We provide a necessary and sufficient condition on the coefficient matrices A,C for the diagonalizability of quadratic fields of the form, \[X =\sum_{i,j=1}^n (A_{i,j}a^+_i a^+_j+\overline{A}_{i,j}a_i a_j+C_{i,j}a^+_i a_j)\] where, the a’s and a†’s are the generators of the multi-dimensional Schr\¨odinger Lie algebra. We also consider the Fock vacuum characteristic function $\langle \Phi, e^{isX}\Phi\rangle$ of X and study its factorizability/decomposability and how it relates to the commutativity of the simple quadratic components of X.
On the Diagonalizability and Factorizability of Quadratic Boson Fields
Luigi Accardi;Yungang Lu;
2022-01-01
Abstract
We provide a necessary and sufficient condition on the coefficient matrices A,C for the diagonalizability of quadratic fields of the form, \[X =\sum_{i,j=1}^n (A_{i,j}a^+_i a^+_j+\overline{A}_{i,j}a_i a_j+C_{i,j}a^+_i a_j)\] where, the a’s and a†’s are the generators of the multi-dimensional Schr\¨odinger Lie algebra. We also consider the Fock vacuum characteristic function $\langle \Phi, e^{isX}\Phi\rangle$ of X and study its factorizability/decomposability and how it relates to the commutativity of the simple quadratic components of X.File in questo prodotto:
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