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

q

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 ...

p

, and if the position and momentum of the apparatus are

Q

and

P

respectively, assuming that the total momentum

p + P

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

q - Q

. The Wigner–Araki–Yanase theorem generalizes this to the case of two arbitrary observables

A

and

B

for the system and an observable

C

for the apparatus, satisfying the condition that

B + C

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 {{quantum-stub