Asymptotic Perturbation Expansion for the S-matrix in P(⌽)2 Quantum Field Theory Models

Abstract A precise link is derived between scalar-graviton S-matrix elements and expectation values of operators in a worldline quantum field theory (WQFT), both used to describe classical scattering of black holes. The link is formally provided by a worldline path integral representation of the graviton-dressed scalar propagator, which may be inserted into a traditional definition of the S-matrix in terms of time-ordered correlators. To calculate expectation values in the WQFT a new set of Feynman rules is introduced which treats the gravitational field hμν(x) and position $$ {x}_i^{\mu}\left({\tau}_i\right) $$ x i μ τ i of each black hole on equal footing. Using these both the 3PM three-body gravitational radiation 〈hμv(k)〉 and 2PM two-body deflection $$ \Delta {p}_i^{\mu } $$ Δ p i μ from classical black hole scattering events are obtained. The latter can also be obtained from the eikonal phase of a 2 → 2 scalar S-matrix, which we show corresponds to the free energy of the WQFT.

Download Full-text

Interacting Fields

10.1093/oso/9780192844200.003.0011 ◽

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.

Download Full-text

The Principles of Renormalisation

10.1093/oso/9780192844200.003.0015 ◽

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.

Download Full-text

Universal bounds for large determinants from non-commutative Hölder inequalities in fermionic constructive quantum field theory

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202517500361 ◽

2017 ◽

Vol 27 (10) ◽

pp. 1963-1992 ◽

Cited By ~ 3

Author(s):

J.-B. Bru ◽

W. de Siqueira Pedra

Keyword(s):

Field Theory ◽

Quantum Field Theory ◽

Von Neumann Algebras ◽

Perturbation Expansion ◽

Perturbation Series ◽

Interparticle Interaction ◽

Quantum Field ◽

Von Neumann ◽

Constructive Quantum Field Theory ◽

Strength Formula

Efficiently bounding large determinants is an essential step in non-relativistic constructive quantum field theory to prove the absolute convergence of the perturbation expansion of correlation functions in terms of powers of the strength [Formula: see text] of the interparticle interaction. We provide, for large determinants of fermionic covariances, sharp bounds which hold for all (bounded and unbounded, the latter not being limited to semibounded) one-particle Hamiltonians. We find the smallest universal determinant bound to be exactly [Formula: see text]. In particular, the convergence of perturbation series at [Formula: see text] of any fermionic quantum field theory is ensured if the matrix entries (with respect to some fixed orthonormal basis) of the covariance and the interparticle interaction decay sufficiently fast. Our proofs use Hölder inequalities for general non-commutative [Formula: see text]-spaces derived by Araki and Masuda [Positive cones and [Formula: see text]-spaces for von Neumann algebras, Publ. RIMS[Formula: see text] Kyoto Univ. 18 (1982) 339–411].

Download Full-text