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Eightfold way (physics)
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===Connection to representation theory=== {{main|Particle physics and representation theory}} {{see also|Compact group#Representation theory of a connected compact Lie group}}Assume we have a certain particle—for example, a proton—in a quantum state <math>|\psi\rangle</math>. If we apply one of the flavour rotations ''A'' to our particle, it enters a new quantum state which we can call <math>A|\psi\rangle</math>. Depending on ''A'', this new state might be a proton, or a neutron, or a superposition of a proton and a neutron, or various other possibilities. The set of all possible quantum states spans a vector space. [[Representation theory]] is a mathematical theory that describes the situation where elements of a group (here, the flavour rotations ''A'' in the group SU(3)) are [[automorphism]]s of a vector space (here, the set of all possible quantum states that you get from flavour-rotating a proton). Therefore, by studying the representation theory of SU(3), we can learn the possibilities for what the vector space is and how it is affected by flavour symmetry. Since the flavour rotations ''A'' are approximate, not exact, symmetries, each orthogonal state in the vector space corresponds to a different particle species. In the example above, when a proton is transformed by every possible flavour rotation ''A'', it turns out that it moves around an 8 dimensional vector space. Those 8 dimensions correspond to the 8 particles in the so-called "baryon octet" (proton, neutron, [[Sigma baryon|{{SubatomicParticle|Sigma+}}, {{SubatomicParticle|Sigma0}}, {{SubatomicParticle|Sigma-}}]], [[Xi baryon|{{SubatomicParticle|Xi-}}, {{SubatomicParticle|Xi0}}]], [[Lambda baryon|{{SubatomicParticle|Lambda}}]]). This corresponds to an 8-dimensional ("octet") representation of the group SU(3). Since ''A'' is an approximate symmetry, all the particles in this octet have similar mass.<ref name=Griffiths-2008/> Every [[Lie group]] has a corresponding [[Lie algebra]], and each [[group representation]] of the Lie group can be mapped to a corresponding [[Lie algebra representation]] on the same vector space. The Lie algebra <math>\mathfrak{su}</math>(3) can be written as the set of 3×3 traceless [[Hermitian matrices]]. Physicists generally discuss the representation theory of the Lie algebra <math>\mathfrak{su}</math>(3) instead of the Lie group SU(3), since the former is simpler and the two are ultimately equivalent.
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