In classical physics, physical observables are typically associated with real-valued functions on the phase space, such as position, momentum, or energy. These observables are not typically described in terms of operators, and there is no strict requirement for them to be Hermitian operators.
However, in quantum theory, physical observables are represented by Hermitian operators. This is a fundamental aspect of quantum mechanics. The Hermitian property ensures that the eigenvalues of the operator are real, which corresponds to the possible outcomes of measurements. The eigenvectors of a Hermitian operator form an orthonormal basis, which allows for the probabilistic interpretation of quantum states and measurement outcomes.
In quantum mechanics, the expectation value of an observable is calculated by taking the inner product of the state vector with the corresponding eigenstate of the operator. The Hermiticity of the operator guarantees that the expectation value is a real quantity, consistent with experimental observations.
So, while in classical physics observables are not typically described in terms of operators, in quantum theory, the mathematical framework requires physical observables to be represented by Hermitian operators.