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Twisted Galilean symmetry and the Pauli principle at low energies. (English) Zbl 1100.81040

Let $g$ be an element of $𝒫$, $𝒫$ is the acting on ${ℝ}^{d+1}$. Then $g$ acts on $\alpha \in {C}^{\infty }\left({ℝ}^{d+1}\right)$ by $\left(g\alpha \right)\left(x\right)=\alpha \left({g}^{-1}x\right)$. But this action does not satisfy $\left(g\alpha \right){*}_{\theta }\left(g\beta \right)=g\left({\alpha }_{\theta }\beta \right)$ in general. Here $\alpha {*}_{\theta }$, $\theta =\left({\theta }^{\mu \nu }\right)$, ${\theta }^{\mu \nu }=-{\theta }^{\nu \mu }\in ℝ$, is the Moyal product which express the space-time noncommutativity. To overcome this difficulty, twisted implemetation of the Poincaré group, which is covariant with the Moyal product, was discovered [M. Chaichian, P. P. Kulish, K. Nishijima and A. Tureanu, Phys. Lett. B 604, 98–102 (2004), hereafter referred to as [1], M. Dimitrijevic, J. Wess, Deformed bialgebra of diffeomorphisms, talk given by M. Dimitrijevic at 1st Vienna Central European Seminar on Particle Physics and Quantum Field Theory, 26–28 November 2004, hep-th/0411224]. The authors say this approach seems to be promising. But it violates Pauli’s principle [A. P. Balachandran, G. Mangano, A. Pinzul and S. Vaidaya, Int. J. Mod. Phys. A 21, 3111–3126 (2006; Zbl 1104.81067), see also hep-th/0508002].

Since there are no experimental observation of the effect of violation of the Pauli principle, it is expected any violation of the Pauli principle can only appear at high energies. To address this issue, the low temperature limit of the two-particle correlation function in a twisted implementation of the Poincaré group should be studied. Since this is a problem at low-energy, the authors study the twisted implementation of the Galilean symmetry and conclude the implied violation of Pauli’s principle will have no observable effect at current energies.

Since the twisted implementation is done via the deforming (twisting) coproduct, mathematical preliminaries on the Hopf algebra associated to a group $G$ and its group algebra ${G}^{*}$, and deformation of the coproduct are given in §2. The twisted coproduct of Lorentz generators is described in §3.1 according to [1]. This twist is adapted to the nonrelativistic case (Galilean boost transformation) in §3.2. §4 discusses the non-relativistic reduction of the Klein-Gordan field to the Schrödinger field in $\left(2+1\right)$ dimensions in commutative space. It is used to obtain the action of the twisted Galilean transformation on the Fourier coefficients. Then the action of the twisted Galilean transformation on the non-relativistic Schrödinger fields is obtained in §6. Applying these results, the two-particle correlation function of a free gas in two spatial dimensions is computed and the above conclusion on observation of the effects of the implied violation of Pauli’s principle is obtained in §7. In the Appendix, the Wigner-Inönü group contraction of the Poincaré group to the Galilean group is derived.

##### MSC:
 81T75 Noncommutative geometry methods (quantum field theory) 81R60 Noncommutative geometry (quantum theory) 17B81 Applications of Lie algebras to physics 16W30 Hopf algebras (assoc. rings and algebras) (MSC2000)