I am a math bachelor student studying Quantum Mechanics and I was very briefly introduced to the Heisenberg picture. (Hence many of the following may be trivial)

In particular what I know is that:

  1. the exponentiation of the Hamiltonian $H$ yields (the adjoint of) the propagator operator $U=e^{\frac{itH}{h}}$
  2. To any operator $A$ we can associate its evolved $A_t=U^+AU$ so that given an initial state $\psi_0$ the average of the expected value of A at time t is $<A>_t=(\psi_0,A_t\psi_0)$ and it holds the Heisenberg equation $\frac{d}{dt}A_t=\frac{1}{ih}[A_t,H]+\partial_t A_t$.
  3. The exponantiation of operators other than the Hamiltonian, such as angular momemtum L, momentum p yields propagations in'dimensions' other than time, in the sense that if $T=e^{\frac{itL}{h}}$ then $T\psi(\rho,\phi,\theta)=\psi(\rho,\phi,\theta+t)$ (where L has been expressed in z-axis spherical coordinates). Analogously $ T=e^{\frac{itp}{h}}$ then $T\psi(x)=\psi(x+t)$.

Now, in Hamiltonian mechanics we define Hamiltonian Flows as the group action $\phi_t$ associated to the solution $(x(t),p(t))$ of the 'generalized' Hamilton equations $$\frac{dx_k}{dt}=\partial_{p_k}F$$ $$\frac{dp_k}{dt}=-\partial_{x_k}F$$ so that $\phi_t(x(0),p(0))=(x(t),p(t))$.

$F$ may be the Hamiltonian, in which case we have time evolution of the system, but it may also be an affine function of position, so that we have (negative) momentum evolution (in the sense that $\phi_t(x,p)=(x,p-t)$) or of momentum and we have spatial evolution (in the sense that $\phi_t(x,p)=(x+t,p)$). If finally F equals L we have rotations.

Now, I cannot fully and deeply explain, if it is possible, the parrallelism I notice. In particular:

  1. Can we identify in some sense the exponentiation of operators with the group action (of $\mathbb{S}^1$?) thus connecting the two scenarios?
  2. What is the precise notion of 'dimensions' other than time, if any, in which the system evolves?
  3. In the Schroedinger picture, Ehrenfest theorem allows to find an analog to the Hamilton equations, is this possible also in the Heisenber picture (it may be through the Heisenberg Equation , bu I cannot make this precise), in particular, also in the cases $F\neq H$?
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    $\begingroup$ To your question number one, I'm not sure, but it sounds like you are knocking on the same door that led Feynman to develop the path integral fomalism of quantum mechanics. So the answer, if I understand your question, is a qualified "yes". $\endgroup$ – bob.sacamento Jun 12 at 14:29
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    $\begingroup$ I'm pretty sure you are trying to access advective flows of Abelian groups. Motion is a canonical transformation, so, in classical mechanics, it is a symplectomorphic flow, and in quantum mechanics, it is a quantum flow in phase space. I'm not sure what precise definition of "dimensions" ~ dynamical variables/observables is required. $\endgroup$ – Cosmas Zachos Jun 12 at 15:28
  • $\begingroup$ @bob.sacamento thanks! Could you tell me a bit more about the connection to the little I know in Feynman picture we have an analogous of Action which is minimised by the most probable path. $\endgroup$ – Francesco Bilotta Jun 12 at 16:05
  • $\begingroup$ @CosmasZachos thanks! I will check them out, so basically there exists a universal formalism comprising the two cases? $\endgroup$ – Francesco Bilotta Jun 12 at 16:08
  • $\begingroup$ In phase space you can "segue" from quantum mechanics to classical mechanics, or vice versa... The booklet I linked to outlines the picture... $\endgroup$ – Cosmas Zachos Jun 12 at 16:18

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