Wigner–Araki–Yanase theorem

The Wigner–Araki–Yanase theorem, also known as the WAY theorem, is a result in quantum physics establishing that the presence of a conservation law limits the accuracy with which observables that fail to commute with the conserved quantity can be measured.^[1][2][3] It is named for the physicists Eugene Wigner,^[4] Huzihiro Araki and Mutsuo Yanase.^[5][6]

The theorem can be illustrated with a particle coupled to a measuring apparatus.^[7]: 421 If the position operator of the particle is ${\displaystyle q}$ and its momentum operator is ${\displaystyle p}$, and if the position and momentum of the apparatus are ${\displaystyle Q}$ and ${\displaystyle P}$ respectively, assuming that the total momentum ${\displaystyle p+P}$ 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 ${\displaystyle q-Q}$. The Wigner–Araki–Yanase theorem generalizes this to the case of two arbitrary observables ${\displaystyle A}$ and ${\displaystyle B}$ for the system and an observable ${\displaystyle C}$ for the apparatus, satisfying the condition that ${\displaystyle B+C}$ is conserved.^[8][9]

Mikko Tukiainen gave a generalized version of the WAY theorem, which makes no use of conservation laws, but uses quantum incompatibility instead.^[10]

Yui Kuramochi and Hiroyasu Tajima proved a generalized form of the theorem for possibly unbounded and continuous conserved observables.^[11]

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