B.5 Imaginary Time Operators

(B.25) |

where may be interpreted as a

where includes only the non-interacting part of . It should be noticed that is not the adjoint of as long as is real. If is interpreted as a complex variable, however, it may be analytically continued to a pure imaginary value . The resulting expression then becomes the true adjoint of and is formally identical with the original HEISENBERG picture defined in (B.12), apart from the substitution of for . For this reason (B.26) are sometimes called

The modified HEISENBERG and interaction pictures are related by (compare (B.13) and (B.14))

where the operator is defined by (compare (B.17))

Note that is not unitary, but it still satisfies the group property

(B.29) |

and the boundary condition

(B.30) |

In addition, the equation of motion of is calculated as

where

(B.32) |

It follows that the operator obeys essentially the same differential equation as the unitary operator introduced in (B.15), and one may immediately write down the solution (compare (B.24))

If is set equal to , (B.28) may be rewritten as

which relates the many particle density operator to the single-particle density operator by means of an imaginary time-evolution operator.