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Cluster properties in Noncommutative Quantum Field Theory
Yu. S. Vernov Institute for Nuclear Research, Russian Academy of Sciences, Moscow, Russia based on the joint work with M. N. Mnatsakanova (Skobeltsyn Institute of Nuclear Physics, Moscow State University)

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1 Intro duction

Cluster properties of Wightman functions are very important properties of quantum eld theory (QFT). Let us recall that cluster properties of Wightman functions imply that
W (x1, . . . xk , xk+1 + a, . . W (x1, . . . xk ) W (xk+1 if a2 = -1, , W (x1, . . . xn) where 0 is a vacuum vector. . xn ,... 0 , + a) xn), (1) (x1) . . . (xn) 0 ,

Cluster properties are very natural consequence of Lorentz invariance. In fact, quantum eld theory is dened by its Wightman functions. Owing to Lorentz invariance quantum eld operators are independent in points x1, . . . xk and points xk+1 + a, . . . xn + a.
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Thus (2) In order to come to eq. (1) we have to take into account that Wightman functions are translation invariant. In recent years noncommutative quantum eld theory (NC QFT) attracts a great attention as very natural extension of the commutative theory to high energies. Moreover, in some cases NC QFT is a low energy limit of string theory. Thus it is very interesting to consider cluster properties in NC QFT. In this report we show that cluster properties are valid, if NC QFT is determined by the Heisenberg-like commutative relations:
^^ [xµ, x ] = i µ , W (x1, . . . xk , xk+1 + a, . . . xn + a) W (x1, . . . xk ) W (xk+1 + a, . . . xn + a).

(3)

where µ is a constant antisymmetric matrix.
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It is known that NC QFT can be formulated in commutative space if the usual product of quantum eld operators is substituted by the -product. Precisely, it implies that the usual product of the corresponding test functions is substituted by the -product. NC QFT is a physically important example of such a theory. In fact, the class of theories, in which usual product is substituted by the -product, is wider than NC QFT as Heisenberg-like commutation relations imply the existence of S O(1, 1)S O(2) symmetry of the theory. Evidently, theories with the -product have not to possess such a symmetry in a general case. Here we show that cluster properties are valid in NC QFT as well as in commutative QFT. Moreover, they are valid in any theory with the -product of test functions.

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2 Space-space Noncommutativity

First let us consider the case when time commutes with spatial variables. It is known that in this case there exists also one spatial variable which commutes with all other variables. Thus in this case we have two commutative variables, say x0 and x3 and two noncommutative (x1 and x2) ones. Let us stress that, actually, the number of noncommutative variables can be arbitrary. Let us point out that in the case of space-space noncommutativity there exists one spatial variable, which commutes with all others irrespective of a number of noncommutative variables.

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We admit in correspondence with the standard QFT that Wightman functions are tempered distributions in respect with commutative variables. In fact, after integration over noncommutative variables we, actually, obtain two-dimensional commutative QFT. The cluster properties in this case exist on the same ground as in the fourdimensional theory. Thus we can use the standard methods and obtain that, actually, in this case in the theories with mass gap the dierence between the left-hand member of the eq.(1) and its right-hand member falls more rapidly than -n, where n is an arbitrary positive number. In the theory with massless particals n = 1, a is two-dimensional vector: a = {a0, a3}.

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3 General Case

The general case when all ij = 0 is much more nontrivial. Nevertheless we prove that eq.(1) is valid also in this case. The crucial point in the derivation of eq.(1) is the fact that any test function fi fi (xi) belongs to the Gelfand-Shilov space S with < 1/2. Precisely it was shown in our paper that if fi S with < 1/2, then
fi fi+1 exp
n=0

i µ µ 2 xi x+1 i
n

fi (xi) fi+1 (xi+1) = fi (xi) fi+1 (xi+1)

1 n!

converges. The similar result was obtained also in the paper of Soloviev.
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i µ µ 2 xi x+1 i

(4)


fi+1 converges i, then f1 . . . fn converges as well if fi S with < 1/2. Moreover this function belongs to S with the same . Let us remind that in the simplest case of one variable f (x) belongs to the space S if d q f ( x) Ck B q q q , - < x < , k , q IN, xk d xq
i

If f

(5) where the constants Ck and B depend on the function f (x). In fact, for convergence of the series (4) it is sucient to use the inequality (5) only at k = 0. The extension of the last inequality on the case of multivariable function is straightforward.

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Convergence of the series (4) implies that
N N f1 (x1) · · · fn (xn) = f1 (x1) · · · fn (xn)+ (N ), (N ) 0,

if N .
N

(6)

fiN (xi) 1 n!

fiN (xi+1) = +1
n

i µ fi (xi) fi+1 (xi+1). µ 2 xi xi+1 n=0 Let us stress that as B in inequality (5) depends on function f (x), but does not depend on x, thus estimation (4) is the same for every xi. Now let us point out that as fiN (xi) fiN contains only a +1

nite numbers of derivatives, then the corresponding generalized functions are tempered distributions.
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Thus at any N we have cluster properties. This fact gives us the possibility to prove cluster properties in the general case. Let us point out that W (x1, . . . xn) is a functional in the space of functions f1 (x1) · · · fn (xn). Actually, on the one hand, for arbitrary
W (x1, . . . xk , xk+1 + a, . . . xn + a) W N (x1, . . . xk , xk+1 + a, . . . xn + a) + (N ), (7) where (N ) 0, if N . Here a is a four-dimensional

vector. On the other hand, at any N

W N (x1, . . . xk , xk+1 + a, . . . xn + a) = W N (x1, . . . xk ) W N (xk+1, . . . xn) + (), (8) () 0, if . Combining the eqs. (7) and (8), we come
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to eq. (1):
W (x1, . . . xk , xk+1 + a, . . . xn + a) W (x1, . . . xk ) W (xk+1, . . . xn).

Thus the cluster properties of Wightman functions are proved in any theory, in which the usual product of test functions is substituted by the -product. Physically important example of such a theory is NC QFT. Now let us remark that if time and so one spatial variable commutes with all variables, then it is very natural to assume that Wightman functions are tempered distributions in respect with commuting coordinates. The most important consequence of cluster properties is the uniqueness of vacuum state.

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4 Conclusion

In conclusion let us stress that as assertion that test functions belong to the Gelfand-Shilov space S , < 1/2 does not depend on the dimension of the underlying space, all the results are valid in the space of arbitrary dimensions.

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