Different definitions for the complex inner product.

Question

I have asked this question on P.S.E. and have gotten some nice answers, but I felt I might get even more satisfactory answers if I post it here.

"I have just now noticed that Griffiths (in his book Introduction to Quantum Mechanics) defines the complex inner product as $\big<z,w\big>=\sum_{i=1}^n\overline{z}_iw_i$. In all mathematics books (I study math and physics) I have ever come across, it is defined as $\big<z,w\big>=\sum_{i=1}^nz_i\overline{w}_i$. Maybe there is an answer to this question, maybe there isn't, but why on earth is this defined differently in physics than in math?"

https://physics.stackexchange.com/questions/215783/why-does-griffiths-define-the-complex-inner-product-differently

I'll be glad to see someone make the case for mathematics, and let me know if the question shouldn't be asked here.

Answer 1

It is just a convention.

@Bye_World's explanation is a good justification of the physicists point of view.

We mathematicians probably just thought "We want it to be complex linear in one slot, and complex antilinear in the other slot. Might as well make the first one complex linear."

Answer 2

It was von Neumann who was the first to axiomatically define the inner product as a pairing in the manner it is done today. So he set the standard. Interestingly, von Neumann was the one who coined the term Hilbert Space, and effectively tied Hilbert's name to this abstract development, instead of his own (von Neumann was Hilbert's assistant during that time.) Von Neumann then developed the abstract theory of normal and Hermitian operators in a series of 3 papers from 1929-1932 which remain, in substance, the basis of modern accounts. Dirac introduced bra-ket in 1939.