Physics:Galilei-covariant tensor formulation

Short description: Tensor formulation of non-relativistic physics

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

[math]\displaystyle{ \begin{align} \mathbf{x}' &= R\mathbf{x} - \mathbf{v} t + \mathbf{a} \\ t' &= t + \mathbf{b}. \end{align} }[/math]

where [math]\displaystyle{ R }[/math] stands for the three-dimensional Euclidean rotations, [math]\displaystyle{ \mathbf{v} }[/math] is the relative velocity determining Galilean boosts, a stands for spatial translations and b, for time translations. Consider a free mass particle [math]\displaystyle{ m }[/math]; the mass shell relation is given by [math]\displaystyle{ p^2 - 2mE = 0 }[/math].

We can then define a 5-vector,

[math]\displaystyle{ p^\mu = (p_x, p_y, p_z, m, E) = (p_i, m, E) }[/math],

with [math]\displaystyle{ i = 1, 2, 3 }[/math].

Thus, we can define a scalar product of the type

[math]\displaystyle{ p_\mu p_\nu g^{\mu\nu} = p_i p_i - p_5 p_4 - p_4 p_5 = p^2 - 2mE = k, }[/math]

where

[math]\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}, }[/math]

is the metric of the space-time, and [math]\displaystyle{ p_\nu g^{\mu\nu} = p^\mu }[/math].

Extended Galilei algebra

A five dimensional Poincaré algebra leaves the metric [math]\displaystyle{ g^{\mu\nu} }[/math] invariant,

[math]\displaystyle{ \begin{align}[] [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{align} }[/math]

We can write the generators as

[math]\displaystyle{ \begin{align} J_i &= \frac{1}{2}\epsilon_{ijk}M_{jk}, \\ K_i &= M_{5i}, \\ C_i &= M_{4i}, \\ D &= M_{54}. \end{align} }[/math]

The non-vanishing commutation relations will then be rewritten as

[math]\displaystyle{ \begin{align} \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{align} }[/math]

An important Lie subalgebra is

[math]\displaystyle{ \begin{align}[] [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{align} }[/math]

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

The third Casimir invariant is given by [math]\displaystyle{ W_{\mu\,5}W^\mu{}_5 }[/math], where [math]\displaystyle{ W_{\mu\nu}=\epsilon_{\mu\alpha\beta\rho\nu}P^{\alpha}M^{\beta\rho} }[/math] 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

[math]\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}. }[/math]

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