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It seems that $i$ plays an important role in quantum mechanics (Q.M.). On the other hand, linear algebra plays such an important role in Q.M. too. So would linear algebra, such as a matrix be able to represent the formalism of Q.M. totally?

$i$ is the symmetric decoupling of $-1$, and in matrix representation, the $i$ is also a requisite for Hamiltonian symmetric decoupling of the negative identity matrix.

On the other hand, $i$ could be viewed as a phase factor. Would it be possible to replace $i$ by a $2\times 2$ matrix?

JamalS
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Hansly
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  • Do you mean imaginary unit by "i"? – Ruslan Mar 24 '14 at 18:57
  • Possible duplicates: http://physics.stackexchange.com/q/32422/2451 , http://physics.stackexchange.com/q/11396/2451 , http://physics.stackexchange.com/q/8062/2451 , and links therein. – Qmechanic Mar 24 '14 at 20:38
  • I believe this kind of thing is done in the context of complex manifolds (hence when you try to add gravity to the picture), where this kind of matrix is given by the (almost) complex structure J (http://en.wikipedia.org/wiki/Almost_complex_manifold).

    Now I guess you can eliminate i everywhere which will yield anti-hermitian matrices (in stead of hermitian), but then you get the problem that your observables (which first were eigenvalues of hermitian matrices) are no longer real.

     – Nick Mar 24 '14 at 20:59

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