New Concept of Relativistic Invariance in Noncommutative Space-Time: Twisted Poincaré Symmetry and Its Implications

Abstract

We present a systematic framework for noncommutative (NC) QFT within the new concept of relativistic invariance based on the notion of twisted Poincaré symmetry (with all 10 generators), as proposed in ref. [7]. This allows to formulate and investigate all fundamental issues of relativistic QFT and offers a firm frame for the classification of particles according to the representation theory of the twisted Poincaré symmetry and as a result for the NC versions of CPT and spin-statistics theorems, among others, discussed earlier in the literature. As a further application of this new concept of relativism we prove the NC analog of Haag's theorem.

Twists of quantum groups and noncommutative field theory

“…We could apply the twisting of the Poincaré algebra with the action (20) after quantizing field theory [20]. Using the twist element F = exp( i 2 θ µν P µ ⊗ P ν ), we have to change product of observables according to the general rule…”

Deformed relativistic and nonrelativistic symmetries on canonical noncommutative spaces

“…For the choice of parameters that leads to the undeformed algebra these rules agree with those obtained by an application of the abelian twist function on the primitive comultiplication rule. (For the conformal-Poincaré case this computation of modified coproduct rules using the twist function already exists in the literature [5,6,7,8], but a similar analysis for the nonrelativistic symmetries is new and presented here.) For other choices of the free parameters the deformations cannot be represented by twist functions.…”

“…The coproduct rules for the Poincaré sector were earlier derived in [1,4,5,6] and for the conformal sector in [7,15]. The free parameter appearing inK ρ does not appear explicitly in the coproduct ∆(K ρ ).…”

“…There is an alternative method, based on quantum-group-theoretic arguments, of computing the coproducts [5,6,7]. This is obtained for the particular case when the deformed generators satisfy the undeformed algebra.…”

Charge conservation and the equivalence principle in noncommutative spacetime

“…Recently, groups of physicists have constructed the quantum field theory in the noncommutative spacetime by twisting the quantum space as a module space [8], [9], [10]. Espcially, Bu et.al.…”