# Posts Tagged planck length

## Recent Postings from planck length

### 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.

### 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.

### 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, we show that there is actually 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 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 picture may be consistent with the recent "ER=EPR" proposal for resolving the recent black hole entanglement debates.

### Black Hole as a Wormhole Factory [Cross-Listing]

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, we show that there is actually 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 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 picture may be consistent with the recent "ER=EPR" proposal for resolving the recent black hole entanglement debates.

### 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, we show that there is actually 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 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 picture may be consistent with the recent "ER=EPR" proposal for resolving the recent black hole entanglement debates.

### Black Hole as a Wormhole Factory

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, we show that there is actually 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 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 picture may be consistent with the recent "ER=EPR" proposal for resolving the recent black hole entanglement debates.

### New Constraints on Quantum Gravity from X-ray and Gamma-Ray Observations [Replacement]

One aspect of the quantum nature of spacetime is its "foaminess" at very small scales. Many models for spacetime foam are defined by the accumulation power $\alpha$, which parameterizes the rate at which Planck-scale spatial uncertainties (and thephase shifts they produce) may accumulate over large path-lengths. Here $\alpha$ is defined by theexpression for the path-length fluctuations, $\delta \ell$, of a source at distance $\ell$, wherein $\delta \ell \simeq \ell^{1 – \alpha} \ell_P^{\alpha}$, with $\ell_P$ being the Planck length. We reassess previous proposals to use astronomical observations ofdistant quasars and AGN to test models of spacetime foam. We show explicitly how wavefront distortions on small scales cause the image intensity to decay to the point where distant objects become undetectable when the path-length fluctuations become comparable to the wavelength of the radiation. We use X-ray observations from {\em Chandra} to set the constraint $\alpha \gtrsim 0.58$, which rules out the random walk model (with $\alpha = 1/2$). Much firmer constraints canbe set utilizing detections of quasars at GeV energies with {\em Fermi}, and at TeV energies with ground-based Cherenkovtelescopes: $\alpha \gtrsim 0.67$ and $\alpha \gtrsim 0.72$, respectively. These limits on $\alpha$ seem to rule out $\alpha = 2/3$, the model of some physical interest.

### New Constraints on Quantum Gravity from X-ray and Gamma-Ray Observations [Replacement]

One aspect of the quantum nature of spacetime is its "foaminess" at very small scales. Many models for spacetime foam are defined by the accumulation power $\alpha$, which parameterizes the rate at which Planck-scale spatial uncertainties (and thephase shifts they produce) may accumulate over large path-lengths. Here $\alpha$ is defined by theexpression for the path-length fluctuations, $\delta \ell$, of a source at distance $\ell$, wherein $\delta \ell \simeq \ell^{1 – \alpha} \ell_P^{\alpha}$, with $\ell_P$ being the Planck length. We reassess previous proposals to use astronomical observations ofdistant quasars and AGN to test models of spacetime foam. We show explicitly how wavefront distortions on small scales cause the image intensity to decay to the point where distant objects become undetectable when the path-length fluctuations become comparable to the wavelength of the radiation. We use X-ray observations from {\em Chandra} to set the constraint $\alpha \gtrsim 0.58$, which rules out the random walk model (with $\alpha = 1/2$). Much firmer constraints canbe set utilizing detections of quasars at GeV energies with {\em Fermi}, and at TeV energies with ground-based Cherenkovtelescopes: $\alpha \gtrsim 0.67$ and $\alpha \gtrsim 0.72$, respectively. These limits on $\alpha$ seem to rule out $\alpha = 2/3$, the model of some physical interest.

### New Constraints on Quantum Gravity from X-ray and Gamma-Ray Observations [Replacement]

One aspect of the quantum nature of spacetime is its "foaminess" at very small scales. Many models for spacetime foam are defined by the accumulation power $\alpha$, which parameterizes the rate at which Planck-scale spatial uncertainties (and thephase shifts they produce) may accumulate over large path-lengths. Here $\alpha$ is defined by theexpression for the path-length fluctuations, $\delta \ell$, of a source at distance $\ell$, wherein $\delta \ell \simeq \ell^{1 – \alpha} \ell_P^{\alpha}$, with $\ell_P$ being the Planck length. We reassess previous proposals to use astronomical observations ofdistant quasars and AGN to test models of spacetime foam. We show explicitly how wavefront distortions on small scales cause the image intensity to decay to the point where distant objects become undetectable when the path-length fluctuations become comparable to the wavelength of the radiation. We use X-ray observations from {\em Chandra} to set the constraint $\alpha \gtrsim 0.58$, which rules out the random walk model (with $\alpha = 1/2$). Much firmer constraints canbe set utilizing detections of quasars at GeV energies with {\em Fermi}, and at TeV energies with ground-based Cherenkovtelescopes: $\alpha \gtrsim 0.67$ and $\alpha \gtrsim 0.72$, respectively. These limits on $\alpha$ seem to rule out $\alpha = 2/3$, the model of some physical interest.

### New Constraints on Quantum Gravity from X-ray and Gamma-Ray Observations [Replacement]

One aspect of the quantum nature of spacetime is its "foaminess" at very small scales. Many models for spacetime foam are defined by the accumulation power $\alpha$, which parameterizes the rate at which Planck-scale spatial uncertainties (and thephase shifts they produce) may accumulate over large path-lengths. Here $\alpha$ is defined by theexpression for the path-length fluctuations, $\delta \ell$, of a source at distance $\ell$, wherein $\delta \ell \simeq \ell^{1 – \alpha} \ell_P^{\alpha}$, with $\ell_P$ being the Planck length. We reassess previous proposals to use astronomical observations ofdistant quasars and AGN to test models of spacetime foam. We show explicitly how wavefront distortions on small scales cause the image intensity to decay to the point where distant objects become undetectable when the path-length fluctuations become comparable to the wavelength of the radiation. We use X-ray observations from {\em Chandra} to set the constraint $\alpha \gtrsim 0.58$, which rules out the random walk model (with $\alpha = 1/2$). Much firmer constraints canbe set utilizing detections of quasars at GeV energies with {\em Fermi}, and at TeV energies with ground-based Cherenkovtelescopes: $\alpha \gtrsim 0.67$ and $\alpha \gtrsim 0.72$, respectively. These limits on $\alpha$ seem to rule out $\alpha = 2/3$, the model of some physical interest.

### Silent initial conditions for cosmological perturbations with a change of space-time signature [Cross-Listing]

Recent calculations in loop quantum cosmology suggest that a transition from a Lorentzian to an Euclidean space-time might take place in the very early Universe. The transition point leads to a state of silence, characterized by a vanishing speed of light. This behavior can be interpreted as a decoupling of different space points, similar to the one characterizing the BKL phase. In this study, we address the issue of imposing initial conditions for the cosmological perturbations at the transition point between the Lorentzian and Euclidean phases. Motivated by the decoupling of space points, initial conditions characterized by a lack of correlations are investigated. We show that the "white noise" initial conditions are supported by the analysis of the vacuum state in the Euclidean regime adjacent to the state of silence. Furthermore, the possibility of imposing the silent initial conditions at the trans-Planckian surface, characterized by a vanishing speed for the propagation of modes with wavelengths of the order of the Planck length, is studied. Such initial conditions might result from a loop-deformations of the Poincar\’e algebra. The conversion of the silent initial power spectrum to a scale-invariant one is also examined.

### Silent initial conditions for cosmological perturbations with a change of space-time signature [Cross-Listing]

Recent calculations in loop quantum cosmology suggest that a transition from a Lorentzian to an Euclidean space-time might take place in the very early Universe. The transition point leads to a state of silence, characterized by a vanishing speed of light. This behavior can be interpreted as a decoupling of different space points, similar to the one characterizing the BKL phase. In this study, we address the issue of imposing initial conditions for the cosmological perturbations at the transition point between the Lorentzian and Euclidean phases. Motivated by the decoupling of space points, initial conditions characterized by a lack of correlations are investigated. We show that the "white noise" initial conditions are supported by the analysis of the vacuum state in the Euclidean regime adjacent to the state of silence. Furthermore, the possibility of imposing the silent initial conditions at the trans-Planckian surface, characterized by a vanishing speed for the propagation of modes with wavelengths of the order of the Planck length, is studied. Such initial conditions might result from a loop-deformations of the Poincar\’e algebra. The conversion of the silent initial power spectrum to a scale-invariant one is also examined.

### Silent initial conditions for cosmological perturbations with a change of space-time signature

Recent calculations in loop quantum cosmology suggest that a transition from a Lorentzian to an Euclidean space-time might take place in the very early Universe. The transition point leads to a state of silence, characterized by a vanishing speed of light. This behavior can be interpreted as a decoupling of different space points, similar to the one characterizing the BKL phase. In this study, we address the issue of imposing initial conditions for the cosmological perturbations at the transition point between the Lorentzian and Euclidean phases. Motivated by the decoupling of space points, initial conditions characterized by a lack of correlations are investigated. We show that the "white noise" initial conditions are supported by the analysis of the vacuum state in the Euclidean regime adjacent to the state of silence. Furthermore, the possibility of imposing the silent initial conditions at the trans-Planckian surface, characterized by a vanishing speed for the propagation of modes with wavelengths of the order of the Planck length, is studied. Such initial conditions might result from a loop-deformations of the Poincar\’e algebra. The conversion of the silent initial power spectrum to a scale-invariant one is also examined.

### How universe evolves with cosmological and gravitational constants [Replacement]

With a basic varying space-time cutoff $\tilde\ell$, we study a regularized and quantized Einstein-Cartan gravitational field theory and its domains of ultraviolet-unstable fixed point $g_{\rm ir}\gtrsim 0$ and ultraviolet-stable fixed point $g_{\rm uv}\approx 4/3$ of the gravitational gauge coupling $g=(4/3)G/G_{\rm Newton}$. Because the fundamental operators of quantum gravitational field theory are dimension-2 area operators, the cosmological constant is inversely proportional to the squared correlation length $\Lambda\propto \xi^{-2}$. The correlation length $\xi$ characterizes an infrared size of a causally correlate patch of the universe. The cosmological constant $\Lambda$ and the gravitational constant $G$ are related by a generalized Bianchi identity. As the basic space-time cutoff $\tilde\ell$ decreases and approaches to the Planck length $\ell_{\rm pl}$, the universe undergoes inflation in the domain of the ultraviolet-unstable fixed point $g_{\rm ir}$, then evolves to the low-redshift universe in the domain of ultraviolet-stable fixed point $g_{\rm uv}$. We give the quantitative description of the low-redshift universe in the scaling-invariant domain of the ultraviolet-stable fixed point $g_{\rm uv}$, and its deviation from the $\Lambda$CDM can be examined by low-redshift $(z\lesssim 1)$ cosmological observations, such as supernova Type Ia.

### How universe evolves with cosmological and gravitational constants

We study a quantized Einstein-Cartan gravity and its ultraviolet unstable (stable) fixed point $\bar G_c\approx 0$ ($G_c\approx G_{\rm N}$) of running gravitational constant $G$. The cosmological constant $\Lambda\propto \xi^{-2}$ appears via a dimensional transmutation. The correlation length $\xi$ relates to the gravitational constant by a generalized Bianchi identity. Inflation possibly occurs in the neighborhood of fixed point $\bar G_c$, then universe evolves from $\bar G_c$ to $G_c$ as the space-time cutoff $\tilde a$ approaching to the Planck length $a_{\rm pl}$. The quantitative description of present universe in the scaling region of fixed point $G_c$ is given, and its deviation from the $\Lambda$CDM can be examined by recent cosmological observations, such as supernova Type Ia.

### How universe evolves with cosmological and gravitational constants [Replacement]

With a basic varying space-time cutoff $\tilde\ell$, we study a regularized and quantized Einstein-Cartan gravitational field theory and its domains of ultraviolet-unstable fixed point $g_{\rm ir}\gtrsim 0$ and ultraviolet-stable fixed point $g_{\rm uv}\approx 4/3$ of the gravitational gauge coupling $g=(4/3)G/G_{\rm Newton}$. Because the fundamental operators of quantum gravitational field theory are dimension-2 area operators, the cosmological constant is inversely proportional to the squared correlation length $\Lambda\propto \xi^{-2}$. The correlation length $\xi$ characterizes an infrared size of a causally correlate patch of the universe. The cosmological constant $\Lambda$ and the gravitational constant $G$ are related by a generalized Bianchi identity. As the basic space-time cutoff $\tilde\ell$ decreases and approaches to the Planck length $\ell_{\rm pl}$, the universe undergoes inflation in the domain of the ultraviolet-unstable fixed point $g_{\rm ir}$, then evolves to the low-redshift universe in the domain of ultraviolet-stable fixed point $g_{\rm uv}$. We give the quantitative description of the low-redshift universe in the scaling-invariant domain of the ultraviolet-stable fixed point $g_{\rm uv}$, and its deviation from the $\Lambda$CDM can be examined by low-redshift $(z\lesssim 1)$ cosmological observations, such as supernova Type Ia.

### How universe evolves with cosmological and gravitational constants [Cross-Listing]

We study a quantized Einstein-Cartan gravity and its ultraviolet unstable (stable) fixed point $\bar G_c\approx 0$ ($G_c\approx G_{\rm N}$) of running gravitational constant $G$. The cosmological constant $\Lambda\propto \xi^{-2}$ appears via a dimensional transmutation. The correlation length $\xi$ relates to the gravitational constant by a generalized Bianchi identity. Inflation possibly occurs in the neighborhood of fixed point $\bar G_c$, then universe evolves from $\bar G_c$ to $G_c$ as the space-time cutoff $\tilde a$ approaching to the Planck length $a_{\rm pl}$. The quantitative description of present universe in the scaling region of fixed point $G_c$ is given, and its deviation from the $\Lambda$CDM can be examined by recent cosmological observations, such as supernova Type Ia.

### Semi-classical approach to quantum black holes

In this Chapter we would like to review a "~phenomenological~" approach taking into account the most fundamental feature of string theory or, more in general, of quantum gravity, whatever its origin, which is the existence of a minimal length in the space-time fabric. This length is generally identified with the Planck length, or the string length, but it could be also much longer down to the TeV region. A simple and effective way to keep track of the effects the minimal length in black hole geometries is to solve the Einstein equations with an energy momentum tensor describing non point-like matter. The immediate consequence is the absence of any curvature singularity. Where textbook solutions of the Einstein equations loose any physical meaning because of infinite tidal forces, we find a de Sitter vacuum core of high, but finite, energy density and pressure. An additional improvement regards the final stage of the black hole evaporation leading to a vanishing Hawking temperature even in the neutral, non-rotating, case. In spite of th simplicity of this model we are able to describe the final stage of the black hole evaporation, resulting in a cold remnant with a degenerate, extremal, horizon of radius of the order of the minimal length. In this chapter we shall describe only neutral, spherically symmetric, regular black holes although charged, rotating and higher dimensional black holes can be found in the literature.

### Semi-classical approach to quantum black holes [Cross-Listing]

In this Chapter we would like to review a "~phenomenological~" approach taking into account the most fundamental feature of string theory or, more in general, of quantum gravity, whatever its origin, which is the existence of a minimal length in the space-time fabric. This length is generally identified with the Planck length, or the string length, but it could be also much longer down to the TeV region. A simple and effective way to keep track of the effects the minimal length in black hole geometries is to solve the Einstein equations with an energy momentum tensor describing non point-like matter. The immediate consequence is the absence of any curvature singularity. Where textbook solutions of the Einstein equations loose any physical meaning because of infinite tidal forces, we find a de Sitter vacuum core of high, but finite, energy density and pressure. An additional improvement regards the final stage of the black hole evaporation leading to a vanishing Hawking temperature even in the neutral, non-rotating, case. In spite of th simplicity of this model we are able to describe the final stage of the black hole evaporation, resulting in a cold remnant with a degenerate, extremal, horizon of radius of the order of the minimal length. In this chapter we shall describe only neutral, spherically symmetric, regular black holes although charged, rotating and higher dimensional black holes can be found in the literature.

### Semi-classical approach to quantum black holes [Cross-Listing]

In this Chapter we would like to review a "~phenomenological~" approach taking into account the most fundamental feature of string theory or, more in general, of quantum gravity, whatever its origin, which is the existence of a minimal length in the space-time fabric. This length is generally identified with the Planck length, or the string length, but it could be also much longer down to the TeV region. A simple and effective way to keep track of the effects the minimal length in black hole geometries is to solve the Einstein equations with an energy momentum tensor describing non point-like matter. The immediate consequence is the absence of any curvature singularity. Where textbook solutions of the Einstein equations loose any physical meaning because of infinite tidal forces, we find a de Sitter vacuum core of high, but finite, energy density and pressure. An additional improvement regards the final stage of the black hole evaporation leading to a vanishing Hawking temperature even in the neutral, non-rotating, case. In spite of th simplicity of this model we are able to describe the final stage of the black hole evaporation, resulting in a cold remnant with a degenerate, extremal, horizon of radius of the order of the minimal length. In this chapter we shall describe only neutral, spherically symmetric, regular black holes although charged, rotating and higher dimensional black holes can be found in the literature.

### Particle in a cavity in one-dimensional bandlimited quantum mechanics

The effects of the generalized uncertainty principle (GUP) on the low-energy stationary states of a particle moving in a cavity with no sharp boundaries are determined by means of the perturbation expansion in the framework of one-dimensional bandlimited quantum mechanics. A realization of GUP resulting in the existence of a finite ultraviolet (UV) wave-vector cutoff $K\sim 1/\ell_P$ (with the Planck length $\ell_P$) is considered. The cavity of the size $\ell \gg \ell_P$ is represented by an infinitely deep trapezoid-well potential with boundaries smeared out in a range $R$ satisfying the inequalities $\ell\gg R\gtrsim \ell_P$. In order to determine the energy shifts of the low-lying stationary states, the usual perturbation expansion is reformulated in a manner that enables one to treat consistently order-by-order the direct and indirect GUP effects, i.e., those due to the modification of the Hamiltonian and the lack of the UV modes, respectively. It is shown that the leading terms of the indirect and the direct GUP effects are of the first and second order, respectively, in the small parameter $\ell_P/\ell$ in agreement with our previous finding in a more naive approach [1].

### Neutrino-antineutrino mass splitting in the Standard Model and baryogenesis [Cross-Listing]

On the basis of a previously proposed mechanism of neutrino-antineutrino mass splitting in the Standard Model, which is Lorentz and $SU(2)\times U(1)$ invariant but non-local to evade $CPT$ theorem, we discuss the possible implications of neutrino-antineutrino mass splitting on neutrino physics and baryogenesis. 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.

### Fermions and gravitational gyrotropy [Cross-Listing]

In conventional general relativity without torsion, high-frequency gravitational waves couple to the chiral number density of spin one-half quanta: the polarization of the waves is rotated by $2\pi N_5 {\ell _{\rm Pl}^2}$, where $N_5$ is the chiral column density and $\ell _{\rm Pl}$ is the Planck length. This means that if a primordial distribution of gravitational waves with E-E or B-B correlations passed through a chiral density of fermions in the very early Universe, an E-B correlation in the waves, and one in the cosmic microwave background, would be generated. Less obviously but more primitively, the condition Albrecht called cosmic coherence’ would be violated, changing the restrictions on the class of admissible cosmological gravitational waves.

### Fermions and gravitational gyrotropy

In conventional general relativity without torsion, high-frequency gravitational waves couple to the chiral number density of spin one-half quanta: the polarization of the waves is rotated by $2\pi N_5 {\ell _{\rm Pl}^2}$, where $N_5$ is the chiral column density and $\ell _{\rm Pl}$ is the Planck length. This means that if a primordial distribution of gravitational waves with E-E or B-B correlations passed through a chiral density of fermions in the very early Universe, an E-B correlation in the waves, and one in the cosmic microwave background, would be generated. Less obviously but more primitively, the condition Albrecht called cosmic coherence’ would be violated, changing the restrictions on the class of admissible cosmological gravitational waves.

### The Minimal Length and the Quantum Partition Functions

We study the thermodynamics of various physical systems in the framework of the Generalized Uncertainty Principle that implies a minimal length uncertainty proportional to the Planck length. We present a general scheme to analytically calculate the quantum partition function of the physical systems to first order of the deformation parameter based on the behavior of the modified energy spectrum and compare our results with the classical approach. Also, we find the modified internal energy and heat capacity of the systems for the anti-Snyder framework.

### Emergent gravity by tuning the effective Planck length in Bose-Einstein Condensate

We show that at lowest energy nonlocal interactions, the tuning of s-wave scattering length can enable a systematic control over the quantum pressure term in a Bose-Einstein condensate (BEC). We derive the equation for the massless free excitations in an analogue curved space-time by controlling the effective Planck length. Our controlled derivation indicates a breakdown of this dynamics at length scales comparable to effective Planck length. We also specify the correction that one has to take into account at a larger length scale in a flat space-time due to the emergent gravity at intermediate length scales.

### Spacetime-symmetry violations: motivations, phenomenology, and tests

An important open question in fundamental physics concerns the nature of spacetime at distance scales associated with the Planck length. The widespread belief that probing such distances necessitates Planck-energy particles has impeded phenomenological and experimental research in this context. However, it has been realized that various theoretical approaches to underlying physics can accommodate Planck-scale violations of spacetime symmetries. This talk surveys the motivations for spacetime-symmetry research, the SME test framework, and experimental efforts in this field.

### Cosmological Constant from the Emergent Gravity Perspective

Observations indicate that our universe is characterized by a late-time accelerating phase, possibly driven by a cosmological constant $\Lambda$, with the dimensionless parameter $\Lambda L_P^2 \simeq 10^{-122}$, where $L_P = (G \hbar /c^3)^{1/2}$ is the Planck length. In this review, we describe how the emergent gravity paradigm provides a new insight and a possible solution to the cosmological constant problem. After reviewing the necessary background material, we identify the necessary and sufficient conditions for solving the cosmological constant problem. We show that these conditions are naturally satisfied in the emergent gravity paradigm in which (i) the field equations of gravity are invariant under the addition of a constant to the matter Lagrangian and (ii) the cosmological constant appears as an integration constant in the solution. The numerical value of this integration constant can be related to another dimensionless number (called CosMIn) that counts the number of modes inside a Hubble volume that cross the Hubble radius during the radiation and the matter dominated epochs of the universe. The emergent gravity paradigm suggests that CosMIn has the numerical value $4 \pi$, which, in turn, leads to the correct, observed value of the cosmological constant. Further, the emergent gravity paradigm provides an alternative perspective on cosmology and interprets the expansion of the universe itself as a quest towards holographic equipartition. We discuss the implications of this novel and alternate description of cosmology.

### Cosmological Constant from the Emergent Gravity Perspective [Cross-Listing]

Observations indicate that our universe is characterized by a late-time accelerating phase, possibly driven by a cosmological constant $\Lambda$, with the dimensionless parameter $\Lambda L_P^2 \simeq 10^{-122}$, where $L_P = (G \hbar /c^3)^{1/2}$ is the Planck length. In this review, we describe how the emergent gravity paradigm provides a new insight and a possible solution to the cosmological constant problem. After reviewing the necessary background material, we identify the necessary and sufficient conditions for solving the cosmological constant problem. We show that these conditions are naturally satisfied in the emergent gravity paradigm in which (i) the field equations of gravity are invariant under the addition of a constant to the matter Lagrangian and (ii) the cosmological constant appears as an integration constant in the solution. The numerical value of this integration constant can be related to another dimensionless number (called CosMIn) that counts the number of modes inside a Hubble volume that cross the Hubble radius during the radiation and the matter dominated epochs of the universe. The emergent gravity paradigm suggests that CosMIn has the numerical value $4 \pi$, which, in turn, leads to the correct, observed value of the cosmological constant. Further, the emergent gravity paradigm provides an alternative perspective on cosmology and interprets the expansion of the universe itself as a quest towards holographic equipartition. We discuss the implications of this novel and alternate description of cosmology.

### Cosmological Constant from the Emergent Gravity Perspective [Cross-Listing]

Observations indicate that our universe is characterized by a late-time accelerating phase, possibly driven by a cosmological constant $\Lambda$, with the dimensionless parameter $\Lambda L_P^2 \simeq 10^{-122}$, where $L_P = (G \hbar /c^3)^{1/2}$ is the Planck length. In this review, we describe how the emergent gravity paradigm provides a new insight and a possible solution to the cosmological constant problem. After reviewing the necessary background material, we identify the necessary and sufficient conditions for solving the cosmological constant problem. We show that these conditions are naturally satisfied in the emergent gravity paradigm in which (i) the field equations of gravity are invariant under the addition of a constant to the matter Lagrangian and (ii) the cosmological constant appears as an integration constant in the solution. The numerical value of this integration constant can be related to another dimensionless number (called CosMIn) that counts the number of modes inside a Hubble volume that cross the Hubble radius during the radiation and the matter dominated epochs of the universe. The emergent gravity paradigm suggests that CosMIn has the numerical value $4 \pi$, which, in turn, leads to the correct, observed value of the cosmological constant. Further, the emergent gravity paradigm provides an alternative perspective on cosmology and interprets the expansion of the universe itself as a quest towards holographic equipartition. We discuss the implications of this novel and alternate description of cosmology.

### Towards a Cosmology with Minimal Length and Maximal Energy [Replacement]

The Friedmann-Robertson-Walker (FRW) universe and Bianchi I,II universes are investigated in the framework of the generalized uncertainty principle (GUP) with a linear and a quadratic term in Planck length and momentum, which predicts minimum measurable length as well as maximum measurable momentum. We get a dynamic cosmological bounce for the FRW universe. With Bianchi universe, we found that the universe may be still isotropic by implementing GUP. Moreover, the wall velocity appears to be stationary with respect to the universe velocity which means that when the momentum of the Universe evolves into a maximum measurable energy, the bounce is enhanced against the wall which means no maximum limit angle is manifested anymore.

### Towards a Cosmology with Minimal Length and Maximal Energy [Replacement]

The Friedmann-Robertson-Walker (FRW) universe and Bianchi I,II universes are investigated in the framework of the generalized uncertainty principle (GUP) with a linear and a quadratic term in Planck length and momentum, which predicts minimum measurable length as well as maximum measurable momentum. We get a dynamic cosmological bounce for the FRW universe. With Bianchi universe, we found that the universe may be still isotropic by implementing GUP. Moreover, the wall velocity appears to be stationary with respect to the universe velocity which means that when the momentum of the Universe evolves into a maximum measurable energy, the bounce is enhanced against the wall which means no maximum limit angle is manifested anymore.

### $\lambda\phi^{4}$ Kink and sine-Gordon Soliton in the GUP Framework [Replacement]

We consider $\lambda\phi^{4}$ kink and sine-Gordon soliton in the presence of a minimal length uncertainty proportional to the Planck length. The modified Hamiltonian contains an extra term proportional to $p^4$ and the generalized Schr\"odinger equation is expressed as a forth-order differential equation in quasiposition space. We obtain the modified energy spectrum for the discrete states and compare our results with 1-loop resummed and Hartree approximations for the quantum fluctuations. We finally find some lower bounds for the deformations parameter so that the effects of the minimal length have the dominant role.

### $\lambda\phi^{4}$ Kink and sine-Gordon Soliton in the GUP Framework

We consider $\lambda\phi^{4}$ kink and sine-Gordon soliton in the presence of a minimal length uncertainty proportional to the Planck length. The modified Hamiltonian contains an extra term proportional to $p^4$ and the generalized Schr\"odinger equation is expressed as a forth-order differential equation in quasiposition space. We obtain the modified energy spectrum for the discrete states and compare our results with 1-loop resummed and Hartree approximations for the quantum fluctuations. We finally find some lower bounds for the deformations parameter so that the effects of the minimal length have the dominant role.

### Directional Entanglement of Quantum Fields with Quantum Geometry [Replacement]

It is conjectured that the spatial structure of quantum field states is influenced by a new kind of directional indeterminacy of quantum geometry set by the Planck length, $l_P$, that does not occur in a classical background geometry. Entanglement of fields with geometry modifies the transverse phase of field states at wavelength $\lambda$ and propagation distance $c\tau$ by about $\Delta \phi\approx \sqrt{l_P\tau}/\lambda$. The new effect is not detectable in measurements of propagating states that depend only on longitudinal coordinates. The reduced information content of fields in large systems is consistent with holographic bounds from gravitation theory, and may appear as measurable quantum-geometrical noise in interferometers.

### Directional Entanglement of Quantum Fields with Quantum Geometry [Replacement]

It is conjectured that the spatial structure of quantum field states is influenced by a new kind of directional indeterminacy of quantum geometry set by the Planck length, $l_P$, that does not occur in a classical background geometry. Entanglement of fields with geometry modifies the transverse phase of field states at wavelength $\lambda$ and propagation distance $c\tau$ by about $\Delta \phi\approx \sqrt{l_P\tau}/\lambda$. The new effect is not detectable in measurements of propagating states that depend only on longitudinal coordinates. The reduced information content of fields in large systems is consistent with holographic bounds from gravitation theory, and may appear as measurable quantum-geometrical noise in interferometers.

### Directional Entanglement of Quantum Fields with Quantum Geometry [Replacement]

It is conjectured that the spatial structure of quantum field states is influenced by a new kind of directional indeterminacy of quantum geometry set by the Planck length, $l_P$, that does not occur in the usual classical background geometry. Entanglement of field and geometry states is described in the small angle approximation, using paraxial wave solutions instead of the usual plane waves. It is shown to have almost no effect on local measurements, microscopic particle interactions, or measurements of propagating states that depend only on longitudinal coordinates, but it significantly alter properties of field states at wavelength $\lambda$ that depend on transverse coordinates or direction, in systems larger than $\approx \lambda^2/l_P$. The reduced information content of fields in large systems is consistent with holographic bounds from gravitation theory, significantly reduces the energy density of field vacua, and may appear as measurable quantum-geometrical noise in interferometers.

### Directional Entanglement of Quantum Fields with Quantum Geometry [Replacement]

It is conjectured that the spatial structure of quantum field states is influenced by a new kind of directional indeterminacy of quantum geometry set by the Planck length, $l_P$, that does not occur in the usual classical background geometry. Entanglement of field and geometry states is described in the small angle approximation, using paraxial wave solutions instead of the usual plane waves. It is shown to have almost no effect on local measurements, microscopic particle interactions, or measurements of propagating states that depend only on longitudinal coordinates, but it significantly alter properties of field states at wavelength $\lambda$ that depend on transverse coordinates or direction, in systems larger than $\approx \lambda^2/l_P$. The reduced information content of fields in large systems is consistent with holographic bounds from gravitation theory, significantly reduces the energy density of field vacua, and may appear as measurable quantum-geometrical noise in interferometers.

### Why the length of a quantum string cannot be Lorentz contracted [Cross-Listing]

We propose a quantum gravity-extended form of the classical length contraction law obtained in Special Relativity. More specifically, the framework of our discussion is the UV self-complete theory of quantum gravity. Against this background, we show how our results are consistent with, i) the generalised form of the Uncertainty Principle (GUP), ii) the so called hoop-conjecture which we interpret, presently, as the saturation of a Lorentz boost by the formation of a black hole in a two-body scattering, and iii) the intriguing notion of "classicalization" of trans-Planckian physics. Pushing these ideas to their logical conclusion, we argue that there is a physical limit to the Lorentz contraction rule in the form of some minimal universal length determined by quantum gravity, say the Planck Length, or any of its current embodiments such as the string length, or the TeV quantum gravity length scale. In the latter case, we determine the \emph{critical boost} that separates the ordinary "particle phase," characterized by the Compton wavelength, from the "black hole phase", characterized by the effective Schwarzschild radius of the colliding system. Finally, with the "classicalization" of quantum gravity in mind, we comment on the remarkable identity, to our knowledge never noticed before, between three seemingly independent universal quantities, namely, a) the "string tension", b) the "linear energy density," or \emph{tension} that exists at the core of all Schwarzschild black holes, and c) the "superforce" i.e., the Planckian limit of the static electro-gravitational force and, presumably, the unification point of all fundamental forces.

### Planck's uncertainty principle and the saturation of Lorentz boosts by Planckian black holes

A basic inconsistency arises when the Theory of Special Relativity meets with quantum phenomena at the Planck scale. Specifically, the Planck length is Lorentz invariant and should not be affected by a Lorentz boost. We argue that Planckian relativity must necessarily involve the effect of black hole formation. Recent proposals for resolving the noted inconsistency seem unsatisfactory in that they ignore the crucial role of gravity in the saturation of Lorentz boosts. Furthermore, an invariant length at he Planck scale amounts to a universal quantum of resolution in the fabric of spacetime. We argue, therefore, that the universal Planck length requires an extension of the Uncertainty Principle as well. Thus, the noted inconsistency lies at the core of Quantum Gravity. In this essay we reflect on a possible resolution of these outstanding problems.

### Distribution of Entropy of Bardeen Regular Black Hole with Corrected State Density

We consider corrections to all orders in the Planck length on the quantum state density, and calculate the statistical entropy of the scalar field on the background of the Bardeen regular black hole numerically. We obtain the distribution of entropy which is inside the horizon of black hole and the contribution of the vicinity of horizon takes a great part of the whole entropy.

### Notes on several phenomenological laws of quantum gravity [Cross-Listing]

Phenomenological approaches to quantum gravity try to infer model-independent laws by analyzing thought experiments and combining both quantum, relativistic, and gravitational ingredients. We first review these ingredients -three basic inequalities- and discuss their relationships with the nature of fundamental constants. In particular, we argue for a covariant mass bound conjecture: in a spacetime free of horizon, the mass inside a surface $A$ cannot exceed $16 \pi G^2 m^2< A$, while the reverse holds in a spacetime with horizons. This is given a precise definition using the formalism of light-sheets. We show that $\hbar/c$ may be also given a geometrical interpretation, namely $4 \pi \hbar^2/m^2< A$. We then combine these inequalities and find/review the following: (1) Any system must have a size greater than the Planck length, in the sense that there exists a minimal area (2) We comment on the Minimal Length Scenarios and the fate of Lorentz symmetry near the Planck scale (3) Quanta with transplanckian frequencies are allowed in a large enough boxes (4) There exists a mass-dependent maximal acceleration given by $m c^3/\hbar$ if $m<m_p$ and by $c^4/G m$ if $m>m_p$ (5) There exists a mass dependent maximal force and power (6) There exists a maximal energy density and pressure (7) Physical systems must obey the Holographic Principle (8) Holographic bounds can only be saturated by systems with $m>m_p$; systems lying on the “Compton line” $l \sim 1/m$ are fundamental objects without substructures (9) We speculate on a new bound from above for the action. In passing, we note that the maximal acceleration is of the order of Milgrom’s acceleration $a_0$ for ultra-light particles ($m\sim H_0)$ that could be associated to the Dark Energy fluid. This suggests designing toy-models in which modified gravity in galaxies is driven by the DE field, via the maximal acceleration principle.

### Building non commutative spacetimes at the Planck length for Friedmann flat cosmologies [Replacement]

We propose physically motivated spacetime uncertainty relations (STUR) for flat Friedmann-Lema\^{i}tre cosmologies. We show that the physical features of these STUR crucially depend on whether a particle horizon is present or not. In particular, when this is the case we deduce the existence of a maximal value for the Hubble rate (or equivalently for the matter density), thus providing an indication that quantum effects may rule out a pointlike big bang singularity. Finally, we costruct a concrete realisation of the corresponding quantum Friedmann spacetime in terms of operators on some Hilbert space.

### Building non commutative spacetimes at the Planck length for Friedmann flat cosmologies [Replacement]

We propose physically motivated spacetime uncertainty relations (STUR) for flat Friedmann-Lema\^{i}tre cosmologies. We show that the physical features of these STUR crucially depend on whether a particle horizon is present or not. In particular, when this is the case we deduce the existence of a maximal value for the Hubble rate (or equivalently for the matter density), thus providing an indication that quantum effects may rule out a pointlike big bang singularity. Finally, we costruct a concrete realisation of the corresponding quantum Friedmann spacetime in terms of operators on a Hilbert space.

### Building non commutative spacetimes at the Planck length for Friedmann flat cosmologies [Replacement]

We propose physically motivated spacetime uncertainty relations (STUR) for flat Friedmann-Lema\^{i}tre cosmologies. We show that the physical features of these STUR crucially depend on whether a particle horizon is present or not. In particular, when this is the case we deduce the existence of a maximal value for the Hubble rate (or equivalently for the matter density), thus providing an indication that quantum effects may rule out a pointlike big bang singularity. Finally, we costruct a concrete realisation of the corresponding quantum Friedmann spacetime in terms of operators on a Hilbert space.

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