BRST formalism of f(R) gravity

We present the manifestly covariant quantization of quadratic gravity or higher-derivative gravity in the de Donder gauge condition(or harmonic gauge condition) for general coordinate invariance on the basis of the BRST transformation. We explicitly calculate variousequal-time commutation relations (ETCRs), in particlular, the ETCRs between the metric tensor and its time derivatives in detail,and show that they are identically vanishing. We also clarify global symmetries, the physical content of quadratic gravity, and clearly showthat this theory is not unitary and has a massive scalar, massive ghost and massless graviton as physical modes. Finally, we comment onconfinement of the massive ghost, thereby recovering the unitarity of the physical S-matrix in quadratic gravity.

We perform the manifestly covariant quantization of a scale invariant gravity with a scalar field, which is equivalent to the well-known Brans-Dicke gravity via a field redefinition of the scalar field, in the de Donder gauge condition (or harmonic gauge condition) for general coordinate invariance. First, without specifying the expression of a gravitational theory, we write down various equal-time (anti-)commutation relations (ETCRs), in particular, those involving the Nakanishi-Lautrup field, the FP ghost, and the FP antighost only on the basis of the de Donder gauge condition. It is shown that choral symmetry, which is a Poincar${\rm{\acute{e}}}$-like $IOSp(8|8)$ supersymmetry, can be derived from such a general action with the de Donder gauge. Next, taking the scale invariant gravity with a scalar field as a classical theory, we derive the ETCRs for the gravitational sector involving the metric tensor and scalar fields. Moreover, we account for how scale symmetry is spontaneously broken in quantum gravity, thereby showing that the dilaton is a massless Nambu-Goldstone particle.

We perform a manifestly covariant quantization of a Weyl-invariant (i.e., a locally scale-invariant) scalar-tensor gravity in the extended de Donder gauge condition (or harmonic gauge condition) for general coordinate invariance and a new scalar gauge for Weyl invariance within the framework of the BRST formalism. We show that choral symmetry, which is a Poincar\'e-like $IOSp(8|8)$ supersymmetry in the case of Einstein gravity, is extended to a Poincar\'e-like $IOSp(10|10)$ supersymmetry. We point out that there is a gravitational conformal symmetry in quantum gravity and account for how conventional conformal symmetry in a flat Minkowski space-time is related to the gravitational conformal symmetry. Moreover, we examine the mechanism of the spontaneous symmetry breaking of the choral symmetry, and show that the gravitational conformal symmetry is spontaneously broken to the Poincar\'e symmetry and the corresponding massless Nambu-Goldstone bosons are the graviton and the dilaton. We also prove the unitarity of the physical $S$ matrix on the basis of the BRST quartet mechanism.

We present the BRST formalism of a Weyl conformal gravity in Weyl geometry. Choosing the extended de Donder gauge-fixing condition (or harmonic gauge condition) for the general coordinate invariance and the new scalar gauge-fixing for the Weyl invariance we find that there is a Poincar${\rm{\acute{e}}}$-like $\IOSp(10|10)$ supersymmetry as in a Weyl invariant scalar-tensor gravity in Riemann geometry. We also point out that there is a gravitational conformal symmetry in quantum gravity although there is a massive Weyl gauge field as a result of spontaneous symmetry breakdown of Weyl gauge symmetry and account for how the gravitational conformal symmetry is spontaneously broken to the Poincar\'e symmetry. The corresponding massless Nambu-Goldstone bosons are the graviton and the dilaton. We also prove the unitarity of the physical S-matrix on the basis of the BRST quartet mechanism.

We study the degrees of freedom of the metric in a general class of higher derivative gravity models, which are interesting in the context of quantum gravity as they are (super)renormalizable. First, we linearize the theory for a flat background metric in Teyssandier gauge for an arbitrary number of spacetime dimensions $D$. The higher-order derivative field equations for the metric perturbation can be decomposed into tensorial and scalar field equations resembling massless and massive wave equations. For the massive tensor field in $D$-dimensions we demonstrate that the harmonic gauge condition is induced dynamically and only the transverse modes are excited in the presence of a matter source. For the special case of quadratic gravity in four-dimensional spacetime, we show that only the quadrupole moment contributes to the gravitational radiation from an idealized binary system.

We consider the harmonic gauge condition in linearized gravity, seen as a gauge theory for a symmetric tensor field. Once the harmonic gauge condition is implemented, as customary, according to the Faddeev-Popov procedure, the gauge fixed action still depends on one gauge parameter. Consequently, the harmonic gauge appears to be a class of conditions, rather than a particular one. This allows to give a physical motivation for the covariant harmonic gauge(s), which emerges when the gravitational perturbation is given a mass term. In fact, for a particular choice of harmonic gauge, we find a theory of linearized massive gravity displaying five degrees of freedom, as it should, and which is not affected by the vDVZ discontinuity, differently from what happens in the standard Fierz-Pauli theory.

The one-particle matrix elements of the local equal-time commutators of the isovector currents are derived by applying the Dyson representation to the causal parts of the invariant absorptive parts. Assuming the equal-time commutation relation between the total charge and charge density and also assuming certain asymptotic limits for the Dyson spectral function, we can generate the local equal-time commutation relations between various components of the currents. It is shown that, for a reasonable asymptotic behavior of the spectral function, the charge-space component has no antisymmetric (in isospin) Schwinger terms, but involves two possible $q$-numer symmetric Schwinger terms, and that the space-space components can have no antisymmetric Schwinger terms. It is pointed out that as the asymptotic behavior becomes worse, we can no longer define the equal-time commutation relations uniquely.

The idea of gauge theories of gravity predicts that there should exist not only the massless graviton but also massive particles carrying the gravitational force. We study the cosmology in a quadratic gravity with dynamical torsion where gravity may be interpreted as a gauge force associated with the Poincaré group. In addition to the massless spin-2 graviton, the model contains four non-ghost massive particle species: a couple of spin-0, a spin-1 and a spin-2. Supposing the restoration of the local Weyl invariance in the UV limit and the parity invariance, we find the most general minisuperspace action describing a homogeneous and isotropic universe with a flat spatial geometry. We then transform the minisuperspace action to a quasi-Einstein frame in which the field space is a hyperboloid and the field potential is a combination of those of a Starobinsky-like inflation and a natural inflation. Remarkably, thanks to the multi-field dynamics, the Starobinsky-like inflationary trajectory can be realized even if the initial condition is away from the top of the Starobinsky-like potential. We also study linear tensor perturbations and find qualitatively different features than the Starobinsky inflation, spontaneous parity violation and mixing of the massless and massive spin-2 modes, which might reveal the underlying nature of gravity through inflationary observables.

We perform an explicit one-loop calculation for the gravitational contributions to the two-, three- and four-point gauge Green's functions with paying attention to the quadratic divergences. It is shown for the first time in the diagrammatic calculation that the Slavnov-Taylor identities are preserved even if the quantum graviton effects are included at one-loop level, such a conclusion is independent of the choice of regularization schemes. We also present a regularization scheme independent calculation based on the gauge condition independent background field framework of Vilkovisky-DeWitt's effective action with focusing on both the quadratic divergence and quartic divergence that is not discussed before. With the harmonic gauge condition, the results computed by using the traditional background field method can consistently be recovered from the Vilkovisky-DeWitt's effective action approach by simply taking a limiting case, and are found to be the same as the ones yielded by the diagrammatic calculation. As a consequence, in all the calculations, the symmetry-preserving and divergent-behavior-preserving loop regularization method can consistently lead to a nontrivial gravitational contribution to the gauge coupling constant with an asymptotic free power-law running at one loop near the Planck scale.

Equal time commutation relations and sum rules

In this paper we give a rigorous formulation of Gell-Mann's equal time commutation relations in the framework of general quantum field theory. We show that this can be achieved despite the nonexistence of charge operators for nonconserved currents. Starting from the properly formulated equal time commutation relations of “generalized charges”, we justify the application of the Gauss-Theorem and we discuss the limits for large times of time dependent “generalized charges”. The Jost-Lehmann-Dyson representation is used in order to show that the equal time commutation relations always lead to exactly one, frame independent, sum rule. We discuss the connection between properties of the Jost-Lehmann-Dyson spectral function and the convergence of Adler-Weisberger type sum rules.

We propose that a criterion to characterize interacting theories in a suitable Wightman framework of relativistic quantum field theories which incorporates a “singularity hypothesis”, which has been conjectured for a long time, is supported by renormalization group theory, but has never been formulated mathematically. The (nonperturbative) wave function renormalization Z occurring in these theories is shown not to be necessarily equal to zero, except if the equal time commutation relations (ETCR) are assumed. Since the ETCR are not justified in general (because the interacting fields cannot in general be restricted to sharp times, as is known from model studies), the condition $$Z=0$$ is not of general validity in interacting theories. We conjecture that it characterizes either unstable (composite) particles or the charge-carrying particles, which become infraparticles in the presence of massless particles. In the case of QED, such “dressed” electrons are not expected to be confined, but in QCD, we propose a quark confinement criterion, which follows naturally from lines suggested by the works of Casher, Kogut and Susskind, and Lowenstein and Swieca.

A set of trilinear equal-time commutation relations is proposed as a second-quantization scheme for satisfying conventional statistics. The scheme specifies the bilinear equal-time commutation relations between distinct (a finite number for each spin for the representations considered) in addition to the commutation relations of with themselves. Multiplet schemes for a particular representation of generalized fields satisfying the trilinear equal-time commutation relations are studied. It is shown that vacuum expectation values can be defined, and that the $S$-matrix formalism can be developed for fields. Differences with the conventional $S$-matrix expansions are examined, in particular in connection with possible applications to renormalization procedures. A novel regularization method, based on the bilinear equal-time commutation relations between distinct fields, is considered.

The connection between divergence conditions for currents or generalized currents and equal-time commutation relations is investigated in terms of the response of physical systems to external fields. We give a simple method of constructing physical amplitudes and of obtaining generalized Ward identities and equal-time commutation relations for currents. In order to set up this scheme, we have to make an assumption about the response of the physical system to a change of the external parameters and also consider the gradient terms. In this way, we try to reproduce the current algebra formalism. It is hoped that, when this has been done, we will have made it plausible that a certain representation of the algebra exist. However, we do not discuss the formal reconstruction of the field theory of strong interactions in terms of electromagnetic, weak, and gravitational scattering data. The discussion in this paper is mostly restricted to finite-dimensional algebras, but a generalization of the method to infinite-dimensional situations is briefly outlined.

We examine three overlapping problems in the application of five-dimensional (5D) manifolds to physics. First, we linearize the 5D theory along the lines of the four-dimensional (4D) theory, using the harmonic gauge condition. The resulting wave equations have sources, and can in principle describe gravitons and scalar particles with finite masses, but the natural choice of gauge parameters makes both massless. Second, we generalize the 5D metric by including separate conformal factors on its 4D and extra parts. Then the 5D harmonic gauge gives back the 4D harmonic gauge of gravitational waves and the Lorentz gauge of electromagnetic waves, but both particles are massless. Third, we again take a conformally rescaled metric, but rewrite the field equations in a novel manner. This allows us to interpret the finite masses of ordinary particles in terms of the wavelengths associated with 4D spaces embedded in a 5D space.