VNU Journal of Science: Mathematics – Physics, Vol. 31, No. 1 (2015) 47-51
UV/IR phenomenon of Noncommutative Quantum
Fields in Example
Nguyen Quang Hung*, Bui Quang Tu
Faculty of Physics, VNU University of Science, 334 Nguyễn Trãi, Hanoi, Vietnam
Received 05 December 2014
Revised 18 February 2015; Accepted 20 March 2015
Abstract: Noncommutative Quantum Field (NCQF) is a field defined over a space endowed with
a noncommutative structure. In the last decade, the theory of NCQF has been studied intensively,
and many qualitatively new phenomena have been discovered. In this article we study one of these
phenomena known as UV/IR mixing.
Keywords: Noncommutative quantum field theory.
1. Introduction∗
Noncommutative quantum field theory (NC QFT) is the natural generalization of standard
quantum field theory (QFT). It has been intensively developed during the past years, for reviews, see
[1,2]. The idea of NC QFT was firstly suggested by Heisenberg and the first model of NC QFT was
developed in Snyder’s work [3]. The present development in NC QFT is very strongly connected with
the development of noncommutative geometry in mathematics [4], string theory [5] and physical
arguments of noncommutative space-time [6].
The simplest version of NC field theory is based on the following commutation relations between
coordinates [7]:
[ xˆ µ , xˆν ] = i θ µν ,
where θ
µν
(1)
is a constant antisymmetric matrix.
Since the construction of NC QFT in a general case ( θ 0i ≠ 0 ) has serious difficulties with unitarity
and causality [8-10], we consider a simpler version with θ 0i = 0 (thus space-space noncommutativity
only), in which there do not appear such difficulties. This case is also a low-energy limit of the string
theory [1, 2].
_______
∗
Corresponding author. Tel.: 84- 904886699
Email:
[email protected]
47
N.Q. Hung, B.Q. Tu / VNU Journal of Science: Mathematics – Physics, Vol. 31, No. 1 (2015) 47-51
48
2. Moyal Product
We introduce d -dimensional noncommutative space-time by assuming that time and position are
not c -numbers but self-adjoint operators defined in a Hilbert space and obeying the commutation
algebra
[ xˆ µ , xˆν ] = i θ µν ,
(2)
µν
where the θ are the elements of a real constant d × d antisymmetric matrix θ . Then we define
the Moyal star product
n
∞
i 1
f ( x) g ( x) = f ( x) g ( x) + ∑ θ µ1ν1 …θ µnν n [∂ µ1 …∂ µn f ( x)] [∂ν1 …∂ν n g ( x)]
n =1 2 n!
i ∂ µν ∂
= f ( x) exp
θ
g ( x).
µ
∂xν
2 ∂x
(3)
In particular we have:
e
ipµ x µ
ν
i
i ( p + q )µ x µ
eiqν x = exp − p ∧ q e
,
2
(4)
where we have defined the wedge product
p ∧ q = ∑ pµθ µν qν .
(5)
µ ,ν
The natural generalization of the star product (3) follows:
i
∂ ∂
f1 ( x1 ) f 2 ( x2 ) f n ( xn ) = ∏ exp θ µν µ ν f1 ( x1 ) f n ( xn ), for a, b = 1,…, n. (6)
∂xa ∂xb
a
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