Renormalization of Quantum Field Theory in Curved Space-Time and Renormalization Group Equations

The structure of quantum field theory renormalization in curved space-time is investigated. The equations allowing us to investigate the behaviour of vacuum energy and vertex functions in the limit of small distances in the external gravitational field are established. The behaviour of effective charges corresponding to the parameters of nonminimal coupling of the matter with the gravitational field is studied and the conditions under which asymptotically free theories become asymptotically conformally invariant are found. The examples of asymptotically conformally invariant theories are given. On the basis of a direct solution of renormalization group equations the effective potential in the external gravitational field and the effective action in the gravity with the high derivatives are obtained. The expression for the cosmological constant in terms of R²-gravity Lagrangian parameters is given which does not contradict the observable data. Renormalization and renormalization group equations for the theory in curved space-time with torsion are investigated.     

Asymptotic Freedom in the Conformal Quantum Gravity with Matter

The grand unified theory (GUT) based on the O(N) and SU(N)-gauge groups after the conformally invariant gravity being included is investigated. We calculate the one-loop gravity contributions into the renormalization group equations and study their solutions. The analysis performed show that the asymptotic freedom behaviour early established for all GUT's coupling constants is not broken by taking into account this kind of gravity. However all restrictions imposed on the GUT multiplet composition become less firmly and the physical content of the constructed models is more realistic.     

Predictions of SUSY masses in the minimal supersymmetric grand unified theory

The Minimal Supersymmetric Standard Model (MSSM) distinguishes itself from other GUT's by a successful prediction of many unrelated phenomena with a minimum number of parameters. Among them: a) Unification of the gauge couplings constants; b) Unification of the b-quark and τ-lepton masses; c) Proton stability; d) Electroweak symmetry breaking at a scale far below the unification scale and the corresponding relation between the gauge boson masses and the top quark mass. A combined fit of the free parameters in the MSSM to these low energy constraints shows that the MSSM model can satisfy these constraints simultaneously. From the fitted parameters the masses of the as yet unobserved superpartners of the SM particles are predicted, the top mass is constrained to a range between 140 and 200 GeV, and the second order QCD coupling constant is required to be between 0.108 and 0.132. The complete second order renormalization group equations for the gauge and Yukawa couplings are used and analytical solutions for the neutral gauge boson, the Higgs masses and the sparticle masses are derived, taking into account the one-loop corrections to the Higgs potential.     

Symmetry breaking and gravitational interaction

A scalar field Lagrangian is considered in the curved space-time to which a Hamiltonian determining nonzero vacuum field value is added. The initial Lagrangian can be expressed as a sum of Lagrangians for the constant scalar field component and perturbation. The first Lagrangian can be considered as a Lagrangian for the Einstein gravitational field in vacuum. The problem of renormalization of the constant scalar field component is investigated. It is demonstrated that in the case of conformal relation of the scalar field to the space-time curvature, there exists a unique value of the scalar space curvature for which the field can be considered constant (field perturbations do not result in renormalization of the constant component). This curvature value determines the unique value of the equilibrium nuclide density. A correlation of the examined Lagrangian parameters with the integral parameters of the Solar system is discussed. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 18–34, July, 2006.     

Axioms for renormalization in Euclidean quantum field theory

A set of axioms which fix Euclidean renormalizations up to a finite renormalization is proposed. There exists a one to one correspondence between Euclidean renormalizations and renormalizations in Minkowski space-time satisfying Hepp's axioms. No restrictions on masses are imposed.     

Renormalization of a quantum field theory model in a curved space-time with torsion

A quantum field theory model that contains interacting non-Abelian gauge fields, scalar fields, and spinor fields is considered in a curved space-time with torsion. The cone-loop counterterms are found. It is shown that the multiplicative renormalization condition requires a nonminimal coupling of the matter with the gravitational field.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 94–100, August, 1985.     

Microlocal Analysis and¶Interacting Quantum Field Theories:¶Renormalization on Physical Backgrounds

We present a perturbative construction of interacting quantum field theories on smooth globally hyperbolic (curved) space-times. We develop a purely local version of the Stückelberg–Bogoliubov–Epstein–Glaser method of renormalization by using techniques from microlocal analysis. Relying on recent results of Radzikowski, K?hler and the authors about a formulation of a local spectrum condition in terms of wave front sets of correlation functions of quantum fields on curved space-times, we construct time-ordered operator-valued products of Wick polynomials of free fields. They serve as building blocks for a local (perturbative) definition of interacting fields. Renormalization in this framework amounts to extensions of expectation values of time-ordered products to all points of space-time. The extensions are classified according to a microlocal generalization of Steinmann scaling degree corresponding to the degree of divergence in other renormalization schemes. As a result, we prove that the usual perturbative classification of interacting quantum field theories holds also on curved space-times. Finite renormalizations are deferred to a subsequent paper. As byproducts, we describe a perturbative construction of local algebras of observables, present a new definition of Wick polynomials as operator-valued distributions on a natural domain, and we find a general method for the extension of distributions which were defined on the complement of some surface. Received: 31 March 1999 / Accepted: 10 June 1999     

BRST Renormalization

We consider the renormalization of general gauge theories on curved space-time background, with the main assumption being the existence of a gauge-invariant and diffeomorphism invariant regularization. Using the Batalin-Vilkovisky (BV) formalism one can show that the theory possesses gauge invariant and diffeomorphism invariant renormalizability at quantum level, up to an arbitrary order of the loop expansion.     

The Hadamard Construction of Green's Functions on a Curved Space-Time with Symmetries

The Hadamard constituents of Green's functions for a ζ-parametrized generalization of the massless scalar d'Alembert equation to a curved space-time including the conformally invariant wave equation: the world function of space-time, the transport scalar, and the tail-term coefficients, being simultaneously coefficients in the Schwinger-DeWitt expansion of the Feynman propagator for the corresponding invariant Klein-Gordon equation, are considered on a general static spherically symmetric and (2,2)-decomposable metric. The construction equations determining the Hadamard building elements are cast into a symmetry-adapted form and used to obtain, on a specific model metric, exact explicit solutions.     

Renormalization of non-abelian gauge theories in curved space-time

We use indirect, renormalization group arguments to calculate the gravitational counterterms needed to renormalize an interacting non-abelian gauge theory in curved space-time. This method makes it straightforward to calculate terms in the trace anomaly which first appear at high order in the coupling constant, some of which would need a 4-loop calculation to find directly. The role of gauge invariance in the theory is considered, and we discuss briefly the effect of using coordinate-dependent gauge-fixing terms. We conclude by suggesting possible applications of this work to models of the very early universe.     

Generalized Wilson expansions in dimensional renormalization

Operator product expansions in the framework of dimensional regularization and renormalization are discussed. Following the definition of a subtraction operator for dimensionally regularized and points-split Green's functions, a generalized Wilson expansion pansion is proved. The terms of the expansion are normal products defined via dimensional renormalization, and the coefficients are doubly regularized with singularities in the physical dimension as the spacial separations of the product fields vanish, or at zero separations as the dimension of space-time becomes physical.     

Calculation of the energy-momentum tensor of the vacuum in an anisotropic universe

Subtractive methods (N-wave and adiabatic) which are applicable to the calculation of the energy-momentum tensor of quantum fields in curved space-time are in need of a foundation in terms of renormalizations. In the example of a scalar field in an anisotropic universe of Bianchi type I it is shown that the Pauli-Villars scheme, in which the renormalization is in fact realized separately in each mode, provides such a foundation. The technical difficulty obstructing the explicit regularization of divergent integrals in momentum space is shown. We calculate the polarization of the vacuum of a scalar field with arbitrary coupling to the curvature in a weakly anisotropic universe.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 93–98, July, 1986.     

Grand unified theories in coset space dimensional reduction

We examine particle content of the effective four-dimensional GUT's arising in the coset space dimensional reduction of 10-dimensionalE ₈ supersymmetric Yang-Mills theory.     

Feigenbaum universality in string theory

Brane-like vertex operators, defining backgrounds with ghost-matter mixing in NSR superstring theory, play an important role in the world-sheet formulation of D-branes and M theory as creation operators for extended objects in the second quantized formalism. In this paper, we show that the dilaton beta function in ghost-matter mixing backgrounds becomes stochastic. The renormalization group (RG) equations in ghost-matter mixing backgrounds lead to non-Markovian Fokker-Planck equations whose solutions describe superstrings in curved space-times with brane-like metrics. We show that the Feigenbaum universality constant δ=4.669..., describing transitions from order to chaos in a huge variety of dynamical systems, appears analytically in these RG equations. We find that the appearance of this constant is related to the scaling of relative space-time curvatures at fixed points of the RG flow. In this picture, the fixed points correspond to the period doubling of Feigenbaum iteration schemes.     

Dimensional renormalization of scalar field theory in curved space-time

The regularization and renormalization of an interacting scalar field φ in a curved spacetime background is performed by the method of continuation to n dimensions. In addition to the familiar counter terms of the flat-space theory, c-number, “vacuum” counter terms must also be introduced. These involve zero, first, and second powers of the Reimann curvature tensor R_αβψδ. Moreover, the renormalizability of the theory requires that the Lagrange function couple φ² to the curvature scalar R with a coupling constant η. The coupling η must obey an inhomogeneous renormalization group equation, but otherwise it is an arbitrary, free parameter. All the counter terms obey renormalization group equations which determine the complete structure of these quantities in terms of the residues of their simple poles in n ? 4. The coefficient functions of the counter terms determine the construction of φ² and φ⁴ in terms of renormalized composite operators 1, [φ²], and [φ⁴]. Two of the counter terms vanish in conformally flat space-time. The others may be computed from the theory in purely flat space-time. They are determined, in a rather intricate fashion, by the additive renormalizations for two-point functions of [φ²] and [φ⁴] in Minkowski space-time. In particular, using this method, we compute the leading divergence of the R² interaction which is of fifth order in the coupling constant λ.     

Scaling behavior of semiclassical gravity

Using the idea of metric scaling we examine the scaling behavior of the stress tensor of a scalar quantum field in curved space-time. The renormalization of the stress tensor results in a departure from naive scaling. We view the process of renormalizing the stress tensor as being equivalent to renormalizing the coupling constants in the Lagrangian for gravity (with terms quadratic in the curvature included). Thus the scaling of the stress tensor is interpreted as a nonnaive scaling of these coupling constants. In particular, we find that the cosmological constant and the gravitational constant approach UV fixed points. The constants associated with the terms which are quadratic in the curvature logarithmically diverge. This suggests that quantum gravity is asymptotically scale invariant.     

Effective potential in curved space and cut-off regularizations

We consider derivation of the effective potential for a scalar field in curved space-time within the physical regularization scheme, using two sorts of covariant cut-off regularizations. The first one is based on the local momentum representation and Riemann normal coordinates and the second is operatorial regularization, based on the Fock-Schwinger-DeWitt proper-time representation. We show, on the example of a self-interacting scalar field, that these two methods produce equal results for divergences, but the first one gives more detailed information about the finite part. Furthermore, we calculate the contribution from a massive fermion loop and discuss renormalization group equations and their interpretation for the multi-mass theories.     

The renormalization group and the ϵ expansion

The modern formulation of the renormalization group is explained for both critical phenomena in classical statistical mechanics and quantum field theory. The expansion in ? = 4?d is explained [d is the dimension of space (statistical mechanics) or space-time (quantum field theory)]. The emphasis is on principles, not particular applications. Sections 1–8 provide a self-contained introduction at a fairly elementary level to the statistical mechanical theory. No background is required except for some prior experience with diagrams. In particular, a diagrammatic approximation to an exact renormalization group equation is presented in sections 4 and 5; sections 6–8 include the approximate renormalization group recursion formula and the Feyman graph method for calculating exponents. Sections 10–13 go deeper into renormalization group theory (section 9 presents a calculation of anomalous dimensions). The equivalence of quantum field theory and classical statistical mechanics near the critical point is established in section 10; sections 11–13 concern problems common to both subjects. Specific field theoretic references assume some background in quantum field theory. An exact renormalization group equation is presented in section 11; sections 12 and 13 concern fundamental topological questions.     

The Universe Accelerated Expansion using Extra-dimensions with Metric Components Found by a New Equivalence Principle

Curved multi-dimensional space-times (5D and higher) are constructed by embedding them in one higher-dimensional flat space. The condition that the embedding coordinates have a separable form, plus the demand of an orthogonal resulting space-time, implies that the curved multi-dimensional space-time has 4D de-Sitter subspaces (for constant extra-dimensions) in which the 3D subspace has an accelerated expansion. A complete determination of the curved multi-dimensional spacetime geometry is obtained provided we impose a new type of “equivalence principle”, meaning that there is a geodesic which from the embedding space has a rectliniar motion. According to this new equivalence principle, we can find the extra-dimensions metric components, each curved multi-dimensional spacetime surface’s equation, the energy-momentum tensors and the extra-dimensions as functions of a scalar field. The generic geodesic in each 5D spacetime are studied: they include solutions where particle’s motion along the extra-dimension is periodic and the 3D expansion factor is inflationary (accelerated expansion). Thus, the 3D subspace has an accelerated expansion.     

Renormalization-group improved effective potential for interacting theories with several mass scales in curved spacetime

The renormalization group (RG) is used in order to obtain the RG improved effective potential in curved spacetime. This potential is explicitly calculated for the Yukawa model and for scalar electrodynamics, i.e. theories with several (namely, more than one) mass scales, in a space of constant curvature. Using the λ?⁴-theory on a general curved spacetime as an example, we show how it is possible to find the RG improved effective Lagrangian in curved spacetime. As specific applications, we discuss the possibility of curvature-induced phase transitions in the Yukawa model and the effective equations (back-reaction problem) for the λ?⁴-theory on a De Sitter background.