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Bilinear forms can give us a notion of distance, whether it is the typical Euclidean distance, or the spacetime interval between two events in Minkowski space. But what about skew-symmetric bilinear forms?

Skew-symmetry means that every vector has $B(v,v)=0$. Also, we can always find a basis such that picking any one element of that basis, say $v_i$, we get zero when applied all other basis elements, except exactly one other, $v_j$, where $B(v_i,v_j)=1$ and $B(v_j,v_i)=-1$. Usually $B(v,w)=0$ means some kind of orthogonality or perpendicularity, like perpendicular directions in Euclidean space. So each basis vector is "orthogonal" to all the others except one of them, where those combine together to give $+1$ or $-1$. What's going on here?

koobtseej
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  • My [very uninformed] 2 cents: I don't think that it is a good idea to use orthogonality as intuition, for precisely the first reason ($B(v,v)=0$). But I am also curious to see if there is some other interpretation. – angryavian Oct 07 '16 at 22:17
  • @angryavian Orthogonality is related to skew-symmmetricity via the exp map: every orthogonal matrix is the exponential of an antisymmetric matrix. – Jean Marie Oct 07 '16 at 23:10

3 Answers3

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The keywords are symplectic geometry. A symplectic manifold is a smooth manifold equipped with a non-degenerate closed $2-$form $\omega$. In each tangent space, $\omega$ is a skew-symmetric non-degenerate bilinear form.

Symplectic geometry has lots of applications in classical mechanics. I think this text by Dusa Mcduff might satisfy your curiosity.

Ivo Terek
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Another example would be oriented intersection numbers of (say) curves on an oriented surface. Think of a torus. Let $v_1$ correspond to the "equator" and let $v_2$ correspond to a meridian (one of the circles that rotates about the axis to generate the torus). Depending on how we choose to orient everybody, you'll have $\#(v_1,v_2) = 1 = -\#(v_2,v_1)$. (The intersection number is positive if an oriented tangent vector to the first curve, followed by an oriented tangent vector to the second curve gives you the outward-pointing normal of the torus when you apply the right-hand rule.) On the other hand $\#(v_1,v_1)=0=\#(v_2,v_2)$, because you can easily "push" each one of these curves off itself.

Ted Shifrin
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I am not sure if this correct, hence why I am marking it community wiki.

But based on this Wikipedia article, it seems like every symplectic form induces notions of orientation and volume via its associated volume form. (Or one notion of signed volume.)

The skew-symmetry of the symplectic form certainly is reminiscent of the skew-symmetry of permutations and of alternating multilinear forms, differential forms among them.

The relationship of this to physics again comes from the geometric interpretation of phase space: basically, one way to understand Liouville's theorem -- volumes in phase space are preserved under canonical transformations. Any canonical transformation is a symplectomorphism, i.e. a map preserving symplectic structure, and thus since signed volume is derived from symplectic structure, that signed volumes (in phase space) should be preserved under maps preserving symplectic structure (canonical transformations) becomes somewhat unsurprising.

I wrote a similar answer here: https://math.stackexchange.com/a/1977701/327486

Community
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Chill2Macht
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