My quantum mechanics professor was discussing the properties of Pauli matrices, their being both Hermitian and unitary. Then he made a remark that it is not possible to find three $n \times n$ matrices, where $n > 2$, that are simultaneously Hermitian and unitary. Can someone please explain or give a hint as to why this has to be true?
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Integral multiples of the identity are not unitary (unless the integer multiplier is $\pm1$). – Gerry Myerson Feb 19 '22 at 08:54
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I edited the question. Thanks for pointing out. – Kashish Feb 19 '22 at 10:00
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2It doesn't seem right to me, but maybe your prof had some additional property in mind (or mentioned some additional property which escaped your attention). Anyway, you might find https://math.stackexchange.com/questions/57148/matrices-which-are-both-unitary-and-hermitian enlightening. – Gerry Myerson Feb 19 '22 at 11:32
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1Why three matrices? I strongly suspect he was talking about three such matrices satisfying the su(2) algebra, spin matrix irreps, in which case he'd be right, as per the irreducible representation structure of spin. – Cosmas Zachos Feb 19 '22 at 17:24
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What does "simultaneously" mean in your context? – user1551 Feb 19 '22 at 19:49
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Any thoughts on the comments and the answer, Kashish? – Gerry Myerson Feb 20 '22 at 12:40
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He said that the statement was incorrect and that he meant that spins higher than 1/2 have spin observable matrices hermitian but not unitary. – Kashish Feb 27 '22 at 08:55
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Three of the four Dirac matrices $$ \gamma^1=\left(\begin{matrix}0&0&0&1\\0&0&1&0\\0&-1&0&0\\-1&0&0&0\end{matrix}\right)\,,\quad \gamma^2=\left(\begin{matrix}0&0&0&-i\\0&0&i&0\\0&i&0&0\\-i&0&0&0\end{matrix}\right)\,,\quad \gamma^3=\left(\begin{matrix}0&0&1&0\\0&0&0&-1\\-1&0&0&0\\0&1&0&0\end{matrix}\right) $$ are obviously anti-Hermitian and unitary. Therefore, their multiples with $\pm i$ are Hermitian and unitary.
Kurt G.
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