Topology of the Quantum Dimension – Alice and Omega

Alice has continued to think about Bob’s admittedly somewhat awkward search for the secret passageway, and has figured out the twist she needs in order to find Omega. Alice takes two paper strips, sticking them together on the longitudinal side, to make a strip that is twice as wide. With the “twisting” and “regluing” operations, she forms the well-known Möbius strip.

If we run 360° in the Möbius strip, we get from the front to the back. Only after 720°, do we arrive at where we started. Here is Alice’s trick: She separates the two original paper strips, by cutting the Möbius strip longitudinally again. Alice obtains a strip that is twice as long and twisted four times, which is also known as “Dirac’s belt”, in honour of Paul Dirac, the scientist who discovered anti-matter. That is what a fermion in the 720° world looks like, before the two angle ranges are stuck together, in order to project them into the 360° world.

Come on, Bob – it is your turn! Do the same for a boson! Okay: stick two paper strips together at the longitudinal edge – twist once … twist twice, reglue both ends … Yet, already after 360° Bob the boson comes back again to the starting point …. What happens, then, if we cut the double-twisted strip longitudinally again? … we obtain two copies of the initial state, intertwined once. Oops! Oh, Bob – you will never make it to the 720° world of the quantum dimension this way …

Alice got the twist: she takes her own hair in two thick tresses, making a plait, with one, two, three, four, five twists … Do you like my hairdo? Watch out though, Bob! You need an odd number of twists … Then put your hair back again, and look into the quantum mirror. Turn once through 360° there. Now all you need to do is to superpose with yourself … and, hey presto, …. Alice is gone … Alice?!?

Bob the Boson thinks to himself: What Alice can do, I can do, too! So, off we go – make a lovely plait, one, two…. Oh, you darn wig! The hair is much too short, this is getting me nowhere. … Wow, now I look exactly like Alice! Put the hair back, and look into the mirror. Turn through 360° there…. Now all I will do is just superpose with myself … and …. Nothing happens …?!?

Oh yes, it needs to be an odd number of twists – one, two, three, four – twist once more … This should work – put the hair back… Look into the quantum mirror, Turn once through 360° there … Superpose with myself … and, hey presto, ….

Alice?
Bob, you made it?
That‘s strange – I see everything twice…
No, no – it is very unique!
I don’t know where I am anymore….
Omega, where has the time gone?