Quantum Gravity and String Theory

U(1) X SU(2) X Su(3) Quantum Gravity Successes

Authors: Nige Cook
Comments: 63 Pages.

See paper for technical abstract. Paper covers checked predictions for a theory which modifies the Standard Model's electroweak group representations to include quantum gravity, replacing the Higgs mechanism with checkable predictions. The model correctly predicted the cosmological acceleration in 1996. Full references, analysis, and feedback from peer-reviewed string theory dominated journals is included.
Category: Quantum Gravity and String Theory

On The Zero Point Field

Authors: Paul Karl Hoiland
Comments: 5 Pages.

Via a look at the Ricci and Weyl curvature tensors I show that from a quantum perspective the Zero Point Field or ZPF is the origin of both inertia and gravity. I also point out that accelerated expansion of the cosmos and a observational slow down of C should have been expected.
Category: Quantum Gravity and String Theory

Inertia

Authors: Meir Amiram
Comments: 12 Pages.

I indicate that the key factor in the mechanism of inertia is the proximity of any elementary particle to itself, and consequently show that Newton laws of motion are derivatives of Newton's inverse square law of gravity. Inertia is originated in the microscopic realm, in the particle's diameter scale of reality, and is the response of an elementary particle to the gravitational field of itself, nothing more or less. Experimental evidences and several consequences of the discovery are discussed.
Category: Quantum Gravity and String Theory

Dark Matter, Dark Energy, and Elementary Particles and Forces

Authors: Thomas J. Buckholtz
Comments: 41 Pages.

Patterns link properties of six quarks and three leptons, the set of fundamental forces, and possible properties of dark matter and dark energy.
Category: Quantum Gravity and String Theory

Condensation States And Landscaping With The Theory Of Abstraction

Authors: Subhajit Ganguly
Comments: 11 pages

The Abstraction theory is applied in landscaping. A collection of objects may be made to be vast or meager depending upon the scale of observations. This idea may be developed to unite the worlds of the great vastness of the universe and the minuteness of the sub-atomic realm. Keeping constant a scaling ratio for both worlds, these may actually be converted into two self-same representatives with respect to scaling. The Laws of Physical Transactions are made use of to study Bose-Einstein condensation. As the packing density of concerned constituents increase to a certain critical value, there may be evolution of energy from the system.
Category: Quantum Gravity and String Theory

Quantum Arithmetics and the Relationship Between Real and P-Adic Physics

Authors: Matti Pitkänen
Comments: 37 Pages.

p-Adic physics involves two only partially understood questions.

1. Is there a duality between real and p-adic physics? What is its precice mathematic formulation? In particular, what is the concrete map p-adic physics in long scales (in real sense) to real physics in short scales? Can one find a rigorous mathematical formulation of canonical identification induced by the map p→ 1/p in pinary expansion of p-adic number such that it is both continuous and respects symmetries.

2. What is the origin of the p-adic length scale hypothesis suggesting that primes near power of two are physically preferred? Why Mersenne primes are especially important?

A possible answer to these questions relies on the following ideas inspired by the model of Shnoll effect. The first piece of the puzzle is the notion of quantum arithmetics formulated in non-rigorous manner already in the model of Shnoll effect.

1. Quantum arithmetics is induced by the map of primes to quantum primes by the standard formula. Quantum integer is obtained by mapping the primes in the prime decomposition of integer to quantum primes. Quantum sum is induced by the ordinary sum by requiring that also sum commutes with the quantization.

2. The construction is especially interesting if the integer defining the quantum phase is prime. One can introduce the notion of quantum rational defined as series in powers of the preferred prime defining quantum phase. The coefficients of the series are quantum rationals for which neither numerator and denominator is divisible by the preferred prime.

3. p-Adic--real duality can be identified as the analog of canonical identification induced by the map p→ 1/p in the pinary expansion of quantum rational. This maps maps p-adic and real physics to each other and real long distances to short ones and vice versa. This map is especially interesting as a map defining cognitive representations.

Quantum arithmetics inspires the notion of quantum matrix group as counterpart of quantum group for which matrix elements are ordinary numbers. Quantum classical correspondence and the notion of finite measurement resolution realized at classical level in terms of discretization suggest that these two views about quantum groups are closely related. The preferred prime p defining the quantum matrix group is identified as p-adic prime and canonical identification p→ 1/p is group homomorphism so that symmetries are respected.

1. The quantum counterparts of special linear groups SL(n,F) exists always. For the covering group SL(2,C) of SO(3,1) this is the case so that 4-dimensional Minkowski space is in a very special position. For orthogonal, unitary, and orthogonal groups the quantum counterpart exists only if quantum arithmetics is characterized by a prime rather than general integer and when the number of powers of p for the generating elements of the quantum matrix group satisfies an upper bound characterizing the matrix group.

2. For the quantum counterparts of SO(3) (SU(2)/ SU(3)) the orthogonality conditions state that at least some multiples of the prime characterizing quantum arithmetics is sum of three (four/six) squares. For SO(3) this condition is strongest and satisfied for all integers, which are not of form n= 2^2r(8k+7)). The number r₃(n) of representations as sum of squares is known and r₃(n) is invariant under the scalings n→ 2^2rn. This means scaling by 2 for the integers appearing in the square sum representation.

3. r₃(n) is proportional to the so called class number function h(-n) telling how many non-equivalent decompositions algebraic integers have in the quadratic algebraic extension generated by (-n)^1/2.

The findings about quantum SO(3) suggest a possible explanation for p-adic length scale hypothesis and preferred p-adic primes.

1. The basic idea is that the quantum matrix group which is discrete is very large for preferred p-adic primes. If cognitive representations correspond to the representations of quantum matrix group, the representational capacity of cognitive representations is high and this kind of primes are survivors in the algebraic evolution leading to algebraic extensions with increasing dimension.

2. The preferred primes correspond to a large value of r₃(n). It is enough that some of their multiples do so (the 2^2r multiples of these do so automatically). Indeed, for Mersenne primes and integers one has r₃(n)=0, which was in conflict with the original expectations. For integers n=2M_m however r₃(n) is a local maximum at least for the small integers studied numerically.

3. The requirement that the notion of quantum integer applies also to algebraic integers in quadratic extensions of rationals requires that the preferred primes (p-adic primes) satisfy p=8k+7. Quite generally, for the integers n=2^2r(8k+7) not representable as sum of three integers the decomposition of ordinary integers to algebraic primes in the quadratic extensions defined by (-n)^1/2 is unique. Therefore also the corresponding quantum algebraic integers are unique for preferred ordinary prime if it is prime also in the algebraic extension. If this were not the case two different decompositions of one and same integer would be mapped to different quantum integers. Therefore the generalization of quantum arithmetics defined by any preferred ordinary prime, which does not split to a product of algebraic primes, is well-defined for p=2^2r(8k+7).

4. This argument was for quadratic extensions but also more complex extensions defined by higher polynomials exist. The allowed extensions should allow unique decomposition of integers to algebraic primes. The prime defining the quantum arithmetics should not decompose to algebraic primes. If the algebraic evolution leadis to algebraic extensions of increasing dimension it gradually selects preferred primes as survivors.

Is Kähler Action Expressible in Terms of Areas of Minimal Surfaces?

Authors: Matti Pitkänen
Comments: 5 Pages.

The general form of ansatz for preferred extremals implies that the Coulombic term in Kähler action vanishes so that it reduces to 3-dimensional surface terms in accordance with general coordinate invariance and holography. The weak form of electric-magnetic duality in turn reduces this term to Chern-Simons terms.

The strong form of General Coordinate Invariance implies effective 2-dimensionality (holding true in finite measurement resolution) so that also a strong form of holography emerges. The expectation is that Chern-Simons terms in turn reduces to 2-dimensional surface terms.

The only physically interesting possibility is that these 2-D surface terms correspond to areas for minimal surfaces defined by string world sheets and partonic 2-surfaces appearing in the solution ansatz for the preferred extremals. String world sheets would give to Kähler action an imaginary contribution having interpretation as Morse function. This contribution would be proportional to their total area and assignable with the Minkowskian regions of the space-time surface. Similar but real string world sheet contribution defining Kähler function comes from the Euclidian space-time regions and should be equal to the contribution of the partonic 2-surfaces. A natural conjecture is that the absolute values of all three areas are identical: this would realize duality between string world sheets and partonic 2-surfaces and duality between Euclidian and Minkowskian space-time regions.

Zero energy ontology combined with the TGD analog of large N_c expansion inspires an educated guess about the coefficient of the minimal surface terms and a beautiful connection with p-adic physics and with the notion of finite measurement resolution emerges. The t'Thooft coupling λ should be proportional to p-adic prime p characterizing particle. This means extremely fast convergence of the counterpart of large N_c expansion in TGD since it becomes completely analogous to the pinary expansion of the partition function in p-adic thermodynamics. Also the twistor description and its dual have a nice interpretation in terms of zero energy ontology. This duality permutes massive wormhole contacts which can have off mass shell with wormhole throats which are always massive (also for the internal lines of the generalized Feynman graphs).

An Attempt to Understand Preferred Extremals of Kähler Action

Authors: Matti Pitkänen
Comments: 23 Pages.

There are pressing motivations for understanding the preferred extremals of Kähler action. For instance, the conformal invariance of string models naturally generalizes to 4-D invariance defined by quantum Yangian of quantum affine algebra (Kac-Moody type algebra) characterized by two complex coordinates and therefore explaining naturally the effective 2-dimensionality. One problem is how to assign a complex coordinate with the string world sheet having Minkowskian signature of metric. One can hope that the understanding of preferred extremals could allow to identify two preferred complex coordinates whose existence is also suggested by number theoretical vision giving preferred role for the rational points of partonic 2-surfaces in preferred coordinates. The best one could hope is a general solution of field equations in accordance with the hints that TGD is integrable quantum theory.

A lot is is known about properties of preferred extremals and just by trying to integrate all this understanding, one might gain new visions. The problem is that all these arguments are heuristic and rely heavily on physical intuition. The following considerations relate to the space-time regions having Minkowskian signature of the induced metric. The attempt to generalize the construction also to Euclidian regions could be very rewarding. Only a humble attempt to combine various ideas to a more coherent picture is in question.

The core observations and visions are following.

1. Hamilton-Jacobi coordinates for M⁴ > define natural preferred coordinates for Minkowskian space-time sheet and might allow to identify string world sheets for X⁴ as those for M⁴. Hamilton-Jacobi coordinates consist of light-like coordinate m and its dual defining local 2-plane M²⊂ M⁴ and complex transversal complex coordinates (w,w*) for a plane E²_x orthogonal to M²_x at each point of M⁴. Clearly, hyper-complex analyticity and complex analyticity are in question.

2. Space-time sheets allow a slicing by string world sheets (partonic 2-surfaces) labelled by partonic 2-surfaces (string world sheets).

3. The quaternionic planes of octonion space containing preferred hyper-complex plane are labelled by CP₂, which might be called CP₂^mod. The identification CP₂=CP₂^mod motivates the notion of M⁸--M⁴× CP₂. It also inspires a concrete solution ansatz assuming the equivalence of two different identifications of the quaternionic tangent space of the space-time sheet and implying that string world sheets can be regarded as strings in the 6-D coset space G₂/SU(3). The group G₂ of octonion automorphisms has already earlier appeared in TGD framework.

4. The duality between partonic 2-surfaces and string world sheets in turn suggests that the CP₂=CP₂^mod conditions reduce to string model for partonic 2-surfaces in CP₂=SU(3)/U(2). String model in both cases could mean just hypercomplex/complex analyticity for the coordinates of the coset space as functions of hyper-complex/complex coordinate of string world sheet/partonic 2-surface.

The considerations of this section lead to a revival of an old very ambitious and very romantic number theoretic idea.

1. To begin with express octonions in the form o=q₁+Iq₂, where q_i is quaternion and I is an octonionic imaginary unit in the complement of fixed a quaternionic sub-space of octonions. Map preferred coordinates of H=M⁴× CP₂ to octonionic coordinate, form an arbitrary octonion analytic function having expansion with real Taylor or Laurent coefficients to avoid problems due to non-commutativity and non-associativity. Map the outcome to a point of H to get a map H→ H. This procedure is nothing but a generalization of Wick rotation to get an 8-D generalization of analytic map.

2. Identify the preferred extremals of Kähler action as surfaces obtained by requiring the vanishing of the imaginary part of an octonion analytic function. Partonic 2-surfaces and string world sheets would correspond to commutative sub-manifolds of the space-time surface and of imbedding space and would emerge naturally. The ends of braid strands at partonic 2-surface would naturally correspond to the poles of the octonion analytic functions. This would mean a huge generalization of conformal invariance of string models to octonionic conformal invariance and an exact solution of the field equations of TGD and presumably of quantum TGD itself.

The Master Formula for the U-Matrix Finally Found?

Authors: Matti Pitkänen
Comments: 11 Pages.

In zero energy ontology U-matrix replaces S-matrix as the fundamental object characterizing the predictions of the theory. U-matrix is defined between zero energy states and its orthogonal rows define what I call M-matrices, which are analogous to thermal S-matrices of thermal QFTs. M-matrix defines the time-like entanglement coefficients between positive and negative energy parts of the zero energy state. M-matrices identifiable as hermitian square roots of density matrices. In this article it is shown that M-matrices form in a natural manner a generalization of Kac-Moody type algebra acting as symmetries of M-matrices and U-matrix and that the space of zero energy states has therefore Lie algebra structure so that quantum states act as their own symmetries. The generators of this algebra are multilocal with respect to partonic 2-surfaces just as Yangian algebras are multilocal with respect to points of Minkowski space and therefore define generalization of the Yangian algebra appearing in the Grassmannian twistor approach to N=4 SUSY.
Category: Quantum Gravity and String Theory