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3

If you know the propagator, ie. $\langle x'|e^{itH}|x\rangle\,,$ then you could differentiate with respect to time at $t=0$ to get $\langle x'|H|x\rangle\,.$ From this we have, using the resolution of the identity, $H|x\rangle=\int_{-\infty}^\infty dx'\, |x'\rangle\langle x'|H|x\rangle\,, $ from which we have $V(x)|x\rangle=\int_{-\infty}^\infty \, ...


3

First some terminology: A non-degenerate 2-form $\omega$ is called an almost symplectic structure. A closed 2-form $\omega$ is often called a pre-symplectic structure. If the 2-form $\omega$ is both non-degenerate and closed, it becomes a symplectic structure. In the non-degenerate case, the closedness condition $$\mathrm{d}\omega~=~0\tag{C}$$ is ...


2

The Liouville equation for the $N$ particle system, describes the time evolution of the phase space N-particle probability density, which you can also neatly rewrite with the Liouville operator: $f^{N}(t)= e^{-iLt}f^{N}(0).$ Now almost always we're interested in a smaller subset of only $n$ particles, for which then we have to define a reduced phase space ...


2

I'm going to use Einstein summation notation throughout. You're almost there. You just need to use $$ {\bf B} = \nabla \times {\bf A} $$ or, equivalently, $$ \begin{eqnarray} B_k &=& \frac{\partial}{\partial r_i} A_j \epsilon_{i j k} \\ &=& \frac{1}{2}\left(\frac{\partial}{\partial r_i} A_j \epsilon_{i j k} + \frac{\partial}{\partial r_j} ...


2

You are confusing two definitions - closed system and conservation of energy. I'll clear them up for you. In classical dynamics a closed system is one where no force external to the system acts. In a closed system, the total energy, total momentum and total angular momentum must be conserved. This follows from Noether's theorem. If a has no interaction with ...


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People were excited about the ADM equations for two reasons. One was quantum gravity -- the equations give you a canonical coordinate and its momentum, so you might hope to promote these to quantum operators and be done with it. Decades later, little has come of this. The other reason is numerical relativity -- the study of the evolution of spacetime given ...


2

The propagators themselves are not indicative for the form of the Lagrangian. They only provide information regarding the nature of the field - e.g. scalar / fermion / vector boson, etc (gravity metric?). Things that allude what the Lagrangian looks like are vertices / interactions. As a simple example: if you have a theory of field $\phi$ with a 4-prong ...


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Comments to the question (v2): First of all, be aware that there exist various different definitions of canonical transformations (CT) in the literature, cf. e.g. this Phys.SE post. What OP (v2) above refers to as a CT, we will in this answer call a symplectomorphism for clarity. What we in this answer will refer to as a CT, will just be a CT of type 2. It ...


1

The Hamilton-Jacobi equation is a partial non-linear differential equation. A complete integral depends on $2n+1$ arbitrary integration constants. The complete integral defines an integral surface on which there are characteristics that are solutions to a set of first order coupled ODEs. In this way we have related a 1st order, non-linear PDE to a set of ...


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The key point here is that, any dynamical system that is not completely integrable will exhibit chaotic regimes1. In other words not all orbits will lie on an invariant torus (Liouville's torus is the topological structure of a fully integrable system), in principle a chaotic system can even have closed stable periodic orbits (typical for regular/integrable ...



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