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In physics, particularly in quantum physics, a system observable is a property of the system state that can be determined by some sequence of physical operations. For example, these operations might involve submitting the system to various electromagnetic fields and eventually reading a value off some gauge. In systems governed by classical mechanics, any experimentally observable value can be shown to be given by a real-valued function on the set of all possible system states. In quantum physics, on the other hand, the relation between system state and the value of an observable is more subtle, requiring some basic linear algebra to explain. In the mathematical formulation of quantum mechanics, states are given by non-zero vectors in a Hilbert space V (where two vectors are considered to specify the same state if, and only if, they are scalar multiples of each other) and observables are given by self-adjoint operators on V. However, as indicated below, not every self-adjoint operator corresponds to a physically meaningful observable. For the case of a system of particles, the space V consists of functions called wave functions.

In quantum mechanics, measurement of observables exhibits some seemingly mysterious phenomena. This often leads to many misconceptions about the nature of quantum mechanics itself. The facts of the matter, however, are far more ordinary. Specifically, if a system is in a state described by a wave function, the measurement process affects the state in a non-deterministic, but statistically predictable way. In particular, after a measurement is applied, the state description by a single wave function may be destroyed, being replaced by a statistical ensemble of wave functions. The irreversible nature of measurement operations in quantum physics is sometimes referred to as the measurement problem and is described mathematically by quantum operations. By the structure of quantum operations, this description is mathematically equivalent to that offered by relative state interpretation where the original system is regarded as a subsystem of a larger system and the state of the original system is given by the partial trace of the state of the larger system.

Physically meaningful observables must also satisfy transformation laws which relate observations performed by different observers in different frames of reference. These transformation laws are automorphisms of the state space, that is bijective transformations which preserve some mathematical property. In the case of quantum mechanics, the requisite automorphisms are unitary (or antiunitary) linear transformations of the Hilbert space V. Under Galilean relativity or special relativity, the mathematics of frames of reference is particularly simple, and in fact restricts considerably the set of physically meaningful observables.

• S. Auyang, How is Quantum Field Theory Possible, Oxford University Press, 1995.
• G. Mackey, Mathematical Foundations of Quantum Mechanics, W. A. Benjamin, 1963.
• V. Varadarajan, The Geometry of Quantum Mechanics vols 1 and 2, Springer-Verlag 1985.