As we know eigenfunctions of Hamiltonian (quantum states) constitute a Hilbert space (vector space) which equipped with an inner product.
I wounder if I ask you(Dear Mohamed El Naschie) to explain your answer
"cross product": a special subalgebra of R3: a(ijk)x(j)y(k) = z(i)
this construction works in all dimension, for example:
det(x,a,b,c, ...):
a "cross product" of n-1 n-tupels a,b,c, ... resulting in projection on x (n-tuple)
then there are several extensions possible to R(n), n to infinity.
(here: Hilbert-space as sequence of square summable numbers)
trivial example: this cross-product of the unit-vectors (0, ... , 0,1, 0 ...) is equal 1, as it should be.
When you define a cross product among vectors one gets two kinds of vectors: Polar vectors and axial vectors. Axial vectors are pseudo-vectors: they do not change sign upon inversion of the crossed vectors.
Now in case of states if a cross product were to be defined then one would get pseudo-states too. Now I don't know what they are, they might probably even exist, just emphasizing the fact that a cross product involving states should take this into consideration.
@Mohamed: I didn't get what you were trying to say pertaining to the question, could you please answer in a concise form?
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