The
Eisenhart-Lift
for Field Theories
K. Finn et al. 2018
arxiv:1806.02431
Are
Conservative Forces Fictitious?
K. Finn et al. 2018
arxiv:1806.02431
Title Text
Subtitle
Title Text
Subtitle
Newtonian dynamics in Euclidean space
GR
in space-time
Title Text
Subtitle
Newtonian dynamics in Euclidean space
GR
in space-time
Maxwell's equ.
in space-time
Kaluza-Klein theory
in 5D
Title Text
Subtitle
Newtonian dynamics in Euclidean space
GR
in space-time
Maxwell's equ.
in space-time
Kaluza-Klein theory
in 5D
Eisenhart-lift
L = \frac{1}{2}m\sum_{i=1}^{n} \dot{x}_i^2 + \frac{1}{2}\frac{M^2}{V(\mathbf{x})} \dot{y}^2
L = \frac{1}{2}m\sum_{i=1}^{n} \dot{x}_i^2 - V(\mathbf{x})
Eisenhart-Lift
\mathcal{L} = \sqrt{-g} \left( \frac{1}{2}g^{\mu\nu}k_{ij}(\mathbf{\phi})\partial_\mu\phi^i\partial_\nu\phi^j + \frac{1}{2}\frac{M^4}{V(\mathbf{\phi})} \nabla_\mu B^\mu \nabla_\nu B^\nu \right)
\mathcal{L} = \sqrt{-g} \left( \frac{1}{2}g^{\mu\nu}k_{ij}(\mathbf{\phi})\partial_\mu\phi^i\partial_\nu\phi^j - V(\mathbf{\phi})\right)
field theory with a potential
free theory with extra d.o.f.
Conclusion
- Any scalar field theory can be written in a kinetic-only form
Application
- Multi-field inflation
- Initial conditions in inflation (arXiv:1812.07095)
- Using geometric reasoning for field theory problems Â