BPHZ renormalization of λφ4 field theory in curved spacetime

A proof is given to all orders in perturbation theory of the renormalizability of λφ⁴ field theory in curved spacetime. The proof is based on the BPHZ definition of a renormalized Feynman integrand and uses dimensional regularization to ensure that products of Feynman propagators are well-defined distributions. The explicit structure of the pole terms in the Feynman integrand is obtained using a local momentum space representation of the Feynman propagator and is shown to be of a form which can be cancelled by counterterms in the scalar field Lagrangian. The proof given is, technically, only valid for metrics which have been analytically continued to Euclidean (++++) signature.     

Graviton propagator from background-independent quantum gravity

We study the graviton propagator in Euclidean loop quantum gravity. We use spin foam, boundary-amplitude, and group-field-theory techniques. We compute a component of the propagator to first order, under some approximations, obtaining the correct large-distance behavior. This indicates a way for deriving conventional spacetime quantities from a background-independent theory.     

Space-Time Grains: Roots of Special and Doubly Special Relativity

We show that the special relativistic dynamics when combined with quantum mechanics and the concept of superstatistics can be interpreted as arising from two interlocked non-relativistic stochastic processes that operate at different energy scales. This interpretation leads to Feynman amplitudes that are in the Euclidean regime identical to transition probability of a Brownian particle propagating through a granular space. Some kind of spacetime granularity could be therefore held responsible for the emergence at larger scales of various symmetries. For illustration we consider also the dynamics and the propagator of a spinless relativistic particle. Implications for doubly special relativity, quantum field theory, quantum gravity and cosmology are discussed.     

Renormalization and scaling behavior of non-Abelian gauge fields in curved spacetime

In this article we discuss the one loop renormalization and scaling behavior of non-Abelian gauge field theories in a general curved spacetime. A generating functional is constructed which forms the basis for both the perturbation expansion and the Ward identities. Local momentum space representations for the vector and ghost particles are developed and used to extract the divergent parts of Feynman integrals. The one loop diagram for the ghost propagator and the vector-ghost vertex are shown to have no divergences not present in Minkowski space. The Ward identities insure that this is true for the vector propagator as well. It is shown that the above renormalizations render the three- and four-vector vertices finite. Finally, a renormalization group equation valid in curved spacetimes is derived. Its solution is given and the theory is shown to be asymptotically free as in Minkowski space.     

Instanton-like solutions in chiral models

General two-dimensional Euclidean chiral models of field theory are considered in detail. It is shown that in the case when the field takes its values in an arbitrary Kähler manifold the “duality equations” reduce to the Cauchy- Riemann equations on this manifold. For homogeneous manifolds the solutions of these equations do exist and are given by rational functions.     

Dimensional regularization of massless quantum electrodynamics in spherical spacetime

The quantization and renormalization of massless electrodynamics in a spacetime of constant curvature are discused. A formalism is presented which is valid in an arbitrary number of dimensions and therefore allows the use of dimensional regularization. In the discussion of the photon propagator it is found that anomalous mass terms dependent on the curvature arise, although these vanish in four dimensions. Further, the gauge-fixing term in the Lagrangian has the unconventional feature of not being a perfect square. The renormalizability of the theory is then demonstrated to one loop order, and the renormalization constants are shown to retain their flat spacetime values. Finally, expansions for the renormalized electron and photon propagators in terms of appropriate spherical harmonics are derived.     

On analytic formulas of Feynman propagators in position space

We correct an inaccurate result of previous work on the Feynman propagator in position space of a free Dirac field in(3+1)-dimensional spacetime; we derive the generalized analytic formulas of both the scalar Feynman propagator and the spinor Feynman propagator in position space in arbitrary(D+1)-dimensional spacetime; and we further find a recurrence relation among the spinor Feynman propagator in(D+l)-dimensional spacetime and the scalar Feynman propagators in(D+1)-,(D-1)-and(D+3)-dimensional spacetimes.     

Semiclassical Estimates¶in Asymptotically Euclidean Scattering

We consider long range semiclassical perturbations of the Laplacian on asymptotically Euclidean manifolds. We obtain precise resolvent estimates under non-trapping assumptions. The novelty lies in a systematic use of geometric microlocal methods.     

高频波激发低频磁场和离子声波的重整化强湍动理论

本文给出了高温等离子体中高频波激发低频磁场和离子声波强湍动过程的重整化理论,以便改善通常的弱非线性处理方法,从Vlasov-Maxwell方程组出发,在Fourier表象中得到了包含“最发散”和“次发散”效应互相耦合的高频和低频传播于重整化方程组,从而获得了高、低频振荡粒子重整化分布函数和场的耦合关系。在“最发散”重整化近似下,我们求解了高低频传播子方程组,得到了展开到v₄(高频湍动场能密度与等离子体热能密度之比)一次方的近似解和重整化介电函数等表达式。然后,在Fourier逆变换下导得了我们所要的时空表用中重整化强湍动方程组。最后,作为一个说明重整化作用的例子,在一维稳态下求解了孤立子的形式。 关键词：     

Algebraic approach to quantum field theory on a class of noncommutative curved spacetimes

In this article we study the quantization of a free real scalar field on a class of noncommutative manifolds, obtained via formal deformation quantization using triangular Drinfel’d twists. We construct deformed quadratic action functionals and compute the corresponding equation of motion operators. The Green’s operators and the fundamental solution of the deformed equation of motion are obtained in terms of formal power series. It is shown that, using the deformed fundamental solution, we can define deformed *-algebras of field observables, which in general depend on the spacetime deformation parameter. This dependence is absent in the special case of Killing deformations, which include in particular the Moyal-Weyl deformation of the Minkowski spacetime.     

de Sitter gauge theories and induced gravities

The European Physical Journal C - Pure de Sitter, anti de Sitter, and orthogonal gauge theories in four-dimensional Euclidean spacetime are studied. It is shown that, if the theory is...     

Conformal Operators on Forms and Detour Complexes on Einstein Manifolds

For even dimensional conformal manifolds several new conformally invariant objects were found recently: invariant differential complexes related to, but distinct from, the de Rham complex (these are elliptic in the case of Riemannian signature); the cohomology spaces of these; conformally stable form spaces that we may view as spaces of conformal harmonics; operators that generalise Branson’s Q-curvature; global pairings between differential form bundles that descend to cohomology pairings. Here we show that these operators, spaces, and the theory underlying them, simplify significantly on conformally Einstein manifolds. We give explicit formulae for all the operators concerned. The null spaces for these, the conformal harmonics, and the cohomology spaces are expressed explicitly in terms of direct sums of subspaces of eigenspaces of the form Laplacian. For the case of non-Ricci flat spaces this applies in all signatures and without topological restrictions. In the case of Riemannian signature and compact manifolds, this leads to new results on the global invariant pairings, including for the integral of Q-curvature against the null space of the dimensional order conformal Laplacian of Graham et al.     

Gauge invariance for extended objects

Just as the vector potential (one-form) couples to charged point-particles, antisymmetric tensor fields of higher rank (p-forms) couple to elementary objects of higher dimensionality (strings, membranes, …). It is shown that the only possible gauge invariant interaction of such an extended object with a gauge field in spacetime is based on the abelian group U(1). This is unlike the situation for particles where Yang-Mills actions based on any gauge group may be written down. The properties of the abelian theory are explored. It is pointed out that a compact object is analogous to a particle-antiparticle pair and its quantum rate of production in a constant external field is calculated semiclassically. The analysis is performed keeping generic both the dimension of the object and that of spacetime.     

Magnetic backgrounds from generalised complex manifolds

The magnetic backgrounds that physically give rise to spacetime noncommutativity are generally treated using noncommutative geometry. In this paper we prove that also the theory of generalised complex manifolds contains the necessary elements to generate B-fields geometrically. As an example, the Poisson brackets of the Landau model (electric charges on a plane subject to an external, particularly applied magnetic field) are rederived using the techniques of generalised complex manifolds.     

Classical BV Theories on Manifolds with Boundary

In this paper we extend the classical BV framework to gauge theories on spacetime manifolds with boundary. In particular, we connect the BV construction in the bulk with the BFV construction on the boundary and we develop its extension to strata of higher codimension in the case of manifolds with corners. We present several examples including electrodynamics, Yang-Mills theory and topological field theories coming from the AKSZ construction, in particular, the Chern-Simons theory, the BF theory, and the Poisson sigma model. This paper is the first step towards developing the perturbative quantization of such theories on manifolds with boundary in a way consistent with gluing.     

Colored Flux Tube in Euclidean Spacetime

No Heading The flux tube solution in the Euclidean spacetime with the color longitudinal electric field in the SU(2) Yang-Mills-Higgs theory with broken gauge symmetry is found. Some arguments are given that this flux tube is a pure quantum object in the SU(3) quantum theory reduced to the SU(2) Yang-Mills-Higgs theory.     

Renormalizability of φ3 theory in 6-dimensional curved spacetime

We show that φ³ theory in a general 6-dimensional curved spacetime is renormalizable at the two-loop level by demonstrating the cancellation of non-local divergences. We also calculate the full set of local two-loop counterterms. The heat kernel is used to obtain an expansion for the propagator, in the presence of the background field, in configuration space. This expansion isolates singular terms in the short-distance limit which give rise to ultraviolet divergences.     

The Spinning and Curving of Spacetime: The Electromagnetic and Gravitational Fields in the Evans Field Theory

The unification of the gravitational and electromagnetic fields achieved geometrically in the generally covariant unified field theory of Evans implies that electromagnetism is the spinning of spacetime and gravitation is the curving of spacetime. The homogeneous unified field equation of Evans is a balance of spacetime spin and curvature and governs the influence of electromagnetism on gravitation using the first Bianchi identity of differential geometry. The second Bianchi identity of differential geometry is shown to lead to the conservation law of the Evans unified field, and also to a generalization of the Einstein field equation for the unified field. Rigorous mathematical proofs are given in appendices of the four equations of differential geometry which are the cornerstones of the Evans unified field theory: the first and second Maurer-Cartan structure relations and the first and second Bianchi identities. As an example of the theory, the origin of wavenumber and frequency is traced to elements of the torsion tensor of spinning spacetime.     

Solutions of the Einstein Constraint Equations with Apparent Horizon Boundaries

We construct asymptotically Euclidean solutions of the vacuum Einstein constraint equations with an apparent horizon boundary condition. Specifically, we give sufficient conditions for the constant mean curvature conformal method to generate such solutions. The method of proof is based on the barrier method used by Isenberg for compact manifolds without boundary, suitably extended to accommodate semilinear boundary conditions and low regularity metrics. As a consequence of our results for manifolds with boundary, we also obtain improvements to the theory of the constraint equations on asymptotically Euclidean manifolds without boundary.Acknowledgement I would like to thank D. Pollack, J. Isenberg, and S. Dain for helpful discussions and advice. I would also like to thank an anonymous referee for suggestions that improved the papers style. This research was partially supported by NSF grant DMS-0305048.     

From Euclidean to Minkowski space with the Cauchy–Riemann equations

We present an elementary method to obtain Green’s functions in non-perturbative quantum field theory in Minkowski space from Green’s functions calculated in Euclidean space. Since in non-perturbative field theory the analytical structure of amplitudes often is unknown, especially in the presence of confined fields, dispersive representations suffer from systematic uncertainties. Therefore, we suggest to use the Cauchy–Riemann equations, which perform the analytical continuation without assuming global information on the function in the entire complex plane, but only in the region through which the equations are solved. We use as example the quark propagator in Landau gauge quantum chromodynamics, which is known from lattice and Dyson–Schwinger studies in Euclidean space. The drawback of the method is the instability of the Cauchy–Riemann equations against high-frequency noise,which makes it difficult to achieve good accuracy. We also point out a few curious details related to the Wick rotation.