# Posts Tagged planck length

## Recent Postings from planck length

### From Schwinger Balls to Black Holes [Cross-Listing]

We have shown intriguing similarities between Schwinger balls and black holes. By considering black hole as a gravitational Schwinger ball, we have derived the Bekenstein-Hawking entropy and the first law of black hole thermodynamics as a direct result of the inverse area dependence of the gravitational force. It is also shown that the Planck length is nothing but the gravitational Schwinger length. The relation between the mass and the radius of the black hole is derived by considering the black hole as a Schwinger ball of gravitons. We show how the evolution of the entanglement entropy of the black hole, as Page introduced many years ago, can be obtained by including gravitons in the black hole's evaporation process and using a deformed EPR mechanism. Also this deformed EPR mechanism can solve the information paradox. We show how naive simultaneous usage of Page's argument and equivalence principle leads to firewall problem.

### The Effects of Minimal Length, Maximal Momentum and Minimal Momentum in Entropic Force

In this paper, the modified entropic force law is studied by using a new kind of generalized uncertainty principle which contains a minimal length, a minimal momentum and a maximal momentum. Firstly, the quantum corrections to the thermodynamics of a black hole is investigated. Then, according to Verlinde's theory, the generalized uncertainty principle (GUP) corrected entropic force is obtained. The result shows that the GUP corrected entropic force is related not only to the properties of the black holes, but also to the Planck length and the dimensionless constants \alpha_0\ and \beta_0\. Moreover, based on the GUP corrected entropic force, we also derive the modified Einstein's field equation (EFE) and the modified Friedmann equation.

### The Effects of Minimal Length, Maximal Momentum and Minimal Momentum in Entropic Force [Replacement]

In this paper, the modified entropic force law is studied by using a new kind of generalized uncertainty principle which contains a minimal length, a minimal momentum and a maximal momentum. Firstly, the quantum corrections to the thermodynamics of a black hole is investigated. Then, according to Verlinde's theory, the generalized uncertainty principle (GUP) corrected entropic force is obtained. The result shows that the GUP corrected entropic force is related not only to the properties of the black holes, but also to the Planck length and the dimensionless constants $\alpha _{\rm{0}}$ and $\beta _{\rm{0}}$. Moreover, based on the GUP corrected entropic force, we also derive the modified Einstein's field equation (EFE) and the modified Friedmann equation.

### Lorentz invariant CPT breaking in the Dirac equation

If one modifies the Dirac equation in momentum space to $[\gamma^{\mu}p_{\mu}-m-\Delta m(\theta(p_{0})-\theta(-p_{0})) \theta(p_{\mu}^{2})]\psi(p)=0$, the symmetry of positive and negative energy eigenvalues is lifted by $m\pm \Delta m$ for a small $\Delta m$. The mass degeneracy of the particle and antiparticle is thus lifted in a Lorentz invariant manner since the combinations $\theta(\pm p_{0})\theta(p_{\mu}^{2})$ with step functions are manifestly Lorentz invariant. We explain an explicit construction of this CPT breaking term in coordinate space, which is Lorentz invariant but non-local at a distance scale of the Planck length. The application of this Lorentz invariant CPT breaking mechanism to the possible mass splitting of the neutrino and antineutrino in the Standard Model is briefly discussed.

### Probing quantum commutators [Cross-Listing]

Quantum gravity theories predict a minimal length at the order of magnitude of the Planck length, under which the concepts of space and time lose every physical meaning. In quantum mechanics, the insurgence of such minimal length can be described by introducing a modified position-momentum commutator, which in turn yields a generalized uncertainty principle, where the uncertainty on the position measurement has a lower bound. The value of the minimal length is not predicted by theories and must be evaluated experimentally. In this paper, we address the quantum bound to estimability of the minimal uncertainty length by performing measurements on a harmonic oscillator, which is analytically solvable in the deformed algebra of the Hilbert subspace.

### Probing deformed quantum commutators [Replacement]

Quantum gravity theories predict a minimal length at the order of magnitude of the Planck length, under which the concepts of space and time lose every physical meaning. In quantum mechanics, the insurgence of such minimal length can be described by introducing a modified position-momentum commutator, which in turn yields a generalized uncertainty principle, where the uncertainty on the position measurement has a lower bound. The value of the minimal length is not predicted by theories and must be evaluated experimentally. In this paper, we address the quantum bound to estimability of the minimal uncertainty length by performing measurements on a harmonic oscillator, which is analytically solvable in the deformed algebra of the Hilbert subspace.

### The Variation of Photon Speed with Photon Frequency in Quantum Gravity

Einstein's special relativity is Lorentz invariance; the postulate that all observers measure exactly the same speed of light in vacuum, independent of photon frequency. There is a fundamental scale the Planck scale, at which quantum effects are expected to strongly affect the nature of space-time. The commonly used ideas of space-time should break down at or before the Planck length is reached. It is then natural to question the exactness of the Lorentz invariance that is pervasive in all macroscopic theories. Quantum gravity effect could be seen from the dispersion relations violating Lorentz invariance, because the motivation for the Lorentz invariance violation is quantum gravity. Then it is expected that the energy-momentum dispersion relation could be modified to include the dependence on the ratio of the particle's energy and the quantum gravity energy. In the present work, we have derived an expression of Planck mass or Planck energy by equating the Compton wavelength with Kerr gravitational radius of the Kerr rotating body. Then we derived the modified expression for the photon energy-momentum dispersion relation and hence derived the variation of the photon propagation speed with photon frequency.

### On a nonlinear gravitational wave. Geodesics

An exact, plane wave solution of the gravitational field equations is investigated. The source stress tensor is represented by an anisotropic null fluid with energy flux to which the energy density $\rho$ and the pressure $p_{z}$ are negative but finite throughout the spacetime. They depend on a constant length (taken of the order of the Planck length) and acquire Planck values close to the null surface $t - z = 0$, $Oz$ axis being the direction of propagation. The timelike geodesics of a test particle are contained in a plane whose normal has constant direction and the null trajectories are comoving with a plane of fixed direction.

### On a nonlinear gravitational wave. Geodesics [Cross-Listing]

An exact, plane wave solution of the gravitational field equations is investigated. The source stress tensor is represented by an anisotropic null fluid with energy flux to which the energy density $\rho$ and the pressure $p_{z}$ are negative but finite throughout the spacetime. They depend on a constant length (taken of the order of the Planck length) and acquire Planck values close to the null surface $t - z = 0$, $Oz$ axis being the direction of propagation. The timelike geodesics of a test particle are contained in a plane whose normal has constant direction and the null trajectories are comoving with a plane of fixed direction.

### On a nonlinear gravitational wave. Geodesics [Replacement]

An exact, plane wave solution of the gravitational field equations is investigated. The source stress tensor is represented by an anisotropic null fluid with energy flux to which the energy density $\rho$ and the pressure $p_{z}$ are negative but finite throughout the spacetime. They depend on a constant length (taken of the order of the Planck length) and acquire Planck values close to the null surface $t - z = 0$, the $z$-axis being the direction of propagation. The timelike geodesics of a test particle are contained in a plane whose normal has constant direction and the null trajectories are comoving with a plane of fixed direction.

### On a nonlinear gravitational wave. Geodesics [Replacement]

An exact, plane wave solution of the gravitational field equations is investigated. The source stress tensor is represented by an anisotropic null fluid with energy flux to which the energy density $\rho$ and the pressure $p_{z}$ are negative but finite throughout the spacetime. They depend on a constant length (taken of the order of the Planck length) and acquire Planck values close to the null surface $t - z = 0$, the $z$-axis being the direction of propagation. The timelike geodesics of a test particle are contained in a plane whose normal has constant direction and the null trajectories are comoving with a plane of fixed direction.

### On a nonlinear gravitational wave. Geodesics [Replacement]

An exact, plane wave solution of the gravitational field equations is investigated. The source stress tensor is represented by an anisotropic null fluid with energy flux to which the energy density $\rho$ and all pressures are finite throughout the spacetime. They depend on a constant length (taken of the order of the Planck length) and acquire Planck values close to the null surface $t - z = 0$, the $z$-axis being the direction of propagation. The timelike geodesics of a test particle are contained in a plane whose normal has constant direction and the null trajectories are comoving with a plane of fixed direction.

### On a nonlinear gravitational wave. Geodesics [Replacement]

An exact, plane wave solution of the gravitational field equations is investigated. The source stress tensor is represented by an anisotropic null fluid with energy flux to which the energy density $\rho$ and all pressures are finite throughout the spacetime. They depend on a constant length (taken of the order of the Planck length) and acquire Planck values close to the null surface $t - z = 0$, the $z$-axis being the direction of propagation. The timelike geodesics of a test particle are contained in a plane whose normal has constant direction and the null trajectories are comoving with a plane of fixed direction.

### 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.

### 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.

### 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 [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

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 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 Quantum Geometry [Replacement]

Estimates are presented of correlated large scale rotational fluctuations in a flat space-time assembled from noncommuting quantum elements at the Planck length $l_P$. They are visualized as "quantum twists of space" about any observer: the orientation of a nonrotating inertial frame at separation $R$ fluctuates relative to more distant space at an angular rate $\omega$ with vanishing mean, but with an exotic variance $\langle\omega^2 \rangle \approx c^2 l_PR^{-3}$ on a timescale $\approx R/c$. It is shown that the expected centrifugal acceleration matches observed cosmic acceleration on the scale where rms rotational displacements match the strong interaction scale.

### 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.

### Exotic Rotational Correlations in Quantum Geometry [Replacement]

Estimates are presented of correlated large scale rotational fluctuations in a flat space-time assembled from noncommuting quantum elements at the Planck length $l_P$. They are visualized as "quantum twists of space" about any observer: the orientation of a nonrotating inertial frame at separation $R$ fluctuates relative to more distant space at an angular rate $\omega$ with vanishing mean, but with an exotic variance $\langle\omega^2 \rangle \approx c^2 l_PR^{-3}$ on a timescale $\approx R/c$. It is shown that the expected centrifugal acceleration matches observed cosmic acceleration on the scale where rms rotational displacements match the strong interaction scale.

### 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 correlations of directional motion 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 $R/c$. Entanglement leads to exotic correlations in the phase of transversely-propagating massless fields: "quantum twists of space" that approximate a transverse random walk on causal boundaries around an observer of a Planck step 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 their 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 directional motion 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 $R/c$. Entanglement leads to exotic correlations in the phase of transversely-propagating massless fields: "quantum twists of space" that approximate a transverse random walk on causal boundaries around an observer of a Planck step 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 their 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 directional motion on large scales from the imperfect emergence of inertial frames from quantum geometry at the Planck length $l_P$. Extrapolation of standard quantum theory and gravity suggests that a constant direction in an inertial frame defined on scale $R$ fluctuates relative to distant space at an angular rate $\omega$ with vanishing mean, but with an exotic variance $\langle\omega^2 \rangle \approx c^2 l_PR^{-3}$ on a timescale $\approx R/c$. These are interpreted as "quantum twists of space" from entangled Planck scale quantum elements: a random transverse displacement on each successive light cone by a Planck length every Planck time. Projection of exotic correlation onto an interferometer signal is shown to vanish unless the light path sweeps out a nonzero spatial area. It is conjectured that exotic rotational correlations entangled 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 correlations of directional motion on large scales from the imperfect emergence of inertial frames from quantum geometry at the Planck length $l_P$. Extrapolation of standard quantum theory and gravity suggests that a constant direction in an inertial frame defined on scale $R$ fluctuates relative to distant space at an angular rate $\omega$ with vanishing mean, but with an exotic variance $\langle\omega^2 \rangle \approx c^2 l_PR^{-3}$ on a timescale $\approx R/c$. These are interpreted as "quantum twists of space" from entangled Planck scale quantum elements: a random transverse displacement on each successive light cone by a Planck length every Planck time. Projection of exotic correlation onto an interferometer signal is shown to vanish unless the light path sweeps out a nonzero spatial area. It is conjectured that exotic rotational correlations entangled 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 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.

### 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

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.