Irreducibility of the set of field operators in NC QFT
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ELEMENTARY PARTICLES AND FIELDS Theory
Irreducibility of the Set of Field Operators in NC QFT∗ M. N. Mnatsakanova1)** and Yu. S. Vernov2) Received May 29, 2012
Abstract—Irreducibility of the set of quantum field operators has been proved in noncommutative quantum field theory in the general case when time does not commute with spatial variables. DOI: 10.1134/S106377881309010X
1. INTRODUCTION Irreducibility of the set of quantum field operators ϕ(x) is one of the principal results in axiomatic quantum field theory (QFT) [1, 2]. It implies that, if vacuum vector is cyclic, then the corresponding set of quantum field operators has to be irreducible one. Let us recall that vacuum vector Ψ0 is a cyclic one, if any vector in the space under consideration can be approximated by a finite linear combination of the vectors Ψn = ϕ(x1 ) · · · ϕ(xn )Ψ0 with arbitrary accuracy. In accordance with axiom of vacuum vector cyclicity, any scalar product in the space in question can be approximated by the linear combination of Wightman functions W (x1 , . . . , xn ) ≡ Ψ0 , ϕ(x1 ) · · · ϕ(xn )Ψ0 . Let us prove that the set of quantum field operators is irreducible one in noncommutative quantum field theory (NC QFT) as well. Besides, we prove that in usual commutative QFT the irreducibility of a set of quantum field operators follows from assumptions weaker than standard. Let us recall that NC QFT is defined by the Heisenberg-like commutation relations between coordinates: ˆν ] = iθμν , [ˆ xμ , x
(1)
where θμν is a constant antisymmetric matrix. ∗
The text was submitted by the authors in English. Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, Russia. 2) Institute for Nuclear Research, Russian Academy of Sciences, Moscow, Russia. ** E-mail: [email protected] 1)
It is very important that NC QFT can be also formulated in commutative space, if we replace the usual product of quantum field operators (strictly speaking, of the corresponding test functions) by the - (Moyaltype) product (see [3, 4]). Let us remind that the -product is defined as ϕ(x) ϕ(y) (2) ∂ ∂ i θμν ϕ(x)ϕ(y) = exp 2 ∂xμ ∂yν ∞ 1 i ∂ ∂ n θμν ϕ(x)ϕ(y). ≡ n! 2 ∂xμ ∂yν n=0 Evidently, the set in Eq. (2) has to be convergent. It was proved [5] that this set is a convergent one if f (x) belongs to one of the Gelfand–Shilov spaces S β with β < 1/2. The similar result was obtained also in paper [6]. Noncommutative theories defined by Heisenberglike commutation relations (1) can be divided in two classes. The first of them is the case of only space–space noncommutativity, that is θ0i = 0, time commutes with spatial coordinates. It is known that this case is free from the problems with causality and unitarity [7–9] and in this case the main axiomatic results: CPT and spin-statistics theorems, Haag’s theorem remain valid [10–13]. Besides, this case can be obtained as low-energy limit from string theory [14]. Let us remind that if time commutes with spatial coordinates, then there exists one spatial coordinate, say x3 , which commutes with all
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