I would like to gain a more formal mathematical understanding of a construct relating to spinors.(adsbygoogle = window.adsbygoogle || []).push({});

When I write down Dirac spinors in the Weyl basis, I see why if I multiply the adjoint (conjugate transpose) of a spinor with the original spinor I don't get a SL(2,C) scalar. It just doesn't work if we stick with the conventions of multiplication between (row and column) vectors. To clarify, here I am thinking of column vectors in spinor space as elements of a complex vector space carrying the fundamental representations of the complexified Clifford algebra, Cl(C). Rows are the complex conjugate, transposed object.

Thus one sees that one needs to "rotate" the components of the Dirac spinor (say with [itex] \gamma^{0}[/itex]), as well as complex transpose them, to be able to multiply properly into SL(2,C) scalars.

My question is the following. From the perspective of vector spaces, what goes wrong with defining the usual inner product [itex] x^{\dagger}x[/itex] on this complex vector space (i.e. the spinor space)? Does this construct have a particular name in the maths literature?

Thanks!

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# Inner product on spinor space

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