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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 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 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.
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.
External links
[1] A critical essay by Stephen Barrett on quantum mechanics, healing and consciousness.
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