Noncommutative field theory (Q5907054)

There are many recent works studying noncommutative field theories and corresponding quantum mechanical problems. An attempt to construct self-consistent QFT directly on noncommutative Minkowski space encounters difficulties due to the violation of basic physical principles like Lorentz and gauge invariances, unitarity and causality. In this paper noncommutative space-time and a method to construct noncommutative field theory is proposed in terms of a covariant \(\star\)-product Moyal algebra. The main assumption is that in string theories space-time has more than four dimensions with the additional noncommutative ones. The noncommutative models can be realized in terms of a \(\star\)-product. The commutative algebra of functions (with the usual product) is replaced by the \(\star\)-product Moyal algebra. The Moyal \(\star\)-product is replaced by the covariant \((\star)_c\)-product. Its physical meaning is that noncommutative properties of space-time take place in the \(d\)-dimensional case and the usual four-dimensional space-time and physical fields on it are defined as residual or averaging procedure obtained by taking trace of the Dirac \(\gamma_\mu\)-matrices. The outline of this work is as follows: In Section 2 the definition of the Moyal \(\star\)-product in any \(d\)-dimensional space-time is modified and the trace of its noncommutativity is calculated. Section 3 deals with free fields and their commutation relations, the Pauli-Jordan and Green functions in the noncommutative space-time. Sections 4-7 are dedicated to the construction of the noncommutative quantum electrodynamics and to the calculation of the vacuum polarization. In Section 8 the author studied causality and unitarity conditions for the \(S_\star\)-matrix of the noncommutative field theory. In the last Section some geometrical and physical consequences of the noncommutative theory are given.