Difficulties around the idea of spontaneous breaking of time translation symmetry in a closed quantum mechanical system are identified, and then overcome in a simple model. The possibility of ordering in imaginary time is also discussed.
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A quick check makes me think that one has fallen quite recently(?): https://arxiv.org/pdf/1202.2539v2.pdf
https://en.wikipedia.org/wiki/Time_translation_symmetry
https://en.wikipedia.org/wiki/Time_crystal
Edit: Also:
https://www.reddit.com/r/askscience/comments/52has4/what_is_...
https://arxiv.org/pdf/1603.08001v4.pdf
Periodic, discrete symmetry holds, continuous is toast. Interesting.. Maybe there should be a rule of thumb (XKCD?) that says for any intuition one may hold, some researcher somewhere is, at ``this'' moment (in some measuring system) constructing a contrived situation that will break it via some contrivance that is strange in inverse proportion to the homeliness of the intuition? ....And some talky-site-poster (me in this case) is preparing to make it yak about it?
The seminal papers in this field:
Quantum Time Crystals (Wilczek)
http://arxiv.org/abs/1202.2539
Classical Time Crystals (Shapere, Wilczek)
http://arxiv.org/abs/1202.2537
Space-time Crystals of Trapped Ions ( Li, Gong, Yin, Quan, Yin, Zhang, Duan,Zhang )
http://arxiv.org/abs/1206.4772
All three papers appeared in a single issue of Physical Review Letters (The fancy Physics journal).
PRL also issued this Physics Viewpoint, a popular science article:
http://physics.aps.org/articles/v5/116
Pfft. What if I conceive to build a perpetual motion machine, or conceive to disprove Special Relativity?
First one, liquids with zero viscosity, such as helium superfluids. - http://www.sciencemag.org/content/286/5438/213.summary
Or even 4 dimensional quantum time crystals (which surprisingly, doesn't appear to be completely mental) - http://arxiv.org/abs/1202.2539
Course, you can't take any power off them and you have to put them somewhere very very cold, but they will keep going, apparently.
As for the second, we know there are issues with the current physics and I think that might just be what stuff like CERN is for. Or to prove it really really well.
In honor of Alan Turing's hundredth birthday, I unwisely set out some thoughts about one of Turing's obsessions throughout his life, the question of physics and free will.