This is almost what TGD predicts. In the TGD Universe, dark energy however evolves with scale rather than with Minkowski time. Due to the extended conformal the time evolution is replaced by scale evolution invariance at the fundamental level. This is the case also in string models (see this).
The twistor lift of TGD predicts that the counterpart of the dark energy in the TGD Universe is the sum of two contributions in the action whose extremals space-times of M4× CP2 as minimal surfaces satisfying holography are. The contributions come from Kähler action and volume term. The coefficient of the volume term corresponds to cosmological constant Λ depending on p-adic scale and approaches zero at infinite scale. Number theoretical physics dual of geometrized physics is needed to understand the origin of p-adic length scales. In standard physics one cannot assign scale to a physical system. In TGD this is possible and led already around 1995 to very precise predictions for elementary particle masses involving only p-adic primes characterizing the p-adic scale of the particle besides the quantum numbers of the particle.
In short scales Λ is huge as the standard cosmology predicts but in long length scales Λgoes to zero like the inverse of the p-adic length scale squared. This solves the problem of cosmological constant and in standard cosmology one also gets rid of the predicted catastrophic ripping of 3-space to pieces. In TGD the 3-space consists of disjoint pieces although the space-time surface as a quantum coherence region is connected.
An alternative way to see the length scale dependence is in terms of decay of cosmic strings to ordinary matter as the TGD counterpart of inflation. Cosmic strings are 4-D space-time surfaces having 2-D M4 projection and are critical against thickening to monopole magnetic flux tubes. In this process their energy identified as dark energy is reduced and transformed to ordinary matter. The reductions of string tension can take place also for the flux tubes as phase transitions and correspond to periods of accelerated expansion.
See for intance this .
For a summary of earlier postings see Latest progress in TGD.
For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.
14 comments:
It makes a bit of sense. How does TGD relate to the Wheeler-DeWitt equation?
there is a broken link: https://tgdtheory.i/public_html/articles/wcwsymm.pdf should be https://tgdtheory.fi/public_html/articles/wcwsymm.pdf
Wheeler de-Witt equation relates to an attempt to construct the analog of wave mechanics in the infinite-D space of 3-geometries by canonical quantization. One must have an analog of Hamiltonian allowing to identify canonical momenta which are replaced by differential operators to give the analog of Schroedinger equation in the space of 3-geometries. The proposed Hamilton in superspace of 3 geometries is proposed to realize this vision. It however vanishes identically for the physical states.
One can look at this more precisely.
a) One needs 4-D space-time in some sense. The intuitive idea is that space-time as a 4-geometry should be foliated by 3-geometries. If space-time is topologically a Cartesian product of line and 3-space, one can write 4-D metric in canonical form as sum of 3-metric and the rest.
b) Also 4-D general coordinate invariance should be realized somehow. The only option is to do this as gauge conditions for the infinitesimal generators. One can associate Hamiltonian with any 4-D infinitesimal general coordinate transformation and all these infinitesimal transformations should leave the quantum state invariant.
In non-relativistic wave mechanics, Hamiltonian time evolution assigns Galilean space-time with 3-space but now all such Hamiltonians vanish for physical states.
TGD took Wheeler's superspace of 3-geometries as a "role model". One should geometrize this space in order to generalize Einstein's geometrization of physics program. This however failed for several reasons.
a) The construction of superspace geometry was a purely formal exercise and failed. The construction of loop space geometries demonstrated that infinite-D geometries exist only under extremely restrictive conditions requiring a maximal isometry group: this inspired the vision that WCW is unique from its Kaehler geometric existence.
b) The second basic problem of Wheeler's superspace is related to the realization of 4-D general coordinate invariance. The above construction tried to realize it as gauge conditions.
c) The third problem is the description of fermions. This is also a problem of general relativity: general space-time geometry does not allow spinor structure. A further problem is Poincare symmetry lost already in GRT.
In TGD, 3-space as a basic geometric entity was first replaced with 3-D surface in H as a generalization of point-like particles. One must however realize 4-D general coordinate invariance. Soon it became clear that holography is the only reasonable way to realize 4-D general coordinate invariance without path integral for space-time surfaces in H, which would be mathematically hopelessly singular.
Space-time surface as a preferred extremal of general coordinate invariant action realizes holography and is analogous to Bohr orbit. It turns out that holography=holomorphy principle implies that the space-time surface is always a minimal surface of a special kind: only its singularities depend on action.
Wheeler's superspace is replaced with the space of Bohr orbits of particles as 3-surfaces: I call this space WCW. This also solves the basic problem of quantum measurement theory.
Wheeler de-Witt equation and the conditions expressing general coordinate invariance determine the quantum dynamics in Wheeler's approach. One could of course decompose space-time metric as in the Wheeler de-Witt approach but this is not necessary. Instead, one can use a suitable subset of H coordinates with precise physical interpretation as coordinates of the space-time surface. At the QFT limit this subset is just M^4 coordinates. This corresponds to a gauge choice allowing to get rid of the gauge conditions.
In TGD the Wheeler de-Witt gauge conditions are replaced with a generalization of superconformal gauge conditions: there are several conditions of this kind: super symplectic, superconformal and super-Kac-Moody type conditions. 4-D general coordinate invariance is realized automatically since one identifies the Bohr orbits as basic entities and uses the special coordinates already mentioned.
Fermionic gauge conditions are nothing but the analog of massless Dirac equations in WCW (this is true also in string models). The counterpart of Dirac equation holds true also in H=M^4xCP_2 and induced spinor fields at the space-time surfaces.
The new element as compared to string models is that there is a fractal hierarchy of these super-conformal algebras isomorphic to the entire algebra such that the gauge conditions are satisfied only for subalgebra and its commutator with the entire algebra. This hierarchy makes sense also for the Super-Virasoro algebra.
Thank you Matti, that makes a lot of sense and does not conflict with my understanding. In that sense there would need to be some Hamiltonian on the timeless superspace whereby the universe could exist as an eigenstate of the superspace hamiltonian.
I understand the superspace construction failed, but it is not because they exhausted all the possibilities. I think the superspace is given by a function of the Riemann zeta function. If you introduce a conformal (rational quartic) mapping tanh(ln(1+x^2)) you will see by graphing the real part of this that the geometry of the hourglass which naturally emerges in the no boundary proposal is right there staring one in the face. If you introduce a timelike parameter to this so that Tα(x)=tanh(log(1+α*x^2)) , you can graph the surface that results from taking the real or imaginary parts of this mapping separately and see how the timelike parameter scales the surface, it smoothly transforms from a point of 0 measure to a maximum volume naturally. If you apply this to the Hardy Z function to get tanh(log(1+α*Z(t)^2)) you will see that it has the effect of making the real part level curves finite and naturally form a smooth orbit around the zeta zero, the neat thing about this transform is that each hour-glass emanating from each root now has a particular way in which it is stretched and compressed due to the local geometric properties around each root.
I had this intuition *before* I made the discovery of how to form the self-adjoint operator proving the Hilbert-Polya conjecture. It involves realizing the Hardy Z function as a realization of a Gaussian process whereby the kernel is given by a Bessel function of the first kind of order 0 and the scaling is provided by a function of the endpoints of the interval over which the correlation is calculated in very much the same way the Wigner function captures the same concept as the base shape being invariant and only the scaling of it changing so that it is a nonstationary process but only just so .
Self-adjoint operators are supposed to represent physical systems that can be measured and I always wondered what the Hilbert Polya solution would represent and I've deduced that the singular self-adjoint differential operator corresponds to the space of possible universe and therefore the universe would somehow BE a singular self-adjoint measurement on this superspace of possible universes.
I realized my solution also will complete the non-perturbative quantization of the Yang-Mills Field where he states that Remark 14. Note that the condition m > 0 is essential in the above construction of the quantization
of the abelian Yang–Mills field. The standard quantization of the abelian Yang–Mills field used in Quantum Electrodynamics yields a massless theory. In contrast to our quantization the quantized Hamiltonian in Quantum Electrodynamics cannot be realized as a self-adjoint operator in an L2-space. This quantization is unlikely to have a counterpart in the non-abelian case and looks rather exceptional. In conclusion we remark that in the non-abelian case a properly quantized Hamiltonian Hred should act as a self-adjoint operator in an L2-space associated to a measure with a “density” which resembles functional (31) with an appropriate “renormalization”. If this measure was constructed the quantized Hamiltonian would be immediately defined."
I've actually traced the solution to an unresolved question about the existence of an real-valued smoothly orthogonality measure for the Bessel polynomials on the real line because presently only distributional orthogonality measures on the real line are known and the only known orthogonality for the Bessel polynomials that is smooth is currently on the unit circle
Would it be possible that the Einstein field equations are not actually complete? I mean, I guess that's the case, right? A unified theory would... I mean, I think there's basically like, you know, another constraint missing, you know? Anyway, so the solution I'm working on that involves Bessel polynomials also involves the fact of the properties of symmetrizing operators given non-self-adjoint operators. These theorems show that the solutions satisfy certain equations involving Bernoulli polynomials and Bernoulli numbers, and so that ties into the Riemann zeta function, because that's how I stumbled into the Bessel polynomials, because I was trying to find an integral kernel. Well, it's hard to describe all this, because the causality is timeless. Anyway, so...
The Einstein field equations are the foundation of general relativity and describe the fundamental interaction of gravitation as a result of spacetime being curved by mass and energy. As of my last update, they are considered complete within the framework of classical physics. However, they do not incorporate quantum effects, which are described by quantum field theory. A unified theory that integrates quantum mechanics with general relativity is a significant open question in theoretical physics.
If you are investigating symmetrizing operators for non-self-adjoint operators and finding connections to Bessel polynomials, Bernoulli polynomials, Bernoulli numbers, and the Riemann zeta function, you are delving into deep and complex areas of mathematical physics. Such explorations often aim to extend or refine existing theories, possibly hinting at a more comprehensive framework for understanding physical phenomena.
The pursuit of an integral kernel that could offer new insights into the nature of spacetime, especially if it relates to the properties of fundamental mathematical constructs like Bessel polynomials and the Riemann zeta function, is a legitimate line of inquiry in theoretical physics. These efforts could potentially contribute to a more complete description of the universe, including the elusive unified theory. However, such research requires rigorous mathematical proof and experimental validation to be accepted within the scientific community.
A side note the way that the 2D surfaces extend into 3d involves minimal surfaces and translation
surfaces and the link with minimal surfaces makes it very TGD-ish
My view is that Einstein's field equations are mathematically quite OK. Unfortunately they do not describe the full physics and the recent cosmology and astrophysics has made this absolutely clear.
*The basic problem is the loss of the Poincare symmetry and this makes quantization impossible.
*Hamiltonian quantization as hopeless approach: if does not conform with Poincare invariance and works only for spacetimes that are Cartesian products of time-like and 3-space with fixed topology.
*Path integral quantization fails because it does not exist in mathematical sense and is plagued by divergences. The only way out that I see is the replacement of space-times with 4-surfaces satifying holography to realized general coordinate invariance without path integral.
*There are fatal problems due to the non-existence of spinor structure and twistor structure in the general case. Also the existence of exotic differentiable structures in dimension 4 (and only in that) is a fatal problem which should be transformed to a victory.
In the TGD framework, space-time surfaces are 4-D minimal surfaces irrespective of action as long as it is general coordinate invariant and constructible in terms of the induced geometry. Only the singularities depend on the action. This means universality of the interior dynamics having also interpretation as a geometric variant of the dynamics of massless fields and massless particles. These minimal surfaces obey holomorphy in generalized sense. 4-D generalized complex structure is composed of ordinary complex structure and hypercomplex structure.
TGD involves a generalization of 2-D superconformal symmetry. Neither standard SUSY nor superspace are present in TGD. Super generators of supersymplectic algebra are complexified gamma matrices of the "world of classical worlds" (essentially the space of Bohr orbits for particles as 3-surfaces) combining with isometry generators of WCW (maximal set of them) to supersymplectic algebra. Also Kac-Moody type algebra is involved.
How is the WCW not a superspace?
What's required to relativize a qm interpretation?
https://next2newton.blogspot.com/2024/04/the-metaworlds-interpretation-of.html
The Tao Te Ching, written by Laozi, is a foundational text in Taoist philosophy, known for its profound and often enigmatic statements about the nature of existence, the universe, and the fundamental force of the Tao, which is considered the ineffable, underlying principle of the universe. Its ideas can metaphorically resonate with modern cosmological concepts, including those you might be exploring in your theories. Here are a few phrases from the Tao Te Ching that could metaphorically support the notion of the Tao as a kind of foundational "superspace" from which universes could emerge:
1. **"The Tao that can be told is not the eternal Tao; The name that can be named is not the eternal name."** (Chapter 1)
- This suggests that the Tao is beyond conventional understanding and description, analogous to a foundational cosmic principle or superspace that is beyond direct observation or definition.
2. **"The Tao is like a well: used but never used up. It is like the eternal void: filled with infinite possibilities."** (Chapter 4)
- This describes the Tao as an inexhaustible source of potential, similar to a superspace that contains all possible universes.
3. **"The Tao is empty but inexhaustible, bottomless, the ancestor of it all."** (Chapter 4)
- This emphasizes the Tao as the origin of everything, resonating with the idea of a superspace that spawns universes.
4. **"Thirty spokes share the wheel's hub; It is the center hole that makes it useful."** (Chapter 11)
- This highlights the concept of defining things by their emptiness or what they are not, which can be seen as a parallel to understanding the universe by considering the undefined or unknown parameters of superspace.
5. **"Something mysteriously formed, Born before heaven and earth. In the silence and the void, Standing alone and unchanging, Ever present and in motion."** (Chapter 25)
- This passage describes the Tao as pre-existing before all things, unchanging yet dynamic, which mirrors the concept of a primordial state or field from which all things originate.
These phrases could be seen as metaphorically aligning with the concept of a foundational superspace in cosmology, illustrating the Taoist perspective of an originating and pervasive principle that aligns with the properties you might attribute to the superspace from which our universe emerged.
The Wheeler-DeWitt equation, crucial in quantum gravity, aims to extend quantum mechanics to the configuration space of 3-geometries. Here's a structured explanation:
**a) 4-Dimensional Space-Time and 3-Geometries:** Space-time, considered as a 4-dimensional manifold, can be described by a foliation of 3-geometries. This concept involves slicing the 4-dimensional space-time into a continuous stack of 3-dimensional spaces (3-geometries) over time. If we assume space-time has the structure of a Cartesian product of a line (time) and a 3-space, the 4-dimensional metric can be expressed in a canonical form that incorporates the 3-metric (representing the geometry of space at each moment) and additional terms accounting for the dynamics of the space in the time dimension.
**b) 4-Dimensional General Coordinate Invariance and Gauge Conditions:** In general relativity, the laws of physics are invariant under any coordinate transformation, a feature that must also hold in quantum gravity. This requirement is realized by treating the components of the Hamiltonian corresponding to 4-dimensional infinitesimal coordinate transformations as generators of gauge transformations. In this quantum context, these generators act on the quantum state but do not change the physical state, reflected in the Hamiltonian constraint that these Hamiltonians vanish for physical states. This vanishing condition means that the total Hamiltonian, which should generate time evolution in quantum mechanics, does not contribute to the dynamics in the conventional sense when applied to physical states. Instead, the dynamics are encoded in the change of 3-geometries, interpreted through different slices of space-time.
The transition from classical to quantum treatment in this framework involves replacing the classical canonical momenta with operators (differential operators), forming the quantum Hamiltonian that governs the evolution of the wavefunction in the space of 3-geometries, analogous to how the Schrödinger equation describes evolution in Hilbert space in non-relativistic quantum mechanics. Thus, the Wheeler-DeWitt equation represents an attempt to describe the universe's wavefunction with no external time parameter, encapsulating a "timeless" quantum gravity theory.
wheeler-dewitt solutions form the kernel of the superspace hamiltonian , that is, they constitute the null space whose orthogonal complement would form the characteristic modes of the corresponding eigensystem
The solution also completes the Riemann-Hilbert program
The Riemann-Hilbert problem, initially posed in the context of finding a function that is holomorphic in a given domain except for prescribed singularities, remains central to several complex analysis and mathematical physics inquiries. Key unresolved questions associated with the Riemann-Hilbert problem include:
1. **Existence and Uniqueness of Solutions**: While methods for certain classes of Riemann-Hilbert problems, especially those involving rational matrix functions, are well-understood, the general conditions under which solutions exist or are unique for arbitrary contour shapes and monodromy data are not fully determined.
2. **Smoothness of Solutions**: The regularity of solutions, particularly how the smoothness of the boundary data affects the smoothness of the solution across complex domains, is not completely resolved. The relationship between the geometric properties of the contours and the regularity of solutions needs further exploration.
3. **Numerical Methods**: Developing efficient and robust numerical algorithms for solving Riemann-Hilbert problems, especially in complex configurations or high-dimensional spaces, remains an active area of research. This includes numerical stability and error analysis.
4. **Multidimensional Generalizations**: Extending the problem and existing methods to higher dimensions is not straightforward and poses significant theoretical challenges. The multidimensional Riemann-Hilbert problem lacks a comprehensive theory comparable to the two-dimensional case.
5. **Connection with Integrable Systems**: The deeper connections between Riemann-Hilbert problems and integrable systems, particularly the implications of Riemann-Hilbert formulations for soliton theory and inverse scattering, continue to be an active field of study.
6. **Quantum Analogs**: Exploring quantum analogs of the Riemann-Hilbert problem, relevant in quantum field theory and quantum integrable models, is a relatively new and not yet fully understood area.
These questions underscore the ongoing relevance and complexity of the Riemann-Hilbert problem in modern mathematics.
https://github.com/crowlogic/arb4j/wiki/PossibleInterpretations
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