Noncommutative Φ⁴ theory at two loops

Abstract

We study perturbative aspects of noncommutative field theories. This work is arranged in two parts. First, we review noncommutative field theories in general and discuss both canonical and path integral quantization methods. In the second part, we consider the particular example of noncommutative Φ 4 theory in four dimensions and work out the corresponding effective action and discuss renormalizability of the theory, up to two loops.

Gravitational non-commutativity and Gödel-like spacetimes

“…Finally, it would be interesting to see whether the non-commutativity studied here is related to non-commutativity in string theory, where the Neveu-Schwarz B field is taken to infinity in the presence of D-branes [1]. It is known that for ordinary field theories with quadratic actions, the star-product reduces to the ordinary product [23]. Thus quantities such as the two-point function should remain unchanged, not acquiring corrections from the underlying non-commutativity.…”

Hydrogen Atom Spectrum and the Lamb Shift in Noncommutative QED

“…The noncommutative space can be realized by the coordinate operators satisfying 1) wherex are the coordinate operators and θ µν is the noncommutativity parameter and is of dimension of (length) 2 ; for a review on the string theory side, see [1]. The action for field theories on noncommutative spaces, NCFT's, is then obtained using the Weyl-Moyal correspondence [2][3][4], according to which, in order to find the noncommutative action, the usual product of fields should be replaced by the star-product:…”

“…Performing explicit loop calculations, for θ 0i = 0 cases (noncommutative space), it has been shown that noncommutative φ 4 theory up to two loops [3,5] and NCQED up to one loop [6,4,7], are renormalizable. For noncommutative space-time (θ 0i = 0) it has been shown that the theory is not unitary and hence, as a field theory, it is not appealing [8].…”

Running couplings and triviality of field theories on noncommutative spaces

“…In a noncommutative theory, it appears that masses do not cutoff the running of coupling constants. We note that this is a non-perturbative issue, which occurs in addition to the perturbative renormalizability of non-commutative field theories which has recently been examined in detail [12,13,14].…”