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The idea of generators of transformations is quite useful and quite simple to formulate from a pure mathematical point of view.

For example, in Hamiltonian Mechanics, we have a configuration manifold $Q$ and the phase space $M=T^\ast Q$ which is just the cotangent bundle. In that case, we know that the Poison Brackets $\{\cdot,\cdot\}$ allows us to get a vector field from a function in the following way: given $f\in C^\infty(M)$ we define $X_f$ by

$$X_f g=\{f,g\}.$$

By the properties of $\{\cdot,\cdot\}$ it turns out that $X_f$ is a derivation and thus defines a vector field.

But as always, being a vector field $X_f$ defines one system of infinitesimal transformations which, when integrated, gives rise to a full transformation on the phase space.

We then say that $f$ is the generator of the corresponding transformation.

Following this procedure we can easily say that momentum is the generator of translations, angular momentum is the generator of rotations and the Hamiltonian is the generator of time evolution.

This is the Hamiltonian Mechanics picture, but the same can be formulated in Lagrangian Mechanics by the use of Noether's theorem.

Furthremore, all of this can be also done in Quantum Mechanics, dealing now with observables. In truth, Schrödinger's equation just says that the Hamiltonian should be the generator of time evolution as in Hamiltonian Mechanics.

All of that is nice, but IMHO, this all is based on too much math. This is fine, but I think that the grasping the physical understanding behind this is really worthwhile.

So, based on this discussion, when we have a transformation like translations, rotations, time evolution and so forth, what the corresponding generators are from a physical and conceptual standpoint?

If we want to explain what it means to say that momentum is the generator of translations, how can we do so, from a purely conceptual and physical standpoint?

I do believe there is a nice way to understand this idea of generator from a physical standpoint. Something along the lines that: the particle can only undergo a translation because it has momentum. That is: the position can only change because the particle is moving and momentum measures the "quantity of motion". But this is yet not very good.

I just want to grasp this physical understanding in a more general way for any generator of any transformation.

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  • $\begingroup$ What more "precise" way are you looking for than Noether's theorem and its Hamiltonian inverse? You either get precision with math or you get handwaving without math. This question sounds as if it wants the precision of the mathematical formulation without the mathematical formulation, when the gain of precision is the sole reason we use so much math in the first place. $\endgroup$ – ACuriousMind Aug 20 '16 at 17:13
  • $\begingroup$ Perhaps the wording came out wrong and perhaps this "precise" I said was misleading. What I wanted to know here is how I could explain to someone what physically it means for a quantity to be a generator without explaining all of the rigorous formulation. What I wanted to is to grasp a more conceptual picture of a generator and the relation it has with the transformation it generates. $\endgroup$ – user1620696 Aug 20 '16 at 17:19
  • $\begingroup$ Imo this is more of a mathSE question. But everybody understands that we want use the simplest equations/techniques we can. So the generator is a simple matrix compared to the full say, (still simple) lorentz boost matrix. And everybody follows the idea of small sums adding up to a larger result. I can't see how you can explain it other than something along those lines without assuming a calculus, understanding of exponentials and series expansion background. Apologies if I have misunderstood your point. $\endgroup$ – user108787 Aug 20 '16 at 18:16

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