Types of quantum field thories
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 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 -algebra , 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.
|Poisson algebra||Poisson manifold|
|deformation quantization||geometric quantization|
|algebra of observables||space of states|
|Heisenberg picture||Schrödinger picture|
|higher algebra||higher geometry|
|Poisson n-algebra||n-plectic manifold|
|En-algebras||higher symplectic geometry|
|BD-BV quantization||higher geometric quantization|
|factorization algebra of observables||extended quantum field theory|
|factorization homology||cobordism representation|
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).