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One Rigorous Negative Result in Noncommutative Quantum Field Theory
M. N. Mnatsakanova Institute of Nuclear Physics, Moscow State University, Moscow, Russia based on the joint work with Yu.S. Vernov (Institute for Nuclear Research)

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1 Intro duction

Negative results play important role in axiomatic QFT. They show that relation between asymptotic and interacting elds is very nontrivial. The most important example of such results is Haag's theorem. Here we consider another well-known result and show that it is possible to obtain it at weaker conditions. I mean the following Theorem If any local eld (x) is irreducible and [ (x), (y )] = B (x - y ), (1) then operator B (x) is multiple of unit operator, that is (x) is an asymptotic eld. We show that this results is true also in the theories, in which Lorentz symmetry is broken up to S O(1, 1) S O(2).
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As an example of such class of theories we can name noncommutative quantum eld theory (NC QFT), which is important in the view of physics. In this report we show that, actually, the commutator in question can not be an operator depending on the dierence between one spatial coordinate in points x and y . Our result is most interesting in the case of noncommutative theory, precisely, in the case of space-space noncommutativity, in which time commutes with spatial variables and, as a consequence, one spatial variable, say x3, commutes with others. In what follows we consider just that very case.

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In our proof we use the following general principles of axiomatic eld theory:
i) Local commutativity condition (LCC); ii) Irreducibility of the set of eld operators.

For simplicity we consider the case of neutral scalar elds. Local commutativity means that
[ (x), (y )] = 0,

if x y .

(2)

The condition x y in usual (commutative) theory means that
(x - y )2 < 0.

In noncommutative quantum eld theory (NC QFT) LCC can be fullled with respect to commutative variables only.
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The reason is that test functions, corresponding to noncommutative variables, belong to the one of Gelfand-Shilov spaces S with < 1/2, which does not contain functions with nite support and so corresponding eld operators can not satisfy LCC. Thus in NC QFT we have the following LCC:
[ (x), (y )] = 0,

if (x0 - y0)2 - (x3 - y3)2 < 0.

(3)

Let us stress that our result is valid in any theory, where this condition is fullled. Now let us recall the condition of irreducibility. The set of eld operators (x) is irreducible if the bounded operator, which commutes with all eld operators, has to be C I , where I I I is identical operator and C is some function. Our proof is the modication of the classical proof given in the book of N. N. Bogoliubov, A. A. Logunov and I. T. Todorov.
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2 Pro of

Let us prove that if
[ (x), (y )] = A (x3 - y3, X, Y ), where we denote al l other variables as X, Y , then A ( x3 - y 3 , X , Y ) = C I I

(4)

Let me remind the Jacobi identity:
[ (x), [ (y ), (z )]]+[ (y ), [ (z ), (x)]]+[ (z ), [ (x), (y )]] = 0

If

(5)

(z0 - y0)2 - (z3 - y3)2 < 0, (z0 - x0)2 - (z3 - x3)2 < 0,
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then in accordance with LCC from Jacobi identity it follows that
[ (z ), [ (x), (y )]] = 0.

(6)

The necessary conditions: z - x as well as z - y are space-like vectors in respect with commutative coordinates, are fullled if
x3 = + x3, y3 = x3, y3 are arbitrary, = (0, 0 then x3 - y 3 = + x3 - - y 3 = So, A (x3 - y3, X, Y ), which we have + y3 , , 0) x3 - y 2 -; 3.

in (4), is:

A (x3 - y3, X, Y ) = B (x3 - y3, X, Y ).

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Let me remind eq. (6):
[ (z ), [ (x), (y )]] = [ (z ), A (x3 - y3, X, Y )] = [ (z ), B (x3 - y3, X, Y )] = 0, (7) where z , x and y are arbitrary. So we see that B (x3 - y3, X, Y )] commutes with (z ) at arbitrary z .

Owing to irreducibility of [ (x ), (y )] = C I , where I Thus we have proved that a function. It is known that

the set of quantum eld operators, C is some function. commutator [ (x ), (y )] has to be in this case any Wightman function

0, (x1), . . . (xn) 0

has to be some superposition of two-point Wightman functions and so in this case the set of Wightman functions can not dene any nontrivial theory.
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Let us stress that our result is valid in a space of arbitrary dimensions.

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