Gravity and Spin Forces in Gravitational Quantum Field Theory

In the new framework of gravitational quantum field theory (GQFT) with spin and scaling gauge invariance developed in Phys. Rev. D 93 (2016) 024012-1, we make a perturbative expansion for the full action in a background field which accounts for the early inflationary universe. We decompose the bicovariant vector fields of gravifield and spin gauge field with Lorentz and spin symmetries SO(1,3) and SP(1,3) in biframe spacetime into SO(3) representations for deriving the propagators of the basic quantum fields and extract their interaction terms. The leading order Feynman rules are presented. A tree-level 2 to 2 scattering amplitude of the Dirac fermions, through a gravifield and a spin gauge field, is calculated and compared to the Born approximation of the potential. It is shown that the Newton's gravitational law in the early universe is modified due to the background field. The spin dependence of the gravitational potential is demonstrated.     

Gravity and Spin Forces in Gravitational Quantum Field Theory

In the new framework of gravitational quantum field theory(GQFT) with spin and scaling gauge invariance developed in Phys. Rev. D 93(2016) 024012-1, we make a perturbative expansion for the full action in a background field which accounts for the early inflationary universe. We decompose the bicovariant vector fields of gravifield and spin gauge field with Lorentz and spin symmetries SO(1,3) and SP(1,3) in biframe spacetime into SO(3) representations for deriving the propagators of the basic quantum fields and extract their interaction terms. The leading order Feynman rules are presented. A tree-level 2 to 2 scattering amplitude of the Dirac fermions, through a gravifield and a spin gauge field, is calculated and compared to the Born approximation of the potential. It is shown that the Newton's gravitational law in the early universe is modified due to the background field. The spin dependence of the gravitational potential is demonstrated.     

Three-Dimensional Quantum Gravity,Chern-Simons Theory,and the A-Polynomial

We study three-dimensional Chern-Simons theory with complex gauge group SL(2,), which has many interesting connections with three-dimensional quantum gravity and geometry of hyperbolic 3-manifolds. We show that, in the presence of a single knotted Wilson loop in an infinite-dimensional representation of the gauge group, the classical and quantum properties of such theory are described by an algebraic curve called the A-polynomial of a knot. Using this approach, we find some new and rather surprising relations between the A-polynomial, the colored Jones polynomial, and other invariants of hyperbolic 3-manifolds. These relations generalize the volume conjecture and the Melvin-Morton-Rozansky conjecture, and suggest an intriguing connection between the SL(2,) partition function and the colored Jones polynomial.     

Spin in Relativistic Quantum Theory

We discuss the role of spin in Poincaré invariant formulations of quantum mechanics.     

Some Possible Roles for Topos Theory in Quantum Theory and Quantum Gravity

We discuss some ways in which topos theory (a branch of category theory) can be applied to interpretative problems in quantum theory and quantum gravity. In Sec.1, we introduce these problems. In Sec.2, we introduce topos theory, especially the idea of a topos of presheaves. In Sec.3, we discuss several possible applications of topos theory to the problems in Sec.1. In Sec.4, we draw some conclusions.     

Perturbation Theory for Open Two-Level Nonlinear Quantum Systems

Perturbation theory is an important tool in quantum mechanics. In this paper, we extend the traditional perturbation theory to open nonlinear two-level systems, treating decoherence parameterγ as a perturbation. By this virtue, we give a perturbative solution to the master equation, which describes a nonlinear open quantum system. The results show that for small decoherence rateγ, the ratio of the nonlinear rate C to the tunneling coefficient V (i.e., r=C/V) determines the validity of the perturbation theory. For small ratio r, the perturbation theory is valid, otherwise it yields wrong results.     

Spin Structure of Quantum Transport Theory

Relativistic quantum transport theory is investigated in the Nambu-Jona-Lasinio model. The spin structure of the kinetic equations is systematically analyzed. The covariant transport and constraint equations are derived in a matrix form in the spin space.     

Renormalizable Quantum Gauge Theory of Gravity

The quantum gravity is formulated based on the principle of local gauge invariance. The model discussedin this paper has local gravitational gauge symmetry, and gravitational field is represented by gauge field. In the leading-order approximation, it gives out classical Newton‘s theory of gravity. In the first-order approximation and for vacuum,it gives out Einstein‘s general theory of relativity. This quantum gauge theory of gravity is a renormalizable quantumtheory.     

Renormalizable Quantum Gauge Theory of Gravity

The quantum gravity is formulated based on the principle of local gauge invariance. The model discussed in this paper has local gravitational gauge symmetry, and gravitational field is represented by gauge field. In the leading-order approximation, it gives out classical Newton's theory of gravity. In the first-order approximation and for vacuum, it gives out Einstein's general theory of relativity. This quantum gauge theory of gravity is a renormalizable quantum theory.     

量子场论中的自旋算符

从量子场论的角度对相对论粒子的运动自旋概念作了进一步深入研究.构造了场量子自旋以及场系统运动自旋两个新算符.给出了场量子自旋动量空间的显式表达式以及用Poincar啨群生成元表示的场系统运动自旋的显式表达式.借助这两个算符,可以干净地解决有关场自旋的问题,表明它们才是场自旋的恰当的算符.     

Deterrents to a Theory of Quantum Gravity

As shown previously, quantum mechanics directly violates the weak equivalence principle in general, and thus indirectly violates the strong equivalence principle in all dimensions. The present paper shows that quantum mechanics also directly violates the strong equivalence principle unless it is arbitrarily abetted in hindsight. Vital domains are shown to exist in which quantum gravity would be non-applicable. There are classical subtleties in which the strong equivalence principle appears to be violated, but is not. Neutron free fall interference experiments in a gravitational field are examined, as is Galileo's falling body assertion and the misconception it leads to.     

A Kinetic Theory Approach to Quantum Gravity

We describe a kinetic theory approach to quantum gravity by which we mean a theory of the microscopic structure of space-time, not a theory obtained by quantizing general relativity. A figurative conception of this program is like building a ladder with two knotty poles: quantum matter field on the right and space-time on the left. Each rung connecting the corresponding knots represents a distinct level of structure. The lowest rung is hydrodynamics and general relativity; the next rung is semiclassical gravity, with the expectation value of quantum fields acting as source in the semiclassical Einstein equation. We recall how ideas from the statistical mechanics of interacting quantum fields helped us identify the existence of noise in the matter field and its effect on metric fluctuations, leading to the establishment of the third rung: stochastic gravity, described by the Einstein–Langevin equation. Our pathway from stochastic to quantum gravity is via the correlation hierarchy of noise and induced metric fluctuations. Three essential tasks beckon: (1) deduce the correlations of metric fluctuations from correlation noise in the matter field; (2) reconstituting quantum coherence—this is the reverse of decoherence—from these correlation functions; and (3) use the Boltzmann–Langevin equations to identify distinct collective variables depicting recognizable metastable structures in the kinetic and hydrodynamic regimes of quantum matter fields and how they demand of their corresponding space-time counterparts. This will give us a hierarchy of generalized stochastic equations—call them the Boltzmann–Einstein hierarchy of quantum gravity—for each level of space-time structure, from the the macroscopic (general relativity) through the mesoscopic (stochastic gravity) to the microscopic (quantum gravity).     

Quantum Gravity,Information Theory and the CMB

We review connections between the metric of spacetime and the quantum fluctuations of fields. We start with the finding that the spacetime metric can be expressed entirely in terms of the 2-point correlator of the fluctuations of quantum fields. We then discuss the open question whether the knowledge of only the spectra of the quantum fluctuations of fields also suffices to determine the spacetime metric. This question is of interest because spectra are geometric invariants and their quantization would, therefore, have the benefit of not requiring the modding out of diffeomorphisms. Further, we discuss the fact that spacetime at the Planck scale need not necessarily be either discrete or continuous. Instead, results from information theory show that spacetime may be simultaneously discrete and continuous in the same way that information can. Finally, we review the recent finding that a covariant natural ultraviolet cutoff at the Planck scale implies a signature in the cosmic microwave background (CMB) that may become observable.     

Schwarzschild-de-Sitter Solution in Quantum Gauge Theory of Gravity

We use the theory based on the gravitational gauge group G to obtain a spherical symmetric solution of the field equations for the gravitational potentials on a Minkowski space-time. The gauge group G is defined and then we introduce the gauge-covariant derivative Dμ. The strength tensor of the gravitational gauge field is also obtained and a gauge-invariant Lagrangian including the cosmological constant is constructed. A model whose gravitational gauge potentials A^α μ （x） have spherical symmetry, depending only on the radial coordinate τ is considered and an analytical solution of these equations, which induces the Schwarzschild-de-Sitter metric on the gauge group space, is then determined. All the calculations have been performed by GR Tensor II computer algebra package, running on the Maple V platform, along with several routines that we have written for our model.     

Search for Quantum Gravity

A satisfactory theory of quantum gravity maynecessitate a drastic modification of our perception ofspace-time, by giving it a foamy structure at distancescomparable to the Planck length. It is argued in this essay that the experimental detectionof such structures may be a realistic possibility in theforeseeable future. After a brief review of differenttheoretical approaches to quantum gravity and the relationships between them, we discussvarious possible experimental tests of the quantumnature of space-time. Observations of photons fromdistant astrophysical sources such as Gamma-Ray Bursters and laboratory experiments on neutral kaon decays may be sensitive to quantum-gravitational effectsif they are only minimally suppressed. Experimentallimits from the Whipple Observatory and the CPLEAR Collaboration are already probing close to the Planck scale, and significant increases in sensitivityare feasible.     

Random Matrix Theory and Entanglement in Quantum Spin Chains

We compute the entropy of entanglement in the ground states of a general class of quantum spin-chain Hamiltonians — those that are related to quadratic forms of Fermi operators — between the first N spins and the rest of the system in the limit of infinite total chain length. We show that the entropy can be expressed in terms of averages over the classical compact groups and establish an explicit correspondence between the symmetries of a given Hamiltonian and those characterizing the Haar measure of the associated group. These averages are either Toeplitz determinants or determinants of combinations of Toeplitz and Hankel matrices. Recent generalizations of the Fisher-Hartwig conjecture are used to compute the leading order asymptotics of the entropy as N. This is shown to grow logarithmically with N. The constant of proportionality is determined explicitly, as is the next (constant) term in the asymptotic expansion. The logarithmic growth of the entropy was previously predicted on the basis of numerical computations and conformal-field-theoretic calculations. In these calculations the constant of proportionality was determined in terms of the central charge of the Virasoro algebra. Our results therefore lead to an explicit formula for this charge. We also show that the entropy is related to solutions of ordinary differential equations of Painlevé type. In some cases these solutions can be evaluated to all orders using recurrence relations.Acknowledgement We gratefully acknowledge stimulating discussions with Estelle Basor, Peter Forrester and Noah Linden. We are also grateful for the kind hospitality of the Isaac Newton Institute for the Mathematical Sciences, Cambridge, while this research was completed. Francesco Mezzadri was supported by a Royal Society Dorothy Hodgkin Research Fellowship.     

Algebraic Quantum Field Theory, Perturbation Theory, and the Loop Expansion

The perturbative treatment of quantum field theory is formulated within the framework of algebraic quantum field theory. We show that the algebra of interacting fields is additive, i.e. fully determined by its subalgebras associated to arbitrary small subregions of Minkowski space. We also give an algebraic formulation of the loop expansion by introducing a projective system ?^{( n )} of observables “up to n loops”, where ?⁽⁰⁾ is the Poisson algebra of the classical field theory. Finally we give a local algebraic formulation for two cases of the quantum action principle and compare it with the usual formulation in terms of Green's functions. Received: 9 February 2000 / Accepted: 21 March 2000