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
x ′ = R x − v t + a t ′ = t + b . {\displaystyle {\begin{aligned}\mathbf {x} '&=R\mathbf {x} -\mathbf {v} t+\mathbf {a} \\t'&=t+\mathbf {b} .\end{aligned}}}
where R {\displaystyle R} stands for the three-dimensional Euclidean rotations, v {\displaystyle \mathbf {v} } is the relative velocity determining Galilean boosts, a stands for spatial translations and b, for time translations. Consider a free mass particle m {\displaystyle m} ; the mass shell relation is given by p 2 − 2 m E = 0 {\displaystyle p^{2}-2mE=0} . We can then define a 5-vector,
p μ = ( p x , p y , p z , m , E ) = ( p i , m , E ) {\displaystyle p^{\mu }=(p_{x},p_{y},p_{z},m,E)=(p_{i},m,E)} , with i = 1 , 2 , 3 {\displaystyle i=1,2,3} . Thus, we can define a scalar product of the type
p μ p ν g μ ν = p i p i − p 5 p 4 − p 4 p 5 = p 2 − 2 m E = k , {\displaystyle 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 μ ν = ± ( 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 ) , {\displaystyle 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 ν g μ ν = p μ {\displaystyle p_{\nu }g^{\mu \nu }=p^{\mu }} .
Extended Galilei algebra A five dimensional Poincaré algebra leaves the metric g μ ν {\displaystyle g^{\mu \nu }} invariant,
[ P μ , P ν ] = 0 , 1 i [ M μ ν , P ρ ] = g μ ρ P ν − g ν ρ P μ , 1 i [ M μ ν , M ρ σ ] = g μ ρ M ν σ − g μ σ M ν ρ − g ν ρ M μ σ + g ν σ M μ ρ , {\displaystyle {\begin{aligned}[][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 }+g_{\nu \sigma }M_{\mu \rho },\end{aligned}}}
We can write the generators as
J i = 1 2 ϵ i j k M j k , K i = M 5 i , C i = M 4 i , D = M 54 . {\displaystyle {\begin{aligned}J_{i}&={\frac {1}{2}}\epsilon _{ijk}M_{jk},\\K_{i}&=M_{5i},\\C_{i}&=M_{4i},\\D&=M_{54}.\end{aligned}}}
The non-vanishing commutation relations will then be rewritten as
[ J i , J j ] = i ϵ i j k J k , [ J i , C j ] = i ϵ i j k C k , [ D , K i ] = i K i , [ P 4 , D ] = i P 4 , [ P i , K j ] = i δ i j P 5 , [ P 4 , K i ] = i P i , [ P 5 , D ] = − i P 5 , [ J i , K j ] = i ϵ i j k K k , [ K i , C j ] = i δ i j D − i ϵ i j k J k , [ C i , D ] = i C i , [ J i , P j ] = i ϵ i j k P k , [ P i , C j ] = i δ i j P 4 , [ P 5 , C i ] = i P i . {\displaystyle {\begin{aligned}\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},\\[4pt]\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}.\end{aligned}}}
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 [ J i , J j ] = i ϵ i j k J k , [ J i , P j ] = i ϵ i j k P k , [ J i , K j ] = i ϵ i j k K k , [ P 4 , K i ] = i P i , [ P i , K j ] = i δ i j P 5 , {\displaystyle {\begin{aligned}[][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},\end{aligned}}}
P 4 {\displaystyle P_{4}} is the generator of time translations (Hamiltonian), Pi is the generator of spatial translations (momentum operator), K i {\displaystyle K_{i}} is the generator of Galilean boosts, and J i {\displaystyle J_{i}} stands for a generator of rotations (angular momentum operator). The generator P 5 {\displaystyle P_{5}} is a Casimir invariant and P 2 − 2 P 4 P 5 {\displaystyle 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 {\displaystyle P_{5}=-M} , The central charge, interpreted as mass, and P 4 = − H {\displaystyle P_{4}=-H} . The third Casimir invariant is given by W μ 5 W μ
5 {\displaystyle W_{\mu \,5}W^{\mu }{}_{5}} , where W μ ν = ϵ μ α β ρ ν P α M β ρ {\displaystyle W_{\mu \nu }=\epsilon _{\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 μ ν = ( 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 ) . {\displaystyle g^{\mu \nu }={\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}}.}
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
References