scholarly journals Fractional Quantum Field Theory: From Lattice to Continuum

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Vasily E. Tarasov
Keyword(s):  
Field Theory ◽  
Fractional Order ◽  
Differential Operators ◽  
Dimensional Space ◽  
Space Time ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Field Equations ◽  
Lattice Space

An approach to formulate fractional field theories on unbounded lattice space-time is suggested. A fractional-order analog of the lattice quantum field theories is considered. Lattice analogs of the fractional-order 4-dimensional differential operators are proposed. We prove that continuum limit of the suggested lattice field theory gives a fractional field theory for the continuum 4-dimensional space-time. The fractional field equations, which are derived from equations for lattice space-time with long-range properties of power-law type, contain the Riesz type derivatives on noninteger orders with respect to space-time coordinates.

scholarly journals HIDDEN SYMMETRIES IN QUANTUM FIELD THEORIES FROM EXTENDED COMPLEX NUMBERS

1999 ◽  
Vol 14 (26) ◽  
pp. 4201-4235 ◽  
Author(s):  
PASCAL BASEILHAC
Keyword(s):  
Field Theory ◽  
Dimensional Space ◽  
Field Theories ◽  
Complex Numbers ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Hidden Symmetries ◽  
Dimensional Space Time ◽  
Extended Complex ◽  
Dual Models

The two-dimensional space–time sine–Gordon field theory is extended algebraically within the n-dimensional space of extended complex numbers. This field theory is constructed in terms of an adapted extension of standard vertex operators. A whole set of nonlocal conserved charges is constructed and studied in this framework. Thereby, an algebraic nonperturbative description is possible for this n-1 parameters family of quantum field theories. Known results are obtained for specific values of the parameters, especially in relation to affine Toda field theories. Different (dual)-models can then be described in this formalism.

scholarly journals REPRESENTATION DEPENDENCE OF SUPERFICIAL DEGREE OF DIVERGENCES IN QUANTUM FIELD THEORY

2011 ◽  
Vol 26 (17) ◽  
pp. 2913-2925 ◽  
Author(s):  
ABOUZEID M. SHALABY
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Vacuum Energy ◽  
Space Time ◽  
Feynman Diagrams ◽  
Coupling Constants ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Metric Operator

In this work, we investigate a very important but unstressed result in the work of C. M. Bender, J.-H. Chen, and K. A. Milton, J. Phys. A39, 1657 (2006). These authors have calculated the vacuum energy of the iϕ3 scalar field theory and its Hermitian equivalent theory up to g4 order of calculations. While all the Feynman diagrams of the iϕ3 theory are finite in 0+1 space–time dimensions, some of the corresponding Feynman diagrams in the equivalent Hermitian theory are divergent. In this work, we show that the divergences in the Hermitian theory originate from superrenormalizable, renormalizable and nonrenormalizable terms in the interaction Hamiltonian even though the calculations are carried out in the 0+1 space–time dimensions. Relying on this interesting result, we raise a question: Is the superficial degree of divergence of a theory is representation dependent? To answer this question, we introduce and study a class of non-Hermitian quantum field theories characterized by a field derivative interaction Hamiltonian. We showed that the class is physically acceptable by finding the corresponding class of metric operators in a closed form. We realized that the obtained equivalent Hermitian and the introduced non-Hermitian representations have coupling constants of different mass dimensions which may be considered as a clue for the possibility of considering nonrenormalizability of a field theory as a nongenuine problem. Besides, the metric operator is supposed to disappear from path integral calculations which means that physical amplitudes can be fully obtained in the simpler non-Hermitian representation.

scholarly journals Quantum Field Theory over $\mathbb{F}_q$

10.37236/589 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Oliver Schnetz
Keyword(s):  
Field Theory ◽  
Perturbation Series ◽  
The Other ◽  
Field Theories ◽  
Roots Of Unity ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Feynman Rules ◽  
Graph Hypersurfaces ◽  
Prime 2

We consider the number $\bar N(q)$ of points in the projective complement of graph hypersurfaces over $\mathbb{F}_q$ and show that the smallest graphs with non-polynomial $\bar N(q)$ have 14 edges. We give six examples which fall into two classes. One class has an exceptional prime 2 whereas in the other class $\bar N(q)$ depends on the number of cube roots of unity in $\mathbb{F}_q$. At graphs with 16 edges we find examples where $\bar N(q)$ is given by a polynomial in $q$ plus $q^2$ times the number of points in the projective complement of a singular K3 in $\mathbb{P}^3$. In the second part of the paper we show that applying momentum space Feynman-rules over $\mathbb{F}_q$ lets the perturbation series terminate for renormalizable and non-renormalizable bosonic quantum field theories.

scholarly journals ON ALGEBRAIC STRUCTURES IMPLICIT IN TOPOLOGICAL QUANTUM FIELD THEORIES

1999 ◽  
Vol 08 (02) ◽  
pp. 125-163 ◽  
Author(s):  
Louis Crane ◽  
David Yetter
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Hopf Algebra ◽  
Tensor Category ◽  
Topological Quantum Field Theory ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Topological Quantum ◽  
Natural Way

We show that any 3D topological quantum field theory satisfying physically reasonable factorizability conditions has associated to it in a natural way a Hopf algebra object in a suitable tensor category. We also show that all objects in the tensor category have the structure of left-left crossed bimodules over the Hopf algebra object. For 4D factorizable topological quantum filed theories, we provide by analogous methods a construction of a Hopf algebra category.

Integrable Quantum Field Theories

2020 ◽  
pp. 575-621
Author(s):  
Giuseppe Mussardo
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Two Dimensional ◽  
Conformal Theory ◽  
Integrable Quantum ◽  
Set Up ◽  
Integrable Quantum Field Theory

Chapter 16 covers the general properties of the integrable quantum field theories, including how an integrable quantum field theory is characterized by an infinite number of conserved charges. These theories are illustrated by means of significant examples, such as the Sine–Gordon model or the Toda field theories based on the simple roots of a Lie algebra. For the deformations of a conformal theory, it shown how to set up an efficient counting algorithm to prove the integrability of the corresponding model. The chapter focuses on two-dimensional models, and uses the term ‘two-dimensional’ to denote both a generic two-dimensional quantum field theory as well as its Euclidean version.

The Principles of Renormalisation

2021 ◽  
pp. 304-328
Author(s):  
J. Iliopoulos ◽  
T.N. Tomaras
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Perturbation Expansion ◽  
Power Counting ◽  
Field Theories ◽  
Asymptotic Freedom ◽  
Green Functions ◽  
Quantum Field Theories ◽  
Quantum Field

Loop diagrams often yield ultraviolet divergent integrals. We introduce the concept of one-particle irreducible diagrams and develop the power counting argument which makes possible the classification of quantum field theories into non-renormalisable, renormalisable and super-renormalisable. We describe some regularisation schemes with particular emphasis on dimensional regularisation. The renormalisation programme is described at one loop order for φ‎4 and QED. We argue, without presenting the detailed proof, that the programme can be extended to any finite order in the perturbation expansion for every renormalisable (or super-renormalisable) quantum field theory. We derive the equation of the renormalisation group and explain how it can be used in order to study the asymptotic behaviour of Green functions. This makes it possible to introduce the concept of asymptotic freedom.

scholarly journals DIFF(Σ) AND METRICS FROM HAMILTONIAN-TQFT'S IN 2+1 DIMENSIONS

1993 ◽  
Vol 08 (24) ◽  
pp. 2277-2283 ◽  
Author(s):  
ROGER BROOKS
Keyword(s):  
Topological Field Theories ◽  
Gauge Theories ◽  
Dimensional Space ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Two Dimensional ◽  
Canonical Gravity ◽  
Topological Quantum ◽  
Topological Field

The constraints of BF topological gauge theories are used to construct Hamiltonians which are anti-commutators of the BRST and anti-BRST operators. Such Hamiltonians are a signature of topological quantum field theories (TQFTs). By construction, both classes of topological field theories share the same phase spaces and constraints. We find that, for (2+1)- and (1+1)-dimensional space-times foliated as M=Σ × ℝ, a homomorphism exists between the constraint algebras of our TQFT and those of canonical gravity. The metrics on the two-dimensional hypersurfaces are also obtained.

scholarly journals UNITARITY OF NONCOMMUTATIVE FIELD THEORIES FROM STRING THEORY

2003 ◽  
Vol 18 (33n35) ◽  
pp. 2525-2532 ◽  
Author(s):  
ALESSANDRO TORRIELLI
Keyword(s):  
String Theory ◽  
Open String ◽  
Space Time ◽  
Bosonic String ◽  
Field Theories ◽  
Noncommutative Space ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Superstring Theory ◽  
Time Quantum

We improve the study of the lack of perturbative unitarity of noncommutative space-time quantum field theories derived from open string theory in electric backgrounds, enforcing the universality of the mechanism by which a tachyonic branch cut appears when the Seiberg-Witten limit freezes the string in an unstable vacuum. The main example is realized in the context of the on-shell four-tachyon amplitude of the bosonic string, and the dependence of the phenomenon on the brane-worldvolume dimension is analysed. We discuss the possibility of a proof in superstring theory, and finally mention the NCOS limit in this framework.

DOES STOCHASTIC ANALYTIC REGULARIZATION PROVIDE QUANTUM FIELD THEORIES?

1992 ◽  
Vol 07 (04) ◽  
pp. 777-794
Author(s):  
C. P. MARTIN
Keyword(s):  
Field Theory ◽  
Regularization Method ◽  
Gauge Theories ◽  
Field Theories ◽  
Intermediate Step ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Point Function ◽  
Four Dimensions ◽  
Analytic Regularization

We analyze whether the so-called method of stochastic analytic regularization is suitable as an intermediate step for constructing perturbative renormalized quantum field theories. We choose a λϕ3 in six dimensions to prove that this regularization method does not in general provide a quantum field theory. This result seems to apply to any field theory with a quadratically UV-divergent stochastic two-point function, for instance λϕ4 and gauge theories in four dimensions.

scholarly journals EXISTENCE OF ASYMPTOTIC EXPANSIONS IN NONCOMMUTATIVE QUANTUM FIELD THEORIES

2008 ◽  
Vol 20 (08) ◽  
pp. 933-949
Author(s):  
C. A. LINHARES ◽  
A. P. C. MALBOUISSON ◽  
I. RODITI
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Asymptotic Behaviors ◽  
Scalar Quantum ◽  
Feynman Amplitudes ◽  
Mellin Representation ◽  
Noncommutative Quantum

Starting from the complete Mellin representation of Feynman amplitudes for noncommutative vulcanized scalar quantum field theory, introduced in a previous publication, we generalize to this theory the study of asymptotic behaviors under scaling of arbitrary subsets of external invariants of any Feynman amplitude. This is accomplished in both convergent and renormalized amplitudes.

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