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In physics and in the theory of dynamical systems (deterministic, stochastic, quantum, autonomous, nonautonomous, open, closed, discrete, continuous, with finite or infinite number of degrees of freedom…), an observable is a quantity in some theoretical framework whose value can be measured and observed in principle. Any good theoretical framework of physical phenomena should come with carefully established notion of an observable.
In classical mechanics an observable is any smooth function on the phase space of the system, and of time. The value of the observable is just the value of the function for fixed argument.
In quantum mechanics an observable is a Hermitean operator on the physical Hilbert space of the theory. See quantum observable for more details.
In this case, one distinguishes the concepts of the expectation value of the observable and the concept of the measured value; they are evaluated in some state of the system. The expectation value can be taken in any state of the system, while the measured value is always in some eigenstate of the observable operator. The process of measurement results in the quantum mechanical collapse or reduction, in which the system passes to an eigenstate of the measured operator. The probability of taking a given eigenstate depends on the the transition matrix element from the previously prepared state to the given eigenstate.
In relativistic quantum mechanics and relativistic quantum field theory the question of observables is more complicated: issues like causality and superselection sectors are involved.
In the AQFT approach to quantum field theory the observables are the very starting point of the theory: At the beginning one is handed an abstract $C^*$-algebra $C$, see C-star algebra (to be more precise: a net of such algebras). The selfadjoined elements of the algebras of the net are defined to be the observables of the theory.
duality between algebra and geometry in physics:
Careful discussion of local gauge invariant observables in gravity/general relativity is in
showing that, while there are no globally defined local gauge invariant observables, they do exist on an open cover of the space of field configuration and form something like a sheaf of observables (but, hence, one without global sections).
Careful discussion of observables in abelian gauge theory (electromagnetism) is in