atyy said:
Is this an example of the difference you are talking about, or is it yet another difference?
http://mathworld.wolfram.com/BerryConjecture.html:"...physicists define this operator to be Hermitian, mathematicians do not."
Let A be a densely defined linear operator on a Hilbert space. One of the defining properties, *, for all cases is
\left<Ax,y\right> = \left<x,Ay\right>, \ *
where x and y are elements of the Hilbert space.
The text that was used for the functional analysis course that I took as a student makes the following definitions:
A is Hermitian if A is bounded and * is true for every x and y in the Hilbert space;
A is symmetric if * holds for for every x and y in the domain of A;
A is self-adjoint if A is symmetric and the domain of A equals the domain of A^\dagger.
According to these defintions, every Hermitian operator is and self-adjoint, but not all self-adjoint operators are Hemitian. Some books leave off the first definition and call symmetric operators Hermitian. Then, every self-adjoint operator is Hermitian, but not all Hermitian operators are self-adjoint.
MathWorld said:
Note that A is symmetric but might have nontrivial deficiency indices, so while physicists define this operator to be Hermitian, mathematicians do not.
A symmetric (or Hemitian, depending on the terminology used) operator is self-adjoint iff it has trivial deficiency indices. Some physicists and physics references do not distinguish between the above defintions, and use the terms Hermitian and self-adjoint interchangeably to refer to all cases.
If A is unbounded, then the Hilbert space is necessarily infinite-dimensional, and the domain of A need not be all of the Hilbert space. If this is the case and if A is symmetric, then the domain of A is a subset of the domain of A^\dagger, which I have not defined.
In physics, the canonical commutation relation is important. If self-adjoint operators A and B satisfy such a relation, then it is easy to show that at least one of the operators must be unbounded. Suppose it is A. The Hellinger-Toeplitz theorem states any linear operator that satisfies * for all elements of the Hilbert space must bounded. Since A is self-adjoint and unbounded, the domain of physical observable A cannot be all of Hilbert space!
For one consequence of these concepts, see
https://www.physicsforums.com/showthread.php?t=122063&highlight=dirac.
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