Galilei-covariant tensor formulation

Template:Short description The Galilei-covariant tensor formulation is a method for treating non-relativistic physics using the extended Galilei group as the representation group of the theory. It is constructed in the light cone of a five dimensional manifold.

Takahashi et al., in 1988, began a study of Galilean symmetry, where an explicitly covariant non-relativistic field theory could be developed. The theory is constructed in the light cone of a (4,1) Minkowski space. Previously, in 1985, Duval et al. constructed a similar tensor formulation in the context of Newton–Cartan theory. Some other authors also have developed a similar Galilean tensor formalism.

Galilean manifold

The Galilei transformations are

$\begin{array}{cc}{𝐱}^{\prime }& =R𝐱-𝐯t+𝐚\\ t^{\prime }& =t+𝐛.\end{array}$

where $R$ stands for the three-dimensional Euclidean rotations, $𝐯$ is the relative velocity determining Galilean boosts, a stands for spatial 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

$p_{\mu }{p}_{\nu }{g}^{\mu \nu }=p_i{p}_i-p_5{p}_4-p_4{p}_5=p^2-2mE=k,$

where

$g^{\mu \nu }=\pm \left(\begin{array}{ccccc}1& 0& 0& 0& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 0& -1\\ 0& 0& 0& -1& 0\end{array}\right),$

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,

$\begin{array}{cc}[][P_{\mu },P_{\nu }]& =0,\\ \frac{1}{i}[M_{\mu \nu },P_{\rho }]& =g_{\mu \rho }{P}_{\nu }-g_{\nu \rho }{P}_{\mu },\\ \frac{1}{i}[M_{\mu \nu },M_{\rho \sigma }]& =g_{\mu \rho }{M}_{\nu \sigma }-g_{\mu \sigma }{M}_{\nu \rho }-g_{\nu \rho }{M}_{\mu \sigma }+{\eta }_{\nu \sigma }{M}_{\mu \rho },\end{array}$

We can write the generators as

$\begin{array}{cc}{J}_i& =\frac{1}{2}{ϵ}_{ijk}{M}_{jk},\\ K_i& =M_{5i},\\ C_i& =M_{4i},\\ D& =M_{54}.\end{array}$

The non-vanishing commutation relations will then be rewritten as

$\begin{array}{cc}[J_i,J_j]& =i{ϵ}_{ijk}{J}_k,\\ [J_i,C_j]& =i{ϵ}_{ijk}{C}_k,\\ [D,K_i]& =i{K}_i,\\ [P_4,D]& =i{P}_4,\\ [P_i,K_j]& =i{\delta }_{ij}{P}_5,\\ [P_4,K_i]& =i{P}_i,\\ [P_5,D]& =-i{P}_5,\\ [J_i,K_j]& =i{ϵ}_{ijk}{K}_k,\\ [K_i,C_j]& =i{\delta }_{ij}D+i{ϵ}_{ijk}{J}_k,\\ [C_i,D]& =i{C}_i,\\ [J_i,P_j]& =i{ϵ}_{ijk}{P}_k,\\ [P_i,C_j]& =i{\delta }_{ij}{P}_4,\\ [P_5,C_i]& =i{P}_i.\end{array}$

An important Lie subalgebra is

$\begin{array}{cc}[][P_4,P_i]& =0\\ [][P_i,P_j]& =0\\ [][J_i,P_4]& =0\\ [][K_i,K_j]& =0\\ [J_i,J_j]& =i{ϵ}_{ijk}{J}_k,\\ [J_i,P_j]& =i{ϵ}_{ijk}{P}_k,\\ [J_i,K_j]& =i{ϵ}_{ijk}{K}_k,\\ [P_4,K_i]& =i{P}_i,\\ [P_i,K_j]& =i{\delta }_{ij}{P}_5,\end{array}$

$P_4$ is the generator of time translations (Hamiltonian), P_i is the generator of spatial translations (momentum operator), $K_i$ is the generator of Galilean boosts, and $J_i$ stands for a generator of rotations (angular momentum operator). The generator $P_5$ is a Casimir invariant and $P^2-2P_4{P}_5$ 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 $P_4=-H$.Template:Citation needed

The third Casimir invariant is given by $W_{\mu 5}{W}^{\mu }{}_5$, where $W_{\mu \nu }={ϵ}_{\mu \alpha \beta \rho \nu }{P}^{\alpha }{M}^{\beta \rho }$ is a 5-dimensional analog of the Pauli–Lubanski pseudovector.

Bargmann structures

In 1985 Duval, Burdet and Kunzle showed that four-dimensional Newton–Cartan theory of gravitation can be reformulated as Kaluza–Klein reduction of five-dimensional Einstein gravity along a null-like direction. The metric used is the same as the Galilean metric but with all positive entries

$g^{\mu \nu }=\left(\begin{array}{ccccc}1& 0& 0& 0& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1\\ 0& 0& 0& 1& 0\end{array}\right).$

This lifting is considered to be useful for non-relativistic holographic models. Gravitational models in this framework have been shown to precisely calculate the Mercury precession.

See also

• Galilean group
• Representation theory of the Galilean group
• Lorentz group
• Poincaré group
• Pauli–Lubanski pseudovector