided a representation of an arbitrary two-feature pattern x in the terms of a point on the surface of the Bloch sphere S2, i.e. a density operator x. A geometrical extension of this model to the case of n-feature patterns inspired by quantum framework is possible.
What does this mean? This corresponds to the quantization? https://arxiv.org/pdf/chao-dyn/9904002.pdf
...strange attractor shrinks more and more towards the south pole of the Bloch sphere https://books.google.fi/books?id=llluCQAAQBAJ&pg=PA113&lpg=PA113&dq=bloch+sphere+primes&source=bl&ots=VyRCyaznAT&sig=JBKqetOQ2Rg3TzO0unUSh6Y9ebY&hl=sv&sa=X&ved=0ahUKEwiG4sqpi8LRAhWGNJoKHXiFDJwQ6AEIWjAI#v=onepage&q=bloch%20sphere%20primes&f=false
Finiteness, 'reality' decoherence, density matrix?
http://math.stackexchange.com/questions/1483650/how-to-plot-a-qubit-on-the-bloch-sphere talk about the Bloch sphere, but how to plot states on the sphere due to both states being a complex number, thus resulting in 4 "coordinates".
Mixed (inner) states and uncertainty? A 'collapse' gives a pure state (a Surface), that is an integer? This is what GR requires.
Hope I could make myself understood :)
From the 4 "coordinates," the normalisation of the state vector eliminates one, and the arbitrary phase eliminates a second one. The Bloch sphere is actually the projective space CP^1 of all the complex lines through 0 in C^2. If you want 4 coordinates, the projective space HP^1 (H is the field of quaternions) is required. Then the equivalent of the Riemann sphere (or rather its stereographic projection) is H, and the "hypersphere" would be S^7, the seven dimensional sphere.
A mixed state is simply a classical superposition (with classical probabilities and not probability amplitudes) of pure quantum states.
Thanks. interesting.
A mixed state is not superposition, though, (?) if you read the texts. A density matrix, see the states with opposite signs.
The system A described by ρA is not a pure quantum mechanical state although |ψiAB is. It can be considered as an ensemble, prepared with probability of |a|2 in quantum state |+i and probability of |b|2 in quantum state |−i. These probabilities are classical in the sense that there is no interference. In particular, the relative phase between |+i and |−i is completely arbitrary! If one measures Sz, one gets 1/2 and 1/2 probabilities for Sz = ・}¯h/2. If one measures Sx, one gets the same result. A pure state |+ix, although the Sz measurement also gives the same result, but the Sx measurement always gives +¯h/2, without uncertainty. Therefore, we can talk about a pure quantum state |ψi as a vector in the state space. We must also consider the mixed states described by a density matrix... Show its hermiticity, positive-definiteness, and unity-trace. If the density matrix has 1 nonzero eigenvalue, it is a pure state. Then it must satisfy ρ2 = ρ. If the density matrix has more than 1 nonzero eigenvalues, it is a mixed state. The ensemble is said to be an incoherent superposition of the states |ψai without any fixed-phase relation. All possible ~P form a solid sphere which is called the Bloch sphere. All states with P2 = 1 are pure. Otherwise, they are mixed states. Any points inside the sphere can be written as a sum of two pure states. If we start with a large system A+B which is pure, but if A is B is entangled, and if one measures system A only, one finds that the coherence in A is destroyed. We call this situation a collapse of the state in system A. State collapse allows us to understand the measurement of quantum system and decoherence. http://www.glue.umd.edu/afs/glue.umd.edu/department/phys/courses/Phys622/public_html/ji/lecture6.pdf
http://slideplayer.com/slide/4937082/
http://www2.fisica.unlp.edu.ar/materias/qc/disc.pdf
Note that this is a two state system, but quantum states are often seen as two -states or 2D. If we Think of GR we have more states, and the 4-coordinates are a clue, just as you say... Note that only one half is what we call 'real' or finite ( positive-definiteness) and this is compared to GR.
The density matrix of the classical superposition of two states |a> and |b> is
1/2(|a>+|b>).1/2(
a detailed analysis of the structure of the generalized Bloch sphere 3, the state space of the qutrit. We based our analysis on closed form expressions for 3 and its boundary ∂3. That these expressions are nearly as economical in form as the case of qubit was noted. Three special concentric spheres of radii 1, 1/√3, and 1/2 in R8 were noted along with their relevance to the structure of 3. We classified the 28 standard two-sections of 3 into five unitary equivalence classes of only four different geometrically inequivalent shapes. That there are two different sets of circular two-sections that are unitarily inequivalent but geometrically equivalent was noted, a feature missed by earlier authors. The 56 standard three-sections were classified into ten unitary equivalence classes of only seven different geometrically inequivalent shapes, correcting and completing an earlier classification by Mendas 8]. The obese-tetrahedral three-section was examined in considerable detail, clarifying how its tetrahedral symmetry consisting of both proper and improper rotations arises from the SO(3) ⊂ SU(3) group of proper rotations. https://arxiv.org/pdf/1111.4427v1.pdf
The metric dependence of energy https://arxiv.org/pdf/math/0211159v1.pdf
The evolution equation for the metric tensor implies the evolution equation for the curvature tensor of the form Rmt = △Rm + Q, where Q is a certain quadratic expression of the curvatures. In particular, the scalar curvature R satisfies Rt = △R + 2|Ric|2, so by the maximum principle its minimum is non-decreasing along the flow. By developing a maximum principle for tensors, Hamilton [H 1,H 2] proved that Ricci flow preserves the positivity of the Ricci tensor in dimension three and of the curvature operator in all dimensions.
the evolving metrics (on a closed manifold) of positive Ricci curvature in dimension three, or positive curvature operator in dimension four converge, modulo scaling, to metrics of constant positive curvature. (=Hamilton-Perelmans/(Poincare's conjencture) on a 3-sphere in quantum field theory?).
https://books.google.fi/books?id=Pk9yyC239scC&pg=PA346&lpg=PA346&dq=Perelman+conjecture+on+a+3-sphere+in+quantum+field+theory&source=bl&ots=_aQZUpd-Mf&sig=zNBaUBcAa9KP4qT6dAiYMsRQGsc&hl=sv&sa=X&ved=0ahUKEwiFgqLFiMrRAhXLjiwKHSNoBVQQ6AEIJTAB#v=onepage&q=Perelman%20conjecture%20on%20a%203-sphere%20in%20quantum%20field%20theory&f=false
http://www.sci-techuniverse.com/2018/07/every-black-hole-contains-another.html
Article Incompatibility of QED/QCD and repulsive gravity, and implic...
Technically yes. Then you get two possible quantum states of space. Maybe it is a step in the right direction, but probably not enough quantum states to represent the extreme high energy range.
A spherical model is not necessarily the fundamental one, just the one that researchers can relate to in Earth based experience. A polynomial is closer to be physical space time.
With polynomial you can get the folded sheets and layers like TGD but with special cases to simplify the math that a larger number of researchers can understand.
What kind of polynomial describes a sphere, and exactly why did Einstein end up with Riemann spheres?
As here, two vacuas, p 5,6. At the critical point α = ` 2 4( D − 3)( D − 4) , there is only one degenerate vacuum of multiplicity two.Article Vacuum Degeneracy and Conformal Mass in Lovelock AdS Gravity
Einstein started with Riemann differentials, but contracted them to Ricci differentials for agreement with 4D indices. Field equations are written with Ricci differentials. Other contributors gave spheres and ellipses as special case metric solutions.
Maybe a Taylor's series polynomial could be used considering that it can approximate almost any symbolic function. Coefficients would be energy related to make the higher order terms significant at high energy.
Sphere results from second order approximation and cannot be the fundamental geometry of space time general case.
https://math.stackexchange.com/questions/2267421/ricci-flow-on-a-sphere-of-radius-r
Article Curvature, Sphere Theorems, and the Ricci flow
The most elementary metric structure is the one obtained by restricting the distance function from R3 to M. This distance function is called the chord (or extrinsic) distance, and it depends on the embedding of M in R3....
can the minimal surface be a sphere? We usually accept the Planck length as minimal, even if it is from dimensional analysis from a scale that looks very peculiar with a big mass and tiny length. If it is a BH it should be 'outside spacetime' in GR, hence of no use for us in diff. geometry.
https://www.johndcook.com/blog/2018/01/31/integrating-polynomials-over-a-sphere-or-ball/
Note that if we integrate the constant polynomial 1 over the unit sphere, we get the surface area of the unit sphere...
GR is fundamentally a geometry on the right side of field equations and a physics on the left side. QFT attempts to unite GR with QM and has made progress, but not reached completion because of the high energy problem.
In other threads I proposed a way QFT to agree with GR on near flatness of space. When the Lagrangian Density is used to formulate the curvature in in QFT and in GR there is no disagreement, since the Lagrangian Density is small although the Kinetic and Potential Energies are large.
There is some value in describing space time as quantum states, especially in the unusual case where local kinetic energy greatly exceeds potential energy like gravity. Squeezed quantum states are reported in laser experiments and are helpful in decreasing uncertainty of measure in LIGO program.
Scale relativity is suggesting that GR should be should be modified in the high energy range as represented by fast moving vehicles under prolonged acceleration. Reference is made to Hierarchy of Plancks for exchange of kinetic energy between the vehicle and local space.
dE/df = n*h
In present formulations GR represents only the familiar ground state (n=1) in which h is a constant. This h is just the spin angular momentum of the ground state ZPE oscillators and n is the quantum number of the spin angular momentum in vacuum space.
Somehow n must be brought into Einstein field equations as a multiplier on the curvature. which may be either positive like excess gravity or negative like excess kinetic energy. The geometric significance of n is that space time resistance to curvature from differential stress energy increase undergoes step change of decrease resistance as critical high energy densities are reached.
Whether n is brought into GR through Bloch Speer or by other method, a description of space time is needed at the higher energies and faster speeds of prolonged acceleration.
Either way GR is modified it still is a truncated approximation leaving out the higher order terms that represent layers of folded space in TGD theory.
https://www.epj-conferences.org/articles/epjconf/pdf/2019/09/epjconf_ishepp2019_08004.pdf
Can n be in a loop function?
See how angular momentum behaves on the sphere with +/- energy added. https://courses.edx.org/c4x/BerkeleyX/CS191x/asset/chap10.pdf
https://www.nature.com/articles/srep11591
Whether n is brought into GR through Bloch Sphere or by other method, a description of space time is needed at the higher energies and faster speeds of prolonged acceleration.
Elongation? What happen when we accelerate the sphere? Gravitational waves tell us this can happen also in spacetime itself, so constants are what?
I glue here a comment of Matti, TGD: Matti Pitkänen Adelic physics generalizes the notion of space-time. Space-time can be said to be like book or hierarchy of books labelled by extensions of rationals. Each book has sheets labeled by real numbers and extensions of various p-adic numbers induced by this extension of rationals. Evolution corresponds to reading this book: the higher the page number is, the more complex the extension is: the dimension n=h_eff/h_0 of the extension measures the complexity and evolutionary level.
There is also M^8-H duality somewhat analogous to compactification. One can identify imbedding space as either 8-D octonionic Minkowski space M^8 (or rather, its complexification) or H=M^4xCP_2. At the level of M^8 one has purely algebraic physics without any partial differental equations. At the level of H physics based on action and partial differential equations plus infinite number of "gauge conditions" selectring preferred extremals. These conditions guarantee that the two descriptions are equivalent.
Article Universal Quantum Computing and Three-Manifolds
A single qubit may be represented on the Bloch sphere or similarly on the 3-sphere S 3 S3 . Our goal is to dress this correspondence by converting the language of universal quantum computing (UQC) to that of 3-manifolds. A magic state and the Pauli group acting on it define a model of UQC as a positive operator-valued measure (POVM) that one recognizes to be a 3-manifold M 3 M3 . More precisely, the d-dimensional POVMs defined from subgroups of finite index of the modular group PSL(2,Z) PSL(2,Z) correspond to d-fold M 3 M3 - coverings over the trefoil knot. In this paper, we also investigate quantum information on a few ‘universal’ knots and links such as the figure-of-eight knot, the Whitehead link and Borromean rings, making use of the catalog of platonic manifolds available on the software SnapPy. Further connections between POVMs based UQC and M 3 M3 ’s obtained from Dehn fillings are explored.
State reduction is usually expressed in phase space instead of space time.
In hierarchy of Plancks, n=h_eff/h_0 the n represents a quantum number of vacuum spin angular momentum in space time. Increase of n represents a state reduction R in related phase space following a unitary evolution of state U1 and followed by another unitary evolution of state U2.
Your question is related to extreme high energy where GR becomes inaccurate and QM can not be ignored. This is the range TGD was originally intended to describe.
The phase space and space time are mapped into each other, allowing either to be used for GR while remembering GR applies only when (n=1) but TGD describes any value of n up to at least 5 in dynamic systems and possibly 8 in static systems.
The book-like structure must be Minkowskian spacetime, or a phase space maybe? It vanishes so only a rest of magnetic field remain (as reals?)? So complex spaces are 'bubbles' like Fermi bubble?
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