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Consequently the matrix S is an involution 6 and given by:
S =
I 0
0 −I
. (2.33)
With I the L-dimensional identity matrix. This leads to the first condition
for the transformation 7 U:
[ B, B † ] = U[ P , P † ]U † = USU † ! = S . (2.34)
At this point it is worth mentioning, that the commutator of two vectors
of creation and annihilation operators has to be defined first, what is done
in equation E.2. Furthermore U has to be a real transformation for this
condition.
Using S 2 = I, this relation can be rewritten as:
USU † = S
USU † S = I
U −1 USU † SU = U −1 U
U † SU = S (2.35)
The second condition is, that the Hamiltonian has to be diagonal:
6 S ◦ S = I or S = S −1
B † M B = P † U † MU P ! = P † D P
=⇒ U † MU = D
7 See Appendix E for a derivation for this formula.
21
(2.36)