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. ^{[1] [2] [3] [4]} Previously, in 1985, Duval et al. constructed a similar tensor formulation in the context of Newton–Cartan theory. ^[5] Some other authors also have developed a similar Galilean tensor formalism. ^{[6] [7]}

Galilean manifold

The Galilei transformations are

${\displaystyle {\begin{aligned}\mathbf {x} '&=R\mathbf {x} -\mathbf {v} t+\mathbf {a} \\t'&=t+\mathbf {b} .\end{aligned}}}$

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

We can then define a 5-vector,

${\displaystyle p^{\mu }=(p_{x},p_{y},p_{z},m,E)=(p_{i},m,E)}$,

with ${\displaystyle i=1,2,3}$.

Thus, we can define a scalar product of the type

${\displaystyle p_{\mu }p_{\nu }g^{\mu \nu }=p_{i}p_{i}-p_{5}p_{4}-p_{4}p_{5}=p^{2}-2mE=k,}$

where

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

is the metric of the space-time, and ${\displaystyle p_{\nu }g^{\mu \nu }=p^{\mu }}$. ^[3]

Extended Galilei algebra

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

${\displaystyle {\begin{aligned}[][P_{\mu },P_{\nu }]&=0,\\{\frac {1}{i}}~[M_{\mu \nu },P_{\rho }]&=g_{\mu \rho }P_{\nu }-g_{\nu \rho }P_{\mu },\\{\frac {1}{i}}~[M_{\mu \nu },M_{\rho \sigma }]&=g_{\mu \rho }M_{\nu \sigma }-g_{\mu \sigma }M_{\nu \rho }-g_{\nu \rho }M_{\mu \sigma }+\eta _{\nu \sigma }M_{\mu \rho },\end{aligned}}}$

We can write the generators as

${\displaystyle {\begin{aligned}J_{i}&={\frac {1}{2}}\epsilon _{ijk}M_{jk},\\K_{i}&=M_{5i},\\C_{i}&=M_{4i},\\D&=M_{54}.\end{aligned}}}$

The non-vanishing commutation relations will then be rewritten as

${\displaystyle {\begin{aligned}\left[J_{i},J_{j}\right]&=i\epsilon _{ijk}J_{k},\\\left[J_{i},C_{j}\right]&=i\epsilon _{ijk}C_{k},\\\left[D,K_{i}\right]&=iK_{i},\\\left[P_{4},D\right]&=iP_{4},\\\left[P_{i},K_{j}\right]&=i\delta _{ij}P_{5},\\\left[P_{4},K_{i}\right]&=iP_{i},\\\left[P_{5},D\right]&=-iP_{5},\\[4pt]\left[J_{i},K_{j}\right]&=i\epsilon _{ijk}K_{k},\\\left[K_{i},C_{j}\right]&=i\delta _{ij}D+i\epsilon _{ijk}J_{k},\\\left[C_{i},D\right]&=iC_{i},\\\left[J_{i},P_{j}\right]&=i\epsilon _{ijk}P_{k},\\\left[P_{i},C_{j}\right]&=i\delta _{ij}P_{4},\\\left[P_{5},C_{i}\right]&=iP_{i}.\end{aligned}}}$

An important Lie subalgebra is

${\displaystyle {\begin{aligned}[][P_{4},P_{i}]&=0\\[][P_{i},P_{j}]&=0\\[][J_{i},P_{4}]&=0\\[][K_{i},K_{j}]&=0\\\left[J_{i},J_{j}\right]&=i\epsilon _{ijk}J_{k},\\\left[J_{i},P_{j}\right]&=i\epsilon _{ijk}P_{k},\\\left[J_{i},K_{j}\right]&=i\epsilon _{ijk}K_{k},\\\left[P_{4},K_{i}\right]&=iP_{i},\\\left[P_{i},K_{j}\right]&=i\delta _{ij}P_{5},\end{aligned}}}$

${\displaystyle P_{4}}$ is the generator of time translations (Hamiltonian), P_i is the generator of spatial translations (momentum operator), ${\displaystyle K_{i}}$ is the generator of Galilean boosts, and ${\displaystyle J_{i}}$ stands for a generator of rotations (angular momentum operator). The generator ${\displaystyle P_{5}}$ is a Casimir invariant and ${\displaystyle P^{2}-2P_{4}P_{5}}$ is an additional Casimir invariant. This algebra is isomorphic to the extended Galilean Algebra in (3+1) dimensions with ${\displaystyle P_{5}=-M}$, The central charge, interpreted as mass, and ${\displaystyle P_{4}=-H}$.^{[ citation needed ]}

The third Casimir invariant is given by ${\displaystyle W_{\mu \,5}W^{\mu }{}_{5}}$, where ${\displaystyle W_{\mu \nu }=\epsilon _{\mu \alpha \beta \rho \nu }P^{\alpha }M^{\beta \rho }}$ is a 5-dimensional analog of the Pauli–Lubanski pseudovector. ^[4]

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

${\displaystyle g^{\mu \nu }={\begin{pmatrix}1&0&0&0&0\\0&1&0&0&0\\0&0&1&0&0\\0&0&0&0&1\\0&0&0&1&0\end{pmatrix}}.}$

This lifting is considered to be useful for non-relativistic holographic models. ^[8] Gravitational models in this framework have been shown to precisely calculate the Mercury precession. ^[9]

See also

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

References

1. Takahashi, Yasushi (1988). "Towards the Many-Body Theory with the Galilei Invariance as a Guide: Part I". Fortschritte der Physik/Progress of Physics. 36 (1): 63–81. Bibcode:1988ForPh..36...63T. doi:10.1002/prop.2190360105. eISSN   1521-3978.
2. Takahashi, Yasushi (1988). "Towards the Many-Body Theory with the Galilei invariance as a Gluide Part II". Fortschritte der Physik/Progress of Physics. 36 (1): 83–96. Bibcode:1988ForPh..36...83T. doi:10.1002/prop.2190360106. eISSN   1521-3978.
3. Omote, M.; Kamefuchi, S.; Takahashi, Y.; Ohnuki, Y. (1989). "Galilean Covariance and the Schrödinger Equation". Fortschritte der Physik/Progress of Physics (in German). 37 (12): 933–950. Bibcode:1989ForPh..37..933O. doi:10.1002/prop.2190371203. eISSN   1521-3978.
4. Santana, A. E.; Khanna, F. C.; Takahashi, Y. (1998-03-01). "Galilei Covariance and (4,1)-de Sitter Space". Progress of Theoretical Physics. 99 (3): 327–336. arXiv: hep-th/9812223 . Bibcode:1998PThPh..99..327S. doi:10.1143/PTP.99.327. ISSN   0033-068X. S2CID   17091575.
5. Duval, C.; Burdet, G.; Künzle, H. P.; Perrin, M. (1985). "Bargmann structures and Newton–Cartan theory". Physical Review D. 31 (8): 1841–1853. Bibcode:1985PhRvD..31.1841D. doi:10.1103/PhysRevD.31.1841. PMID   9955910.
6. Pinski, G. (1968-11-01). "Galilean Tensor Calculus". Journal of Mathematical Physics. 9 (11): 1927–1930. Bibcode:1968JMP.....9.1927P. doi:10.1063/1.1664527. ISSN   0022-2488.
7. Kapuścik, Edward. (1985). On the relation between Galilean, Poincaré and Euclidean field equations. IFJ. OCLC   835885918.
8. Goldberger, Walter D. (2009). "AdS/CFT duality for non-relativistic field theory". Journal of High Energy Physics. 2009 (3): 069. arXiv: 0806.2867 . Bibcode:2009JHEP...03..069G. doi:10.1088/1126-6708/2009/03/069. S2CID   118553009.
9. Ulhoa, Sérgio C.; Khanna, Faqir C.; Santana, Ademir E. (2009-11-20). "Galilean covariance and the gravitational field". International Journal of Modern Physics A. 24 (28n29): 5287–5297. arXiv: 0902.2023 . Bibcode:2009IJMPA..24.5287U. doi:10.1142/S0217751X09046333. ISSN   0217-751X. S2CID   119195397.