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In the book Condensed Matter Field Theory of Altland and Simons, page 63 eq 2.30 in the second edition, they are doing the following transformation for the Hamiltonian:

$$\hat{H} \rightarrow \hat{H'} = e^{-t\hat{O}}\hat{H}e^{t\hat{O}}.\tag{2.30}$$

They are claiming that this transformation is canonical therefore doesn't change the physics of the system. $\hat{O}$ is a general operator.

My question is why it this true? Did I missed something from their discussion?

Qmechanic
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Mr. J
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1 Answers1

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  1. A quantum canonical transformation is an adjoint group action $\hat{F}\mapsto {\rm Ad}(\hat{U})\hat{F}=\hat{U}\hat{F}\hat{U}^{-1}$ with a unitary operator $\hat{U}$, cf. e.g. this Phys.SE post.

  2. Comparing the unitary operator $\hat{U}$ with eq. (2.30) in A&S, the operator $t\hat{O}$ should apparently be anti-selfadjoint.

Qmechanic
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