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We use Right Hand Rule in calculating Torque not because that's the direction torque is pointing in the real, physical world, but because it's a convenient way to indicate the "sign" of the rotation and its axis, and as long as we pick one coordinate system and stick to it, everything works out.

I think (I might be incorrect) we use the Right Hand Rule in Electromagnetism arbitrarily, but it happens to be very convenient only because of the right combination of Benjamin Franklin's arbitrary choice for designating current flow and the way our magnetic field was aligned at the time of the convention choice for North. But essentially, it is arbitrary -- if Benjamin Franklin had chosen current flow to match the flow of electrons, we'd either use another choice of axes or say that the magnetic field is negative.

What I'm saying is that I don't think that nature itself is inherently "right handed" in this situation.

Is there any case/situation in physics where the Right Hand Rule and choice of coordinate axes is not arbitrary, but rather inherent in nature itself? How so?

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Weak interaction? –  KennyTM Nov 23 '10 at 6:28
    
@KennyTM I've heard that answer mentioned when I've asked this question more casually in the past, but could you explain the deeper reasons why weak interaction fits this description? I don't understand it too much. –  Justin L. Nov 23 '10 at 6:37
    
Your question goes by the name "symmetry under parity transformation". As KennyTM pointed out, the weak interactions violate this symmetry. I'm not too knowledgeable, but you can read about it here: en.wikipedia.org/wiki/Parity_(physics) or watch a lecture covering the topic by Richard Feynman here: research.microsoft.com/apps/tools/tuva/index.html (Messenger lectures, lecture 4 "Symmetry in Physical Law") –  Mark Eichenlaub Nov 23 '10 at 7:24
    
BTW, when you learned Physics in the UK, under the "A-level" system (at least 20years ago), you did use the left-hand-rule for determining force/current relationships, as well as cross products. You just have to label the finger differently. –  ja72 Nov 23 '10 at 19:20
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I read in a book recently a theoretical conversation between an alien and and an earthling where the earthling was trying to explain which hand is the right hand (without any visual contact) and which the left. The alien would have to perform experiments to determine a reference left or right. It turns out after a whole chapter showing how there isn't anything in nature inheritly differentiating between left and right, there was an experiment involving beta decay and the spin of neutrinos that came out. I don't remeber the details, but apparently deep in the bowels of the standard model in the universe there is sense of left and right, at least when it comes to subatomic particles.

Now if I could find this book again in my library ... :-)

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Probably "The Character of Physical Law" amazon.com/Character-Physical-Law-Modern-Library/dp/0679601279 –  Mark Eichenlaub Nov 23 '10 at 19:21
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No, it really is arbitrary. The reason we use the right hand rule today (although it may have been chosen for different reasons of convenience in the past) is simply that our coordinate system of choice is right-handed. Mathematically, this means that we define the directions of the axes so that you have to use the right-hand rule to evaluate this cross product:

$\hat{x}\times\hat{y} = \hat{z}$

and everything flows from there.

It would be just as valid to define a left-handed coordinate system, in which the $\hat{z}$-axis points the other way. In that coordinate system, you would use the left-hand rule to evaluate cross products, but physics would still work the same way.

In the experiment that discovered parity violation, the researchers found that muons were emitted parallel (or antiparallel, I forget which - but it was one and not the other) to the angular momentum of decaying cobalt nuclei. If physics had been developed with a left-handed coordinate system, I think they would have found the opposite result, since angular momentum is something that we use the right-hand rule to define - but for exactly that reason, there's no fundamental distinction between a "parallel" result and an "antiparallel" result. The direction of angular momentum, and thus the word we use for the result of the cobalt decay experiment, is an artifact of our coordinate system.

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