Group Contraction in Quantum Field Theory

Group contraction plays a relevant rôle in spontaneously broken symmetry theories. Its physical meaning in connection with Bose condensation and the origin of macroscopic quantum systems is discussed.     

Group Field Theory: An Overview

We give a brief overview of the properties of a higher-dimensional generalization of matrix model which arise naturally in the context of a background approach to quantum gravity, the so-called group field theory. We show in which sense this theory provides a third quantization point-of-view on quantum gravity. Prepared for the proceedings of Peyresq Physics 9Meeting: Micro and Macro Structureof Spacetime, Peyresq, France, 19-26 June 2004. 5 Except in 2 + 1 dimension (Witten, 1988).     

Melonic Phase Transition in Group Field Theory

Group field theories have recently been shown to admit a 1/N expansion dominated by so-called ‘melonic graphs’, dual to triangulated spheres. In this note, we deepen the analysis of this melonic sector. We obtain a combinatorial formula for the melonic amplitudes in terms of a graph polynomial related to a higher-dimensional generalization of the Kirchhoff tree-matrix theorem. Simple bounds on these amplitudes show the existence of a phase transition driven by melonic interaction processes. We restrict our study to the Boulatov–Ooguri models, which describe topological BF theories and are the basis for the construction of 4-dimensional models of quantum gravity.     

Summing Logarithms in Quantum Field Theory: The Renormalization Group

The process of renormalization in quantum fieldtheory necessarily involves the introduction of anarbitrary mass scale ² into the theory.The effect of having this parameter appear due toquantum effects can be analyzed from many points of view; thegeneral topic is usually called therenormalization group. In this paper, oneaspect of this feature of quantum field theory isdiscussed in some detail. It is shown how the appearance of thisarbitrary mass scale imposes consistency conditions onquantum-induced corrections to the classical action ofa model. This has the effect of determining higher order corrections in terms of lower ordercorrections in the perturbative expansion of theeffective action, which in turn permits at least partialsummation of all terms in the perturbative expansion. This is illustrated in the context of twosimple, well-understood models; a ⁴model in four dimensions and a ³ modelin six dimensions. The technicalities associated withthe renormalization procedure itself are not discussed.     

Unified Field Theory from Enlarged Transformation Group. The Consistent Hamiltonian

A theory has been presented previously in which the geometrical structure of a real four-dimensional space time manifold is expressed by a real orthonormal tetrad, and the group of diffeomorphisms is replaced by a larger group. The group enlargement was accomplished by including those transformations to anholonomic coordinates under which conservation laws are covariant statements. Field equations have been obtained from a variational principle which is invariant under the larger group. These field equations imply the validity of the Einstein equations of general relativity with a stress-energy tensor that is just what one expects for the electroweak field and associated currents. In this paper, as a first step toward quantization, a consistent Hamiltonian for the theory is obtained. Some concluding remarks are given concerning the need for further development of the theory. These remarks include discussion of a possible method for extending the theory to include the strong interaction.     

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.     

Renormalization Group Invariance in Pionless Effective Field Theory for the NN System

We consider the NN interaction in pionless effective field theory (EFT) up to next-to-next-to-leading order (NNLO) and use a recursive subtractive renormalization scheme to describe NN scattering in the ¹ S ₀ channel. We fix the strengths of the contact interactions at a reference scale, chosen to be the one that provides the best fit for the phase-shifts, and then slide the renormalization scale by evolving the driving terms of the subtracted Lippmann?CSchwinger equation through a non-relativistic Callan?CSymanzik equation. The results show that such a systematic renormalization scheme with multiple subtractions is fully renormalization group invariant.     

Models in Boundary Quantum Field Theory Associated with Lattices and Loop Group Models

In this article we give new examples of models in boundary quantum field theory, i.e. local time-translation covariant nets of von Neumann algebras, using a recent construction of Longo and Witten, which uses a local conformal net on the real line together with an element of a unitary semigroup associated with . Namely, we compute elements of this semigroup coming from H?lder continuous symmetric inner functions for a family of (completely rational) conformal nets which can be obtained by starting with nets of real subspaces, passing to its second quantization nets and taking local extensions of the former. This family is precisely the family of conformal nets associated with lattices, which as we show contains as a special case the level 1 loop group nets of simply connected, simply laced groups. Further examples come from the loop group net of at level 2 using the orbifold construction.     

激光场中两维色噪声的近似计算

从色散型介质中三次激光模型出发，通过泛函导数，应用小于τ近拟计算两维色噪声，得到激光场定态强度分布，研究了定态强度分布的极值点随抽运参量和强度噪声相关时间的变化情况。应用平移1／N展开方法，得到激光场瞬态解，即本征值随抽运参量的变化情况。研究结果表明，抽运参量和噪声相关时间是激光系统出现一级相变类比的关键因素，相位噪声相关时间对强度谱分布中的线宽影响较大。     

Renormalizable Models in Rank {d \geq 2} Tensorial Group Field Theory

Classes of renormalizable models in the Tensorial Group Field Theory framework are investigated. The rank d tensor fields are defined over d copies of a group manifold \({G_D=U(1)^D}\) or \({G_D= SU(2)^D}\) with no symmetry and no gauge invariance assumed on the fields. In particular, we explore the space of renormalizable models endowed with a kinetic term corresponding to a sum of momenta of the form \({p^{2a}, a\in (0,1]}\) . This study is tailored for models equipped with Laplacian dynamics on G _D (case a = 1) but also for more exotic nonlocal models in quantum topology (case 0 < a < 1). A generic model can be written \({(_{\dim G_D}\Phi^{k}_{d}, a)}\) , where k is the maximal valence of its interactions. Using a multi-scale analysis for the generic situation, we identify several classes of renormalizable actions, including matrix model actions. In this specific instance, we find a tower of renormalizable matrix models parametrized by \({k \geq 4}\) . In a second part of this work, we study the UV behavior of the models up to maximal valence of interaction k = 6. All rank \({d \geq 3}\) tensor models proved renormalizable are asymptotically free in the UV. All matrix models with k = 4 have a vanishing β-function at one-loop and, very likely, reproduce the same feature of the Grosse–Wulkenhaar model (Commun Math Phys 256:305, 2005).     

Renormalization of a SU(2) Tensorial Group Field Theory in Three Dimensions

We address in this paper the issue of renormalizability for SU(2) Tensorial Group Field Theories (TGFT) with geometric Boulatov-type conditions in three dimensions. We prove that interactions up to ? ⁶-tensorial type are just renormalizable without any anomaly. Our new models define the renormalizable TGFT version of the Boulatov model and provide therefore a new approach to quantum gravity in three dimensions. Among the many new technical results established in this paper are a general classification of just renormalizable models with gauge invariance condition, and in particular concerning properties of melonic graphs, the second order expansion of melonic two point subgraphs needed for wave-function renormalization.     

Qubit Systems from Colored Toric Geometry and Hypercube Graph Theory

We develop a new geometric approach to deal with qubit information systems using colored graph theory.More precisely, we present a one to one correspondence between graph theory, and qubit systems, which may be explored to attack qubit information problems using toric geometry considered as a powerful tool to understand modern physics including string theory. Concretely, we examine in some details the cases of one, two, and three qubits, and we find that they are associated with CP~1, CP~1× CP~1 and CP~1× CP~1× CP~1 toric varieties respectively. Using a geometric procedure referred to as a colored toric geometry, we show that the qubit physics can be converted into a scenario handling toric data of such manifolds by help of hypercube graph theory. Operations on toric information can produce universal quantum gates.     

Parametrized Field Theory

A theory is presented in which a field depends not only on spacetime coordinates x, but also on a Lorentz-invariant parameter . Such a theory is conceptually and technically simple and manifestly covariant at every step. The generator of evolution and the generator of spacetime translations and Lorentz transformations are obtained in a straightforward way. In the quantized theory the Heisenberg equation of motion is written in a covariant form and is equivalent to the field equation. The equal commutator between the field and its canonically conjugate momentum is just proportional to the spacetime function. Finally comparison with the conventional field theory is done, and it is found that the expectation value of the momentum operator in the on shell states is the same.     

Supermulticonformal Field Theory

In this paper we develop the superspacestructure of the multicomplex space MC_n forn = 2ⁿ. We extend the basic properties of themulticomplex analysis to n the case ofSMC_n ^N, called supermulticomplex spaces: this goes from the n superanalyticitycondition to the residue theorem. The formalism of 2Dsuperconformal field theory is also developed onSMC_n ^N. We then show that the nassociated superconformal symmetry is infinite dimensional and leads ton copies of super-Virasoro algebra. This results can beapplied to construct a free-field theory on the volumeof the (n – 1)-super-brane. A model of field theory describing a bosonic case is alsopresented.     

Braided Quantum Field Theory

We develop a general framework for quantum field theory on noncommutative spaces, i.e., spaces with quantum group symmetry. We use the path integral approach to obtain expressions for n-point functions. Perturbation theory leads us to generalised Feynman diagrams which are braided, i.e., they have non-trivial over- and under-crossings. We demonstrate the power of our approach by applying it to φ⁴-theory on the quantum 2-sphere. We find that the basic divergent diagram of the theory is regularised. Received: 3 July 1999 / Accepted: 10 November 2000     

Field Theory on Kappa-spacetime

A general formalism is developed, that allows the construction of field theory on quantum spaces which are deformations of ordinary spacetime. The symmetry group of spacetime is replaced by a quantum group. This formalism is demonstrated for the -deformed Poincaré algebra and its quantum space. The algebraic setting is mapped to the algebra of functions of commuting variables with a suitable -product. Fields are elements of this function algebra. As an example, the Klein-Gordon equation is defined and derived from an action.     

PT-Symmetric Quantum Field Theory

In the context of the PT-symmetric version of quantum electrodynamics, it is argued that the C-operator introduced in order to define a unitary inner product has nothing to do with charge conjugation.     

Shuffling Quantum Field Theory

We discuss shuffle identities between Feynman graphs using the Hopf algebra structure of perturbative quantum field theory. For concrete exposition, we discuss vertex function in massless Yukawa theory.