User:GustavoPetronilo/sandbox

Galilean symmetry is the cornerstone of classical physics and today, even with the advent of discovery of others, more general symmetries, it is relevant in the research of various low-speed phenomena mainly associated with fluid theory.

In 1988 Takahashi ''et. al.'' began a study of Galilean covariance, where it was possible to develop an explicitly covariant non-relativistic field theory. With this formalism, the Schr\"{o}dinger equation takes a similar form as Klein-Gordon equation in the light-cone of a  (4,1) de Sitter Space . With the advent of the Galilean covariance, it was possible to derive  the non-relativistic version of the Dirac theory, which is known in its usual form as the Pauli-Schrödinger equation. The goal in the present work is to derive a Wigner representation for such covariant theory.

Galilean Manifold
The Galilei transformations are given by


 * $$x' = Rx - v t+ \textbf{a} $$
 * $$t' = t+\textbf{b} .$$

where $$R$$ stands for the three-dimensional Euclidean rotations, '$$v$$ is the relative velocity defining the Galilean boosts, a stands for spacial translations and b, for time translations. Consider a free mass particle $$m$$; the mass shell relation is given by $$p^2-2mE=0$$.

We can then define a 5-vector, $$p^\mu=(p_x,p_y,p_z,m,E)=(p^i,m,E)$$, with $$i=1,2,3$$.

Thus, we can define a scalar product of the type


 * $$p_\mu p_\nu g^{\mu\nu}=p_ip_i-p_5p_4-p_4p_5=p^2-2mE=k,$$

where


 * $$g^{\mu\nu} = \pm \begin{pmatrix}1&0&0&0&0\\0&1&0&0&0\\0&0&1&0&0\\0&0&0&0&-1\\0&0&0&-1&0\end{pmatrix},$$

is the metric of the space-time, and $$p_\nu g^{\mu\nu}=p^\mu$$.

Extended Galilei Algebra
A five dimensional Poincaré algebra leaves the metric $$g^{\mu\nu}$$ invariant,


 * $$~[P_\mu, P_\nu] = 0,$$
 * $$\frac{ 1 }{ i }~[M_{\mu\nu}, P_\rho] = g_{\mu\rho} P_\nu - g_{\nu\rho} P_\mu\,$$
 * $$\frac{ 1 }{ i }~[M_{\mu\nu}, M_{\rho\sigma}] = g_{\mu\rho} M_{\nu\sigma} - g_{\mu\sigma} M_{\nu\rho} - g_{\nu\rho} M_{\mu\sigma} + \eta_{\nu\sigma} M_{\mu\rho}\, ,$$

We can write the generators as


 * $$~J_i=\frac{1}{2}\epsilon_{ijk}M_{jk},$$
 * $$~K_i=M_{5i},$$
 * $$~C_i=M_{4i},$$
 * $$~D=M_{54}.$$

Hence, the non-vanishing commutation relations can be rewritten as


 * $$~\left[J_i,J_j\right]=i\epsilon_{ijk}J_k,$$
 * $$~\left[J_i,C_j\right]=i\epsilon_{ijk}C_k,$$
 * $$~\left[D,K_i\right]=iK_i,$$
 * $$~\left[P_4,D\right]=iP_4,$$
 * $$~\left[P_i,K_j\right]=i\delta_{ij}P_5,$$
 * $$~\left[P_4,K_i\right]=iP_i,$$
 * $$~\left[P_5,D\right]=-iP_5,$$


 * $$~\left[J_i,K_j\right]=i\epsilon_{ijk}K_k,$$
 * $$~\left[K_i,C_j\right]=i\delta_{ij}D+i\epsilon_{ijk}J_k,$$
 * $$~\left[C_i,D\right]=iC_i,$$
 * $$~\left[J_i,P_j\right]=i\epsilon_{ijk}P_k,$$
 * $$~\left[P_i,C_j\right]=i\delta_{ij}P_4,$$
 * $$~\left[P_5,C_i\right]=iP_i.$$

An important Lie subalgebra is


 * $$[P_4,P_i]=0$$
 * $$[P_i,P_j]=0$$
 * $$[J_i,P_4]=0$$
 * $$[K_i,K_j]=0$$
 * $$\left[J_i,J_j\right]=i\epsilon_{ijk}J_k,$$
 * $$\left[J_i,P_j\right]=i\epsilon_{ijk}P_k,$$
 * $$\left[J_i,K_j\right]=i\epsilon_{ijk}K_k,$$
 * $$\left[P_4,K_i\right]=iP_i,$$
 * $$\left[P_i,K_j\right]=i\delta_{ij}P_5,$$

$$P_4$$ is the generator of time translations (Hamiltonian), Pi is the generator of translations (momentum operator), $$K_i$$ is the generator of Galileian boosts, and $$J_i$$ stands for a generator of rotations (angular momentum operator). The generator $$P_5$$ is a Casimir invariant and $$P^2-2P_4P_5$$ is an additional Casimir invariant. This algebra is isomorphic to the extended Galilean Algebra in (3+1) dimensions with $$P_5=M$$, The central charge, interpreted as mass, and $$P_4=H$$.