My previous post was *awfully* long, so I must assume many of my readers may have started to read it, but… Well… Gave up halfway or even sooner. đ I added a footnote, though, which is interesting to reflect upon. Also, I know many of my readers aren’t interested in the mathâeven if they understand one cannot really appreciate quantum theory without the math. But… Yes. I may have left some readers behind. Let me, therefore, pick up the most interesting bit of all of the stories in my last posts in as easy a language as I can find.

We have that weird 360/720Â° symmetry in quantum physics orâto be preciseâwe have it for elementary matter-particles (think of electrons, for example). In order to, hopefully, help you understand what it’s all about, I had to explain the often-confused but substantially different concepts of aÂ *reference frameÂ *and a *representational baseÂ *(or representationÂ *tout court*). I won’t repeat that explanation, but think of the following.

If we just rotate the *reference frame* over 360Â°, we’re just using the same reference frame and so we see the same thing: some object which we, vaguely, describe by someÂ *e*^{i}^{ÂˇÎ¸}Â function. To be precise, I should say that we describe it by some *Fourier* sum of such functions. Now, if its spin direction is… Well… In the other direction, then we’ll describe it by by someÂ *e*^{âi}^{ÂˇÎ¸}Â function (again, you should read: aÂ *FourierÂ *sum of such functions). Now, the weird thing is is the following: if we rotate *the object itself*, over the sameÂ 360Â°, we get aÂ *differentÂ *object: ourÂ *e*^{i}^{ÂˇÎ¸}Â andÂ *e*^{âi}^{ÂˇÎ¸}Â function (again: think of aÂ *FourierÂ *sum, so that’s a waveÂ *packet*, really) becomes aÂ â*e*^{Âąi}^{ÂˇÎ¸}Â thing. We get aÂ *minusÂ *sign in front of it.Â So what happened here? What’s the difference, *really*?

Well… I don’t know. It’s very deep. Think of you and me as two electrons who are watching each other. If I do nothing, and you keep watching me *while turning around me*, for a fullÂ 360Â° (so that’s a rotation of your reference frame over 360Â°), then you’ll end up where you were when you started and, importantly, you’ll see the same thing: *me*. đ I mean… You’ll seeÂ *exactlyÂ *the same thing: if I was anÂ *e*^{+i}^{ÂˇÎ¸}Â wave packet, I am still anÂ anÂ *e*^{+i}^{ÂˇÎ¸}Â wave packet now. OrÂ if I was an *e*^{âi}^{ÂˇÎ¸}Â wave packet, then I am still anÂ an *e*^{âi}^{ÂˇÎ¸}Â wave packet now. Easy. Logical. *Obvious*, right?

But so now we try something different:Â *IÂ *turn around, over a fullÂ 360Â° turn, and *youÂ *stay where you are and watch *meÂ *while I am turning around. What happens? Classically, nothing should happen but… Well… This is the weird world of quantum mechanics: when I am back where I wasâlooking at you again, so to speakâthen… Well… I am not quite the same any more. Or… Well… Perhaps I am but youÂ *seeÂ *me differently. If I wasÂ *e*^{+i}^{ÂˇÎ¸}Â wave packet, then I’ve become aÂ â*e*^{+i}^{ÂˇÎ¸}Â wave packet now.

Not *hugely* different but… Well… ThatÂ *minusÂ *sign matters, right? OrÂ If I wasÂ wave packet built up from elementaryÂ *a*Âˇ*e*^{âi}^{ÂˇÎ¸}Â waves, then I’ve become aÂ â*e*^{âi}^{ÂˇÎ¸}Â wave packet now. What happened?

It makes me think of the twin paradox in special relativity. We know it’s aÂ *paradox*âso that’s anÂ *apparentÂ *contradiction only: we know which twin stayed on Earth and which one traveled because of the gravitational forces on the traveling twin. The one who stays on Earth does not experience any acceleration or deceleration. Is it the same here? I mean… The one who’s turning around must experience someÂ *force*.

Can we relate this to the twin paradox? Maybe. Note that aÂ *minusÂ *sign in front of theÂ *e*^{âÂąi}^{ÂˇÎ¸}Â functions amounts a minus sign in front of both the sine and cosine components. So… Well… The negative of a sine and cosine is the sine and cosine but with a phase shift of 180Â°: â*cos*Î¸ =Â *cos*(Î¸ Âą Ď) andÂ â*sin*Î¸ =Â *sin*(Î¸ Âą Ď). Now, adding or subtracting aÂ *commonÂ *phase factor to/from the argument of the wavefunction amounts toÂ *changingÂ *the origin of time. So… Well… I do think the twin paradox and this rather weird business of 360Â° and 720Â° symmetries are, effectively, related. đ

**Post scriptum**:Â *GoogleÂ *honors Max Born’s 135th birthday today. đ I think that’s a great coincidence in light of the stuff I’ve been writing about lately (possible interpretations of the wavefunction). đ