Modified propagators in field theory (with application to the anomalous magnetic moment of the nucleon)

1954 ◽  
Vol 223 (1152) ◽  
pp. 112-129 ◽  
Keyword(s):  
Field Theory ◽  
Asymptotic Expansion ◽  
Magnetic Moment ◽  
Anomalous Magnetic Moment ◽  
Perturbation Expansion ◽  
Field Theories ◽  
Meson Field ◽  
Coupling Constant ◽  
Zero Coupling ◽  
Better Than

Many attempts have been made to improve upon the perturbation expansion in meson-field theories, one such attempt being the introduction of modified propagators S' F and ∆' F . It is shown in this paper that the introduction of these new propagators (or at least in the form that has been proposed) creates new infinities which cannot be removed by renormalization. These new infinities are due to new complex poles of the modified propagators. A tentative prescription is put forth to get over these new difficulties, but it is still intimately connected with the perturbation expansion. Unfortunately, the prescription does not give an unambiguous answer. A particular S' F is used in the calculation of the anomalous magnetic moment of the nucleon. The results obtained are no better than those of other workers. However, there are many reasons why this may be the case. It is also shown that the subseries which arises in this case leads, at best, to an asymptotic expansion in the coupling constant. The nature of the singularity at zero coupling is found.

An example of a divergent perturbation expansion in field theory

1952 ◽  
Vol 48 (4) ◽  
pp. 625-639 ◽  
Author(s):  
C. A. Hurst
Keyword(s):  
Field Theory ◽  
Lower Bound ◽  
Contact Interaction ◽  
General Problem ◽  
Rest Mass ◽  
Perturbation Expansion ◽  
Field Theories ◽  
Coupling Constant ◽  
Perturbation Expansions ◽  
Irreducible Graphs

AbstractThe convergence of the perturbation expansion for the S-matrix in interaction representation of a three-boson contact interaction is investigated. A lower bound is obtained for the integrals corresponding to irreducible graphs when the total rest mass of the system is insufficient for bare particles to be created. It is shown that in this case the perturbation expansion cannot converge no matter what value the coupling constant has. A discussion is given of the bearing of this result on the general problem of the convergence of perturbation expansions for the S-matrix in renormalized field theories.

Interacting Fields

2021 ◽  
pp. 237-252
Author(s):  
J. Iliopoulos ◽  
T.N. Tomaras
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Perturbation Expansion ◽  
Field Theories ◽  
Green Functions ◽  
Quantum Field ◽  
Minkowski Space Time ◽  
Dimensional Minkowski Space ◽  
Feynman Rules ◽  
Wightman Axioms

We present a simple form of the Wightman axioms in a four-dimensional Minkowski space-time which are supposed to define a physically interesting interacting quantum field theory. Two important consequences follow from these axioms. The first is the invariance under CPT which implies, in particular, the equality of masses and lifetimes for particles and anti-particles. The second is the connection between spin and statistics. We give examples of interacting field theories and develop the perturbation expansion for Green functions. We derive the Feynman rules, both in configuration and in momentum space, for some simple interacting theories. The rules are unambiguous and allow, in principle, to compute any Green function at any order in perturbation.

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 Theory of the Anomalous Magnetic Moment of the Electron

Atoms ◽  
2019 ◽  
Vol 7 (1) ◽  
pp. 28 ◽  
Author(s):  
Tatsumi Aoyama ◽  
Toichiro Kinoshita ◽  
Makiko Nio
Keyword(s):  
Fine Structure ◽  
Magnetic Moment ◽  
Anomalous Magnetic Moment ◽  
Quantum Electrodynamics ◽  
Penning Trap ◽  
Weak Interactions ◽  
Perturbation Expansion ◽  
Structure Constant ◽  
Fine Structure Constant ◽  
Best Value

The anomalous magnetic moment of the electron a e measured in a Penning trap occupies a unique position among high precision measurements of physical constants in the sense that it can be compared directly with the theoretical calculation based on the renormalized quantum electrodynamics (QED) to high orders of perturbation expansion in the fine structure constant α , with an effective parameter α / π . Both numerical and analytic evaluations of a e up to ( α / π ) 4 are firmly established. The coefficient of ( α / π ) 5 has been obtained recently by an extensive numerical integration. The contributions of hadronic and weak interactions have also been estimated. The sum of all these terms leads to a e ( theory ) = 1 159 652 181.606 ( 11 ) ( 12 ) ( 229 ) × 10 − 12 , where the first two uncertainties are from the tenth-order QED term and the hadronic term, respectively. The third and largest uncertainty comes from the current best value of the fine-structure constant derived from the cesium recoil measurement: α − 1 ( Cs ) = 137.035 999 046 ( 27 ) . The discrepancy between a e ( theory ) and a e ( ( experiment ) ) is 2.4 σ . Assuming that the standard model is valid so that a e (theory) = a e (experiment) holds, we obtain α − 1 ( a e ) = 137.035 999 1496 ( 13 ) ( 14 ) ( 330 ) , which is nearly as accurate as α − 1 ( Cs ) . The uncertainties are from the tenth-order QED term, hadronic term, and the best measurement of a e , in this order.

scholarly journals A study of the nature of the field theories of the electron and positron and of the meson

1946 ◽  
Vol 185 (1000) ◽  
pp. 14-34 ◽  
Keyword(s):  
Current Density ◽  
Magnetic Moment ◽  
Poynting Vector ◽  
Field Theories ◽  
Meson Field ◽  
Theory Of Relativity ◽  
Magnetic Current ◽  
Additional Degree ◽  
Additional Dimension ◽  
A Current

The field theories of the electron and positron and also of the meson are developed by means of a close analogy with the photon. The analogy consists in the representation of the tracks of these particles by means of null-geodesics. The choice of notation is guided by the attempt to arrive at a theory in which the lengths (h/m 0 c) and (e 2 /m 0 c 2 ) occur naturally without reference to the structure of the particles, and in which the concept of quantization of electric charge is included. It is found that these objects can be attained by assuming that an additional degree of freedom is necessary for the description of the particles. If this is regarded as an additional dimension, it is found that an exact analogy can be made with the field theories familiar in the theory of relativity. An important feature is the union, in a single tensor, of energy, momentum and current density. A certain arbitrariness, not unlike that associated with the Poynting vector, is revealed, and it is shown that if this is removed by making a definite choice of the magnitude of the magnetic moment of the electron and positron, the spin angular momefttum is ^hereby fixed at the value 1/2h. In the development of the meson field the analogy shows* that the nuclear sources of the field act as if contributing a current density analogous to a magnetic current density in the electromagnetic case. The use of the additional degreb of freedom in the sinusoidal form indicates that the ratio of the constants g 1 and g 2 introduced into field theories as measures of the strengths of the sources is determined by the mass of the particle emitted in the neutron-proton transition.

scholarly journals GEOMETRICALLY RELATING MOMENTUM CUT-OFF AND DIMENSIONAL REGULARIZATION

2012 ◽  
Vol 10 (02) ◽  
pp. 1250081 ◽  
Author(s):  
SUSAMA AGARWALA
Keyword(s):  
Field Theory ◽  
Scalar Field ◽  
Hopf Algebras ◽  
Dimensional Regularization ◽  
Mass Scale ◽  
Feynman Diagrams ◽  
Field Theories ◽  
Coupling Constant ◽  
Regularization Scheme ◽  
Β Function

The β function for a scalar field theory describes the dependence of the coupling constant on the renormalization mass scale. This dependence is affected by the choice of regularization scheme. I explicitly relate the β functions of momentum cut-off regularization and dimensional regularization on scalar field theories by a gauge transformation using the Hopf algebras of the Feynman diagrams of the theories.

A Parenthesis on “R2QFTs” ⋆⋆

2018 ◽  
pp. 70-75
Author(s):  
Alvaro De Rújula
Keyword(s):  
Standard Model ◽  
Magnetic Moment ◽  
Anomalous Magnetic Moment ◽  
Relativistic Quantum ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Quantum Fields ◽  
The Standard Model

Renormalizable Relativistic Quantum Field Theories (R2QFTs) are theories In which a few parameters must be taken from observations but otherwise make predictions on the phenomena they describe. The ingreedients of the Standard Model of particles are examples. The most precise of them is Quantum Electro-Dynamics (QED). The QED prediction of the “anomalous” magnetic moment of the muon is discussed as a detailed example. Relativistic quantum fields describe all aspects and properties of particles and their interactions, they are “the mother of all concepts.”

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