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| In [[quantum physics]], '''unitarity''' is a restriction on the allowed evolution of [[quantum system]]s that ensures the sum of [[probabilities]] of all possible outcomes of any event is always 1.
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| More precisely, the operator which describes the progress of a physical system in time must be a [[unitary operator]]. When the [[Hamiltonian (quantum mechanics)|Hamiltonian]] is time-independent the unitary operator is <math>e^{-i \hat{H} t}</math>.
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| Similarly, the [[S-matrix]] that describes how the physical system changes in a [[scattering]] process must be a [[unitary operator]] as well; this implies the [[optical theorem]].
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| In [[quantum field theory]] one usually uses a mathematical description which includes unphysical [[fundamental particle]]s, such as [[Longitudinal wave|longitudinal]] [[photon]]s. These particles must not appear as the end-states of a [[scattering]] process. Unitarity of the [[S-matrix]] and the [[optical theorem]] in particular implies that such unphysical particles must not appear as [[virtual particle]]s in intermediate states. The mathematical machinery which is used to ensure this includes [[gauge symmetry]] and sometimes also [[Faddeev–Popov ghost]]s.
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| Since unitarity of a theory is necessary for its consistency, the term is sometimes also used as a synonym for consistency, and is sometimes used for other necessary conditions for consistency, in particular the condition that the Hamiltonian is bounded from below. This means that there is a state of minimal [[energy]] (called the [[ground state]] or [[vacuum state]]). This is needed for the [[second law of thermodynamics]] to hold.
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| In [[theoretical physics]], a ''unitarity bound'' is any inequality that follows from the [[unitarity]] of the [[evolution operator]], i.e. from the statement that probabilities are numbers between 0 and 1 whose sum is conserved. Unitarity implies, among other things, the [[optical theorem]]. According to the optical theorem, the imaginary part of a [[probability amplitude]] Im(''M'') of a 2-body forward scattering is related to the total [[cross section (physics)|cross section]], up to some numerical factors. Because <math>|M|^2</math> for the forward scattering process is one of the terms that contributes to the total cross section, it cannot exceed the total cross section i.e. Im(''M''). The inequality
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| : <math>|M|^2 \leq \mbox{Im}(M)</math> | |
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| implies that the [[complex number]] ''M'' must belong to a certain disk in the complex plane. Similar unitarity bounds imply that the amplitudes and cross section cannot increase too much with energy or they must decrease as quickly as a certain formula dictates.
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| ==See also==
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| *[[Stone's theorem on one-parameter unitary groups]]
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| *[[Probability axioms]]
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| *[[Antiunitary operator]]
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| *[[Wigner's theorem]]
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| ==References== | |
| {{reflist}}
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| {{physics stub}}
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| [[Category:Quantum mechanics]]
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