Exploring the Math Behind Quantum Mechanics

[plmokn2]

Not really a problem but was reading a review of a book on amazon and came across this:

.) It is mathematically sloppy. Dirac introduces the Bra and Ket notation (for which he is responsible, by the way) without mentioning the dual space, and sometimes even reasons wrongly; i.e., he writes "let us postulate that for each ket, there exists a corresponding bra" - this is not a postulate. This is ALWAYS true for finite dimensional vector spaces, and NEVER true for infinite dimensional vector spaces, and can be proven mathematically. In short, there is little attention given to the mathematics behind QM.

Is the bit in bold right? Isn't <psi| normally infinite dimensions or am I confused?

Thanks

 

[Hurkyl]

Bra-ket notation doesn't make sense for an abstract vector space -- you can't transpose a vector unless you've chosen an inner product. Once you've done so, every ket can be transposed to a bra. But in general, some bras cannot be transposed into a ket.

For inner-product spaces, every bra can be transposed if and only if the vector space is finite-dimensional

However, we consider a Hilbert space not as an inner-product space, but as a topological inner-product space -- and so the space of bras consists only of continuous linear functionals. By the Riesz representation theorem, every bra can be transposed into a ket.

 

[genneth]

Actually even the bit about it always possible in a finite dimensional vector space is off -- as stated by Hurkyl, you need an inner product. It is true that we sloppy physicists aren't always so hot on mathematical accuracy -- but let's just say that Dirac's words have been read with scrutiny for a while...

 

[plmokn2]

Thanks