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Article

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

1
Advanced Computational Biology Center (ACBC), University Avenue 2120, Berkeley, CA 94704, USA
2
Moscow Institute of Physics and Technology (MIPT), Institutskiy Pereulok 9, Dolgoprudny 141701, Moscow Region, Russia
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Authors to whom correspondence should be addressed.
Particles 2019, 2(3), 385-410; https://doi.org/10.3390/particles2030024
Received: 26 July 2019 / Revised: 15 August 2019 / Accepted: 26 August 2019 / Published: 29 August 2019
(This article belongs to the Special Issue QCD and Hadron Structure)
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. View Full-Text
Keywords: quantum field theory (QFT); scalar QFT; nonlocal QFT; nonpolynomial QFT; Euclidean QFT; generating functional ?; abstract (formal) functional integral; functional integration measure; basis functions representation quantum field theory (QFT); scalar QFT; nonlocal QFT; nonpolynomial QFT; Euclidean QFT; generating functional ?; abstract (formal) functional integral; functional integration measure; basis functions representation
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MDPI and ACS Style

Bernard, M.; Guskov, V.A.; Ivanov, M.G.; Kalugin, A.E.; Ogarkov, S.L. Nonlocal Scalar Quantum Field Theory—Functional Integration, Basis Functions Representation and Strong Coupling Expansion. Particles 2019, 2, 385-410. https://doi.org/10.3390/particles2030024

AMA Style

Bernard M, Guskov VA, Ivanov MG, Kalugin AE, Ogarkov SL. Nonlocal Scalar Quantum Field Theory—Functional Integration, Basis Functions Representation and Strong Coupling Expansion. Particles. 2019; 2(3):385-410. https://doi.org/10.3390/particles2030024

Chicago/Turabian Style

Bernard, Matthew, Vladislav A. Guskov, Mikhail G. Ivanov, Alexey E. Kalugin, and Stanislav L. Ogarkov 2019. "Nonlocal Scalar Quantum Field Theory—Functional Integration, Basis Functions Representation and Strong Coupling Expansion" Particles 2, no. 3: 385-410. https://doi.org/10.3390/particles2030024

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