Geometro-stochastic quantization of gravity. II

A second-quantized bundleE, called the quantum gravitational bundle, is constructed from graviton bundles in accordance with outlined general principles for geometro-stochastic second quantization, and quantum gravitational frame fields are introduced in it. The gravitational bundleE is the carrier of a semiclassical connection that can be used as a stepping stone in the construction of a second-quantized gravitational connection. The geometro-stochastic quantum propagations governed by such connections can be expressed in terms of path integrals over causal stochastic paths which embody stochastic parallel transport under free-fall conditions. Their epistemic implications for the quantum theory of measurement are discussed.1. Supported in part by the NSERC Research Grant No. A5206.     

Inertial and gravitational mass in quantum mechanics

We show that in complete agreement with classical mechanics, the dynamics of any quantum mechanical wave packet in a linear gravitational potential involves the gravitational and the inertial mass only as their ratio. In contrast, the spatial modulation of the corresponding energy wave function is determined by the third root of the product of the two masses. Moreover, the discrete energy spectrum of a particle constrained in its motion by a linear gravitational potential and an infinitely steep wall depends on the inertial as well as the gravitational mass with different fractional powers. This feature might open a new avenue in quantum tests of the universality of free fall.     

On the impossibility of a box for holding gravitational radiation in thermal equilibrium

In a recent paper David Garfinkle and Robert Wald argue that it is possible to build a box which will confine and thermalize gravitational radiation. Using the results of their calculations I will show that the Garfinkle-Wald (GW) box will fail to isolate and thermalize gravitational radiation in a universe with external gravitational radiation. The absence of alocal equilibrium distribution of gravitational radiation in this model is further evidence that an operational interpretation of a quantum theory of gravity based on General Relativity and traditional matter couplings does not exist.     

A new type of quantum interferometer for gravitational wave detection

We present an orientational quantum interferometer sensitive to gravitational waves that is based on orienting quantum objects like molecules, atoms, or nuclei in space. The detection principle is based on inducing non-sphericity to the corresponding wave functions by light-pulses. In the field of a gravitational wave these objects then possess spectra that depend on their orientation in space. In our measurement scheme we investigate the adiabatic influence of a monochromatic gravitational wave over a quarter gravitational wave period and compare the corresponding frequencies at instances with maximal and vanishing gravitational wave elongation. We therefore explore the effect over a quarter gravitational wave period (or wavelength) and the resulting frequency shift scales with the binding energy of the system times the amplitude of the gravitational wave. In particular, a gravitational wave with amplitude h = 10⁻²³ will induce a frequency shift of the order of 110 μHz for an atom interferometer based on a 91-fold charged uranium ion.     

On Some Processes of Quantum Gravitation (Radiative Corrections)

Within the framework of the quantum theory of interactions of nonpoint (smeared out) particles [1], the radiative corrections due to gravitational interactions are examined. The gravitational masses are calculated for a photon, a graviton, and some other particles together with the renormalized gravitational interaction constant K(m) depending on the interacting particle mass m. The approximate character of the equivalence principle is confirmed.     

Repulsive gravitational effect of a quantum wave packet and experimental scheme with superfluid helium

We consider the gravitational effect of quantum wave packets when quantum mechanics, gravity, and thermodynamics are simultaneously considered. Under the assumption of a thermodynamic origin of gravity, we propose a general equation to describe the gravitational effect of quantum wave packets. In the classical limit, this equation agrees with Newton’s law of gravitation. For quantum wave packets, however, it predicts a repulsive gravitational effect. We propose an experimental scheme using superfluid helium to test this repulsive gravitational effect. Our studies show that, with present technology such as superconducting gravimetry and cold atom interferometry, tests of the repulsive gravitational effect for superfluid helium are within experimental reach.     

Matter and geometry in a unified theory

The prediction of general relativity on the gravitational collapse of matter ending in a point is viewed as an absurdity of the kind to be expected in any consistent physical theory due to ultimate conflicts of the axioms of geometry with the properties of physical objects. The necessity to introduce a probability interpretation for the solution of partial differential equations in space time for quantum theory points to similar roots. It is pointed out that quantum theory in the very small is not going to eliminate the problem, but macroscopic quantum effects in the large, modifying the properties of the horizon, may achieve it. Solutions such as wormholes allow as much empirical evidence as any science fiction. The present approach considers successive modifications of the field equations and equations of motion of gravitational theory by admixture of terms with higher derivatives. The rigorous application of a gauge principle combines Einstein's equations with the tensor analog of Maxwell's equations which are of third order for the metric. It is speculated that the natural presence of torsion in such a gauge theory is related to matter sources.     

On high frequency background quantization of gravity

Considering background quantization of gravitational fields, it is generally assumed that the classical background satisfies Einstein's gravitational equations. However, there exist arguments showing that, for high frequency (quantum) fluctuations, this assumption has to be replaced by a condition describing the back reaction of fluctuations on the background. It is shown that such an approach leads to limitations for the quantum procedure which occur at distances larger than Planck's elementary lengthl=(Gh/c ³)^1/2.     

Electrostatic potential in a gravitational field

The electrostatic potential in a gravitational field is estimated up to the order ofe ² G ² in the framework of the conventional quantum field theory. It is shown that the electrostatic potential is different from the classical one. We find that this discrepancy is attributable to the process in which a particle emits three massless ones which are absorbed by three other particles.     

On the intrinsic entropy of the gravitational field

I show that in linearized general relativity it is impossible to construct a detector by the use of which the quantum state of the linearized gravitational field could be reliably determined. This is because there is no material satisfying the positive energy condition which can serve as a good conductor or absorber of gravitational radiation over a finite range of frequencies. If this property is true of the full theory then one can conclude that a certain proportion of both the energy and information carried by a gravitational wave is irreversibly lost, and that there is a correspondingintrinsic entropy associated with any distribution of gravitational radiation.     

Entropy of a Black Hole with Distinct Surface Gravities

In gravitational thermodynamics, the entropy of a black hole with distinct surface gravities can be evaluated in a microcanonical ensemble. At the WKB level, the entropy becomes the negative of the Euclidean action of the constrained instanton, which is the seed for the black hole creation in the no-boundary universe. Using the Gauss-Bonnet theorem, we prove the quite universal formula in Euclidean quantum gravity that the entropy of a nonrotating black hole is one quarter the sum of the products of the Euler characteristics and the areas of the horizons. For Lovelock gravity, the entropy and quantum creation of a black hole are also studied.     

“Superconducting” causal nets

The world is described as a relativistic quantum neural net with a quantum condensation akin to superconductivity. The sole dynamical variable is an operator representing immediate causal connection. The net enjoys a quantum principle of equivalence implying local LorentzSL(2,C) invariance and causality. The past-future asymmetry of its cell is similar to that of the neutrino. A net phase transition is expected at temperatures on the order of theW mass rather than the Planck mass, and near gravitational singularities.     

Rotational degeneracy breaking of atomic substates: A composite quantum system in a noninertial reference frame

The degenerate magnetic substates of a field-free atomic system are split in a rotating reference frame. The splitting and ordering of the states should be experimentally demonstrable by means of laser-induced quantum interference spectroscopy of hydrogenic Rydberg states. This would manifest dynamical consequences of a noninertial reference frame on theinternal structure of a composite quantum system-in contrast to the observed neutron Sagnac effect, which involves the relative phase of essentially point particles. The predicted level splitting is independent of constituent particle masses; a composite quantum system in a rotating reference frame exhibits effects inequivalent to those that would be engendered in an inertial frame by a gravitational field.The original essay upon which this article is based received honorable mention from the Gravity Research Foundation for 1988.     

Quantum Non-Gravity and Stellar Collapse

Observational indications combined with analyses of analogue and emergent gravity in condensed matter systems support the possibility that there might be two distinct energy scales related to quantum gravity: the scale that sets the onset of quantum gravitational effects E_BE_{\rm B} (related to the Planck scale) and the much higher scale E_LE_{\rm L} signalling the breaking of Lorentz symmetry. We suggest a natural interpretation for these two scales: E_LE_{\rm L} is the energy scale below which a special relativistic spacetime emerges, E_BE_{\rm B} is the scale below which this spacetime geometry becomes curved. This implies that the first ‘quantum’ gravitational effect around E_BE_{\rm B} could simply be that gravity is progressively switched off, leaving an effective Minkowski quantum field theory up to much higher energies of the order of E_LE_{\rm L}. This scenario may have important consequences for gravitational collapse, inasmuch as it opens up new possibilities for the final state of stellar collapse other than an evaporating black hole.     

On the general covariance and strong equivalence principles in quantum general relativity

The various physical aspects of the general relativistic principles of covariance and strong equivalence are discussed, and their mathematical formulations are analyzed. All these aspects are shown to be present in classical general relativity, although no contemporary formulation of canonical or covariant quantum gravity has succeeded to incorporate them all. This has, in part, motivated the recent introduction of a geometro-stochastic framework for quantum general relativity, in which the classical frame bundles that underlie the formulation of parallel transport in classical general relativity are replaced by quantum frame bundles. It is shown that quantum frames can take over the role played by complete sets of observables in conventional quantum theory, so that they can mediate the natural transference of the general covariance and the strong equivalence principles from the classical to the quantum general relativistic regime. This results in a geometrostochastic mode of quantum propagation in general relativistic quantum bundles, which is mathematically implemented by path integration methods based on parallel transport along horizontal lifts of geodesics for the vacuum expectation values of a quantum gravitational field in a quantum spacetime supermanifold. The covariance features of this field are embedded in a quantum gravitational supergroup, which incorporates Poincaré as well as diffeomorphism invariance, and resolves the issue of time in quantum gravity.     

Quantum phenomena in gravitational field

The subjects presented here are very different. Their common feature is that they all involve quantum phenomena in a gravitational field: gravitational quantum states of ultracold antihydrogen above a material surface and measuring a gravitational interaction of antihydrogen in AEGIS, a quantum trampoline for ultracold atoms, and a hypothesis on naturally occurring gravitational quantum states, an Eötvös-type experiment with cold neutrons and others. Considering them together, however, we could learn that they have many common points both in physics and in methodology.     

Pauli equation for a particle in metric and electromagnetic fields

A Pauli theory (Pauli equation and definition of probability current and density) for a particle in weak metric and arbitrary electromagnetic fields is treated. To formulate non-relativistic quantum mechanical problems in arbitrary electromagnetic fields and weak metrics (non-inertial systems, gravitational fields which are distant fields of arbitrary distribution of masses, gravitational waves) it is not necessary to make use of the general-relativistic Dirac equation. Close analogies to the known Pauli theory with electromagnetic fields exist. For different metric fields the corresponding Hamiltonians are given. For quantum systems (H-atoms) which are disturbed by a homogeneous gravitational field and a gravitational wave the resulting shift of energy levels and the transition probability is calculated.     

Stochastic and quantum space-time metrics and the weak-field limit

In this review we present a simple method of introducing stochastic and quantum metrics into gravitational theory at short distances in terms of small fluctuations around a classical background space-time. We consider only residual effects due to the stochastic (or quantum) theory of gravity and use a perturbative stochastization (or quantization) method. By using the general covariance and correspondence principles, we reconstruct the theory of gravitational, mechanical, electromagnetic, and quantum mechanical processes and tensor algebra in the space-time with stochastic and quantum metrics. Some consequences of the theory are also considered, in particular, it indicates that the value of the fundamental lengthl lies in the interval 10^–23l10^–22 cm.     

Quantum Incompressibility of a Falling Rydberg Atom, and a Gravitationally-Induced Charge Separation Effect in Superconducting Systems

Freely falling point-like objects converge toward the center of the Earth. Hence the gravitational field of the Earth is inhomogeneous, and possesses a tidal component. The free fall of an extended quantum mechanical object such as a hydrogen atom prepared in a high principal-quantum-number state, i.e. a circular Rydberg atom, is predicted to fall more slowly than a classical point-like object, when both objects are dropped from the same height above the Earth’s surface. This indicates that, apart from transitions between quantum states, the atom exhibits a kind of quantum mechanical incompressibility during free fall in inhomogeneous, tidal gravitational fields like those of the Earth.     

A Gravitational Potential with Extra-dimensions and Spin Effects in Hadronic Reactions

The impact of the KK-modes in d-brane models of gravity with large compactification radii and TeV-scale quantum gravity on the hadronic potential at small impact parameters is examined. The effects of the gravitational hadron form factors obtained from the hadron generalized parton distributions (GPDs) on the behavior of the gravitational potential and the possible spin correlation effects are also analysed.