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First off, I don't entirely understand what constitutes a symmetric quantity. If someone could provide a general definition of what this means I would greatly appreciate it. Also, how is the conservation law derived from said symmetric quantity? i.e. how do you make the leap that because time is symmetric, energy is conserved?

And lastly, how would you prove this theorem? It seems entirely too abstract to be able to apply any mathematical rigor to it.

Any help would be appreciated.