Galilei-covariant Tensor Formulation

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

: $\begin \mathbf' &= R\mathbf - \mathbf t + \mathbf \\ t' &= t + \mathbf. \end$

where $R$ stands for the three-dimensional Euclidean rotations, $\mathbf$ 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

: $p_\mu p_\nu g^ = p_i p_i - p_5 p_4 - p_4 p_5 = p^2 - 2mE = k,$

where

: $g^ = \pm \begin 1 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & -1\\ 0 & 0 & 0 & -1 & 0 \end,$

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

Extended Galilei algebra

A five dimensional Poincaré algebra leaves the metric $g^$ invariant,

: $\begin[] [P_\mu, P_\nu] &= 0, \\ \frac~[M_, P_\rho] &= g_ P_\nu - g_ P_\mu, \\ \frac~[M_, M_] &= g_ M_ - g_ M_ - g_ M_ + \eta_ M_, \end$

We can write the generators as

: $\begin J_i &= \frac\epsilon_M_, \\ K_i &= M_, \\ C_i &= M_, \\ D &= M_. \end$

The non-vanishing commutation relations will then be rewritten as

: \begin
\left _i,J_j\right&= i\epsilon_J_k, \\
\left _i,C_j\right&= i\epsilon_C_k, \\
\left ,K_i\right&= iK_i, \\
\left _4,D\right&= iP_4, \\
\left _i,K_j\right&= i\delta_P_5, \\
\left _4,K_i\right&= iP_i, \\
\left _5,D\right&= -iP_5, \\ pt
\left _i,K_j\right&= i\epsilon_K_k, \\
\left _i,C_j\right&= i\delta_D+i\epsilon_J_k, \\
\left _i,D\right&= iC_i, \\
\left _i,P_j\right&= i\epsilon_P_k, \\
\left _i,C_j\right&= i\delta_P_4, \\
\left _5,C_i\right&= iP_i.
\end

An important Lie subalgebra is

: \begin[]
[P_4,P_i] &= 0 \\[]
[P_i,P_j] &= 0 \\[]
[J_i,P_4] &= 0 \\[]
[K_i,K_j] &= 0 \\
\left _i,J_j\right&= i\epsilon_J_k, \\
\left _i,P_j\right&= i\epsilon_P_k, \\
\left _i,K_j\right&= i\epsilon_K_k, \\
\left _4,K_i\right&= iP_i, \\
\left _i,K_j\right&= i\delta_P_5,
\end

$P_4$ 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 $P^2-2P_4P_5$ 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 $P_4=-H$.

The third Casimir invariant is given by $W_W^\mu_5$, where $W_=\epsilon_P^M^$ 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

:$g^ = \begin1&0&0&0&0\\0&1&0&0&0\\0&0&1&0&0\\0&0&0&0&1\\0&0&0&1&0\end.$

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

{{DEFAULTSORT:Galilean Covariance
Rotational symmetry
Quantum mechanics
Representation theory of Lie groups