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Grassmann-odd variables provide a classical description of Grassmann-odd quantum operators in the same way that Grassmann-even variables provide a classical description of Grassmann-even quantum operators. The classical super-Poisson bracket $$\{\psi^i,\psi^j\}_{PB} ~=~ -i (T^{-1})^{ij} \tag{A}$$ is related to the super-commutator$^1$ $$\hat{\psi}^i\hat{\... 2 The quick answer is: no. The Hamiltonian operator is a unitary operator that maps state vectors to other state vectors in a given Hilbert Space, regardless of time. Lubos's answer in this thread discusses this distinction very clearly: Why \displaystyle i\hbar\frac{\partial}{\partial t} can not be considered as the Hamiltonian operator? Another point ... 0 The evolution of systems in the Hamiltonian formalism is called a flow, not because it can be described by a mapping, but because it is described by a particular mapping: one whose evolution in (q,p)-space resembles fluid flow. This resemblance gives rise to Liouville's theorem, where the Hamiltonian flow, like certain fluid flows, is shown to be ... 2 There is a well-known isomorphism between the linear space {\mathcal M}_{m, n} of m\times n matrices and typical (vectorial) linear spaces {\mathcal L}_{m\times n} of dimension \text{dim}({\mathcal L}_{m\times n}) = m\times n. Everything that is valid in {\mathcal L}_{m\times n} has an equivalent in {\mathcal M}_{m, n} and conversely. For this ... 1 In general, \frac{\partial L}{\partial \dot{q}} is the canonical (or generalized or conjugate*) momentum, and m\dot x, for x the actual position, is kinetic momentum. Likewise, the cross product of the former with the generalized coordinate vector q might be called "canonical angular momentum", and the cross product of the latter "kinetic angular ... 1 This is more or less an exercise in chasing definitions. The adiabatic invariant I is defined as$$ I\equiv \oint p \frac{\mathrm{d}q}{2\pi}\tag{49.7} where the integral is taken over the path for given $E$ and $\lambda$. The external parameter $\lambda(t)$ is a slowly varying function of time $t$ in $\S49$, but is assumed to be a constant in $\S50$. ...