The Generalized Cayley — Hamilton Theorem

Abstract

With the given commutative vessel one can associate the following two sets of polynomials in the complex variables z1,...,zn:

$$\Delta _\Gamma ^{in}\left( z \right) = \det \sum\limits_{j,k} {{\Gamma ^{jk}}} \left( {{z_j}{\sigma _k} - {z_k}{\sigma _j} + \gamma _{jk}^{in}} \right)$$

((4.1))

and

$$\Delta _\Gamma ^{out}\left( z \right) = \det \sum\limits_{j,k} {{\Gamma ^{jk}}} \left( {{z_j}{\sigma _k} - {z_k}{\sigma _j} + \gamma _{jk}^{out}} \right)$$

((4.2))

where Γ^jk = —Γ^kj are arbitrary operators in the coupling space E.