- #1
andresB
- 626
- 374
Consider a set of ##n## position operators and ##n## momentum operator such that
$$\left[q_{i},p_{j}\right]=i\delta_{ij}.$$
Lets now perform a linear symplectic transformation
$$q'_{i} =A_{ij}q_{j}+B_{ij}p_{j},$$
$$p'_{i} =C_{ij}q_{j}+D_{ij}p_{j}.$$
such that the canonical commutation relations are maintained
$$\left[q'_{i},p'_{j}\right]=i\delta_{ij}.$$
Any such symplectic transformation should be unitarily implemented due to the Stone-von Neumann theorem (right?)
$$U^{-1}q_{i}U =q'_{i},$$
$$U^{-1}p_{i}U =p'_{i}.$$
The question is: Assuming the coefficients A,B,C and D are given , is there a systematic way to calculate the generator ##G## of the unitary transformation ##U=e^{-iG}##?
$$\left[q_{i},p_{j}\right]=i\delta_{ij}.$$
Lets now perform a linear symplectic transformation
$$q'_{i} =A_{ij}q_{j}+B_{ij}p_{j},$$
$$p'_{i} =C_{ij}q_{j}+D_{ij}p_{j}.$$
such that the canonical commutation relations are maintained
$$\left[q'_{i},p'_{j}\right]=i\delta_{ij}.$$
Any such symplectic transformation should be unitarily implemented due to the Stone-von Neumann theorem (right?)
$$U^{-1}q_{i}U =q'_{i},$$
$$U^{-1}p_{i}U =p'_{i}.$$
The question is: Assuming the coefficients A,B,C and D are given , is there a systematic way to calculate the generator ##G## of the unitary transformation ##U=e^{-iG}##?
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