Quantizing gravity and spacetime: Where do we stand ?

We review five possible solutions to the riddle posed by Quantum Gravity: (1) Gravity should stay as a classical theory (L. Rosenfeld); (2) Quantum Gravity requires a formalism which will take the human mind (or the intelligent observer) into account, resolving at the same time the riddle of the collapse of the wave function/state vector in Quantum Mechanics in general (Penrose); (3) Perturbative Quantization; (4) Hamiltonian Quantization (Dirac, Ashtekar); (5) String Theory. We also discuss the quantization of spacetime.     

Massive Gauge Fields and the Planck Scale

No Heading The present work is devoted to massive gauge fields in special relativity with two fundamental constants- the velocity of light, and the Planck length, so called doubly special relativity (DSR). The two invariant scales are accounted for by properly modified boost parameters. Within above framework we construct the vector potential as the (1/2, 0) (0, 1/2) direct product, build the associated field strength tensor together with the Dirac spinors and use them to calculate various observables as functions of the Planck length.     

Quantenphysikalischer Ursprung der Eichidee

The Quantum Physical Origin of the Gauge Idea To consider quantum physics as an interplay of creation and annihilation processes has the consequence that gauge field theories are not only possible but necessary. Since the complex conjugate phase factors of each pair of fermion creators and annihilators can be arbitrary chosen, quantum field theories must be completely phase invariant. Unfortunately, even globally the Dirac equation for systems of free fermions is not phase invariant. The Dirac matrices are namely transformed, if we multiply the spinor components by different constant phase factors. The Dirac equations before and after the transformation are however physically equivalent. We may therefore say: Systems of free fermions will be completely described, only if we consider the class of all equivalent Dirac equations. Since Dirac's commutation relations are unitarily invariant, the class equivalent Dirac equations is invariant under all transformations of the group U 4. Unitary diagonal matrices yield arbitrary phase transformations. Hence, gauge fields of the group U 4 are compatible with the postulate of general phase invariance. These gauge file are so similar to the QED that we may speak of an “extended quantum electrodynamics”, EQE. Here, we will show that EQE exists. The invariant subgroup U 1 U 4 yields QED. The complementary subgroup SU 4 includes four subgroups SU 3, there subgroups O 4, and six subgroups SU 2. The latter ones may yield three pairs of quarks and three pairs of leptons, where the quarks form a group SU 3. More than two times three pairs of elementary fermions does not exist in in EQE Probably, EQE is different from the United EQD and QCD. However, it should be a promising version of a field theory in elementary particle physics, because it follows from an existing symmetry of the empirically wel founded Dirac theory. EQE is therefore free from hypothesis in the Newtonian sense of the word. Whatever it will finally mean, it cannot be rejected, since phase invariance must be required. The invention of new symmetries and the acception of a bie number of independent spinor components is dispensable or must be postponed at least.     

Gauge invariant theories of the generalized chiral Schwinger model

The gauge invariant theories of the generalized chiral Schwinger model are constructed in terms of two schemes with and without Wess-Zumino terms, respectively. Following the former scheme, we calculate the Wess-Zumino term which cancels the gauge anomaly, and then constitute the gauge invariant theory by adding the Wess-Zumino term to the original Lagrangian of the model. According to the latter, we modify the original Hamiltonian by adding a term composed of constraints of the model. It is so designed that the theory described by the modified Hamiltonian and its corresponding first-order Lagrangian maintains gauge invariance. We show by the canonical Dirac method that each of the two gauge invariant theories has the same physical spectrum as that of the original gauge noninvariant formulation.     

Gauge invariant theories of the generalized chiral Schwinger model

The gauge invariant theories of the generalized chiral Schwinger model are constructed in terms of two schemes with and without Wess-Zumino terms, respectively. Following the former scheme, we calculate the Wess-Zumino term which cancels the gauge anomaly, and then constitute the gauge invariant theory by adding the Wess-Zumino term to the original Lagrangian of the model. According to the latter, we modify the original Hamiltonian by adding a term composed of constraints of the model. It is so designed that the theory described by the modified Hamiltonian and its corresponding first-order Lagrangian maintains gauge invariance. We show by the canonical Dirac method that each of the two gauge invariant theories has the same physical spectrum as that of the original gauge noninvariant formulation.     

Spin and Statistics and First Principles

It was shown in the early seventies that, in Local Quantum Theory (that is the most general formulation of Quantum Field Theory, if we leave out only the unknown scenario of Quantum Gravity) the notion of Statistics can be grounded solely on the local observable quantities (without assuming neither the commutation relations nor even the existence of unobservable charged field operators); one finds that only the well known (para)statistics of Bose/Fermi type are allowed by the key principle of local commutativity of observables. In this frame it was possible to formulate and prove the Spin and Statistics Theorem purely on the basis of First Principles.     

Review: Dirac's Observables for the Rest-Frame Instant Form of Tetrad Gravity in a Completely Fixed 3-Orthogonal Gauge

We define the rest-frame instant form of tetrad gravity restricted to Christodoulou-Klainermann spacetimes. After a study of the Hamiltonian group of gauge transformations generated by the 14 first class constraints of the theory, we define and solve the multitemporal equations associated with the rotation and space diffeomorphism constraints, finding how the cotriads and their momenta depend on the corresponding gauge variables. This allows to find a quasi-Shanmugadhasan canonical transformation to the class of 3-orthogonal gauges and to find the Dirac observables for superspace in these gauges. The construction of the explicit form of the transformation and of the solution of the rotation and supermomentum constraints is reduced to solve a system of elliptic linear and quasi-linear partial differential equations. We then show that the superhamiltonian constraint becomes the Lichnerowicz equation for the conformal factor of the 3-metric and that the last gauge variable is the momentum conjugated to the conformal factor. The gauge transformations generated by the superhamiltonian constraint perform the transitions among the allowed foliations of spacetime, so that the theory is independent from its 3+1 splittings. In the special 3-orthogonal gauge defined by the vanishing of the conformal factor momentum we determine the final Dirac observables for the gravitational field even if we are not able to solve the Lichnerowicz equation. The final Hamiltonian is the weak ADM energy restricted to this completely fixed gauge.     

Chiral Asymmetry and the Spectral Action

We consider orthogonal connections with arbitrary torsion on compact Riemannian manifolds. For the induced Dirac operators, twisted Dirac operators and Dirac operators of Chamseddine-Connes type we compute the spectral action. In addition to the Einstein-Hilbert action and the bosonic part of the Standard Model Lagrangian we find the Holst term from Loop Quantum Gravity, a coupling of the Holst term to the scalar curvature and a prediction for the value of the Barbero-Immirzi parameter.     

Cornwall-Jackiw-Tomboulis Effective Potential for Quark Propagator in Real-Time Thermal Field Theory and Landau Gauge

We complete the derivation of the Cornwall-Jackiw-Tomboulis effective potentiM for quark propagator at finite temperature and finite quark chemical potential in the real-time formalism of thermal field theory and in Landau gauge. In the approximation that the function A（p^2） in inverse quark propagator is replaced by unity, by means of the running gauge coupling and the quark mass function invariant under the renormalization group in zero temperature Quantum Chromadynamics （QCD）, we obtain a calculable expression for the thermal effective potential, which will be a useful means to research chiral phase transition in QCD in the real-time formalism.     

Cornwall-Jackiw-Tomboulis Effective Potential for Quark Propagator in Real-Time Thermal Field Theory and Landau Gauge

We complete the derivation of the Cornwall-Jackiw-Tomboulis effective potential for quark propagator at finite temperature and finite quark chemical potential in the real-time formalism of thermal field theory and in Landau gauge. In the approximation that the function A(p2) in inverse quark propagator is replaced by unity, by means of the running gauge coupling and the quark mass function invariant under the renormalization group in zero temperature Quantum Chromadynamics (QCD), we obtain a calculable expression for the thermal effective potential, which will be a useful means to research chiral phase transition in QCD in the real-time formalism.     

Gauge invariance and the dirac equation

The gauge invariance of the Dirac equation is reviewed and gauge-invariant operators are defined. The Hamiltonian is shown to be gauge dependent, and an energy operator is defined which is gauge invariant. Gauge-invariant operators corresponding to observables are shown to satisfy generalized Ehrenfest theorems. The time rate of change of the expectation value of the energy operator is equal to the expectation value of the power operator. The virial theorem is proved for a relativistic electron in a time-varying electromagnetic field. The conventional approach to probability amplitudes, using the eigenstates of the unperturbed Hamiltonian, is shown in general to be gauge dependent. A gaugeinvariant procedure for probability amplitudes is given, in which eigenstates of the energy operator are used. The two methods are compared by applying them to an electron in a zero electromagnetic field in an arbitrary gauge. Presented at the Dirac Symposium, Loyola University, New Orleans, May 1981.     

On the Mathematical Basis of the Dirac Formulation of Quantum Mechanics

We discuss various attempts to implement mathematically the Dirac formulation of Quantum Mechanics. A first attempt used Hilbert space. This formalization realizes the Dirac formalism if and only if the spectra of the observables under consideration is purely discrete. Therefore, generalized spectral decompositions are needed. These spectral decompositions can be constructed in the framework of rigged Hilbert spaces. We construct generalized spectral decompositions for self-adjoint operators using their spectral measures. We review the previous work by Marlow (in Hilbert spaces), Antoine, Roberts, and Melsheimer and complete it. We show that these generalized spectral decompositions fit well in the framework of a theory constructed by Kato and Kuroda and that all the results can be reproduced in this framework.     

Mach Like Principle from Conserved Charges

We study models where the gauge coupling constants, masses and the gravitational constant are functions of some conserved charge in the universe, and furthermore a cosmological constant that depends on the total charge of the universe. We first consider the standard Dirac action, but where the mass and the electromagnetic coupling constant are a function of the charge in the universe and afterwards extend this to curved spacetime and consider gauge coupling constants, the gravitational constant and the mass as a function of the charge of the universe, which represent a sort of Mach principle for all the constants of nature. In the flat space formulation, the formalism is not manifestly Lorentz invariant, however Lorentz invariance can be restored by performing a phase transformation of the Dirac field. One interesting model of this type is one where the action is invariant under rescalings of the Dirac wave function. In the curved space time formulation, there is the additional feature that some of the equations of motion break the general coordinate invariance also, but in a way that can be understood as a coordinate choice only, so the equations are still of the General Relativity type, but with a certain natural coordinate choice, where there is no current of the charge. We have generalized what we have done and also constructed a cosmological constant which depends on the total charge of the universe. We discuss how these ideas work when the space where the charges live is finite. If we were to use some only approximately conserved charge for these constructions, like say baryon number (in the context of the standard model), this will lead to corresponding violations of Lorentz symmetry in the early universe for example. We also briefly discuss another non-local formulations where the coupling constants are functions of the Pontryagin index of some non-abelian gauge field configurations. The construction of charge dependent contributions can also be motivated from the structure of the “infra-red counter terms” needed to cancel infra red divergences for example in three dimensions.     

Separability,gauge invariance and nonsuperluminality in direct interaction dynamics

We discuss the classical mechanics of relativistic systems with direct interaction. We show that various desiderata can all be accommodated in the single time approach by restricting the observables to the gauge invariant variables. We show how such observables can be constructed in general. We explicitly construct position observables in a general system and show that they lead to separable, invariant world lines. Nonsuperluminality is explicitly demonstrated for two body systems interacting via central forces of semibounded magnitude provided they ensure timelike canonical momenta. For two particles, our results reproduce the usual solution in covariant equal-time gauge.     

Greens functions of 2-dimensional Yang-Mills theories on nonorientable surfaces

By using the path integral method, we calculate the Green functions of the field strength of Yang-Mills theories on arbitrary nonorientable surfaces in Schwinger-Fock gauge.We show that the non-gauge invariant correlators consist of a free part and an almost x-independent part. We also show that the gauge invariant n-point functions are those corresponding to the free part, as in the case of orientable surfaces.     

Invariant information and quantum state estimation

The invariant information, introduced by C. Brukner and A. Zeilinger [Phys. Rev. Lett. 83, 3354 (1999)], is reconsidered from the point of view of quantum state estimation. We show that this quantity is directly related to the mean error of the standard reconstruction from the measurement of a complete set of mutually complementary observables. We give its generalization in terms of the Fisher information. Provided that the optimum reconstruction is adopted, the information loses its invariant character.     

Dynamics of Two-Photon Lasers with

[1]C.O.Weiss and R.Vilaseca,Dynamics of Lasers,VCH,Weinheim (1991); Instabilities and Chaos in Quantum Optics,eds.F.T.Arecchi and R.G.Harrison,Springer-Verlag,Berlin (1987). [2]H.Haken,Phys.Lett.A 53 (1975) 77. [3]Ju Rui,Huang Hong-Bin,Yang Peng,Xie Xia,and Zhao Huan,Commun.Theor.Phys.(Beijing,China) 44 (2005) 65; Ju Rui,Zhang Ya-Jun,Huang Hong-Bin,and Zhao Huan,Acta Phys.Sin.53 (2004) 2191 (in Chinese). [4]C.Z.Ning and H.Haken,Z.Phys.B 77 (1989) 247; B 77 (1989) 157; B 77 (1989) 163; J.Zakrenwski and M.Lewenstein,Phys.Rev.A 45 (1992) 2057. [5]G.J.deValearcel,E.Roldan,and R.Vilaseca,Phys.Rev.A 45 (1992) R2674; Phys.Rev.A 49 (1994) 1243. [6]X.Xie,H.B.Huang,F.Qian,Y.J.Zhang,P.Yang,and G.X.Qi,Commun.Theor.Phys.(Beijing,China) 46 (2006) 1042. [7]X.L.Deng,H.Q.Ma,B.D.Chen,and H.B.Huang,Phys.Lett.A 290 (2001) 77. [8]C.Benkert,and M.O.Scully,Phys.Rev.A 42 (1990) 2817. [9]M.O.Scully and M.S.Zubairy,Quantum Optics,Cambridge University Press,Cambridge (1997).     

On the Higgs feature of gravity

The gauge gravitation theory, based on the equivalence principle besides the gauge principle, is formulated in the fibre bundle terms. The correlation between gauge geometry on spinor bundles describing Dirac fermion fields and space-time geometry on a tangent bundle is investigated. We show that field functions of fermion fields in presence of different gravitational fields are always written with respect to different reference frames. Therefore, the conventional quantization procedure is applicable to fermion fields only if gravitational field is fixed. Quantum gravitational fields violate the above mentioned correlation between two geometries.     

Observables in 3d spinfoam quantum gravity with fermions

We study expectation values of observables in three-dimensional spinfoam quantum gravity coupled to Dirac fermions. We revisit the model introduced by one of the authors and extend it to the case of massless fermionic fields. We introduce observables, analyse their symmetries and the corresponding proper gauge fixing. The Berezin integral over the fermionic fields is performed and the fermionic observables are expanded in open paths and closed loops associated to pure quantum gravity observables. We obtain the vertex amplitudes for gauge-invariant observables, while the expectation values of gauge-variant observables, such as the fermion propagator, are given by the evaluation of particular spin networks.     

Quantum Hall Effect on the Hyperbolic Plane

In this paper, we study both the continuous model and the discrete model of the Quantum Hall Effect (QHE) on the hyperbolic plane. The Hall conductivity is identified as a geometric invariant associated to an imprimitivity algebra of observables. We define a twisted analogue of the Kasparov map, which enables us to use the pairing between K-theory and cyclic cohomology theory, to identify this geometric invariant with a topological index, thereby proving the integrality of the Hall conductivity in this case. Received: 17 March 1997 / Accepted: 24 April 1997