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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. 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 mutiples 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, the measurement process exhibits some seemingly mysterious phenomena. This often leads to many misconceptions about the nature of quantum mechanics itself. Many of these misconceptions lead to frivolous speculations about the relation between consciousness and the material world (see external links). The facts of the matter, however, are far more prosaic. 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 ensembleIn physics, a statistical ensemble is a very large set of similar systems, considered all at once. The topic of statistical ensembles is important in thermodynamics, statistical mechanics and quantum physics. Putting aside for the moment the question of h of wave functions. The irreversible nature of measurement operations in quantum physics is sometimes referred to as the measurement problemQuantum mechanics The measurement problem is a supposed problem with quantum mechanics (exemplified by the Schrodinger's cat "paradox") which asks why there appears to be something "special" about measurement (as opposed to all other physical interactions and is described mathematically by quantum operationIn quantum mechanics, a quantum operation is a mathematical formalism used to describe a broad class of transformations that a quantum mechanical system can undergo. This formalism describes not only time evolution or symmetry transformations of isolateds. 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 traceLinear algebra Functional analysis In linear algebra and functional analysis, the partial trace is a generalization of the trace. Whereas the trace is a scalar valued function on operators, the partial trace is an operator-valued function. The partial tra of the state of the larger system.
Physically meaningful observables must also satisfy transformation law s which relate observations performed by different observerPossible meanings: #In general, an observer is any system which receives information from an object. An observer in an organization is a delegate sent to observe and report on the proceedings of an assembly or a meeting but not vote or otherwise participas 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 anti-unitary ) 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.
