scholarly journals Nonlocal Scalar Quantum Field Theory—Functional Integration, Basis Functions Representation and Strong Coupling Expansion

Particles ◽  
2019 ◽  
Vol 2 (3) ◽  
pp. 385-410 ◽  
Author(s):  
Matthew Bernard ◽  
Vladislav A. Guskov ◽  
Mikhail G. Ivanov ◽  
Alexey E. Kalugin ◽  
Stanislav L. Ogarkov

Nonlocal quantum field theory (QFT) of one-component scalar field φ in D-dimensional Euclidean spacetime is considered. The generating functional (GF) of complete Green functions Z as a functional of external source j, coupling constant g and spatial measure d μ is studied. An expression for GF Z in terms of the abstract integral over the primary field φ is given. An expression for GF Z in terms of integrals over the primary field and separable Hilbert space (HS) is obtained by means of a separable expansion of the free theory inverse propagator L ^ over the separable HS basis. The classification of functional integration measures D φ is formulated, according to which trivial and two nontrivial versions of GF Z are obtained. Nontrivial versions of GF Z are expressed in terms of 1-norm and 0-norm, respectively. In the 1-norm case in terms of the original symbol for the product integral, the definition for the functional integration measure D φ over the primary field is suggested. In the 0-norm case, the definition and the meaning of 0-norm are given in terms of the replica-functional Taylor series. The definition of the 0-norm generator Ψ is suggested. Simple cases of sharp and smooth generators are considered. An alternative derivation of GF Z in terms of 0-norm is also given. All these definitions allow to calculate corresponding functional integrals over φ in quadratures. Expressions for GF Z in terms of integrals over the separable HS, aka the basis functions representation, with new integrands are obtained. For polynomial theories φ 2 n , n = 2 , 3 , 4 , … , and for the nonpolynomial theory sinh 4 φ , integrals over the separable HS in terms of a power series over the inverse coupling constant 1 / g for both norms (1-norm and 0-norm) are calculated. Thus, the strong coupling expansion in all theories considered is given. “Phase transitions” and critical values of model parameters are found numerically. A generalization of the theory to the case of the uncountable integral over HS is formulated—GF Z for an arbitrary QFT and the strong coupling expansion for the theory φ 4 are derived. Finally a comparison of two GFs Z , one on the continuous lattice of functions and one obtained using the Parseval–Plancherel identity, is given.

1998 ◽  
Vol 13 (12) ◽  
pp. 937-951 ◽  
Author(s):  
P. BASEILHAC ◽  
P. GRANGÉ ◽  
V. A. FATEEV

Integrable deformations of two affine Toda field theories based on BCn and [Formula: see text] non-simply laced algebras are studied. The proof of quantum integrability of the two theories is given and it is shown that the two theories are dual, corresponding respectively to the weak and strong coupling regimes of an intermediate theory. Perturbative calculations and the bootstrap principle permit the construction of the intermediate quantum field theory which flows from the first to the second with the coupling constant.


1979 ◽  
Vol 43 (8) ◽  
pp. 537-540 ◽  
Author(s):  
Carl M. Bender ◽  
Fred Cooper ◽  
G. S. Guralnik ◽  
Ralph Roskies ◽  
David H. Sharp

1979 ◽  
Vol 19 (6) ◽  
pp. 1865-1881 ◽  
Author(s):  
Carl M. Bender ◽  
Fred Cooper ◽  
G. S. Guralnik ◽  
David H. Sharp

1981 ◽  
Vol 24 (10) ◽  
pp. 2693-2705 ◽  
Author(s):  
Carl Bender ◽  
Fred Cooper ◽  
Richard Kenway ◽  
L. M. Simmons

2014 ◽  
Vol 14 (11&12) ◽  
pp. 1014-1080 ◽  
Author(s):  
Stephen P. Jordan ◽  
Keith S. M. Lee ◽  
John Preskill

Quantum field theory provides the framework for the most fundamental physical theories to be confirmed experimentally and has enabled predictions of unprecedented precision. However, calculations of physical observables often require great computational complexity and can generally be performed only when the interaction strength is weak. A full understanding of the foundations and rich consequences of quantum field theory remains an outstanding challenge. We develop a quantum algorithm to compute relativistic scattering amplitudes in massive $\phi^4$ theory in spacetime of four and fewer dimensions. The algorithm runs in a time that is polynomial in the number of particles, their energy, and the desired precision, and applies at both weak and strong coupling. Thus, it offers exponential speedup over existing classical methods at high precision or strong coupling.


2021 ◽  
Vol 2021 (10) ◽  
Author(s):  
Hidehiko Shimada ◽  
Hirohiko Shimada

Abstract We identify a nontrivial yet tractable quantum field theory model with space/time anisotropic scale invariance, for which one can exactly compute certain four-point correlation functions and their decompositions via the operator-product expansion(OPE). The model is the Calogero model, non-relativistic particles interacting with a pair potential $$ \frac{g}{{\left|x-y\right|}^2} $$ g x − y 2 in one dimension, considered as a quantum field theory in one space and one time dimension via the second quantisation. This model has the anisotropic scale symmetry with the anisotropy exponent z = 2. The symmetry is also enhanced to the Schrödinger symmetry. The model has one coupling constant g and thus provides an example of a fixed line in the renormalisation group flow of anisotropic theories.We exactly compute a nontrivial four-point function of the fundamental fields of the theory. We decompose the four-point function via OPE in two different ways, thereby explicitly verifying the associativity of OPE for the first time for an interacting quantum field theory with anisotropic scale invariance. From the decompositions, one can read off the OPE coefficients and the scaling dimensions of the operators appearing in the intermediate channels. One of the decompositions is given by a convergent series, and only one primary operator and its descendants appear in the OPE. The scaling dimension of the primary operator we computed depends on the coupling constant. The dimension correctly reproduces the value expected from the well-known spectrum of the Calogero model combined with the so-called state-operator map which is valid for theories with the Schrödinger symmetry. The other decomposition is given by an asymptotic series. The asymptotic series comes with exponentially small correction terms, which also have a natural interpretation in terms of OPE.


1956 ◽  
Vol 4 (2) ◽  
pp. 254-269 ◽  
Author(s):  
W. K. Burton ◽  
A. H. de Borde

2019 ◽  
Vol 49 ◽  
pp. 1960006
Author(s):  
B. A. Fayzullaev

The equations for the QED effective action derived in Ref. 3 are considered using singular perturbation theory. The effective action is divided into regular and singular (in coupling constant) parts. It is shown that expression for the regular part coincides with usual Feynman perturbation series over coupling constant, while the remainder has essential singularity at the vanishing coupling constant: [Formula: see text]. This means that in the frame of quantum field theory it is impossible “to switch off” electromagnetic interaction in general and pass on to “free electron”.


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