Posts Tagged planck length

Recent Postings from planck length

Density matrix of radiation of black hole with fluctuating horizon [Replacement]

The density matrix of Hawking radiation is calculated in the model of black hole with fluctuating horizon. Quantum fluctuations smear the classical horizon of black hole and modify the density matrix of radiation producing the off-diagonal elements. The off-diagonal elements may store information of correlations between radiation and black hole. The smeared density matrix was constructed by convolution of the density matrix calculated with the instantaneous horizon with the Gaussian distribution over the instantaneous horizons. The distribution has the extremum at the classical radius of the black hole and the width of order of the Planck length. Calculations were performed in the model of black hole formed by the thin collapsing shell which follows a trajectory which is a solution of the matching equations connecting the interior and exterior geometries.

Density matrix of radiation of black hole with fluctuating horizon [Replacement]

The density matrix of Hawking radiation is calculated in the model of black hole with fluctuating horizon. Quantum fluctuations smear the classical horizon of black hole and modify the density matrix of radiation producing the off-diagonal elements. The off-diagonal elements may store information of correlations between radiation and black hole. The smeared density matrix was constructed by convolution of the density matrix calculated with the instantaneous horizon with the Gaussian distribution over the instantaneous horizons. The distribution has the extremum at the classical radius of the black hole and the width of order of the Planck length. Calculations were performed in the model of black hole formed by the thin collapsing shell which follows a trajectory which is a solution of the matching equations connecting the interior and exterior geometries.

Density matrix of radiation of black hole with fluctuating horizon [Replacement]

The density matrix of Hawking radiation is calculated in the model of black hole with fluctuating horizon. Quantum fluctuations smear the classical horizon of black hole and modify the density matrix of radiation producing the off-diagonal elements. The off-diagonal elements may store information of correlations between radiation and black hole. The smeared density matrix was constructed by convolution of the density matrix calculated with the instantaneous horizon with the Gaussian distribution over the instantaneous horizons. The distribution has the extremum at the classical radius of the black hole and the width of order of the Planck length. Calculations were performed in the model of black hole formed by the thin collapsing shell which follows a trajectory which is a solution of the matching equations connecting the interior and exterior geometries.

Density matrix of radiation of black hole with fluctuating horizon [Replacement]

The density matrix of Hawking radiation is calculated in the model of black hole with fluctuating horizon. Quantum fluctuations smear the classical horizon of black hole and modify the density matrix of radiation producing the off-diagonal elements. The off-diagonal elements may store information of correlations between radiation and black hole. The smeared density matrix was constructed by convolution of the density matrix calculated with the instantaneous horizon with the Gaussian distribution over the instantaneous horizons. The distribution has the extremum at the classical radius of the black hole and the width of order of the Planck length. Calculations were performed in the model of black hole formed by the thin collapsing shell which follows a trajectory which is a solution of the matching equations connecting the interior and exterior geometries.

Density matrix of radiation of black hole with fluctuating horizon

The density matrix of Hawking radiation is calculated in the model of black hole with fluctuating horizon. Quantum fluctuations smear the classical horizon of black hole and modify the density matrix of radiation producing the off-diagonal elements. The off-diagonal elements may store information of correlations between radiation and black hole. The smeared density matrix was constructed by convolution of the density matrix calculated with the instantaneous horizon with the Gaussian distribution over the instantaneous horizons. The distribution has the extremum at the classical radius of the black hole and the width of order of the Planck length. Calculations were performed in the model of black hole formed by the thin collapsing shell which follows a trajectory which is a solution of the matching equations connecting the interior and exterior geometries.

Density matrix of radiation of black hole with fluctuating horizon [Cross-Listing]

The density matrix of Hawking radiation is calculated in the model of black hole with fluctuating horizon. Quantum fluctuations smear the classical horizon of black hole and modify the density matrix of radiation producing the off-diagonal elements. The off-diagonal elements may store information of correlations between radiation and black hole. The smeared density matrix was constructed by convolution of the density matrix calculated with the instantaneous horizon with the Gaussian distribution over the instantaneous horizons. The distribution has the extremum at the classical radius of the black hole and the width of order of the Planck length. Calculations were performed in the model of black hole formed by the thin collapsing shell which follows a trajectory which is a solution of the matching equations connecting the interior and exterior geometries.

Dirac Equation and Planck-Scale Quantities

This work investigates in which form quantities with Planck dimensions occur already in the common quantum theory with local Lorentz symmetry. Since such Planck quantities as Planck length or Planck mass involve the Planck constant h, the velocity of light c and the Newton gravitational constant G, the relativistic Dirac equation (h, c) in the Newtonian gravitational potential (G) can be considered as a test theory. The evaluation of the break-off condition of the power series of the radial energy eigenfunctions of a purely gravitational atom leads to exact terms for the energy eigenvalues E for various special cases of the quantum numbers N, k and n = N + |k|. It turns out that a meaningful atom model, based solely on Newtonian gravitational forces, can result if, inter alia, the test mass m in the gravitational field of the mass M is selected to be smaller than the Planck mass.

Dirac Equation and Planck-Scale Quantities [Cross-Listing]

This work investigates in which form quantities with Planck dimensions occur already in the common quantum theory with local Lorentz symmetry. Since such Planck quantities as Planck length or Planck mass involve the Planck constant h, the velocity of light c and the Newton gravitational constant G, the relativistic Dirac equation (h, c) in the Newtonian gravitational potential (G) can be considered as a test theory. The evaluation of the break-off condition of the power series of the radial energy eigenfunctions of a purely gravitational atom leads to exact terms for the energy eigenvalues E for various special cases of the quantum numbers N, k and n = N + |k|. It turns out that a meaningful atom model, based solely on Newtonian gravitational forces, can result if, inter alia, the test mass m in the gravitational field of the mass M is selected to be smaller than the Planck mass.

Bounds on quantum gravity parameter from the $SU(2)$ NJL effective model of QCD

Existence of a minimal measurable length, as an effective cutoff in the ultraviolet regime, is a common feature of all approaches to the quantum gravity proposal. It is widely believed that this length scale will be of the order of the Planck length $\lambda=\lambda_0\,l_{_{\rm Pl}}$, where $\lambda_0\sim{\mathcal O}(1)$ is a dimensionless parameter that should be fixed only by the experiments. This issue can be taken into account through the deformed momentum spaces with compact topologies. In this paper, we consider minimum length effects on the physical quantities related to three parameters of the $SU(2)$ Nambu-Jona-Lasinio effective model of QCD by means of the deformed measure which is defined on compact momentum space with ${\mathbf S}^3$ topology. This measure is suggested by the doubly special relativity theories, Snyder deformed spaces, and the deformed algebra that is obtained in the light of the stability theory of Lie algebras. Using the current experimental data of the particle physics collaboration, we constraint quantum gravity parameter $\lambda_0$ and we compare our results with bounds that are arisen from the other experimental setups.

Kinematics of particles with quantum de Sitter symmetries

We present the first detailed study of the kinematics of free relativistic particles whose symmetries are described by a quantum deformation of the de Sitter algebra, known as $q$-de Sitter Hopf algebra. The quantum deformation parameter is a function of the Planck length $\ell$ and the de Sitter radius $H^{-1}$, such that when the Planck length vanishes, the algebra reduces to the de Sitter algebra, while when the de Sitter radius is sent to infinity one recovers the $\kappa$-Poincar\'e Hopf algebra. In the first limit the picture is that of a particle with trivial momentum space geometry moving on de Sitter spacetime, in the second one the picture is that of a particle with de Sitter momentum space geometry moving on Minkowski spacetime. When both the Planck length and the inverse of the de Sitter radius are non-zero, effects due to spacetime curvature and non-trivial momentum space geometry are both present and affect each other. The particles' motion is then described in a full phase space picture. We find that redshift effects that are usually associated to spacetime curvature become energy-dependent. Also, the energy dependence of particles' travel times that is usually associated to momentum space non-trivial properties is modified in a curvature-dependent way.

String theory to the rescue

The search for a theory of quantum gravity faces two great challenges: the incredibly small scales of the Planck length and time, and the possibility that the observed constants of nature are in part the result of random processes. A priori, one might have expected these to be insuperable obstacles. However, clues from observed physics, and the discovery of string theory, raise the hope that the unification of quantum mechanics and general relativity is within reach.

String theory to the rescue [Replacement]

The search for a theory of quantum gravity faces two great challenges: the incredibly small scales of the Planck length and time, and the possibility that the observed constants of nature are in part the result of random processes. A priori, one might have expected these to be insuperable obstacles. However, clues from observed physics, and the discovery of string theory, raise the hope that the unification of quantum mechanics and general relativity is within reach.

String theory to the rescue [Replacement]

The search for a theory of quantum gravity faces two great challenges: the incredibly small scales of the Planck length and time, and the possibility that the observed constants of nature are in part the result of random processes. A priori, one might have expected these to be insuperable obstacles. However, clues from observed physics, and the discovery of string theory, raise the hope that the unification of quantum mechanics and general relativity is within reach.

String theory to the rescue [Replacement]

The search for a theory of quantum gravity faces two great challenges: the incredibly small scales of the Planck length and time, and the possibility that the observed constants of nature are in part the result of random processes. A priori, one might have expected these to be insuperable obstacles. However, clues from observed physics, and the discovery of string theory, raise the hope that the unification of quantum mechanics and general relativity is within reach.

String theory to the rescue [Replacement]

The search for a theory of quantum gravity faces two great challenges: the incredibly small scales of the Planck length and time, and the possibility that the observed constants of nature are in part the result of random processes. A priori, one might have expected these to be insuperable obstacles. However, clues from observed physics, and the discovery of string theory, raise the hope that the unification of quantum mechanics and general relativity is within reach.

Hawking Effect as Quantum Inertial Effect

We show that "particle production" by gravitational field, especially the Hawking effect, may be treated as some quantum inertial effect, with the energy of Hawking radiation as some vacuum energy shift. This quantum inertial effect is mainly resulted from some intrinsical energy fluctuation $\hbar\kappa/c$ for a black hole. In particular, there is an extreme case in which $\hbar\kappa/c$ is the Planck energy, giving a "Planck black hole" whose event horizon's diameter is one Planck length. Moreover, we also provide a possibility to obtain some positive cosmological constant for an expanding universe, which is induced from the vacuum energy shift caused by quantum inertial effect.

Exotic Rotational Correlations in Emergent Quantum Geometry [Replacement]

Estimates are presented of exotic rotational correlations that arise if space emerges from Planck scale quantum elements with no fixed classical background. Directions in the classical inertial frame at separation $R$ from any world line coherently fluctuate on a timescale $R/c$ and angular scale $(R/\ell_P)^{-1/2}$, where $\ell_P$ denotes the Planck length. The projection of the exotic correlations onto an interferometer signal correlation function is predicted exactly, for a light path of arbitrary shape: the signal variance is equal to twice the enclosed area divided by the perimeter or free spectral range, in Planck units. An experiment concept is sketched, based on reconfiguration of the Fermilab Holometer into an area-enclosing light path. It is conjectured that exotic rotational correlations, entangled with the Standard Model field vacuum, could account for the value of the cosmological constant.

Exotic Rotational Correlations in Quantum Geometry [Replacement]

Estimates are presented of exotic quantum correlations of rotational motion on large scales that could emerge from quantum geometry at the Planck length $\ell_P$. Extrapolations of standard quantum theory and gravity suggest that the inertial frame within separation $R$ from any world line coherently fluctuates on a timescale $R/c$ with angular variance $\approx \ell_P/R$. The projection of the exotic correlations onto an interferometer signal correlation function is proportional to the area encompassed by the path of light in an interferometer arm. It is conjectured that exotic rotational correlations could resolve conflicts of field theory with gravity on large scales, and that entanglement with the Standard Model field vacuum might account for the value of the cosmological constant.

Exotic Rotational Correlations in Quantum Geometry [Replacement]

Estimates are presented of exotic quantum correlations of rotational motion on large scales that could emerge from quantum geometry at the Planck length $\ell_P$. Extrapolations of standard quantum theory and gravity suggest that the inertial frame within separation $R$ from any world line coherently fluctuates on a timescale $R/c$ with angular variance $\approx \ell_P/R$. The projection of the exotic correlations onto an interferometer signal correlation function is proportional to the area encompassed by the path of light in an interferometer arm. It is conjectured that exotic rotational correlations could resolve conflicts of field theory with gravity on large scales, and that entanglement with the Standard Model field vacuum might account for the value of the cosmological constant.

Exotic Rotational Correlations in Quantum Geometry [Replacement]

Estimates are presented of exotic correlations of rotation on large scales that could emerge from effects of quantum geometry at the Planck length $\ell_P$. Extrapolation of standard quantum theory and gravity suggests that a constant direction in the inertial frame at separation $R$ from any world line in nearly-flat space-time fluctuates relative to distant space at an angular rate $\omega$ with vanishing mean, but with an exotic variance $ \langle\omega^2 \rangle \approx c^2 \ell_PR^{-3}$ on a timescale $R/c$. A semiclassical statistical model is proposed, based on random quantum twists of space on causal boundaries of about one transverse Planck length every Planck time. Projection of exotic correlation onto an interferometer signal correlation function is estimated, and shown to vanish unless the light path sweeps out a nonzero spatial area. It is conjectured that exotic rotational correlations could resolve conflicts of field theory with gravity on large scales, and that entanglement with the Standard Model field vacuum might account for the value of the cosmological constant.

Exotic Rotational Correlations in Emergent Quantum Geometry [Replacement]

Estimates are presented of exotic rotational correlations that arise if space emerges from Planck scale quantum elements with no fixed classical background. Directions in the classical inertial frame at separation $R$ from any world line coherently fluctuate on a timescale $R/c$ and angular scale $(R/\ell_P)^{-1/2}$, where $\ell_P$ denotes the Planck length. The projection of the exotic correlations onto an interferometer signal correlation function is predicted exactly, for a light path of arbitrary shape: the signal variance is equal to twice the enclosed area divided by the perimeter or free spectral range, in Planck units. An experiment concept is sketched, based on reconfiguration of the Fermilab Holometer into an area-enclosing light path. It is conjectured that exotic rotational correlations, entangled with the Standard Model field vacuum, could account for the value of the cosmological constant.

Exotic Rotational Correlations in Quantum Geometry [Replacement]

Estimates are presented of exotic quantum correlations of rotational motion on large scales that could emerge from quantum geometry at the Planck length $\ell_P$. Extrapolations of standard quantum theory and gravity suggest that the inertial frame within separation $R$ from any world line fluctuates on a timescale $R/c$ with angular variance $\approx \ell_P/R$. A specific model of quantum twists on causal boundaries is presented. The projection of the exotic correlations onto an interferometer signal correlation function is estimated, and shown to vanish unless the light path sweeps out a nonzero spatial area. It is conjectured that exotic rotational correlations could resolve conflicts of field theory with gravity on large scales, and that entanglement with the Standard Model field vacuum might account for the value of the cosmological constant.

Exotic Rotational Correlations in Quantum Geometry [Replacement]

Estimates are presented of exotic correlations of rotation on large scales that could emerge from effects of quantum geometry at the Planck length $\ell_P$. Extrapolation of standard quantum theory and gravity suggests that a constant direction in the inertial frame at separation $R$ from any world line in nearly-flat space-time fluctuates relative to distant space at an angular rate $\omega$ with vanishing mean, but with an exotic variance $ \langle\omega^2 \rangle \approx c^2 \ell_PR^{-3}$ on a timescale $R/c$. A semiclassical statistical model is proposed, based on random quantum twists of space on causal boundaries of about one transverse Planck length every Planck time. Projection of exotic correlation onto an interferometer signal correlation function is estimated, and shown to vanish unless the light path sweeps out a nonzero spatial area. It is conjectured that exotic rotational correlations could resolve conflicts of field theory with gravity on large scales, and that entanglement with the Standard Model field vacuum might account for the value of the cosmological constant.

Exotic Rotational Correlations in Emergent Quantum Geometry [Replacement]

Estimates are presented of exotic rotational correlations that arise if space emerges from Planck scale quantum elements with no fixed classical background. Directions in the classical inertial frame at separation $R$ from any world line coherently fluctuate on a timescale $R/c$ and angular scale $(R/\ell_P)^{-1/2}$, where $\ell_P$ denotes the Planck length. The projection of the exotic correlations onto an interferometer signal correlation function is predicted exactly, for a light path of arbitrary shape: the signal variance is equal to twice the enclosed area divided by the perimeter or free spectral range, in Planck units. An experiment concept is sketched, based on reconfiguration of the Fermilab Holometer into an area-enclosing light path. It is conjectured that exotic rotational correlations, entangled with the Standard Model field vacuum, could account for the value of the cosmological constant.

Exotic Rotational Correlations in Quantum Geometry [Replacement]

Estimates are presented of exotic quantum correlations of rotational motion on large scales that could emerge from quantum geometry at the Planck length $\ell_P$. Extrapolations of standard quantum theory and gravity suggest that the inertial frame within separation $R$ from any world line fluctuates on a timescale $R/c$ with angular variance $\approx \ell_P/R$. A specific model of quantum twists on causal boundaries is presented. The projection of the exotic correlations onto an interferometer signal correlation function is estimated, and shown to vanish unless the light path sweeps out a nonzero spatial area. It is conjectured that exotic rotational correlations could resolve conflicts of field theory with gravity on large scales, and that entanglement with the Standard Model field vacuum might account for the value of the cosmological constant.

Exotic Rotational Correlations in Emergent Quantum Geometry [Replacement]

Estimates are presented of exotic quantum correlations in transverse position that arise in quantum gravity if space emerges from Planck scale quantum elements with no fixed classical background. Directions between world lines at separation $R$ coherently fluctuate on a timescale $R/c$ and angular scale $(R/\ell_P)^{-1/2}$ in the classical ($R\rightarrow \infty$) inertial non-rotating frame, where $\ell_P$ denotes the Planck length. The projection of these rotational correlations onto an interferometer signal correlation function is computed exactly, for a light path of arbitrary shape. The signal variance in a closed circuit is equal to twice the enclosed area divided by the perimeter, in Planck units. It is conjectured that exotic rotational correlations, entangled with the strong interactions, determine the value of the cosmological constant. Cosmic acceleration may be viewed heuristically as centrifugal acceleration by rotational fluctuations of the vacuum. An experiment concept is sketched, based on reconfiguration of the Fermilab Holometer into an area-enclosing light path.

Exotic Rotational Correlations in Quantum Geometry [Replacement]

Estimates are presented of exotic rotational correlations that arise if space emerges from Planck scale quantum elements with no fixed classical background. Directions in the classical inertial frame at separation $R$ from any world line coherently fluctuate on a timescale $R/c$ and angular scale $(R/\ell_P)^{-1/2}$, where $\ell_P$ denotes the Planck length. The projection of the exotic correlations onto an interferometer signal correlation function is predicted exactly, for a light path of arbitrary shape; the signal variance from each arm is equal to the enclosed area divided by the perimeter or free spectral range, in Planck units. An experiment concept is sketched, based on reconfiguration of the Fermilab Holometer into an area-enclosing light path. It is conjectured that exotic rotational correlations, entangled with the Standard Model field vacuum, could account for the value of the cosmological constant.

Exotic Rotational Correlations in Emergent Quantum Geometry [Replacement]

Estimates are presented of exotic rotational correlations that arise if space emerges from Planck scale quantum elements with no fixed classical background. Directions in the classical inertial frame at separation $R$ from any world line coherently fluctuate on a timescale $R/c$ and angular scale $(R/\ell_P)^{-1/2}$, where $\ell_P$ denotes the Planck length. The projection of the exotic correlations onto an interferometer signal correlation function is predicted exactly, for a light path of arbitrary shape: the signal variance is equal to twice the enclosed area divided by the perimeter or free spectral range, in Planck units. An experiment concept is sketched, based on reconfiguration of the Fermilab Holometer into an area-enclosing light path. It is conjectured that exotic rotational correlations, entangled with the Standard Model field vacuum, could account for the value of the cosmological constant.

Exotic Rotational Correlations in Quantum Geometry [Replacement]

Estimates are presented of exotic rotational correlations that arise if space emerges from Planck scale quantum elements with no fixed classical background. Directions in the classical inertial frame at separation $R$ from any world line coherently fluctuate on a timescale $R/c$ and angular scale $(R/\ell_P)^{-1/2}$, where $\ell_P$ denotes the Planck length. The projection of the exotic correlations onto an interferometer signal correlation function is predicted exactly, for a light path of arbitrary shape; the signal variance from each arm is equal to the enclosed area divided by the perimeter or free spectral range, in Planck units. An experiment concept is sketched, based on reconfiguration of the Fermilab Holometer into an area-enclosing light path. It is conjectured that exotic rotational correlations, entangled with the Standard Model field vacuum, could account for the value of the cosmological constant.

Target Space $\neq$ Space [Cross-Listing]

This paper investigates the significance of T-duality in string theory: the indistinguishability with respect to all observables, of models attributing radically different radii to space -- larger than the observable universe, or far smaller than the Planck length, say. Two interpretational branch points are identified and discussed. First, whether duals are physically equivalent or not: by considering a duality of the familiar simple harmonic oscillator, I argue that they are. Unlike the oscillator, there are no measurements 'outside' string theory that could distinguish the duals. Second, whether duals agree or disagree on the radius of 'target space', the space in which strings evolve according to string theory. I argue for the latter position, because the alternative leaves it unknown what the radius is. Since duals are physically equivalent yet disagree on the radius of target space, it follows that the radius is indeterminate between them. Using an analysis of Brandenberger and Vafa (1989), I explain why -- even so -- space is observed to have a determinate, large radius. The conclusion is that observed, 'phenomenal' space is not target space, since a space cannot have both a determinate and indeterminate radius: instead phenomenal space must be a higher-level phenomenon, not fundamental.

Renormalized spacetime is two-dimensional at the Planck scale [Replacement]

Quantum field theory distinguishes between the bare variables -- which we introduce in the Lagrangian -- and the renormalized variables which incorporate the effects of interactions. This suggests that the renormalized, physical, metric tensor of spacetime (and all the geometrical quantities derived from it) will also be different from the bare, classical, metric tensor in terms of which the bare gravitational Lagrangian is expressed. We provide a physical ansatz to relate the renormalized metric tensor to the bare metric tensor such that the spacetime acquires a zero-point-length $\ell _{0}$ of the order of the Planck length $L_{P}$. This prescription leads to several remarkable consequences. In particular, the Euclidean volume $V_D(\ell,\ell _{0})$ in a $D$-dimensional spacetime of a region of size $\ell $ scales as $V_D(\ell, \ell_{0}) \propto \ell _{0}^{D-2} \ell^2$ when $\ell \sim \ell _{0}$, while it reduces to the standard result $V_D(\ell,\ell _{0}) \propto \ell^D$ at large scales ($\ell \gg \ell _{0}$). The appropriately defined effective dimension, $D_{\rm eff} $, decreases continuously from $D_{\rm eff}=D$ (at $\ell \gg \ell _{0}$) to $D_{\rm eff}=2$ (at $\ell \sim \ell _{0}$). This suggests that the physical spacetime becomes essentially 2-dimensional near Planck scale.

Renormalized spacetime is two-dimensional at the Planck scale [Replacement]

Quantum field theory distinguishes between the bare variables -- which we introduce in the Lagrangian -- and the renormalized variables which incorporate the effects of interactions. This suggests that the renormalized, physical, metric tensor of spacetime (and all the geometrical quantities derived from it) will also be different from the bare, classical, metric tensor in terms of which the bare gravitational Lagrangian is expressed. We provide a physical ansatz to relate the renormalized metric tensor to the bare metric tensor such that the spacetime acquires a zero-point-length $\ell _{0}$ of the order of the Planck length $L_{P}$. This prescription leads to several remarkable consequences. In particular, the Euclidean volume $V_D(\ell,\ell _{0})$ in a $D$-dimensional spacetime of a region of size $\ell $ scales as $V_D(\ell, \ell_{0}) \propto \ell _{0}^{D-2} \ell^2$ when $\ell \sim \ell _{0}$, while it reduces to the standard result $V_D(\ell,\ell _{0}) \propto \ell^D$ at large scales ($\ell \gg \ell _{0}$). The appropriately defined effective dimension, $D_{\rm eff} $, decreases continuously from $D_{\rm eff}=D$ (at $\ell \gg \ell _{0}$) to $D_{\rm eff}=2$ (at $\ell \sim \ell _{0}$). This suggests that the physical spacetime becomes essentially 2-dimensional near Planck scale.

Spacetime with zero point length is two-dimensional at the Planck scale [Replacement]

It is generally believed that any quantum theory of gravity should have a generic feature --- a quantum of length. We provide a physical ansatz to obtain an effective non-local metric tensor starting from the standard metric tensor such that the spacetime acquires a zero-point-length $\ell _{0}$ of the order of the Planck length $L_{P}$. This prescription leads to several remarkable consequences. In particular, the Euclidean volume $V_D(\ell,\ell_{0})$ in a $D$-dimensional spacetime of a region of size $\ell $ scales as $V_D(\ell, \ell _{0}) \propto \ell_{0}^{D-2} \ell^2$ when $\ell \sim \ell_{0}$, while it reduces to the standard result $V_D(\ell,\ell_{0}) \propto \ell^D$ at large scales ($\ell \gg \ell_{0}$). The appropriately defined effective dimension, $D_{\rm eff} $, decreases continuously from $D_{\rm eff}=D$ (at $\ell \gg \ell_{0}$) to $D_{\rm eff}=2$ (at $\ell \sim \ell_{0}$). This suggests that the physical spacetime becomes essentially 2-dimensional near Planck scale.

Spacetime with zero point length is two-dimensional at the Planck scale [Replacement]

It is generally believed that any quantum theory of gravity should have a generic feature --- a quantum of length. We provide a physical ansatz to obtain an effective non-local metric tensor starting from the standard metric tensor such that the spacetime acquires a zero-point-length $\ell _{0}$ of the order of the Planck length $L_{P}$. This prescription leads to several remarkable consequences. In particular, the Euclidean volume $V_D(\ell,\ell_{0})$ in a $D$-dimensional spacetime of a region of size $\ell $ scales as $V_D(\ell, \ell _{0}) \propto \ell_{0}^{D-2} \ell^2$ when $\ell \sim \ell_{0}$, while it reduces to the standard result $V_D(\ell,\ell_{0}) \propto \ell^D$ at large scales ($\ell \gg \ell_{0}$). The appropriately defined effective dimension, $D_{\rm eff} $, decreases continuously from $D_{\rm eff}=D$ (at $\ell \gg \ell_{0}$) to $D_{\rm eff}=2$ (at $\ell \sim \ell_{0}$). This suggests that the physical spacetime becomes essentially 2-dimensional near Planck scale.

Renormalized spacetime is two-dimensional at the Planck scale

Quantum field theory distinguishes between the bare variables -- which we introduce in the Lagrangian -- and the renormalized variables which incorporate the effects of interactions. This suggests that the renormalized, physical, metric tensor of spacetime (and all the geometrical quantities derived from it) will also be different from the bare, classical, metric tensor in terms of which the bare gravitational Lagrangian is expressed. We provide a physical ansatz to relate the renormalized metric tensor to the bare metric tensor such that the spacetime acquires a zero-point-length $\ell _{0}$ of the order of the Planck length $L_{P}$. This prescription leads to several remarkable consequences. In particular, the Euclidean volume $V_D(\ell,\ell _{0})$ in a $D$-dimensional spacetime of a region of size $\ell $ scales as $V_D(\ell, \ell_{0}) \propto \ell _{0}^{D-2} \ell^2$ when $\ell \sim \ell _{0}$, while it reduces to the standard result $V_D(\ell,\ell _{0}) \propto \ell^D$ at large scales ($\ell \gg \ell _{0}$). The appropriately defined effective dimension, $D_{\rm eff} $, decreases continuously from $D_{\rm eff}=D$ (at $\ell \gg \ell _{0}$) to $D_{\rm eff}=2$ (at $\ell \sim \ell _{0}$). This suggests that the physical spacetime becomes essentially 2-dimensional near Planck scale.

Renormalized spacetime is two-dimensional at the Planck scale [Cross-Listing]

Quantum field theory distinguishes between the bare variables -- which we introduce in the Lagrangian -- and the renormalized variables which incorporate the effects of interactions. This suggests that the renormalized, physical, metric tensor of spacetime (and all the geometrical quantities derived from it) will also be different from the bare, classical, metric tensor in terms of which the bare gravitational Lagrangian is expressed. We provide a physical ansatz to relate the renormalized metric tensor to the bare metric tensor such that the spacetime acquires a zero-point-length $\ell _{0}$ of the order of the Planck length $L_{P}$. This prescription leads to several remarkable consequences. In particular, the Euclidean volume $V_D(\ell,\ell _{0})$ in a $D$-dimensional spacetime of a region of size $\ell $ scales as $V_D(\ell, \ell_{0}) \propto \ell _{0}^{D-2} \ell^2$ when $\ell \sim \ell _{0}$, while it reduces to the standard result $V_D(\ell,\ell _{0}) \propto \ell^D$ at large scales ($\ell \gg \ell _{0}$). The appropriately defined effective dimension, $D_{\rm eff} $, decreases continuously from $D_{\rm eff}=D$ (at $\ell \gg \ell _{0}$) to $D_{\rm eff}=2$ (at $\ell \sim \ell _{0}$). This suggests that the physical spacetime becomes essentially 2-dimensional near Planck scale.

Neutrino-antineutrino Mass Splitting in the Standard Model: Neutrino Oscillation and Baryogenesis

By adding a neutrino mass term to the Standard Model, which is Lorentz and $SU(2)\times U(1)$ invariant but non-local to evade $CPT$ theorem, it is shown that non-locality within a distance scale of the Planck length, that may not be fatal to unitarity in generic effective theory, can generate the neutrino-antineutrino mass splitting of the order of observed neutrino mass differences, which is tested in oscillation experiments, and non-negligible baryon asymmetry depending on the estimate of sphaleron dynamics. The one-loop order induced electron-positron mass splitting in the Standard Model is shown to be finite and estimated at $\sim 10^{-20}$ eV, well below the experimental bound $< 10^{-2}$ eV. The induced $CPT$ violation in the $K$-meson in the Standard Model is expected to be even smaller and well below the experimental bound $|m_{K}-m_{\bar{K}}|<0.44\times 10^{-18}$ GeV.

Neutrino-antineutrino Mass Splitting in the Standard Model: Neutrino Oscillation and Baryogenesis [Cross-Listing]

By adding a neutrino mass term to the Standard Model, which is Lorentz and $SU(2)\times U(1)$ invariant but non-local to evade $CPT$ theorem, it is shown that non-locality within a distance scale of the Planck length, that may not be fatal to unitarity in generic effective theory, can generate the neutrino-antineutrino mass splitting of the order of observed neutrino mass differences, which is tested in oscillation experiments, and non-negligible baryon asymmetry depending on the estimate of sphaleron dynamics. The one-loop order induced electron-positron mass splitting in the Standard Model is shown to be finite and estimated at $\sim 10^{-20}$ eV, well below the experimental bound $< 10^{-2}$ eV. The induced $CPT$ violation in the $K$-meson in the Standard Model is expected to be even smaller and well below the experimental bound $|m_{K}-m_{\bar{K}}|<0.44\times 10^{-18}$ GeV.

Can gravitational microlensing by vacuum fluctuations be observed?

Although the prospect is more plausible than it might appear, the answer to the title question is, unfortunately, "probably not." Quantum fluctuations of vacuum energy can focus light, and while the effect is tiny, the distribution of fluctuations is highly non-Gaussian, offering hope that relatively rare "large" fluctuations might be observable. I show that although gravitational microlensing by such fluctuations become important at scales much larger than the Planck length, the possibility of direct observation remains remote, although there is a small chance that cumulative effects over cosmological distances might be detectable. The effect is sensitive to the size of the Planck scale, however, and could offer a new test of TeV-scale gravity.

Can gravitational microlensing by vacuum fluctuations be observed? [Cross-Listing]

Although the prospect is more plausible than it might appear, the answer to the title question is, unfortunately, "probably not." Quantum fluctuations of vacuum energy can focus light, and while the effect is tiny, the distribution of fluctuations is highly non-Gaussian, offering hope that relatively rare "large" fluctuations might be observable. I show that although gravitational microlensing by such fluctuations become important at scales much larger than the Planck length, the possibility of direct observation remains remote, although there is a small chance that cumulative effects over cosmological distances might be detectable. The effect is sensitive to the size of the Planck scale, however, and could offer a new test of TeV-scale gravity.

Maximal boost and energy of elementary particles as a manifestation of the limit of localizability of elementary quantum systems [Cross-Listing]

I discuss an upper bound on the boost and the energy of elementary particles. The limit is derived utilizing the core principle of relativistic quantum mechanics stating that there is a lower limit for localization of an elementary quantum system and the suggestion that when the localization scale reaches the Planck length, elementary particles are removed from observables. The limit for the boost and energy, $M_{Planck}/m$ and $M_{Planck}c^{2}\approx\,8.6* 10^{27}$ eV, is defined in terms of fundamental constants and the mass of elementary particle and does not involve any dynamic scale. These bounds imply that the cosmic ray flux of any flavor may stretch up to energies of order $10^{18}$ GeV and will cut off at this value.

Maximal boost and energy of elementary particles as a manifestation of the limit of localizability of elementary quantum systems [Replacement]

I discuss an upper bound on the boost and the energy of elementary particles. The limit is derived utilizing the core principle of relativistic quantum mechanics stating that there is a lower bound for localization of an elementary quantum system and the assumption that when the localization scale reaches the Planck length, elementary particles are removed from the S-matrix observables. The limits for the boost and energy, M_{Planck}/m and M_{Planck}c^{2}\approx\,8.6 * 10^{27} eV, are defined in terms of fundamental constants and the mass of elementary particle and does not involve any dynamic scale. These bounds imply that the cosmic ray flux of any flavor may stretch up to energies of order 10^{18} GeV and will cut off around this value.

Maximal boost and energy of elementary particles as a manifestation of the limit of localizability of elementary quantum systems [Replacement]

I discuss an upper bound on the boost and the energy of elementary particles. The limit is derived utilizing the core principle of relativistic quantum mechanics stating that there is a lower bound for localization of an elementary quantum system and the assumption that when the localization scale reaches the Planck length, elementary particles are removed from the S-matrix observables. The limits for the boost and energy, M_{Planck}/m and M_{Planck}c^{2}\approx\,8.6 * 10^{27} eV, are defined in terms of fundamental constants and the mass of elementary particle and does not involve any dynamic scale. These bounds imply that the cosmic ray flux of any flavor may stretch up to energies of order 10^{18} GeV and will cut off around this value.

Maximal boost and energy of elementary particles as a manifestation of the limit of localizability of elementary quantum systems

I discuss an upper bound on the boost and the energy of elementary particles. The limit is derived utilizing the core principle of relativistic quantum mechanics stating that there is a lower limit for localization of an elementary quantum system and the suggestion that when the localization scale reaches the Planck length, elementary particles are removed from observables. The limit for the boost and energy, $M_{Planck}/m$ and $M_{Planck}c^{2}\approx\,8.6* 10^{27}$ eV, is defined in terms of fundamental constants and the mass of elementary particle and does not involve any dynamic scale. These bounds imply that the cosmic ray flux of any flavor may stretch up to energies of order $10^{18}$ GeV and will cut off at this value.

The hierarchy problem and the cosmological constant problem in the Standard Model

We argue that the SM in the Higgs phase does not suffer form a "hierarchy problem" and that similarly the "cosmological constant problem" resolves itself if we understand the SM as a low energy effective theory emerging from a cut-off medium at the Planck scale. We discuss these issues under the condition of a stable Higgs vacuum, which allows to extend the SM up to the Planck length. The bare Higgs boson mass then changes sign below the Planck scale, such the the SM in the early universe is in the symmetric phase. The cut-off enhanced Higgs mass term as well as the quartically enhanced cosmological constant term trigger the inflation of the early universe. The coefficients of the shift between bare and renormalized Higgs mass as well as of the shift between bare and renormalized vacuum energy density exhibit close-by zeros at some point below the Planck scale. The zeros are matching points between short distance and the renormalized low energy quantities. Since inflation tunes the total energy density to take the critical value of a flat universe Omega_tot=rho_tot/rho_crit=Omega_Lambda+Omega_matter+Omega_radiation}=1 it is obvious that Omega_Lambda today is of order Omega_tot given that 1>Omega_matter, Omega_radiation>0, which saturate the total density to about 26 % only, the dominant part being dark matter(21 %).

Generalized uncertainty principle and the conformally coupled scalar field quantum cosmology

We exactly solve the Wheeler-DeWitt equation for the closed homogeneous and isotropic quantum cosmology in the presence of a conformally coupled scalar field and in the context of the generalized uncertainty principle. This form of generalized uncertainty principle is motivated by the black hole physics and it predicts a minimal length uncertainty proportional to the Planck length. We construct wave packets in momentum minisuperspace which closely follow classical trajectories and strongly peak on them upon choosing appropriate initial conditions. Moreover, based on the DeWitt criterion, we obtain wave packets that exhibit singularity-free behavior.

Generalized uncertainty principle and the conformally coupled scalar field quantum cosmology [Cross-Listing]

We exactly solve the Wheeler-DeWitt equation for the closed homogeneous and isotropic quantum cosmology in the presence of a conformally coupled scalar field and in the context of the generalized uncertainty principle. This form of generalized uncertainty principle is motivated by the black hole physics and it predicts a minimal length uncertainty proportional to the Planck length. We construct wave packets in momentum minisuperspace which closely follow classical trajectories and strongly peak on them upon choosing appropriate initial conditions. Moreover, based on the DeWitt criterion, we obtain wave packets that exhibit singularity-free behavior.

Einstein static Universe in non-minimal kinetic coupled gravity

Non-minimal kinetic coupled gravity is one the novel modification to the general relativity, which includes the non-minimal coupling of kinetic term of a scalar field $\phi$ with the curvature tensor by the coupling $\kappa$, and the minimal coupling with the metric tensor by the coupling $\varepsilon$. It has already been shown that such modified gravity model provides an essentially new inflationary mechanism. In this work, we show that the Universe might have been started out in an asymptotically Einstein static state as a initial state before the inflationary stage of the Universe. We study the stability of Einstein static Universe, with Friedmann-Lema\^{\i}tre-Robertson-Walker metric, by considering linear homogeneous perturbations in the kinetic coupled gravity. By taking linear homogeneous perturbations, we find that the stability of Einstein static Universe, in the kinetic coupled gravity with quadratic scalar field potential, for closed ($K=1$) isotropic and homogeneous Friedmann-Lema\^{\i}tre-Robertson-Walker Universe depends on the coupling parameters $\kappa$ and $\varepsilon$. Specifically, for $\kappa=L_P^2$ and $\varepsilon=1$ we find that the stability condition imposes the inequality $a_0>\sqrt{3}L_P$ on the initial size $a_0$ of the closed Einstein static Universe before the inflation. Such inequality asserts that the initial size of the Einstein static Universe must be greater than the Planck length $L_P$, in consistency with the quantum gravity and quantum cosmology requirements. In this way, we have succeeded to fix the non-minimal coupling parameter $\kappa$ in the context of Einstein static Universe.

Einstein static Universe in non-minimal kinetic coupled gravity [Replacement]

We study the stability of Einstein static Universe, with FLRW metric, by considering linear homogeneous perturbations in the kinetic coupled gravity. By taking linear homogeneous perturbations, we find that the stability of Einstein static Universe, in the kinetic coupled gravity with quadratic scalar field potential, for closed ($K=1$) isotropic and homogeneous FLRW Universe depends on the coupling parameters $\kappa$ and $\varepsilon$. Specifically, for $\kappa=L_P^2$ and $\varepsilon=1$ we find that the stability condition imposes the inequality $a_0>\sqrt{3}L_P$ on the initial size $a_0$ of the closed Einstein static Universe before the inflation. Such inequality asserts that the initial size of the Einstein static Universe must be greater than the Planck length $L_P$, in consistency with the quantum gravity and quantum cosmology requirements. In this way, we have determined the non-minimal coupling parameter $\kappa$ in the context of Einstein static Universe. Such a very small parameter is favored in the inflationary models constructed in the kinetic coupled gravity. We have also studied the stability against the vector and tensor perturbations and discussed on the acceptable values of the equation of state parameter.

Black Hole as a Wormhole Factory [Replacement]

On general grounds, one may argue that a black hole stops radiation at the Planck mass, where the radiated energy is comparable to the black hole's mass. And also, it has been argued that there would be a "wormhole-like" structure, known as "space-time foam", due to large fluctuations below the Planck length. In this paper, as an explicit example, we consider an exact classical solution which represents nicely those two properties in a recently proposed quantum gravity model based on different scaling dimensions between space and time coordinates. The solution, called "Black Wormhole", consists of two different states, depending on its mass M and an IR parameter omega: For the black hole state, a non-traversable wormhole occupies the interior region of the black hole around the singularity at the origin, whereas for the wormhole state, the interior wormhole is exposed to an outside observer as the black hole horizon is disappeared from evaporation. The black hole state becomes thermodynamically stable as it approaches to the merge point where the interior wormhole throat and the black hole horizon merges, and the Hawking temperature vanishes at the exact merge point. This solution suggests the "Generalized Cosmic Censorship" by the existence of a wormhole-like structure which protects the naked singularity even after the black hole evaporation. One could understand the would-be wormholes inside the black hole horizon as the results of microscopic wormholes created by "negative" energy quanta which have entered the black hole horizon in Hawking radiation processes: The quantum black hole could be a wormhole factory. It is found that this speculative picture may be consistent with the recent "ER=EPR" proposal for resolving the recent black hole entanglement debates.

 

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