A quantum gravity theory needs to address the two
challenges of gravity theory and quantum theory: general relativity
cannot account for quanta and quantum mechanics
cannot deal with the curvature of space-time ((RoC2) p.126).
.. Space-time are
manifestations of a physical field, the gravitational field. At the same
time physical fields have quantum character: granular, probabilistic,
manifesting through interactions. The solution concept of LQT is basically about "A
spin network represents a quantum state of a gravitational field: a
quantum state of space; it is characterized by a volume "v" for every
node and a half-integer "j" for every line. ... The crucial difference
between photons (the quanta of the electromagnetic field) and the nodes
of the graph (the quanta of gravity) is, that photons exists in space,
whereas the quanta of gravity constitute space themselves ((RoC2) p.
"The nature of the elementary particles of
the SMEP, and the way they move, is desribed by quantum mechanics
((RoC3) fourth lesson). They are elementary excitations of moving
substratum similar to the Maxwell field: minuscule moving wavelets".
The proposed distributional quantum
state H(-1/2) with corresponding inner product admits and requires infinite linear
combinations of LQT "loop states" (which we "promoted"
becoming "quantum fluid / quantum element / fermion & boson / rotating differential / ideal point / monad" states). The physical LQT
space (which is a quantum superposition of the QLT "spin
networks") corresponds to an orthogonal projection of H(-1/2) onto
H(0). This othogonal projection can be interpreted as a general model for a
"spontaneous symmetry break down".
With respect to the two parameters characterizing a spin network we refer to a corresponding wavelet properties (BrK6): The (Calderón) wavelet reproducing
("duality") formula provides
an additional (second) degree of freedom (compared to a Fourier (transform)
wave and its related Fourier transform inverse; see also (BrK7) Notes
9/10) to apply wavelet analysis with appropriately
(problem specific) defined wavelets in a (distributional) Hilbert scale
framework, where the "microscope
observations" of two
wavelet (optics) functions can
be compared with each other. We note that for
a convenient choice of the two wavelet functions the Gibbs phenomenon
disappears (see also (RoC) 5.5, "Complements").
1. Einstein's geometrodynamics
(CiI) 2.8: Einstein's "general relativity" or ""geometric geometry of gravitation" or "geometrodynamics", has two central ideas: (1) Space-time geometry "tells" mass-energy how to move, (2) mass-energy "tells" space-time geometry how to curve. The concept (1) is automatically obtained by the Einstein field equations, (CiI) (2.3.14), basically as the covariant divergence of the Einstein tensor is zero. At the same point in time there are multiple tests of the geometrical structure and of the geodesic equation of motion, e.g. gravitational deflection and delay of electromagnetic waves, de Sitter and Lense-Thirring effect, perihelion advance of Mercury, Lunar Laser Ranging with its relativistic parameters: time dilation or gravitational redshift, periastron advance, time delay in propagation of pulse, and rate of change of orbital period, (CiI) 3.4.
(CiI) 3.5: "Hilbert used a variational principle and Einstein the requirement that the conservation laws for momentum and energy for both, gravitational field and mass-energy, be satisfied as a direct consequence of the field equations. ... Einstein geometrodynamics, ..., has the important and beautiful property the the equations of motion are a direct mathematical consequence of the Bianchi identities."
With respect to the overall conceptual idea of this homepage a Hilbert space based geometrodynamics is built on "space-time states", which are represented by elements of H(-1/2), while their corresponding "space-time energy" elements are represented by the corresponding "dual" (wavelets) elements in H(1/2). We emphasis that if ((u,v)) denotes the inner product of H(-1/2) the following relationships hold true: ((div(u),v)) ~ (u,v) ~ ((u,grad(v))).
The LQT is a modern version of the Wheeler/deWitt theory ((CiI), (WhJ)). The H(-1/2) Hilbert space of this page is proposed as a model for the "LQT spin network state/quanta of gravity" Hilbert space, enabling an (unified Dirac+Yang-Mills+Higgs+Einstein) hamiltonian operator ((RoC) 7.2.3), including the 4th matter (plasma) state. It is known from general relativity
and quantum theory that all of them are fakes resp. interim
specific mathematical model items.
The loop quantum theory (LQT) (C.
Rovelli) is the choice of a different algebra of basic field functions: a
noncanonical algebra based on the holonomics of the gravitational connections
((RoC) 1.2.1). The holonomy (or the "Wilson loop") is the matrix of
the parallel transport along a closed curve and spacetime itself
is formed by loop-like states. Therefore the position of a loop
state is relevant only with respect to other loops, and not with respect to the
background. The state space of the theory is a separable Hilbert
space spanned by loop states, admitting an orthogonal basis of spin network
states, which are formed by finite linear combinations of loop states,
and are defined precisely as the spin network states of a lattice Yang-Mills
We briefly sketch the central conceptual differentiators of LQT to other GUT theory attempts ((RoC) p. 10, p.
14, p. 140) with its relationship to the topic of this page (related conceptual elements of the Wheeler theory are sketched below). "The theory of LQT combines general relativity with quantum mechanics in a rather conservative way, because it does not employ any other hypothesis apart from those of the two theories themselves, suitable rewritten to render them compatible" ((RoC2) p. 144).
(RoC1): "the key differentiator to Einstein's field theory is the absence of the familar "space-time" stage (background independence),
which is technically realized by the gauge invariance of the action
under (active) diffeomorphisms (or diffeomorphism invariance). It is the
combination of two properties of the action: its invariance under
arbitrary changes of coordinates and the fact that there is no
nondynamical "background" field. ... The notions of space and time is
given up. The space continuum "on which" things are located and the time
"along which" evolution happens are semiclassical approximation
notions; the LQT makes only use of the general tools of quantum theory: a
Hilbert space of states, operators related to the measurements of
physical quantities, and transition amplitudes that determine the probability outcome of measurements of these quantities. ... In the macroscopic world, the physical variable t measured by a clock has peculiar properties. The fact, that time is not a special variable at the fundamental level needs to be reconciled, leading to
the thermal time hypothesis ((Roc) 3.4, "the thermal hypothesis", (RoC1)): "In Nature, there is no
preferred physical time variable t. There are no equilibrium state "r"
preferred a priori. Rather, all variables are equivalent: we can find
the system in an arbitrary state "r"; if the system is in a state "r",
then a preferred variable is singled out by the state of the system.
This variable is what we call time."
The three conceptual elements of the quantum mechanics
(remaining in LQT) are (RoC1): "(1) granularity (2) indeterminism (3) fluctuation ((RoC2), p. 116):
(1) Granularity: the information in the state of a system is finite, and limited by Planck's constant
With respect to the physical
phenomenon "time" this means that for all physical phenoma there is (1)
granularity: a smallest "time" unit, the Planck time (2) indeterminism:
quantum super position of time (3) fluctuation (Heisenberg), when trying
to determine the position of an electron today and tomorrow".
In (RoC) 5.5.2, the relationship of the naturally defined "physical" scalar product of a Hilbert space H (defined based on the solutions of the Wheeler-DeWitt equations) and the related "kinematical" inner product of the (kinematical state) Hilbert space K is considered ((Roc) 5.1.2). This relation depends on the hamiltonian H. The space H is the eigenspace of the hamiltonian H corresponding to the eigenvalue zero ("zero point energy").
By definition, an indefinite inner product space is a real or complex vector space together with a symmetric (in a complex case: hermitian) bilinear form
prescribed on it so that the corresponding quadratic form assumes both
positive and negative values. The most important special case arises
when a Hilbert space is considered as an orthogonal direct sum of two subspaces, one equipped with the original inner product, and the other with -1 times the original inner product (BoJ).
mechanics "time" and "energy" are conjugated variables linked by the concept of "action" ((HeW) II, 2c). Therefore, the conceptual
elements above find its counterpart with respect to the proposed quantum
state Hilbert space H(-1/2) by the facts, that the Hilbert sub-space L(2)=H(0) is
compactly embedded into H(-1/2), where physical quantum mechanics phenomena are "measured" by corresponding
hermitian (projection) operators onto L(2). This property is proposed as the mathematical model for quantum mechanics "granularity" states in H(0). At the same point in time this embedded "granular" Hilbert space
(with respect to the norm of H(-1/2)) is the standard L(2) framework of probability theory, statistical analysis and quantum mechanics. Some related aspects related to Schrödinger's view on statistical thermodynamics are given in (BrK7) p. 25. In other words, the physical (H(0), state resp. H(1), energy Hilbert) spaces are made of quanta (H(-1/2), state resp. H(1/2) energy Hilbert) spaces.
An analogue situation regarding the compactly embedded Hilbert space H(a) into H(b) for a>b, is given by the rational numbers
(~H(0), H(1)) as subset of the real or hyperreal numbers (~H(-1/2), H(1/2)): the
rational numbers are embedded into the ordered field of real numbers, which is a subset of the ordered field of hyper-real (ideal) numbers. The field of hyper-real numbers (or ideal points)
contains infinitely great and small numbers. It is constructed
abstractly using Zorn's lemma. The key differentiator of both fields is, that the Archimedian axiom (which is valid for the real numbers) is no longer valid for the ordered hyper-real numbers.
From a purely mathematical point of view the baseline of all mathematical models are "axioms"; the very first one to be mentioned in the context of the above is the "well-ordering theorem" (which is NOT a "theorem" as such). It is equivalent to the "axiom of choice" and "Zorn's lemma" (which is NOT a "lemma", as such). At the same time all "physical" gravity and quantum theory models are a purely mathematical models building on those kind of axioms. With respect to an appropriate definition of a "mathematical time" beyond the ""physical"/ thermodynamical time" ((RoC) III.9) one could decide for a hyper-real number (which is nothing else than a Leibniz monad), where the corresponding standard part of it (in case the hyper-real number is finite) is the "thermodynamical time" variable. If this option is beeing seen as too sophisticated, please note that already each irrational number is its own mystery/universe, as EACH irrational number is only "existing" (i.e. purely mathematically defined) as the limit of a sequence of INFINITE rational numbers.
The last section of (RoC1) is related to philosophical aspects (including the words of Anaximander, (HeM) "Der Spruch des Anaximander"). From (HeM), Die Zeit des Weltbildes, 72) we recall the following: "Die neuzeitliche Physik heisst mathematische, weil sie in einem vorzüglichen Sinne eine ganz bestimmte Mathematik anwendet. Allein, sie kann in solcher Weise nur mathematisch verfahren, weil sie in einem tieferen Sinne bereits mathematisch ist. .... Keineswegs wird aber das Wesen des Mathematischen durch das Zahlenhafte bestimmt. ... Wenn nun die Physik sich ausdrücklich zu einer mathematischen gestaltet, dann heisst das: Durch sie und für sie wird in einer betonten Weise etwas als das Schon-Bekannte im vorhinein ausgemacht. Dieses Ausmachen betrifft nichts Geringeres als den Entwurf dessen, was für das gesuchte Erkennen der Natur künftig Natur sein soll: der in sich geschlossene Bewegungszusammenhang raum-zeitlicher Massenpunkte."
From the below we quote: The least action principle can be also seen as
THE fundamental principle to develop laws of nature in strong alignment
with Kant's philosophy: ((KnA), p. 55, p. 56): (translated) "the least action principle in his most modern general public is a maxime of Kant's reflective judgment. ... Offenbar
haben wir beim Energieprinzip eine typische Entwicklung vor uns: wenn
das Prinzip der reflektierenden Urteilskraft mit einer seiner Maximen
vollen Erfolg gehabt hat, rückt sein Ergebnis aus dem Reich der Vernunft
im Kantischen Sinne, zu welchem die reflektierende Urteilskraft gehört,
in die Sphäre des Verstandes herab und wird zum allgemeinen Naturgesetz
(law of nature)".
The loop quantum theory (LQT) (C. Rovelli) is the choice of a different algebra of basic field functions: a
noncanonical algebra based on the holonomics of the gravitational
connections ((RoC) 1.2.1). The holonomy (or the "Wilson
loop") is the matrix of the parallel transport along a closed curve and spacetime itself is formed by loop-like states.
Therefore the position of a loop state is relevant only with respect to
other loops, and not with respect to the background. The state
space of the theory is a separable Hilbert space spanned by loop states,
admitting an orthogonal basis of spin network states, which are formed
by finite linear combinations of loop states, and are defined precisely
as the spin network states of a lattice Yang-Mills theory."
The central mathematical concepts of the GRT are differentiable manifolds, affine connexions with the underlying covariant derivative definition on corrresponding tangential (linear) vector spaces. Already the "differentiability" condition is w/o any physical justification. The only "affine" connexion concept and its corresponding locally defined metrical space framework jeopardizes a truly infinitesimal geometry, which is compatible with the Hilbert space framework of the quantum theory and the proposed distributional Hilbert space concept in (BrK). In sync with the above we propose a generalized Gateaux differential operator:
let H(1/2) = H(1) + H(*) denote the orthogonal decomposition of the alternatively proposed "energy/momentum/velocity" Hilbert space, whereby H(1) denotes the (compactly embedded) standard energy space with its inner product, the Dirichlet integral; "lim" denotes the limes for t --> 0 for real t. Then for x,y ex H(1/2) the operator VF(x,y) is defined by VF(x,y):=lim((F(x+t*y)-F(x))/t), whereby the limes is understood in a weak H(-1/2) sense. The operator is homogeneous in y; however, it is not always a linear operator in y ((VaM) 3.1).
The main tools used in geometrical theory of gravitation are tensor fields defined on a Lorentzian manifold representing space-time. A Lorentz manifold L is likewise equipped with a metric tensor g, which is a nondegerated symmetric bilinear form on the tangential space at each point p of L. The Minkowski metric is the metric tensor of the (flat space-time) Minkowski space.
The least action principle can refer to the family of variational principles. The most popular among these is Hamilton's principle of least action. It states that the action is stationary under all path variations q(t) that vanishes at the end points of the path. It does not not strictly imply a minimization of the action.
The least action principle can be also seen as THE fundamental principle to develop laws of nature in strong alignment with Kant's philosophy: ((KnA), p. 55, p. 56): (translated) "the least action principle in his most modern general public is a maxime of Kant's reflective judgment. ... Offenbar haben wir beim Energieprinzip eine typische Entwicklung vor uns: wenn das Prinzip der reflektierenden Urteilskraft mit einer seiner Maximen vollen Erfolg gehabt hat, rückt sein Ergebnis aus dem Reich der Vernunft im Kantischen Sinne, zu welchem die reflektierende Urteilskraft gehört, in die Sphäre des Verstandes herab und wird zum allgemeinen Naturgesetz (law of nature)".
The Einstein-Hilbert action functional W(g) is about the scalar curvature S=scal (which is the Ricci scalar of the Ricci tensor "Ric") applied to the metric tensor g. It is the simplest curvature invariant of a Riemannian manifold. The scalar curvature is the Lagrangian density for the Einstein-Hilbert action. The stationary metrics are known as Einstein metrics. The scalar curvature is defined as the trace of the Ricci tensor. We note that the trace-free Ricci tensor for space-time dimension n=4 is given by Z(g):=Ric(g)-(1/4)*S(g)*g, and that Z vanishes identically if and only if Ric = l*g for some constant l. In physics, this equation states that the manifold (M,g) is a solution of Einstein's vacuum field equations with cosmological constant. We further note, that the Ricci tensor corresponds to the Laplacian operator multiplied by the factor (-1/2) plus lower order terms.
The Einstein-Hilbert functional is an invariant integral, which is a must to describe the field-action of graviation ((WeH), §28). From a physical perspective a field-action term should be based on a scalar density G, which is composed of the potentials g(i,k) and of the field-components of the gravitation field (which are the first derivatives of the g(i,k,), i.e., g(i,k);r): "it would seem to us that only under such circumstances do we obtain differential equations of order not higher than the second for our gravitation laws .... Unfortunately a scalar density G, of the type we wish, does not exist at all; for we can make all g(i,k);r vanish at any given point choosing the appropriate co-ordinate system. Yet, the scalar R, the curvature defined by Riemann, has made us familiar with an invariant which involves the second derivatives of the g(i,k)'s only lineary. ... In consequence of this linearity we may use the invariant integral (the Einstein-Hilbert functional) to get the derivatives of the second order by partial integration. ... We then get a sum of a truly field-action functional (with a scalar density G) plus a divergence integral, that is an integral whose integrand is of the form div(w). Hence for the corresponding variations of theh potential functions g(i,k) the variations of both funtionals are identical; therefore the replacement of the physically required scalar density G by the integrand of the W(g) is justified (as the essential feature of the Hamilton's principle is fulfilled with W(g))." This is where a alternative field-action functional of gravitation in a alternative framework (as proposed above) can be defined, based on a "scalar density" function in a "Plemelj" (Stieltjes integral) sense.
The electromagnetic field is built up from the co-efficients of an invariant linear differential form. The potential of the gravitational field is made up of the co-efficients of an invariant quadratic differential form. Replacing the Newtonian law of attraction by the Einstein theory is about discovering the invariant law of gravitation, according to which matter determines the components of the graviation fields. The topic of the chapter above is about the substance-action and the field-action of electricity and gravitation in the context of the least action principle. The substance-action is based on the mathematical concept "density", while the field action is based on the mathematical concept "potential (function)". The substance-action related "tensor density" of electricity can be easily extented to the substance-action related "tensor density" of gravitation ((WeH) §28). A corresponding field-action of gravitation based on an invariant integral and an approporate potential "scalar density" is not possible from a mathematical perspective, as by choosing the appropriate co-ordinates the field components of the gravitational field vanish. The alternatively proposed aproach of this page can be summarized as follows:
- replacing of the mathematical "density" concept by
Plemelj's "mass element" concept, which goes along with an alternative (more
general) "potential" function concept
- replacing the manifold concept by a (semi) Hilbert
space-based concept, where a non-linear invariant integral functional F(V(g)) is defined by a
distributional (semi-) inner product, which is equivalent to a corresponding
functional F(R(g)) of a related inner product (where R denotes the Riesz
operators (which commute with translations & homothesis having nice
properties relative to rotations)) plus a (non-linear) compact disturbance term; the concept enables variational methods of nonlinear operators based on Stieltjes and curvilinear integrals (VaM).
The Yang-Mills functional is of similar structure than the Maxwell functional regarding the underlying constant fundamental tensor. The field has the property of being self-interacting and equations of motions that one obtains are said to be semilinear, as nonlinearities are both with and without derivatives. The YME mass gap problem is about the energy gap for the vacuum state. Therefore, the above proposed model alignments for the "electricity & gravity forces" phenomena covers also the cases of the "weak & strong nuclear forces" phenomena.
To merge two inconstent theories requires changes on both sides. In the above case this is about a newly proposed common "mass/substance element" concept, alternatively to the today's "mass density" concept, while, at the same time, the linear algebra tensor tool (e.g. a "density" tensor) describing classical PDE systems is replaced by non-linear operator equations defined by weak (variational) functional systems. Those (weak) equations provides the mathematical model of physical phenomena, while its correspondin classical PDE systems (requiring purely mathematical additional regularity assumptions) are interpreted as approximation solutions, only.
As a shortcut reference to geometrodynamics is
given by (WhJ). For a review of discoveries in the nonlinear dynamics of curved spacetime, we refer to ((ScM). An introduction to the foundations and tests of
gravitation and geometrodynamics or the meaning and origin of inertia in
Einstein theory is provided in (CiI).
In ((CiI) 4.6) the Gödel model universe is discussed, which is a four-dimensional model universe, homogeneous both in space and time, which admits the whole four-dimensional simply transitive group of isometries, in other words, a space-time that admits all four "simple translations" as independent Killing vectors. As the Gödel model universe is homogeneous both in space and time it is stationary. In other words, in this model the cosmological fluid is characterized by zero expansion and zero shear. Thus the Gödel model runs into difficulty with the expansion of the universe.
"The specification of the relevant features of a
three-geometry and its time rate change on a closed (compact and without
boundary manifolds), initial value, space-like hypersurface, together
with the energy density and density of energy flow (conformal) on that
hypersurface and together with the expansion of the equation of state of
mass-energy, determines the entire space-time geometry, the local
inertial frames, and hence the inertial properties of every test
particle and every field everywhere and for all time."
The related clarifcations regarding the distortion tensor or gravitomagnetic field is provided in ((CiI) §5.2.6, § 5.2.7).
The Laplacian equation for the gravitomagnetic vector
potential W, in terms of the current J of mass-energy is discussed in
((CiI) 5.3). The Neumann problem and its related integral equations with double layer potential leads to the Prandtl operator, defining a well posed integral equation in case of domain H(1/2) with range H(-1/2) ((LiI) theorem 4.3.2).
whereby (((*,*))) defines the H(-1) inner product and
((*,*)) defines the H(-1/2) inner product of the corresponding Hilbert
scales building on the eigen-pair solutions of the Prandtl operator
equation with domain H(1/2).
The proposed alternative Hilbert space based framework provides
also a "variational wave equation/ function" based approach of the "evolution of
geometric structures on 3-manifolds" in the context of Thurston's "geometrization conjecture" and its underlying Poincare conjecture (which have been established by Perelman),
where the Ricci flows play a central conceptual solution element to build "nice behavior" metrics in manifolds.
"The hypothesis that the universe is infinite and Euclidean at infinity, is, from a relativistic point of view, a complicated hypothesis. In the language of the general theory of relativity it demands that the Riemann tensor of the fourth rank shall vanish at infinity, which furnishes twenty independent conditions, while only ten curvarture components enter the laws of the gravitational field. It is certainly unsatisfactory to postulate such far-reaching limitation without any physical basis for it.
The possibility seems to be particularly satisfying
that the universe is spatially bounded and thus, in accordance with our
assumption of the constancy of the mass-energy density, is of constant
curvature, being either spherical or elliptical; for then the boundary
conditions at infinity which are so inconvenient from the standpoint of
the general theory of relativity, may be replaced by the much more
natural conditions for a closed surface" ((CiI) 5.2.1)
The wave equation can be derived from the Maxwell equations by applying the rot-operator. It results into the "light" phenomenon. A similar tranformation is not possible for Einstein equations, which results into the "gravitation" phnomenon. The "approximation" approach is about the split g(i,k)=m(i,k)+h(i,k), where m(i,k) denotes the flat Minkowski metric. The perturbance term h(i,k) admits a retarded (only) potential representation, representing a gravitational perturbance propagating at the speed of light ((CiI) 2.10). An alternative splitting with defined distortion tensor enabling an analogue approach with electrodynamics is provided in ((CiI) 5.2.7).
In ((CiI) (2.7.10)) an „energy-momentum pseudotensor for the gravity field“ is introduced representing the energy and momentum of the gravitation field. Then, using the corresponding "effective energy-momentum pseudotensor for matter, fields and gravity field", in analogy with special relativity and electromagnetism, the conserved quantities on an asymptotically flat spacelike hypersurface are defined by the sum of four-momentum, energy and angular momentum operators (2.7.19-21). Following an analogue approach, which lead to the modified Maxwell equation (as proposed in the above paper), leads to an alternative effective energy-momentum tensor for matter, fields and gravity field". As the Einstein (gravity) tensor is derived from the condition of a divergence-free energy-momentum tensor, this results to an alternative Einstein tensor. The additional term of this alternative Einstein tensor could be interpreted as "cosmologic term", not to ensure a static state of the universe (which is not the case due to Hubbles observations), but to model the "vacuum energy" properly. This then would also be in sync with the physical interpretation of the corresponding term in the modified Maxwell equations with its underlying split of divergence-free and rotation-free tensors. At the same point in time the approach avoids the affine connexion concept and the "differentiable" manifolds regularity requirement, which is w/o any physical justification.
There are eight 3-dimensional geometries in the context of "nice"
metrics. The nicest metrics are those with a constant curvature, but
there are other ones. Their classification in dimension three is due to
In (GrJ) philosophical aspects of the geometrodynamics are considered. We quote from the cover letter summary:
The above questions concerning
singularities and non-geometric manifolds can be revisited based on the above alternative conceptual framework; the corresponding physical interpretation of the geometrodynamics are in line with Schrödinger's vision (resp. critique about the common handicap of all western philosophy baseline assumptions, propagating instead a purely monoism) of a truly quantum field theory (see also www.quantum-gravitation.de).
In (CoR) there is a
conjecture formulated, that distortion-free families of progressing, spherical waves
of higher order exist if and only if the Huyghens’ principle is valid, and that
families of spherical, progressing waves only exist for space-time dimension
n=2 and n=4 ((CoR) VI, §10.2, 10.3). In combination with Hadamard conjecture
(that the wave equations for even space-time dimension are the only partial
differential equations, where the Huyghens’ principle is valid) this would lead
to an essential characterization of the four-dimension space-time space with
its underlying Maxwell field theory.
We mention that the existing
electromagnetic phenomena on earth are the result of plasma physics phenomena
underneath the earth crust. Those “activities” are all triggered by gravitation "forces".
The above (distributional) Hilbert space based
alternative geometrodynamic modelling framework provides an alternative
approach to Penrose's "cycles of time" concept of a "conformal cyclic cosmology",
addressing e.g. the "collapsing of matter" of an over-massive star to a
black hole problem (PeR) and "the problem of time" (AnE).
"What characterizes the loop quantum theory (LQT) is the choice of a different algebra of basic field functions: a noncanonical algebra based on the holonomics of the gravitational connections ((RoC) 1.2.1). The holonomy (or the "Wilson loop") is the matrix of the parallel transport along a closed curve. ... In LQT, the holonomy becomes a quantum operator that creates "loop states" (to overcome the issue of current dynamics model of coupled gravity + matter system, simply defined by adding the terms defining the matter dynamics to the gravitational relativistic hamiltonian ((RoC) 7.3)). ... Space-time itself is formed by loop-like states. Therefore the position of a loop state is relevant only with respect to other loops, and not with respect to the background. ... The state space of the theory is a separable Hilbert space spanned by loop states, admitting an orthogonal basis of spin network states, which are formed by finite linear combinations of loop states, and are defined precisely as the spin network states of a lattice Yang-Mills theory." The proposed distributional quantum state H(-1/2) above admits and requires infinite linear combinations of those "loop states" (which we call "quantum fluid" state), i.e. overcomes the current challenge of LQT defining the scalar product of the spin network state Hilbert space ((RoC) 7.2.3). The physical space is a quantum superposition of "spin networks" in LQT corresponds to an orthogonal projection of H(-1/2) onto H(0). This othogonal projection can be interpreted as a general model for a "spontaneous symmetry break down".
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