The inner product in quantum mechanics (QM) is defined as positive definite in the context of Hilbert spaces, satisfying the condition <a|a> ≥ 0 with equality if and only if |a> = |0>, the null vector. This definition aligns with standard mathematical conventions, although some authors, like Shankar, refer to it as positive semidefinite, which has led to confusion. The discussion clarifies that while the inner product is indeed positive semidefinite, it is also positive definite when the null vector condition is included. The distinction between positive definite and positive semidefinite is crucial for understanding the properties of inner products in QM.
Quantum mechanics students, physicists, mathematicians, and anyone interested in the mathematical foundations of quantum theory and the properties of inner products in Hilbert spaces.
You mean except on a zero measure set.vanhees71 said:In the standard definition of a Hilbert space the scalar product is positive definite, i.e., if ##\langle a|a \rangle=0##, the vector ##|a \rangle=|0 \rangle## is the null vector.
That's why, e.g., in its realization as the Hilbert space of square integrable functions (as used in "wave mechanics" a la Schrödinger) a function represents the null vector if it is zero except on a zero set.
Ah, i see! It is null set.vanhees71 said:Thanks, I wasn't sure about the correct english term for "Nullmenge" in the sense of Lebesgue-integration theory.
The definition given here shows that in terms of QM the inner product can be defined as positive definite or positive semi-definite which might account for the fact that Shankar calls it positive semidefinite while Sakurai calls it positive definiteDemystifier said:
https://www.physicsforums.com/threads/collection-of-science-jokes-p2.847743/post-5988541vanhees71 said:BTW does anybody know a good one for mathematics/physics vocabulary?
You mix the positive definiteness of the scalar product of a Hilbert space, and by definition a Hilbert space has a positive definite scalar product and the positive definiteness of a self-adjoint operator.dyn said:The definition given here shows that in terms of QM the inner product can be defined as positive definite or positive semi-definite which might account for the fact that Shankar calls it positive semidefinite while Sakurai calls it positive definite
I looked at Shankar. He does call it semidefinite.vanhees71 said:You mix the positive definiteness of the scalar product of a Hilbert space, and by definition a Hilbert space has a positive definite scalar product and the positive definiteness of a self-adjoint operator.
Yes.vanhees71 said:What does Shankar call semidefinite? The scalar product? I'd be surprised!
No, his inner product is the same, he just calls it semidefinite.vanhees71 said:That's very strange. Then QT doesn't work the usual way!
Yes, but you are not listening to what I am saying. His inner product is positive definite, he includes the condition ##\langle v|v\rangle = 0## iff ##v=0##. He just calls it positive semi-definite.vanhees71 said:Ok, an inner product is positive semidefinite if for all vectors ##|v \rangle## you have ##\langle v|v \rangle \geq 0##. Take a finite-dimensional space. Then you can choose a basis such that
$$\langle v|v \rangle=\sum_{j=1}^d g_{jk} v_j^* v_k,$$
and you must have ##(g_{jk})=\mathrm{diag}(1,1,\ldots,1,0,0,0\ldots)##. In that case the inner product does not induce a metric.
That's why, for a scalar product you define it to be positive definite, i.e., it's positive semidefinite and in addition you have ##\langle v|v \rangle = 0 \Leftrightarrow |v \rangle=|0 \rangle## (where ##|0 \rangle## is the null vector).