Wightman axiomatic approach in noncommutative field theory

Vernov, Yu. S.; Mnatsakanova, M. N.

doi:10.1007/s11232-005-0016-y

Cited by 8 publications

(15 citation statements)

References 24 publications

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“…Nevertheless in a series of papers (see [10] - [12] and references therein), the axiomatic approach to QFT was developed for other spaces of test functions. Wightman approach in NC QFT was formulated in [13], [14] (see also [15]). For a theory described by the Hermitian field ϕ(x) and with the vacuum state denoted by Ψ 0 , the Wightman functions can be formally written down as follows :…”

Section: Introductionmentioning

confidence: 99%

Test functions space in noncommutative quantum field theory

Chaichian¹,

Mnatsakanova²,

Tureanu³

et al. 2008

J. High Energy Phys.

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It is proven that the ⋆-product of field operators implies that the space of test functions in the Wightman approach to noncommutative quantum field theory is one of the Gel'fand-Shilov spaces S β with β < 1/2. This class of test functions smears the noncommutative Wightman functions, which are in this case generalized distributions, sometimes called hyperfunctions. The existence and determination of the class of the test function spaces in NC QFT is important for any rigorous treatment in the Wightman approach.

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Section: Introductionmentioning

confidence: 99%

Test functions space in noncommutative quantum field theory

Chaichian¹,

Mnatsakanova²,

Tureanu³

et al. 2008

J. High Energy Phys.

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show abstract

“…However, in the commutative case, from CPT invariance the standard condition of WLC follows, [1]- [4]. The equivalence of LCC (22) with the standard one follows from the fact that, for the validity of usual LCC its validity on arbitrary small spatially divided domains is sufficient (see [4], Proposal 9.12). Indeed, validity of "noncommutative" LCC (22) in the commutative case means validity of standard LCC in the domain (x 0 − y 0 ) 2 − (x 3 − y 3 ) 2 < 0, x k , y k , k = 1, 2 are arbitrary.…”

Section: Equivalence Of Various Conditions Of Local Commutativity In Qftmentioning

confidence: 99%

“…This domain satisfies the requirements of the above mentioned statement. Besides we can replace (22) with the formally weaker condition, requiring that it is valid only when…”

Section: Equivalence Of Various Conditions Of Local Commutativity In Qftmentioning

confidence: 99%

“…In [22] it was shown that, besides the above-mentioned theorems, in NC QFT (with θ 0i = 0) a number of other classical results of the axiomatic theory remain valid. In [20] the Haag's theorem [28,29] (see also [1] and references therein) was obtained on the basis of the twisted Poincaré invariance of the theory.…”

Section: Introductionmentioning

confidence: 99%

“…Here we obtain the close result in the SO(1, 1) symmetric theory using the spectral conditions and translational invariance only with respect to the commutating coordinates. In this case the equality of the two-point Wightman functions in the two theories leads to the conclusion that if LCC (22) is fulfilled and the current in one of the theories is equal to zero, for example, j 1 f = 0, then j 2 f = 0 as well;…”

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confidence: 99%

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Towards an axiomatic formulation of noncommutative quantum field theory. II

2020

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Classical results of the axiomatic quantum field theory, namely the irreducibility of the set of field operators, Reeh and Schlieder's theorems and generalized Haag's theorem, are proven in SO(1, 1) invariant quantum field theory, of which an important example is noncommutative quantum field theory. New consequences of generalized Haag's theorem are obtained in SO(1, 3) invariant theories. It has been proven that the equality of fourpoint Wightman functions in two theories leads to the equality of elastic scattering amplitudes and thus the total cross-sections in these theories. † Professor Yuri Vernov passed away in Moscow on 27 May 2015.

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The ultrahyperfunctional approach to non-commutative quantum field theory

Franco¹,

Lourenço²,

Renoldi³

2008

J. Phys. A: Math. Theor.

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In the present paper, we intend to enlarge the axiomatic framework of non-commutative quantum field theories (QFT). We consider QFT on non-commutative spacetimes in terms of the tempered ultrahyperfunctions of Sebastião e Silva corresponding to a convex cone, within the framework formulated by Wightman. Tempered ultrahyperfunctions are representable by means of holomorphic functions. As is well known there are certain advantages to be gained from the representation of distributions in terms of holomorphic functions. In particular, for non-commutative theories the Wightman functions involving the ⋆-product, , have the same form as the standard form . We conjecture that the functions satisfy a set of properties which actually will characterize a non-commutative QFT in terms of tempered ultrahyperfunctions. In order to support this conjecture, we prove for this setting the validity of some important theorems, of which the CPT theorem and the theorem on the spin-statistics connection are the best known. We assume the validity of these theorems for non-commutative QFT in the case of spatial non-commutativity only.

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Wightman axiomatic approach in noncommutative field theory

Abstract: The axiomatic approach based on Wightman functions is developed in noncommutative field theory. We prove that the main results of the axiomatic approach remain valid if the noncommutativity affects only the spatial variables.

Cited by 8 publications

References 24 publications

Test functions space in noncommutative quantum field theory

Test functions space in noncommutative quantum field theory

Towards an axiomatic formulation of noncommutative quantum field theory. II

The ultrahyperfunctional approach to non-commutative quantum field theory

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