... it isnecessary that the quantities on each side of an equation transform in the same way underchange of coordinates. As a matter of jargon, we will sometimes refer to quantities whichare written ... is the space of bras, φ|, and theaction gives the number φ|ψ. (This is a complex number in quantum mechanics, but theidea is precisely the same.)In spacetime the simplest example of a dual ... ωn)V1V2···Vn= ωiVi. (1.39)Another familiar example occurs in quantum mechanics, where vectors in the Hilbert spaceare represented by kets, |ψ. In this case...