Relativistic corrections to elementary Galilean dynamics and deformations of Poisson brackets (Q2724630)

The Lie algebra \(g\) of Poincaré group, the Poincaré algebra, is the semidirect sum of Lie algebra \(o(3,1)\) of Lorentz group and a 4-dimensional abelian algebra. So, \(g\) is 10-dimensional vector space of all vectors of the form \((A,v,x,t)\) with \(A \in o(3)\), \(v,x \in \mathbb R^3\) and \(t \in \mathbb R\), provided with natural Lie bracket. Let \(\varepsilon > 0\). The vector space automorphism \(U_{\varepsilon}: V \rightarrow V\), \((A,v,x,t) \mapsto (A,\varepsilon^{1/2} v, \varepsilon^{1/2} x,t)\) transports the Lie algebra structure of \(V\) to an isomorphic one, depending from the parameter \(\varepsilon\). For \(\varepsilon \rightarrow 0\) this Lie algebra \(g_{\varepsilon}\) contracts to the Galilei algebra \(g_0\). The dual \(V^*\) of \(V\) can be equipped with standard linear Poisson bracket \(\{\;,\;\}_{\varepsilon}\) associated with Lie algebra \(g_{\varepsilon}\). With the help of 2-cocycle \(\lambda\) of \(g_{\varepsilon}\), defined by \(\lambda(X_1,X_2) = v_1 \cdot x_2 - v_2 \cdot x_1\), this one-parameter family of linear Poisson brackets can be modified in a two-parameter family of affine Poisson brackets \(\{\;,\;\}_{\varepsilon,m}\), the Poisson tensor of which has the form \(\Gamma_{\varepsilon,m} = \Gamma + m \Lambda + \varepsilon {\mathcal R}\), where \(\Lambda\) corresponds to the cocycle \(\lambda\). \({\mathcal R}\) defines the relativistic deformation of Galilean Poisson structure \(\Gamma_{\varepsilon,0}\). NEWLINENEWLINENEWLINEThe main result of this paper is that the infinitesimal relativistic deformation of this structure is trivial, which for the case \(m=0\) is not evident a priori, as the Poisson structures \(\Gamma\) and \(\Gamma_{\varepsilon,0}\) are not isomorphic and the second Poisson cohomology space of \(\Gamma\) is not trivial. This result is applied to the investigation of elementary relativistic dynamics.NEWLINENEWLINEFor the entire collection see [Zbl 0958.00039].