- #1
thatboi
- 132
- 18
Hi all,
I am considering the following paper: https://arxiv.org/abs/quant-ph/0302164. Specifically the discussion of the quantum dynamical semigroup equations in section 6.6, where just above equation (6.66), there was the following condition on the jump operators ##A_{i}##: ##\sum_{i\in K_{0}}A_{i}^{\dagger}A_{i} \leq 1## for all ##K_{0} \subset K##. I then looked at equation (8.12), where the author identified ##A_{i}## with ##N(x) = \int d^{3}y \hspace{0.25cm} g(y-x)a^{\dagger}(y)a(y) ## where ##g(x) = \left(\frac{\alpha}{2\pi}\right)e^{-\frac{\alpha}{2}x^{2}}## and ##a^{\dagger}## is the creation operator of a particle (either fermionic or bosonic), and identified the index ##i## with the position in space ##x##. My confusion is why equation (8.12) is still quantum-dynamical-semigroup type when ##\sum_{i\in K_{0}}A_{i}^{\dagger}A_{i} \leq 1## no longer seems to be satisfied. After all, the number operator itself is not a bounded operator right?
I am considering the following paper: https://arxiv.org/abs/quant-ph/0302164. Specifically the discussion of the quantum dynamical semigroup equations in section 6.6, where just above equation (6.66), there was the following condition on the jump operators ##A_{i}##: ##\sum_{i\in K_{0}}A_{i}^{\dagger}A_{i} \leq 1## for all ##K_{0} \subset K##. I then looked at equation (8.12), where the author identified ##A_{i}## with ##N(x) = \int d^{3}y \hspace{0.25cm} g(y-x)a^{\dagger}(y)a(y) ## where ##g(x) = \left(\frac{\alpha}{2\pi}\right)e^{-\frac{\alpha}{2}x^{2}}## and ##a^{\dagger}## is the creation operator of a particle (either fermionic or bosonic), and identified the index ##i## with the position in space ##x##. My confusion is why equation (8.12) is still quantum-dynamical-semigroup type when ##\sum_{i\in K_{0}}A_{i}^{\dagger}A_{i} \leq 1## no longer seems to be satisfied. After all, the number operator itself is not a bounded operator right?