NC QFT Research Articles

Irreducibility of the set of field operators in NC QFT

Irreducibility of the set of quantum field operators has been proved in noncommutative quantum field theory in the general case when time does not commute with spatial variables.

Irreducibility of the Set of Field Operators in NC QFT

Irreducibility of the Set of Field Operators in NC QFT

BRAIDED FIELD QUANTIZATION FROM QUANTUM POINCARÉ COVARIANCE

We demonstrate that the covariance of the algebra of quantum NC fields under quantum-deformed Poincaré symmetries implies the appearence of braided algebra of fields and the notion of braided locality in NC QFT. We briefly recall the historical development of NC QFT which was firstly formulated in the framework using classical relativistic symmetries but further it was described as generated by the quantum-deformed symmetries. We argue that consistent covariant quantum-deformed formalism requires "braiding all the way," in particular braided commutator of deformed field oscillators as well as the braid between the field oscillators and noncommutative Fourier exponentials. As example of braided quantum-deformed NC QFT we describe the NC scalar free fields on noncommutative canonical (Moyal–Weyl) space–time with braided c-number field commutator which implies braided locality.

BRAIDED FIELD QUANTIZATION FROM QUANTUM POINCARE COVARIANCE

We demonstrate that the covariance of the algebra of quantum NC fields under quantum-deformed Poincare symmetries implies the appearence of braided algebra of fields and the notion of braided locality in NC QFT. We briefly recall the historical development of NC QFT which was firstly formulated in the framework using classical relativistic symmetries but further it was described as generated by the quantum-deformed symmetries. We argue that consistent covariant quantum-deformed formalism requires "braiding all the way", in particular braided commutator of deformed field oscillators as well as the braid between the field oscillators and noncommutative Fourier exponentials. As example of braided quantum-deformed NC QFT we describe the NC scalar free fields on noncommutative canonical (Moyal-Weyl) space-time with braided c-number field commutator which implies braided locality.

Toward an axiomatic formulation of noncommutative quantum field theory

We propose new Wightman functions as vacuum expectation values of products of field operators in the noncommutative space–time. These Wightman functions involve the ⋆-product among the fields, compatible with the twisted Poincaré symmetry of the noncommutative quantum field theory (NC QFT). In the case of only space–space noncommutativity (θ0i = 0), we prove the CPT theorem using the noncommutative form of the Wightman functions. We also show that the spin-statistics theorem, demonstrated for the simplest case of a scalar field, holds in NC QFT within this formalism.

Test functions space in noncommutative quantum field theory

It is proven that the -product of field operators implies that the space of test functions in the Wightman approach to noncommutative quantum field theory is one of the Gel'fand-Shilov spaces Sβ with β < 1/2. This class of test functions smears the noncommutative Wightman functions, which are in this case generalized distributions, sometimes called hyperfunctions. The existence and determination of the class of the test function spaces in NC QFT is important for any rigorous treatment in the Wightman approach.

On a Lorentz-invariant interpretation of noncommutative space–time and its implications on noncommutative QFT

By invoking the concept of twisted Poincaré symmetry of the algebra of functions on a Minkowski space–time, we demonstrate that the noncommutative space–time with the commutation relations [xμ,xν]=iθμν, where θμν is a constant real antisymmetric matrix, can be interpreted in a Lorentz-invariant way. The implications of the twisted Poincaré symmetry on QFT on such a space–time is briefly discussed. The presence of the twisted symmetry gives justification to all the previous treatments within NC QFT using Lorentz invariant quantities and the representations of the usual Poincaré symmetry.

Noncommutative Quantum Field Theories, Spin-Statistics and CPT Theorems

In the framework of noncommutative quantum field theories (NC QFT), we show the general validity of the CPT and spin-statistics theorems, with the exception of some peculiar situations in the latter case.

Spin-statistics and CPT theorems in noncommutative field theory

We show that Pauli's spin-statistics relation remains valid in noncommutative quantum field theories (NC QFT), with the exception of some peculiar cases of noncommutativity between space and time. We also prove that, while the individual symmetries C and T, and in some cases also P, are broken, the CPT theorem still holds in general for noncommutative field theories, in spite of the inherent nonlocality and violation of Lorentz invariance.