Hi all,(adsbygoogle = window.adsbygoogle || []).push({});

I've been fiddling around with this problem for a while. I intuitively understand that the parallel propagator is the path integral of the connection. I would like to be able to show the converse (connection is derivative of parallel propagator) mathematically, and I am having a little trouble.

I've been thinking of the parallel propagator as in http://en.wikipedia.org/wiki/Parall...ng_the_connection_from_the_parallel_transport. I understand how to formulate thecovariant derivative

[tex]

\nabla_X V = \lim_{h\to 0}\frac{\Gamma(\gamma)_h^0V_{\gamma(h)} - V_{\gamma(0)}}{h} = \frac{d}{dt}\Gamma(\gamma)_t^0V_{\gamma(t)}\Big|_{t=0}.

[/tex]

(Actually not really the second equality -- is this letting [itex]V_{\gamma(0)}=\Gamma(\gamma)_0^0V_{\gamma(0)}[/itex]?)

However, the above link doesn't really show what theconnectionis. Yet, if you evaluate the last term of the above equation, then using the product rule you've got a term with the derivative of [itex]V[/itex] and a term with the derivative of the parallel propagator. This is what you'd expect for the covariant derivative of a vector, where the connecton coefficients are the derivatives of the parallel propagator.

Nonetheless, I'm unable to make this all work out mathematically, so I was wondering if anyone could give me a hint? That is, on how the connection coefficients are the derivatives of the parallel propagator.

Cheers.

**Physics Forums - The Fusion of Science and Community**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Connection coefficients as derivatives of parallel propagator

Loading...

Similar Threads - Connection coefficients derivatives | Date |
---|---|

I Connections on principal bundles | Jan 22, 2018 |

A Is the Berry connection a Levi-Civita connection? | Jan 1, 2018 |

A Can you give an example of a non-Levi Civita connection? | Oct 30, 2017 |

Symmetry of connection coefficients? (simple question) | May 22, 2013 |

Connection coefficients entering differential operators | Apr 10, 2007 |

**Physics Forums - The Fusion of Science and Community**