A continuation of the Born Reciprocal Relativity Theory (BRRT) program in phase space shows that ... more A continuation of the Born Reciprocal Relativity Theory (BRRT) program in phase space shows that a natural temperature-dependence of mass occurs after recurring to the Fulling-Davies-Unruh effect. The temperature dependence of the mass m(T) resemblances the energy-scale dependence of mass and other physical parameters in the renormalization (group) program of QFT. It is found in a special case that the effective photon mass is no longer zero, which may have far reaching consequences in the resolution of the dark matter problem. The Fulling-Davies-Unruh effect in a D = 1 + 1-dim spacetime is analyzed entirely from the perspective of BRRT, and we explain how it may be interpreted in terms of a linear superposition of an infinite number of states resulting from the action of the group U (1, 1) on the Lorentz non-invariant vacuum |\tilde 0⟩ of the relativistic oscillator studied by Bars [8].
Recently we have argued [1] that the noncommutativity of the spacetime coordinates is the answer ... more Recently we have argued [1] that the noncommutativity of the spacetime coordinates is the answer to the question : Why is area, mass, entropy quantized ? Furthermore, it casts light into a deep interplay among black hole entropy, discrete calculus, number theory, theory of partitions, random matrix theory, fuzzy spheres,. . .. We extend our previous construction of Schwarzschild black holes and derive the corrections to the Kerr-Newman temperature and black hole entropy, to all orders, from the discrete mass transitions taken place among different mass states. The mass spectrum for Kerr, Kerr-Newman, and Reissner-Nordstrom black holes is explicitly obtained which reduces to the Schwarzschild case when the angular momentum and charge is set to zero. One of the most salient features in the expansion of the modif ied temperature T = T + c_1 T/ N + c_2 T/N^2 +. .. is that it spells a correspondence between the loop expansion in QFT in powers of hbar, after setting hbar ↔ (1/N). N is the principal quantum number labeling the spectrum of mass states and which is given by N = l_3(l_3 + 2) − l_2(l_2 + 1) + l_1^2 , with l_3 ≥ l_2 ≥ |l_1| being the quantum numbers associated with the hyper-spherical harmonics of the three-sphere S^3. These results can be extended to higher dimensions. To finalize, we should add that the deviation from a full thermal spectrum and the corrections to the Hawking temperature might be relevant to the solution of the Black Hole Information paradox.
It is shown how a Noncommutative spacetime leads to an area, mass and entropy quantization condit... more It is shown how a Noncommutative spacetime leads to an area, mass and entropy quantization condition which allows to derive the Schwarzschild black hole entropy A/4G , the logarithmic corrections, and further corrections, from the discrete mass transitions taken place among different mass states in D = 4. The higher dimensional generalization of the results in D = 4 follow. The discretization of the entropy-mass relation S = S(M) leads to an entropy quantization of the form S = S(M_n) = n, and such that one may always assign n "bits" to the discrete entropy, and in doing so, make contact with quantum information. The physical applications of mass quantization, like the counting of states contributing to the black hole entropy, black hole evaporation, and the direct connection to the black holes-string correspondence [23] via the asymptotic behavior of the number of partitions of integers, follows. To conclude, it is shown how the recent large N Matrix model (fuzzy sphere) of [20] leads to very similar results for the black hole entropy as the physical model described in this work and which based on the discrete mass transitions originating from the noncommutativity of the spacetime coordinates.
We revisit the nonlinear Klein-Gordon-like equation that was proposed by us which capture how qua... more We revisit the nonlinear Klein-Gordon-like equation that was proposed by us which capture how quantum mechanical probability densities curve spacetime, and find an exact solution that may appear to be "trivial" but with important physical implications related to the physics of frozen stars and with Mach's principle. The nonlinear Klein-Gordon-like equation is essentially the static spherically symmetric relativistic analog of the Newton-Schrödinger equation. We finalize by studying the higher dimensional generalizations of the nonlinear Klein-Gordon-like equation and examine the relativistic Bohm-Poisson equation as yet another equation capturing the interplay between quantum mechanical probability densities and gravity.
A discrete Hopf fibration of S^15 over S^8 with S^7 (unit octonions) as fibers leads to a 16D Pol... more A discrete Hopf fibration of S^15 over S^8 with S^7 (unit octonions) as fibers leads to a 16D Polytope P_{16} with 4320 vertices obtained from the convex hull of the 16D Barnes-Wall lattice Lambda_{16}. It is argued how a subsequent 2 − 1 mapping (projection) of P_{16} onto a 8D-hyperplane might furnish the 2160 vertices of the uniform 2_{41} polytope in 8-dimensions, and such that one can capture the chain sequence of polytopes 2_{41}, 2_{31}, 2_{21}, 2_{11} in D = 8, 7, 6, 5 dimensions, leading, respectively, to the sequence of Coxeter groups E_8, E_7, E_6, SO(10) which are putative GUT group candidates. An embedding of the E8 ⊕ E8 and E8 ⊕ E8 ⊕ E8 lattice into the Barnes-Wall Λ_{16} and Leech Λ_{24} lattices, respectively, is explicitly shown. From the 16D lattice E8 ⊕ E8 one can generate two separate families of Elser-Sloane 4D quasicrystals (QC's) with H_4 (icosahedral) symmetry via the "cut-and-project" method from 8D to 4D in each separate E_8 lattice. Therefore, one obtains in this fashion the Cartesian product of two Elser-Sloane QC's Q × Q spanning an 8D space. Similarly, from the 24D lattice E8 ⊕E8 ⊕E8 one can generate the Cartesian product of three Elser-Sloane 4D quasicrystals (QC's) Q × Q × Q with H_4 symmetry and spanning a 12D space.
One of the consequences of Fermat's last theorem is the existence of a countable infinite number ... more One of the consequences of Fermat's last theorem is the existence of a countable infinite number of rational points on the unit circle, which allows in turn, to find the rational points on the unit sphere via the inverse stereographic projection of the homothecies of the rational points on the unit circle. We proceed to iterate this process and obtain the rational points on the unit S^3 via the inverse stereographic projection of the homothecies of the rational points on the previous unit S^2. One may continue this iteration/recursion process ad infinitum in order to find the rational points on unit hyper-spheres of arbitrary dimension S^4 , S^5 , • • • , S^N. As an example, it is shown how to obtain the rational points of the unit S^{24} that is associated with the Leech lattice. The physical applications of our construction follow and one finds a direct relation among the N + 1 quantum states of a spin-N/2 particle and the rational points of a unit S^N hyper-sphere embedded in a flat Euclidean R^{N +1} space.
Recently we have shown how the Schwarzschild Black Hole Entropy in all dimensions emerges from tr... more Recently we have shown how the Schwarzschild Black Hole Entropy in all dimensions emerges from truly point mass sources at r = 0 due to a non-vanishing scalar curvature R involving the Dirac delta distribution in the computation of the Euclidean Einstein-Hilbert action. As usual, it is required to take the inverse Hawking temperature β as the length of the circle S^1_β obtained from a compactification of the Euclidean time in thermal field theory which results after a Wick rotation, it = τ , to imaginary time. In this work we extend our novel procedure to evaluate both the Reissner-Nordstrom and Kerr-Newman black hole entropy from truly charge spinning point mass sources.
It is explicitly shown how the Schwarzschild Black Hole Entropy (in all dimensions) emerges from ... more It is explicitly shown how the Schwarzschild Black Hole Entropy (in all dimensions) emerges from truly point mass sources at r = 0 due to a non-vanishing scalar curvature involving the Dirac delta distribution. It is the density and anisotropic pressure components associated with the point mass delta function source at the origin r = 0 which furnish the Schwarzschild black hole entropy in all dimensions D ≥ 4 after evaluating the Euclidean Einstein-Hilbert action. As usual, it is required to take the inverse Hawking temperature β_H as the length of the circle obtained from a compactification of the Euclidean time in thermal field theory which results after a Wick rotation, it = τ , to imaginary time. The appealing and salient result is that there is no need to introduce the Gibbons-Hawking-York boundary term in order to arrive at the black hole entropy because in our case one has that R is not 0. Furthermore, there is no need to introduce a complex integration contour to avoid the singularity as shown by Gibbons and Hawking. On the contrary, the source of the black hole entropy stems entirely from the scalar curvature singularity at the origin r = 0. We conclude by explaining how to generalize our construction to the Kerr-Newman metric by exploiting the Newman-Janis algorithm. The physical implications of this finding warrants further investigation since it suggests a profound connection between the notion of gravitational entropy and spacetime singularities.
A brief introduction of the Extended Relativity Theory in Clifford Spaces (Cspace) paves the way ... more A brief introduction of the Extended Relativity Theory in Clifford Spaces (Cspace) paves the way to the explicit construction of the generalized relativistic transformations of the Clifford multivector-valued coordinates in C-spaces. The most general transformations furnish a full mixing of the grades of the multivectorvalued coordinates. The transformations of the multivector-valued momenta follow leading to an invariant generalized mass M in C-spaces which differs from m. No longer the proper mass appearing in the relativistic dispersion relation E^2 −p^2 = m^2 remains invariant under the generalized relativistic transformations. It is argued how this finding might shed some light into the cosmological constant problem, dark energy, and dark matter. We finalize with some concluding remarks about extending these transformations to phase spaces and about Born reciprocal relativity. An appendix is included with the most general (anti) commutators of the Clifford algebra multivector generators.
We explore the construction of a generalized Dirac equation via the introduction of the notion of... more We explore the construction of a generalized Dirac equation via the introduction of the notion of Clifford-valued actions, and which was inspired by the work of [1], [2] on the De Donder-Weyl theory formulation of field theory. Crucial in this construction is the evaluation of the exponentials of multivectors associated with Clifford (hypercomplex) analysis. Exact matrix solutions (instead of spinors) of the generalized Dirac equation in D = 2, 3 spacetime dimensions were found. This formalism can be extended to curved spacetime backgrounds like it happens with the Schroedinger-Dirac equation. We conclude by proposing a wavefunctional equation governing the quantum dynamics of branes living in C-spaces (Clifford spaces), and which is based on the De Donder-Weyl Hamiltonian formulation of field theory.
After a brief introduction of Born's reciprocal relativity theory is presented, we review the con... more After a brief introduction of Born's reciprocal relativity theory is presented, we review the construction of the deformed Quaplectic group that is given by the semi-direct product of U (1, 3) with the deformed (noncommutative) Weyl-Heisenberg group corresponding to noncommutative fiber coordinates and momenta [X_a, X _b ] not 0; [P_a, P_b ] not 0. This construction leads to more general algebras given by a two-parameter family of deformations of the Quaplectic algebra, and to further algebraic extensions involving antisymmetric tensor coordinates and momenta of higher ranks [X_a 1 a_2 •••a_n , X b_1 b_2 •••b_n ] not 0; [Pa_1 a_2 •••a_n , P b_1 b_2 •••b_n ] not 0. We continue by examining algebraic extensions of the Yang algebra in extended noncommutative phase spaces and compare them with the above extensions of the deformed Quaplectic algebra. A solution is found for the exact analytical mapping of the non-commuting x_µ , p_µ operator variables (associated to an 8D curved phase space) to the canonical Y^A , Π^A operator variables of a flat 12D phase space. We explore the geometrical implications of this mapping which provides, in the classical limit, with the embedding functions Y^A (x, p), Π^A (x, p) of an 8D curved phase space into a flat 12D phase space background. The latter embedding functions determine the functional forms of the base spacetime metric g_{µν} (x, p), the fiber metric of the vertical space h_{ab} (x, p), and the nonlinear connection N_{aµ}(x, p) associated with the 8D cotangent space of the 4D spacetime. Consequently, one has found a direct link between noncommutative curved phase spaces in lower dimensions to commutative flat phase spaces in higher dimensions.
We present the deep connections among (Anti) de Sitter geometry, complex conformal gravity-Maxwel... more We present the deep connections among (Anti) de Sitter geometry, complex conformal gravity-Maxwell theory, and grand unification, from a gauge theory of gravity based on the complex Clifford algebra Cl(4, C). Some desirable results are found, like a plausible cancellation mechanism of the cosmological constant involving an algebraic constraint between e^a_µ , b^a_µ (the real and imaginary parts of the complex vierbein).
We begin with a review of the basics of the Yang algebra of noncommutative phase spaces and Born ... more We begin with a review of the basics of the Yang algebra of noncommutative phase spaces and Born Reciprocal Relativity. A solution is provided for the exact analytical mapping of the non-commuting x^µ , p^µ operator variables (associated to an 8D curved phase space) to the canonical Y^A , Π^A operator variables in a flat 12D phase space. We explore the geometrical implications of this mapping which provides, in the classical limit, with the embedding functions Y^A (x, p), Π^A (x, p) of an 8D curved phase space into a flat 12D phase space background. The latter embedding functions determine the functional forms of the base spacetime metric g_{µν} (x, p), the fiber metric of the vertical space h^{ab} (x, p), and the nonlinear connection N_{aµ}(x, p) associated with the 8D cotangent space of the 4D spacetime. A review of the mathematical tools behind curved phase spaces, Lagrange-Finsler, and Hamilton-Cartan geometries follows. This is necessary in order to answer the key question of whether or not the solutions found for g_{µν} , h^{ab} , N_{aµ} as a result of the embedding, also solve the generalized gravitational vacuum field equations in the 8D cotangent space. We finalize with an Appendix with the key calculations involved in solving the exact analytical mapping of the x^µ , p^µ operator variables to the canonical Y^A , Π^A operator ones.
Starting with a brief review of our prior construction of n-ary algebras in noncommutative Cliffo... more Starting with a brief review of our prior construction of n-ary algebras in noncommutative Clifford spaces, we proceed to construct in full detail the Clifford-Yang algebra which is an extension of the Yang algebra in noncommutative phase spaces. The Clifford-Yang algebra allows to write down the commutators of the noncommutative polyvectorvalued coordinates and momenta and which are compatible with the Jacobi identities, the Weyl-Heisenberg algebra, and paves the way for a formulation of Quantum Mechanics in Noncommutative Clifford spaces. We continue with a detail study of the isotropic 3D quantum oscillator in noncommutative spaces and find the energy eigenvalues and eigenfunctions. These findings differ considerably from the ordinary quantum oscillator in commutative spaces. We find that QM in noncommutative spaces leads to very different solutions, eigenvalues, and uncertainty relations than ordinary QM in commutative spaces. The generalization of QM to noncommutative Clifford (phase) spaces is attained via the Clifford-Yang algebra. The operators are now given by the generalized angular momentum operators involving polyvector coordinates and momenta. The eigenfunctions (wave functions) are now more complicated functions of the polyvector coordinates. We conclude with some important remarks.
Starting with a brief review of our prior construction of n-ary algebras, based on the relation a... more Starting with a brief review of our prior construction of n-ary algebras, based on the relation among the n-ary commutators of noncommuting spacetime coordinates [X 1 , X 2 , ......, X n ] with the polyvector valued coordinates X 123...n in noncommutative Clifford spaces, [X 1 , X 2 , ......, X n ] = n! X 123...n , we proceed to construct generalized brane actions in noncommutative matrix coordinates backgrounds in Clifford-spaces (C-spaces). An instrumental role is played by the Clifford-valued scalar field Φ(σ A) which provides the functional form of the noncommutative matrix coordinates in C-space, X M ≡ Φ −1 (σ A)Γ M Φ(σ A), and that is given in terms of the world manifold's σ A polyvector-valued coordinates of the generalized brane, and which by construction, satisf y the n-ary algebra. We finalize with an extension of coherent states in C-spaces and provide a preliminary study of strings in target C-space backgrounds.
A brief review of the essentials of asymptotic safety and the renormalization group (RG) improvem... more A brief review of the essentials of asymptotic safety and the renormalization group (RG) improvement of the Schwarzschild black hole that removes the r = 0 singularity is presented. It is followed with a RG improvement of the Kantowski–Sachs metric associated with a Schwarzschild black hole interior such that there is no singularity at t = 0 due to the running Newtonian coupling G(t) vanishing at t = 0. Two temporal horizons at [Formula: see text] and [Formula: see text] are found. For times below the Planck scale t < t P, and above the Hubble time t > t H, the components of the Kantowski–Sachs metric exhibit a key sign change, so the roles of the spatial z and temporal t coordinates are exchanged, and one recovers a repulsive inflationary de Sitter-like core around z = 0, and a Schwarzschild-like metric in the exterior region z > R H = 2G o M. The inclusion of a running cosmological constant Λ(t) follows. We proceed with the study of a dilaton-gravity (scalar–tensor theory...
The $nonlinear$ Bohm-Poisson-Schroedinger equation is studied further. It has solutions leading t... more The $nonlinear$ Bohm-Poisson-Schroedinger equation is studied further. It has solutions leading to $repulsive$ gravitational behavior. An exact analytical expression for the observed vacuum energy density is obtained. Further results are provided which include two possible extensions of the Bohm-Poisson equation to the full relativistic regime. Two specific solutions to the novel Relativistic Bohm-Poisson equation (associated to a real scalar field) are provided encoding the repulsive nature of dark energy. One solution leads to an exact cancellation of the cosmological constant, but an expanding decelerating cosmos; while the other solution leads to an exponential accelerated cosmos consistent with a de Sitter phase, and whose extremely small cosmological constant is $ \Lambda = { 3 \over R_H^2}$, consistent with current observations. We conclude with some final remarks about Weyl's geometry.
A continuation of the Born Reciprocal Relativity Theory (BRRT) program in phase space shows that ... more A continuation of the Born Reciprocal Relativity Theory (BRRT) program in phase space shows that a natural temperature-dependence of mass occurs after recurring to the Fulling-Davies-Unruh effect. The temperature dependence of the mass m(T) resemblances the energy-scale dependence of mass and other physical parameters in the renormalization (group) program of QFT. It is found in a special case that the effective photon mass is no longer zero, which may have far reaching consequences in the resolution of the dark matter problem. The Fulling-Davies-Unruh effect in a D = 1 + 1-dim spacetime is analyzed entirely from the perspective of BRRT, and we explain how it may be interpreted in terms of a linear superposition of an infinite number of states resulting from the action of the group U (1, 1) on the Lorentz non-invariant vacuum |\tilde 0⟩ of the relativistic oscillator studied by Bars [8].
Recently we have argued [1] that the noncommutativity of the spacetime coordinates is the answer ... more Recently we have argued [1] that the noncommutativity of the spacetime coordinates is the answer to the question : Why is area, mass, entropy quantized ? Furthermore, it casts light into a deep interplay among black hole entropy, discrete calculus, number theory, theory of partitions, random matrix theory, fuzzy spheres,. . .. We extend our previous construction of Schwarzschild black holes and derive the corrections to the Kerr-Newman temperature and black hole entropy, to all orders, from the discrete mass transitions taken place among different mass states. The mass spectrum for Kerr, Kerr-Newman, and Reissner-Nordstrom black holes is explicitly obtained which reduces to the Schwarzschild case when the angular momentum and charge is set to zero. One of the most salient features in the expansion of the modif ied temperature T = T + c_1 T/ N + c_2 T/N^2 +. .. is that it spells a correspondence between the loop expansion in QFT in powers of hbar, after setting hbar ↔ (1/N). N is the principal quantum number labeling the spectrum of mass states and which is given by N = l_3(l_3 + 2) − l_2(l_2 + 1) + l_1^2 , with l_3 ≥ l_2 ≥ |l_1| being the quantum numbers associated with the hyper-spherical harmonics of the three-sphere S^3. These results can be extended to higher dimensions. To finalize, we should add that the deviation from a full thermal spectrum and the corrections to the Hawking temperature might be relevant to the solution of the Black Hole Information paradox.
It is shown how a Noncommutative spacetime leads to an area, mass and entropy quantization condit... more It is shown how a Noncommutative spacetime leads to an area, mass and entropy quantization condition which allows to derive the Schwarzschild black hole entropy A/4G , the logarithmic corrections, and further corrections, from the discrete mass transitions taken place among different mass states in D = 4. The higher dimensional generalization of the results in D = 4 follow. The discretization of the entropy-mass relation S = S(M) leads to an entropy quantization of the form S = S(M_n) = n, and such that one may always assign n "bits" to the discrete entropy, and in doing so, make contact with quantum information. The physical applications of mass quantization, like the counting of states contributing to the black hole entropy, black hole evaporation, and the direct connection to the black holes-string correspondence [23] via the asymptotic behavior of the number of partitions of integers, follows. To conclude, it is shown how the recent large N Matrix model (fuzzy sphere) of [20] leads to very similar results for the black hole entropy as the physical model described in this work and which based on the discrete mass transitions originating from the noncommutativity of the spacetime coordinates.
We revisit the nonlinear Klein-Gordon-like equation that was proposed by us which capture how qua... more We revisit the nonlinear Klein-Gordon-like equation that was proposed by us which capture how quantum mechanical probability densities curve spacetime, and find an exact solution that may appear to be "trivial" but with important physical implications related to the physics of frozen stars and with Mach's principle. The nonlinear Klein-Gordon-like equation is essentially the static spherically symmetric relativistic analog of the Newton-Schrödinger equation. We finalize by studying the higher dimensional generalizations of the nonlinear Klein-Gordon-like equation and examine the relativistic Bohm-Poisson equation as yet another equation capturing the interplay between quantum mechanical probability densities and gravity.
A discrete Hopf fibration of S^15 over S^8 with S^7 (unit octonions) as fibers leads to a 16D Pol... more A discrete Hopf fibration of S^15 over S^8 with S^7 (unit octonions) as fibers leads to a 16D Polytope P_{16} with 4320 vertices obtained from the convex hull of the 16D Barnes-Wall lattice Lambda_{16}. It is argued how a subsequent 2 − 1 mapping (projection) of P_{16} onto a 8D-hyperplane might furnish the 2160 vertices of the uniform 2_{41} polytope in 8-dimensions, and such that one can capture the chain sequence of polytopes 2_{41}, 2_{31}, 2_{21}, 2_{11} in D = 8, 7, 6, 5 dimensions, leading, respectively, to the sequence of Coxeter groups E_8, E_7, E_6, SO(10) which are putative GUT group candidates. An embedding of the E8 ⊕ E8 and E8 ⊕ E8 ⊕ E8 lattice into the Barnes-Wall Λ_{16} and Leech Λ_{24} lattices, respectively, is explicitly shown. From the 16D lattice E8 ⊕ E8 one can generate two separate families of Elser-Sloane 4D quasicrystals (QC's) with H_4 (icosahedral) symmetry via the "cut-and-project" method from 8D to 4D in each separate E_8 lattice. Therefore, one obtains in this fashion the Cartesian product of two Elser-Sloane QC's Q × Q spanning an 8D space. Similarly, from the 24D lattice E8 ⊕E8 ⊕E8 one can generate the Cartesian product of three Elser-Sloane 4D quasicrystals (QC's) Q × Q × Q with H_4 symmetry and spanning a 12D space.
One of the consequences of Fermat's last theorem is the existence of a countable infinite number ... more One of the consequences of Fermat's last theorem is the existence of a countable infinite number of rational points on the unit circle, which allows in turn, to find the rational points on the unit sphere via the inverse stereographic projection of the homothecies of the rational points on the unit circle. We proceed to iterate this process and obtain the rational points on the unit S^3 via the inverse stereographic projection of the homothecies of the rational points on the previous unit S^2. One may continue this iteration/recursion process ad infinitum in order to find the rational points on unit hyper-spheres of arbitrary dimension S^4 , S^5 , • • • , S^N. As an example, it is shown how to obtain the rational points of the unit S^{24} that is associated with the Leech lattice. The physical applications of our construction follow and one finds a direct relation among the N + 1 quantum states of a spin-N/2 particle and the rational points of a unit S^N hyper-sphere embedded in a flat Euclidean R^{N +1} space.
Recently we have shown how the Schwarzschild Black Hole Entropy in all dimensions emerges from tr... more Recently we have shown how the Schwarzschild Black Hole Entropy in all dimensions emerges from truly point mass sources at r = 0 due to a non-vanishing scalar curvature R involving the Dirac delta distribution in the computation of the Euclidean Einstein-Hilbert action. As usual, it is required to take the inverse Hawking temperature β as the length of the circle S^1_β obtained from a compactification of the Euclidean time in thermal field theory which results after a Wick rotation, it = τ , to imaginary time. In this work we extend our novel procedure to evaluate both the Reissner-Nordstrom and Kerr-Newman black hole entropy from truly charge spinning point mass sources.
It is explicitly shown how the Schwarzschild Black Hole Entropy (in all dimensions) emerges from ... more It is explicitly shown how the Schwarzschild Black Hole Entropy (in all dimensions) emerges from truly point mass sources at r = 0 due to a non-vanishing scalar curvature involving the Dirac delta distribution. It is the density and anisotropic pressure components associated with the point mass delta function source at the origin r = 0 which furnish the Schwarzschild black hole entropy in all dimensions D ≥ 4 after evaluating the Euclidean Einstein-Hilbert action. As usual, it is required to take the inverse Hawking temperature β_H as the length of the circle obtained from a compactification of the Euclidean time in thermal field theory which results after a Wick rotation, it = τ , to imaginary time. The appealing and salient result is that there is no need to introduce the Gibbons-Hawking-York boundary term in order to arrive at the black hole entropy because in our case one has that R is not 0. Furthermore, there is no need to introduce a complex integration contour to avoid the singularity as shown by Gibbons and Hawking. On the contrary, the source of the black hole entropy stems entirely from the scalar curvature singularity at the origin r = 0. We conclude by explaining how to generalize our construction to the Kerr-Newman metric by exploiting the Newman-Janis algorithm. The physical implications of this finding warrants further investigation since it suggests a profound connection between the notion of gravitational entropy and spacetime singularities.
A brief introduction of the Extended Relativity Theory in Clifford Spaces (Cspace) paves the way ... more A brief introduction of the Extended Relativity Theory in Clifford Spaces (Cspace) paves the way to the explicit construction of the generalized relativistic transformations of the Clifford multivector-valued coordinates in C-spaces. The most general transformations furnish a full mixing of the grades of the multivectorvalued coordinates. The transformations of the multivector-valued momenta follow leading to an invariant generalized mass M in C-spaces which differs from m. No longer the proper mass appearing in the relativistic dispersion relation E^2 −p^2 = m^2 remains invariant under the generalized relativistic transformations. It is argued how this finding might shed some light into the cosmological constant problem, dark energy, and dark matter. We finalize with some concluding remarks about extending these transformations to phase spaces and about Born reciprocal relativity. An appendix is included with the most general (anti) commutators of the Clifford algebra multivector generators.
We explore the construction of a generalized Dirac equation via the introduction of the notion of... more We explore the construction of a generalized Dirac equation via the introduction of the notion of Clifford-valued actions, and which was inspired by the work of [1], [2] on the De Donder-Weyl theory formulation of field theory. Crucial in this construction is the evaluation of the exponentials of multivectors associated with Clifford (hypercomplex) analysis. Exact matrix solutions (instead of spinors) of the generalized Dirac equation in D = 2, 3 spacetime dimensions were found. This formalism can be extended to curved spacetime backgrounds like it happens with the Schroedinger-Dirac equation. We conclude by proposing a wavefunctional equation governing the quantum dynamics of branes living in C-spaces (Clifford spaces), and which is based on the De Donder-Weyl Hamiltonian formulation of field theory.
After a brief introduction of Born's reciprocal relativity theory is presented, we review the con... more After a brief introduction of Born's reciprocal relativity theory is presented, we review the construction of the deformed Quaplectic group that is given by the semi-direct product of U (1, 3) with the deformed (noncommutative) Weyl-Heisenberg group corresponding to noncommutative fiber coordinates and momenta [X_a, X _b ] not 0; [P_a, P_b ] not 0. This construction leads to more general algebras given by a two-parameter family of deformations of the Quaplectic algebra, and to further algebraic extensions involving antisymmetric tensor coordinates and momenta of higher ranks [X_a 1 a_2 •••a_n , X b_1 b_2 •••b_n ] not 0; [Pa_1 a_2 •••a_n , P b_1 b_2 •••b_n ] not 0. We continue by examining algebraic extensions of the Yang algebra in extended noncommutative phase spaces and compare them with the above extensions of the deformed Quaplectic algebra. A solution is found for the exact analytical mapping of the non-commuting x_µ , p_µ operator variables (associated to an 8D curved phase space) to the canonical Y^A , Π^A operator variables of a flat 12D phase space. We explore the geometrical implications of this mapping which provides, in the classical limit, with the embedding functions Y^A (x, p), Π^A (x, p) of an 8D curved phase space into a flat 12D phase space background. The latter embedding functions determine the functional forms of the base spacetime metric g_{µν} (x, p), the fiber metric of the vertical space h_{ab} (x, p), and the nonlinear connection N_{aµ}(x, p) associated with the 8D cotangent space of the 4D spacetime. Consequently, one has found a direct link between noncommutative curved phase spaces in lower dimensions to commutative flat phase spaces in higher dimensions.
We present the deep connections among (Anti) de Sitter geometry, complex conformal gravity-Maxwel... more We present the deep connections among (Anti) de Sitter geometry, complex conformal gravity-Maxwell theory, and grand unification, from a gauge theory of gravity based on the complex Clifford algebra Cl(4, C). Some desirable results are found, like a plausible cancellation mechanism of the cosmological constant involving an algebraic constraint between e^a_µ , b^a_µ (the real and imaginary parts of the complex vierbein).
We begin with a review of the basics of the Yang algebra of noncommutative phase spaces and Born ... more We begin with a review of the basics of the Yang algebra of noncommutative phase spaces and Born Reciprocal Relativity. A solution is provided for the exact analytical mapping of the non-commuting x^µ , p^µ operator variables (associated to an 8D curved phase space) to the canonical Y^A , Π^A operator variables in a flat 12D phase space. We explore the geometrical implications of this mapping which provides, in the classical limit, with the embedding functions Y^A (x, p), Π^A (x, p) of an 8D curved phase space into a flat 12D phase space background. The latter embedding functions determine the functional forms of the base spacetime metric g_{µν} (x, p), the fiber metric of the vertical space h^{ab} (x, p), and the nonlinear connection N_{aµ}(x, p) associated with the 8D cotangent space of the 4D spacetime. A review of the mathematical tools behind curved phase spaces, Lagrange-Finsler, and Hamilton-Cartan geometries follows. This is necessary in order to answer the key question of whether or not the solutions found for g_{µν} , h^{ab} , N_{aµ} as a result of the embedding, also solve the generalized gravitational vacuum field equations in the 8D cotangent space. We finalize with an Appendix with the key calculations involved in solving the exact analytical mapping of the x^µ , p^µ operator variables to the canonical Y^A , Π^A operator ones.
Starting with a brief review of our prior construction of n-ary algebras in noncommutative Cliffo... more Starting with a brief review of our prior construction of n-ary algebras in noncommutative Clifford spaces, we proceed to construct in full detail the Clifford-Yang algebra which is an extension of the Yang algebra in noncommutative phase spaces. The Clifford-Yang algebra allows to write down the commutators of the noncommutative polyvectorvalued coordinates and momenta and which are compatible with the Jacobi identities, the Weyl-Heisenberg algebra, and paves the way for a formulation of Quantum Mechanics in Noncommutative Clifford spaces. We continue with a detail study of the isotropic 3D quantum oscillator in noncommutative spaces and find the energy eigenvalues and eigenfunctions. These findings differ considerably from the ordinary quantum oscillator in commutative spaces. We find that QM in noncommutative spaces leads to very different solutions, eigenvalues, and uncertainty relations than ordinary QM in commutative spaces. The generalization of QM to noncommutative Clifford (phase) spaces is attained via the Clifford-Yang algebra. The operators are now given by the generalized angular momentum operators involving polyvector coordinates and momenta. The eigenfunctions (wave functions) are now more complicated functions of the polyvector coordinates. We conclude with some important remarks.
Starting with a brief review of our prior construction of n-ary algebras, based on the relation a... more Starting with a brief review of our prior construction of n-ary algebras, based on the relation among the n-ary commutators of noncommuting spacetime coordinates [X 1 , X 2 , ......, X n ] with the polyvector valued coordinates X 123...n in noncommutative Clifford spaces, [X 1 , X 2 , ......, X n ] = n! X 123...n , we proceed to construct generalized brane actions in noncommutative matrix coordinates backgrounds in Clifford-spaces (C-spaces). An instrumental role is played by the Clifford-valued scalar field Φ(σ A) which provides the functional form of the noncommutative matrix coordinates in C-space, X M ≡ Φ −1 (σ A)Γ M Φ(σ A), and that is given in terms of the world manifold's σ A polyvector-valued coordinates of the generalized brane, and which by construction, satisf y the n-ary algebra. We finalize with an extension of coherent states in C-spaces and provide a preliminary study of strings in target C-space backgrounds.
A brief review of the essentials of asymptotic safety and the renormalization group (RG) improvem... more A brief review of the essentials of asymptotic safety and the renormalization group (RG) improvement of the Schwarzschild black hole that removes the r = 0 singularity is presented. It is followed with a RG improvement of the Kantowski–Sachs metric associated with a Schwarzschild black hole interior such that there is no singularity at t = 0 due to the running Newtonian coupling G(t) vanishing at t = 0. Two temporal horizons at [Formula: see text] and [Formula: see text] are found. For times below the Planck scale t < t P, and above the Hubble time t > t H, the components of the Kantowski–Sachs metric exhibit a key sign change, so the roles of the spatial z and temporal t coordinates are exchanged, and one recovers a repulsive inflationary de Sitter-like core around z = 0, and a Schwarzschild-like metric in the exterior region z > R H = 2G o M. The inclusion of a running cosmological constant Λ(t) follows. We proceed with the study of a dilaton-gravity (scalar–tensor theory...
The $nonlinear$ Bohm-Poisson-Schroedinger equation is studied further. It has solutions leading t... more The $nonlinear$ Bohm-Poisson-Schroedinger equation is studied further. It has solutions leading to $repulsive$ gravitational behavior. An exact analytical expression for the observed vacuum energy density is obtained. Further results are provided which include two possible extensions of the Bohm-Poisson equation to the full relativistic regime. Two specific solutions to the novel Relativistic Bohm-Poisson equation (associated to a real scalar field) are provided encoding the repulsive nature of dark energy. One solution leads to an exact cancellation of the cosmological constant, but an expanding decelerating cosmos; while the other solution leads to an exponential accelerated cosmos consistent with a de Sitter phase, and whose extremely small cosmological constant is $ \Lambda = { 3 \over R_H^2}$, consistent with current observations. We conclude with some final remarks about Weyl's geometry.
Inspired by the Hilbert-Polya proposal to prove the Riemann Hypothesis we have studied the Schroe... more Inspired by the Hilbert-Polya proposal to prove the Riemann Hypothesis we have studied the Schroedinger QM equation involving a highly non-trivial potential, and whose self-adjoint Hamiltonian operator has for its energy spectrum one which approaches the imaginary parts of the zeta zeroes only in the asymptotic (very large N) region. The ordinates λn are the positive imaginary parts of the nontrivial zeta zeros in the critical line : sn = 1 2 + iλn. The latter results are consistent with the validity of the Bohr-Sommerfeld semi-classical quantization condition. It is shown how one may modify the parameters which define the potential, and fine tune its values, such that the energy spectrum of the (modified) Hamiltonian matches not only the first two zeroes but the other consecutive zeroes. The highly non-trivial functional form of the potential is found via the Bohr-Sommerfeld quantization formula using the full-fledged Riemann-von Mangoldt counting formula (without any truncations) for the number N (E) of zeroes in the critical strip with imaginary part greater than 0 and less than or equal to E.
Inspired by the Hilbert-Polya proposal to prove the Riemann Hypothesis we have studied the Schroe... more Inspired by the Hilbert-Polya proposal to prove the Riemann Hypothesis we have studied the Schroedinger QM equation involving a highly non-trivial potential, and whose self-adjoint Hamiltonian operator has for its energy spectrum one which approaches the imaginary parts of the zeta zeroes only in the asymptotic (very large N) region. The ordinates λ_n are the positive imaginary parts of the nontrivial zeta zeros in the critical line : s_n = 1 2 + iλ_n. The latter results are consistent with the validity of the Bohr-Sommerfeld semi-classical quantization condition. It is shown how one may modify the parameters which define the potential, and fine tune its values, such that the energy spectrum of the (modified) Hamiltonian matches not only the first two zeroes but the other consecutive zeroes. The highly non-trivial functional form of the potential is found via the Bohr-Sommerfeld quantization formula using the full-fledged Riemann-von Mangoldt counting formula (without any truncations) for the number N (E) of zeroes in the critical strip with imaginary part greater than 0 and less than or equal to E.
Uploads
Papers by Carlos Castro Perelman
b^a_µ (the real and imaginary parts of the complex vierbein).
b^a_µ (the real and imaginary parts of the complex vierbein).