The Wigner–Araki–Yanase theorem, also known as the WAY theorem, is a result in
quantum physics
Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
establishing that the presence of a
conservation law
In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves over time. Exact conservation laws include conservation of mass-energy, conservation of linear momen ...
limits the accuracy with which
observable
In physics, an observable is a physical property or physical quantity that can be measured. In classical mechanics, an observable is a real-valued "function" on the set of all possible system states, e.g., position and momentum. In quantum ...
s that fail to
commute with the conserved quantity can be
measured. It is named for the physicists
Eugene Wigner
Eugene Paul Wigner (, ; November 17, 1902 – January 1, 1995) was a Hungarian-American theoretical physicist who also contributed to mathematical physics. He received the Nobel Prize in Physics in 1963 "for his contributions to the theory of th ...
,
Huzihiro Araki and Mutsuo Yanase.
The theorem can be illustrated with a particle coupled to a measuring apparatus. If the
position operator
In quantum mechanics, the position operator is the operator that corresponds to the position observable of a particle.
When the position operator is considered with a wide enough domain (e.g. the space of tempered distributions), its eigenvalues ...
of the particle is
and its
momentum operator
In quantum mechanics, the momentum operator is the operator associated with the linear momentum. The momentum operator is, in the position representation, an example of a differential operator. For the case of one particle in one spatial dimensio ...
is
, and if the position and momentum of the apparatus are
and
respectively, assuming that the total momentum
is conserved implies that, in a suitably quantified sense, the particle's position itself cannot be measured. The measurable quantity is its position ''relative'' to the measuring apparatus, represented by the operator
. The Wigner–Araki–Yanase theorem generalizes this to the case of two arbitrary observables
and
for the system and an observable
for the apparatus, satisfying the condition that
is conserved.
Mikko Tukiainen gave a generalized version of the WAY theorem, which makes no use of conservation laws, but uses quantum incompatibility instead.
Yui Kuramochi and Hiroyasu Tajima proved a generalized form of the theorem for possibly unbounded and continuous conserved observables.
References
Quantum measurement
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