Observable
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{{otheruses4|observables in physics|the use of the term "observable" in [[control theory]]|Observability}}
In [[physics]], particularly in [[quantum physics]], a system '''observable''' is a property of the [[State (physics)|system state]] that can be determined by some sequence of physical [[operational definition|operations]]. For example, these operations might involve submitting the system to various [[electromagnetic field]]s and eventually reading a value off some gauge. In systems governed by [[classical mechanics]], any [[experiment]]ally observable value can be shown to be given by a [[real number|real]]-valued [[function (mathematics)|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 [[vector (spatial)|vector]]s 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 operator]]s on ''V''. However, as indicated below, not every self-adjoint operator corresponds to a physically meaningful observable. For the case of a system of [[Elementary particle|particle]]s, the space ''V'' consists of functions called [[wave function]]s.
In quantum mechanics, measurement of observables exhibits some seemingly unintuitive properties. Specifically, if a system is in a state described by a vector in a [[Hilbert space]], 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 vector may be destroyed, being replaced by a [[statistical ensemble]]. The [[reversible process|irreversible]] nature of measurement operations in quantum physics is sometimes referred to as the [[measurement problem]] and is described mathematically by [[quantum operation]]s. 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 law]]s which relate observations performed by different [[observation|observer]]s in different [[frames of reference]]. These transformation laws are [[automorphism]]s of the state space, that is [[bijective]] [[Transformation (mathematics)|transformation]]s which preserve some mathematical property. In the case of quantum mechanics, the requisite automorphisms are [[unitary operator|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.
== References ==
* 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.
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