A twistor approach to the regularization of divergences

1985 ◽  
Vol 397 (1813) ◽  
pp. 341-374 ◽  
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Infrared Divergence ◽  
Standard Theory ◽  
Quantum Field ◽  
New Formalism ◽  
First Order ◽  
Finite Theory ◽  
Finite Quantity ◽  
Modified Theory

One object of the twistor programme, as developed principally by R. Penrose, is the production of a manifestly finite theory of scattering in quantum field theory. Earlier work has shown that progress towards this goal is obstructed even at the first-order level, by the appearance of an infrared divergence in the standard theory. New studies in many-dimensional contour integration now suggest a simple but very powerful modification to this branch of twistor theory, in which the full (as opposed to the projective) twistor space plays an essential role. In this modified theory there arise natural contour-integral expressions with the effect of eliminating the infrared divergence previously noted, and replacing it by a finite quantity. This regularization can be specified by using a formalism of ‘inhomogeneous twistor diagrams’. The interpretation of this new formalism is not yet wholly clear, but the inhomogeneity can be seen as a means of relinquishing the concept of space-time point, while preserving light-cone structure. It therefore suggests a quite fresh approach to the divergences of quantum field theory.

scholarly journals Logarithmic expansion of field theories: higher orders and resummations

2021 ◽  
Vol 81 (6) ◽  
Author(s):  
Vincenzo Branchina ◽  
Alberto Chiavetta ◽  
Filippo Contino
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Finite Order ◽  
Weak Coupling ◽  
Field Theories ◽  
Quantum Field ◽  
First Order ◽  
Free Theory ◽  
Formal Expansion ◽  
The Way

AbstractA formal expansion for the Green’s functions of a quantum field theory in a parameter $$\delta $$ δ that encodes the “distance” between the interacting and the corresponding free theory was introduced in the late 1980s (and recently reconsidered in connection with non-hermitian theories), and the first order in $$\delta $$ δ was calculated. In this paper we study the $${\mathcal {O}}(\delta ^2)$$ O ( δ 2 ) systematically, and also push the analysis to higher orders. We find that at each finite order in $$\delta $$ δ the theory is non-interacting: sensible physical results are obtained only resorting to resummations. We then perform the resummation of UV leading and subleading diagrams, getting the $${\mathcal {O}}(g)$$ O ( g ) and $${\mathcal {O}}(g^2)$$ O ( g 2 ) weak-coupling results. In this manner we establish a bridge between the two expansions, provide a powerful and unique test of the logarithmic expansion, and pave the way for further studies.

From Classical to Quantum Fields

2017 ◽  
Author(s):  
Laurent Baulieu ◽  
John Iliopoulos ◽  
Roland Sénéor
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Laws Of Nature ◽  
Classical Field Theory ◽  
Functional Integral ◽  
Theoretical Physics ◽  
Standard Theory ◽  
Quantum Field ◽  
Detailed Model ◽  
Universal Language

Quantum field theory has become the universal language of most modern theoretical physics. This book is meant to provide an introduction to this subject with particular emphasis on the physics of the fundamental interactions and elementary particles. It is addressed to advanced undergraduate, or beginning graduate, students, who have majored in physics or mathematics. The ambition is to show how these two disciplines, through their mutual interactions over the past hundred years, have enriched themselves and have both shaped our understanding of the fundamental laws of nature. The subject of this book, the transition from a classical field theory to the corresponding Quantum Field Theory through the use of Feynman’s functional integral, perfectly exemplifies this connection. It is shown how some fundamental physical principles, such as relativistic invariance, locality of the interactions, causality and positivity of the energy, form the basic elements of a modern physical theory. The standard theory of the fundamental forces is a perfect example of this connection. Based on some abstract concepts, such as group theory, gauge symmetries, and differential geometry, it provides for a detailed model whose agreement with experiment has been spectacular. The book starts with a brief description of the field theory axioms and explains the principles of gauge invariance and spontaneous symmetry breaking. It develops the techniques of perturbation theory and renormalisation with some specific examples. The last Chapters contain a presentation of the standard model and its experimental successes, as well as the attempts to go beyond with a discussion of grand unified theories and supersymmetry.

scholarly journals A novel view on successive quantizations, leading to increasingly more “miraculous” states

2019 ◽  
Vol 34 (23) ◽  
pp. 1950186 ◽  
Author(s):  
Matej Pavšič
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Wave Function ◽  
Wave Packet ◽  
Time Derivative ◽  
Point Particle ◽  
Order Equation ◽  
Quantum Field ◽  
First Order ◽  
Complex Wave

A series of successive quantizations is considered, starting with the quantization of a non-relativistic or relativistic point particle: (1) quantization of a particle’s position, (2) quantization of wave function, (3) quantization of wave functional. The latter step implies that the wave packet profiles forming the states of quantum field theory are themselves quantized, which gives new physical states that are configurations of configurations. In the procedure of quantization, instead of the Schrödinger first-order equation in time derivative for complex wave function (or functional), the equivalent second-order equation for its real part was used. In such a way, at each level of quantization, the equation a quantum state satisfies is just like that of a harmonic oscillator, and wave function(al) is composed in terms of the pair of its canonically conjugated variables.

scholarly journals Quantum Ostrogradsky theorem

2020 ◽  
Vol 2020 (9) ◽  
Author(s):  
Hayato Motohashi ◽  
Teruaki Suyama
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Quantum Field ◽  
First Order ◽  
Higher Derivative ◽  
Original Theorem ◽  
Classical Lagrangian

Abstract The Ostrogradsky theorem states that any classical Lagrangian that contains time derivatives higher than the first order and is nondegenerate with respect to the highest-order derivatives leads to an unbounded Hamiltonian which linearly depends on the canonical momenta. Recently, the original theorem has been generalized to nondegeneracy with respect to non-highest-order derivatives. These theorems have been playing a central role in construction of sensible higher-derivative theories. We explore quantization of such non-degenerate theories, and prove that Hamiltonian is still unbounded at the level of quantum field theory.

A stochastic variational treatment of quantum mechanics

1995 ◽  
Vol 450 (1939) ◽  
pp. 417-437 ◽  
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Wave Function ◽  
Quantum Mechanics ◽  
Standard Theory ◽  
Quantum Field ◽  
Extended Form ◽  
Variational Treatment ◽  
Close Relationship ◽  
Methods Of Control

An earlier development of some results in quantum mechanics from a stochastic variational principle is extended in several directions. An outline is first given of the methods of control theory upon which the development is based, and earlier results are briefly described. Extensions are then given to relativistic systems, to Dirac’s equation, and to elementary quantum field theory. The aim thoughout is to show that results in the standard theory can be obtained in a uniform way from an extended form of Hamilton’s principle, which has the advantage of conciseness and a relatively close relationship to the classical theory. The wave function appears as a modified form of the optimal cost function, and the photon can be identified with a singularity in the electromagnetic field. Interference is explained by optimization of an expected value, the ensemble over which the expectation is taken being dependent upon the information available.

Quantum Gravity as a Resource

2020 ◽  
pp. 193-214
Author(s):  
Dean Rickles
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Quantum Gravity ◽  
Gravitational Collapse ◽  
Maximum Energy ◽  
Minimal Length ◽  
Quantum Field ◽  
Small Scales ◽  
Finite Theory ◽  
Primary Problem

This chapter focuses on the central motivation for much of what can be labeled ‘quantum gravity’ in the earliest phases of research, namely that it provides a potentially abundant resource for curing problems in quantum field theory. While it was rare to have fully worked out examples along these lines, it provided a much needed impetus to the study of quantum gravity at a time when there were few other reasons to bother with it. The primary problem was the ubiquitous divergences, which proved extremely stubborn and worrying to field theorists. Not all of the approaches were looked at involved gravitation directly, however, and focused more on ways of generating a discrete structure (with a minimal length or maximum energy) that would provide a physical cutoff, thus grounding a finite theory. These filtered through into gravitational research only later than our timeframe, in a variety of ways, including the small scales necessarily reached in gravitational collapse.

scholarly journals Fixing the non-relativistic expansion of the 1PM potential

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Gianluca Grignani ◽  
Troels Harmark ◽  
Marta Orselli ◽  
Andrea Placidi
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Scattering Amplitudes ◽  
Canonical Transformation ◽  
Effective Potential ◽  
Scattering Amplitude ◽  
Quantum Field ◽  
First Order ◽  
Scalar Matter ◽  
Frame Dependence

Abstract We obtain a first order post-Minkowskian two-body effective potential whose post-Newtonian expansion directly reproduces the Einstein-Infeld-Hoffmann potential. Post-Minkowskian potentials can be extracted from on-shell scattering amplitudes in a quantum field theory of scalar matter coupled to gravity. Previously, such potentials did not reproduce the Einstein-Infeld-Hoffmann potential without employing a suitable canonical transformation. In this work, we resolve this issue by obtaining a new expression for the first-order post-Minkowskian potential. This is accomplished by exploiting the reference frame dependence that arises in the scattering amplitude computation. Finally, as a check on our result, we demonstrate that our new potential gives the correct scattering angle.

scholarly journals Beyond the Variational Principle in Quantum Field Theory

1995 ◽  
Vol 48 (1) ◽  
pp. 39
Author(s):  
Lloyd CL Hollenberg
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Cluster Expansion ◽  
Vacuum Energy ◽  
Gaussian Approximation ◽  
Diagonal Form ◽  
Quantum Field ◽  
Zeroth Order ◽  
First Order ◽  
The Continuum

A method of summing diagrams in quantum field theory beyond the variational Gaussian approximation is proposed using the continuum form of the recently developed plaquette expansion. In the context of >-<j} theory the Hamiltonian, H[�], of the Schrodinger functional equation H[�]\II[�] = E\II[�] can be written down in tri-diagonal form as a cluster expansion in terms of connected moment coefficients derived from Hamiltonian moments (Hn) == !V�VI[�]Hn[�JVd�] with respect to a trial state VI [�]. The usual variational procedure corresponds to minimising the zeroth order of this cluster expansion. At first order in the expansion, the Hamiltonian in this form can be diagonalised analytically. The subsequent expression for the vacuum energy E contains Hamiltonian moments up to fourth order and hence is a summation over multi-loop diagrams, laying the foundation for the calculation of the effective potential beyond the Gaussian approximation.

Twistor diagrams and massless Møller scattering

1983 ◽  
Vol 385 (1788) ◽  
pp. 207-228 ◽  
Keyword(s):  
Field Theory ◽  
Quantum Electrodynamics ◽  
General Type ◽  
Infrared Divergence ◽  
Scattering Data ◽  
Quantum Field ◽  
First Order ◽  
Moller Scattering ◽  
Correct Account ◽  
Massless Fields

The theory of twistor diagrams, as devised by Penrose, is intended to lead to a manifestly finite account of scattering amplitudes in quantum field theory. The theory is here extended to a more general type of interaction between massless fields than has hitherto been described. It is applied to the example of first-order massless Møller scattering in quantum electrodynamics. It is shown that earlier studies of this example have failed to render a correct account, in particular by overlooking an infrared divergence, but that the scattering data can nevertheless be represented within the twistor formalism.

scholarly journals The Snyder Model and Quantum Field Theory

2019 ◽  
Vol 64 (11) ◽  
pp. 991 ◽  
Author(s):  
S. Mignemi
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Quantum Field ◽  
Finite Theory ◽  
Standard Formalism

We review the main features of the relativistic Snyder model and its generalizations. We discuss the quantum field theory on this background using the standard formalism of noncommutative QFT and discuss the possibility of obtaining a finite theory.

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