Tree weights and renormalization in QFT

2021 ◽  
pp. 39-49
Author(s):  
Adrian Tanasa
Keyword(s):  
Partition Function ◽  
Fundamental Theorem ◽  
Functional Integral ◽  
Perturbative Expansion ◽  
Functional Integrals ◽  
Perturbative Series ◽  
Fundamental Step ◽  
Feynman Graphs ◽  
Perturbative Renormalization ◽  
Feynman Amplitudes

In this chapter we define specific tree weights which appear natural when considering a certain approach to non-perturbative renormalization in QFT, namely the constructive renormalization. Several examples of such tree weights are explicitly given in Appendix A. A fundamental step in QFT is to compute the logarithm of functional integrals used to define the partition function of a given model This comes from a fundamental theorem of enumerative combinatorics, stating the logarithm counts the connected objects. The main advantage of the perturbative expansion of a QFT into a sum of Feynman amplitudes is to perform this computation explicitly: the logarithm of the functional integral is the sum of Feynman amplitudes restricted to connected graphs. The main disadvantage is that the perturbative series indexed by Feynman graphs typically diverges.

THE FUNCTIONAL INTEGRAL ON THE HALF-LINE

1990 ◽  
Vol 05 (15) ◽  
pp. 3029-3051 ◽  
Author(s):  
EDWARD FARHI ◽  
SAM GUTMANN
Keyword(s):  
Time Evolution ◽  
Wide Class ◽  
Green's Functions ◽  
Green’S Functions ◽  
Functional Integral ◽  
Half Line ◽  
Functional Integrals ◽  
Quantum Hamiltonian ◽  
Do So

A quantum Hamiltonian, defined on the half-line, will typically not lead to unitary time evolution unless the domain of the Hamiltonian is carefully specified. Different choices of the domain result in different Green’s functions. For a wide class of non-relativistic Hamiltonians we show how to define the functional integral on the half-line in a way which matches the various Green’s functions. To do so we analytically continue, in time, functional integrals constructed with real measures that give weight to paths on the half-line according to how much time they spend near the origin.

scholarly journals Some comments on infinities on quantum field theory: A functional integral approach

2016 ◽  
Vol 24 (2) ◽  
Author(s):  
Luiz C. L. Botelho
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Function Space ◽  
Functional Integral ◽  
Field Theories ◽  
Integral Approach ◽  
Quantum Field ◽  
Functional Integrals ◽  
Euclidean Field ◽  
The Subject

AbstractWe analyze on the formalism of probabilities measures-functional integrals on function space the problem of infinities on Euclidean field theories. We also clarify and generalize our previous published studies on the subject.

Algebraic combinatorics and QFT

2021 ◽  
pp. 76-94
Author(s):  
Adrian Tanasa
Keyword(s):  
Hopf Algebra ◽  
Hochschild Cohomology ◽  
Algebraic Combinatorics ◽  
Fourth Section ◽  
Analytic Combinatorics ◽  
Multi Scale ◽  
Starting Point ◽  
Feynman Graphs ◽  
Perturbative Renormalization

We have seen in the previous chapter some of the non-trivial interplay between analytic combinatorics and QFT. In this chapter, we illustrate how yet another branch of combinatorics, algebraic combinatorics, interferes with QFT. In this chapter, after a brief algebraic reminder in the first section, we introduce in the second section the Connes–Kreimer Hopf algebra of Feynman graphs and we show its relation with the combinatorics of QFT perturbative renormalization. We then study the algebra's Hochschild cohomology in relation with the combinatorial Dyson–Schwinger equation in QFT. In the fourth section we present a Hopf algebraic description of the so-called multi-scale renormalization (the multi-scale approach to the perturbative renormalization being the starting point for the constructive renormalization programme).

scholarly journals PERTURBATIVE EXPANSION OF τ HADRONIC SPECTRAL FUNCTION MOMENTS

2014 ◽  
Vol 35 ◽  
pp. 1460442
Author(s):  
DIOGO BOITO
Keyword(s):  
Perturbation Theory ◽  
Renormalization Group ◽  
Systematic Study ◽  
Higher Order ◽  
Perturbative Expansion ◽  
Main Stream ◽  
Spectral Functions ◽  
Perturbative Series ◽  
Fixed Order ◽  
Adler Function

In the extraction of αs from hadronic τ decay data several moments of the spectral functions have been employed. Furthermore, different renormalization group improvement (RGI) frameworks have been advocated, leading to conflicting values of αs. Recently, we performed a systematic study of the perturbative behavior of these moments in the context of the two main-stream RGI frameworks: Fixed Order Perturbation Theory (FOPT) and Contour Improved Perturbation Theory (CIPT). The yet unknown higher order coefficients of the perturbative series were modelled using the available knowledge of the renormalon singularities of the QCD Adler function. We were able to show that within these RGI frameworks some of the commonly employed moments should be avoided due to their poor perturbative behavior. Furthermore, under reasonable assumptions about the higher order behavior of the perturbative series FOPT provides the preferred RGI framework.

scholarly journals Approximate evaluation of functional integrals with centrifugal potential

2019 ◽  
Vol 55 (2) ◽  
pp. 152-157 ◽  
Author(s):  
V. B. Malyutin
Keyword(s):  
Eigenvalue Problem ◽  
Centrifugal Force ◽  
Computation Time ◽  
Functional Integral ◽  
Computer Memory ◽  
Approximate Evaluation ◽  
Functional Integrals ◽  
Sequence Method

Approximate evaluation of functional integrals containing a centrifugal potential is considered. By a centrifugal potential is understood a potential arising from a centrifugal force. A combination of the method based on expanding into a series of the eigenfunctions of a Hamiltonian generating a functional integral and the Sturm sequence method for the eigenvalue problem is used for approximate evaluation of functional integrals. This combination allows one to significantly reduce a computation time and a used computer memory volume in comparison to other known methods.

scholarly journals A NEW DERIVATION OF THE COLEMAN–WEINBERG MECHANISM IN MASSLESS SCALAR QED

1996 ◽  
Vol 11 (09) ◽  
pp. 749-754 ◽  
Author(s):  
A.P.C. MALBOUISSON ◽  
F.S. NOGUEIRA ◽  
N.F. SVAITER
Keyword(s):  
Phase Transition ◽  
Effective Potential ◽  
Functional Integral ◽  
Massless Scalar ◽  
Integral Formalism ◽  
First Order ◽  
Unitary Gauge ◽  
Feynman Graphs ◽  
Functional Integral Formalism

We present a new derivation of the Coleman–Weinberg expression for the effective potential for massless scalar QED. Our result is obtained using the functional integral formalism, without expansions in Feynman graphs. We perform our calculations in the unitary gauge. The first-order character of the phase transition is established.

scholarly journals INTERACTION HIERARCHY: STRING AND QUANTUM GRAVITY

1996 ◽  
Vol 11 (17) ◽  
pp. 1379-1396 ◽  
Author(s):  
G.K. SAVVIDY ◽  
K.G. SAVVIDY
Keyword(s):  
Partition Function ◽  
Quantum Gravity ◽  
Spin System ◽  
Scaling Limit ◽  
Functional Integral ◽  
Local Interaction ◽  
Linear Action ◽  
Random Surfaces

We have found that the functional integral for quantum gravity can be represented as a superposition of less complicated theory of random surfaces with Euler character as an action. We propose an alternative linear action A(M4) for quantum gravity. On the lattice we constructed spin system with local interaction, which has the equivalent partition function. The scaling limit is discussed.

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