scholarly journals The Use of Power Expansions in Quantum Field Theory

1997 ◽  
Vol 12 (21) ◽  
pp. 3625-3663 ◽  
Author(s):  
Jan Fischer
Keyword(s):  
Operator Product Expansion ◽  
Strong Dependence ◽  
Summation Method ◽  
Operator Product ◽  
Quantum Field ◽  
Coupling Constant ◽  
Perturbation Expansions ◽  
Borel Summation ◽  
Energy Variable ◽  
Euclidean Region

Methods of summation of power series relevant to applications in quantum theory are reviewed, with particular attention to expansions in powers of the coupling constant and in inverse powers of an energy variable. Alternatives to the Borel summation method are considered and their relevance to different physical situations is discussed. Emphasis is placed on quantum chromodynamics. Recent applications of the renormalon language to perturbation expansions (resummation of bubble chains) in various QCD processes are reported and the importance of observing the full renormalization-group invariance in predicting observables is emphasized. News in applications of the Borel-plane formalism to phenomenology are conveyed. The properties of the operator-product expansion along different rays in the complex plane are examined and the problem is studied as to how the remainder after subtraction of the first n terms depends on the distance from the Euclidean region. Estimates of the remainder are explicitly calculated and their strong dependence on the nature of the discontinuity along the cut is shown. Relevance of this subject to calculations of various QCD effects is discussed.

scholarly journals OPERATOR-PRODUCT EXPANSION AND ANALYTICITY

1999 ◽  
Vol 14 (30) ◽  
pp. 4819-4840
Author(s):  
JAN FISCHER ◽  
IVO VRKOČ
Keyword(s):  
Operator Product Expansion ◽  
Deflection Angle ◽  
Vacuum Expectation ◽  
Vacuum Expectation Value ◽  
Operator Product ◽  
Coupling Constant ◽  
Expectation Value ◽  
Euclidean Region ◽  
The Deflection Angle ◽  
Class Of Functions

We discuss the current use of the operator-product expansion in QCD calculations. Treating the OPE as an expansion in inverse powers of an energy-squared variable (with possible exponential terms added), approximating the vacuum expectation value of the operator product by several terms and assuming a bound on the remainder along the Euclidean region, we observe how the bound varies with increasing deflection from the Euclidean ray down to the cut (Minkowski region). We argue that the assumption that the remainder is constant for all angles in the cut complex plane down to the Minkowski region is not justified. Making specific assumptions on the properties of the expanded function, we obtain bounds on the remainder in explicit form and show that they are very sensitive both to the deflection angle and to the class of functions considered. The results obtained are discussed in connection with calculations of the coupling constant αs from the τ decay.

scholarly journals RENORMALIZED QUANTUM YANG–MILLS FIELDS IN CURVED SPACETIME

2008 ◽  
Vol 20 (09) ◽  
pp. 1033-1172 ◽  
Author(s):  
STEFAN HOLLANDS
Keyword(s):  
Operator Product Expansion ◽  
Poisson Algebra ◽  
Technical Difficulty ◽  
Operator Product ◽  
Quantum Field ◽  
Gauge Invariant ◽  
Yang Mills ◽  
Brst Charge ◽  
A New Technique ◽  
Formal Power

We present a proof that the quantum Yang–Mills theory can be consistently defined as a renormalized, perturbative quantum field theory on an arbitrary globally hyperbolic curved, Lorentzian spacetime. To this end, we construct the non-commutative algebra of observables, in the sense of formal power series, as well as a space of corresponding quantum states. The algebra contains all gauge invariant, renormalized, interacting quantum field operators (polynomials in the field strength and its derivatives), and all their relations such as commutation relations or operator product expansion. It can be viewed as a deformation quantization of the Poisson algebra of classical Yang–Mills theory equipped with the Peierls bracket. The algebra is constructed as the cohomology of an auxiliary algebra describing a gauge fixed theory with ghosts and anti-fields. A key technical difficulty is to establish a suitable hierarchy of Ward identities at the renormalized level that ensures conservation of the interacting BRST-current, and that the interacting BRST-charge is nilpotent. The algebra of physical interacting field observables is obtained as the cohomology of this charge. As a consequence of our constructions, we can prove that the operator product expansion closes on the space of gauge invariant operators. Similarly, the renormalization group flow is proved not to leave the space of gauge invariant operators. The key technical tool behind these arguments is a new universal Ward identity that is formulated at the algebraic level, and that is proven to be consistent with a local and covariant renormalization prescription. We also develop a new technique to accomplish this renormalization process, and in particular give a new expression for some of the renormalization constants in terms of cycles.

scholarly journals IR-IMPROVED OPERATOR PRODUCT EXPANSIONS IN NON-ABELIAN GAUGE THEORY

2013 ◽  
Vol 28 (18) ◽  
pp. 1350069 ◽  
Author(s):  
B. F. L. WARD
Keyword(s):  
Gauge Theory ◽  
Operator Product Expansion ◽  
Primary Objective ◽  
Operator Product ◽  
Coupling Constant ◽  
Abelian Gauge ◽  
Abelian Gauge Theory ◽  
Lhc Physics

We present a formulation of the operator product expansion that is infrared finite to all orders in the attendant massless non-Abelian gauge theory coupling constant which we will oftentimes associate with the QCD theory. We actually have the QCD theory as our primary objective in view of the operation of the LHC at CERN. We make contact in this way with the recently introduced IR-improved DGLAP-CS theory and point-out phenomenological implications accordingly, with an eye toward the precision QCD theory for LHC physics.

Dynamical derivation of vacuum operator-product expansion in Euclidean conformal quantum field theory

1976 ◽  
Vol 13 (4) ◽  
pp. 887-912 ◽  
Author(s):  
V. K. Dobrev ◽  
V. B. Petkova ◽  
S. G. Petrova ◽  
I. T. Todorov
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Operator Product Expansion ◽  
Operator Product ◽  
Quantum Field

scholarly journals QUANTUM FIELD THEORY IN CURVED SPACE–TIME, THE OPERATOR PRODUCT EXPANSION, AND DARK ENERGY

2008 ◽  
Vol 17 (13n14) ◽  
pp. 2607-2615 ◽  
Author(s):  
STEFAN HOLLANDS ◽  
ROBERT M. WALD
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Operator Product Expansion ◽  
Vacuum Expectation ◽  
Vacuum State ◽  
Space Time ◽  
Operator Product ◽  
Energy Tensor ◽  
Quantum Field ◽  
Curved Space

To make sense of quantum field theory in an arbitrary (globally hyperbolic) curved space–time, the theory must be formulated in a local and covariant manner in terms of locally measureable field observables. Since a generic curved space–time does not possess symmetries or a unique notion of a vacuum state, the theory also must be formulated in a manner that does not require symmetries or a preferred notion of a "vacuum state" and "particles". We propose such a formulation of quantum field theory, wherein the operator product expansion (OPE) of the quantum fields is elevated to a fundamental status, and the quantum field theory is viewed as being defined by its OPE. Since the OPE coefficients may be better behaved than any quantities having to do with states, we suggest that it may be possible to perturbatively construct the OPE coefficients — and, thus, the quantum field theory. By contrast, ground/vacuum states — in space–times, such as Minkowski space–time, where they may be defined — cannot vary analytically with the parameters of the theory. We argue that this implies that composite fields may acquire nonvanishing vacuum state expectation values due to nonperturbative effects. We speculate that this could account for the existence of a nonvanishing vacuum expectation value of the stress-energy tensor of a quantum field occurring at a scale much smaller than the natural scales of the theory.

scholarly journals OPE for XXX

2018 ◽  
Vol 30 (06) ◽  
pp. 1840006 ◽  
Author(s):  
Philippe Di Francesco ◽  
Fedor Smirnov
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Experimental Data ◽  
Correlation Functions ◽  
Spin Chain ◽  
Operator Product Expansion ◽  
The Other ◽  
Operator Product ◽  
Quantum Field ◽  
Xxx Model

We explain a new method for finding the correlation functions for the XXX model which is based on the concepts of Operator Product Expansion of Quantum Field Theory on one hand and of fermionic bases for the XXX spin chain on the other. With this method, we are able to perform computations for up to 11 lattice sites. We show that these “experimental” data allow to guess exact formulae for the OPE coefficients. In memory of Ludwig Dmitrievich Faddeev

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