Transformations of observables generated by an observable

In Classical Mechanics, any function \(f \in \mathcal{C}^{\infty}(T^*Q)\) is called an observable. They constitute a commutative algebra that is, in fact, a Poisson algebra due to the existence of the Poisson bracket \(\{-,-\}\). If you fix any \(f\in \mathcal{C}^{\infty}(T^*Q)\) then you get a 1-parameter local tranformation group for \( \mathcal{C}^{\infty}(T^*Q)\) in the following way.

First, you have the vector field \(X_f\) given by
\[
X_f=\{-,f\}
\]

Then, the fundamental theorem of ODEs let you assure that there exists the local flow
\[
\phi_s:T^*Q \mapsto T^*Q
\]
such that
  1. \(\phi_0=id\)
  2. \(\frac{d}{ds} \phi_s (m)|_{s=0}=X_f \)
  3. \(\phi_s \circ \phi_t=\phi_{s+t}\)

Now, we can define a family of transformations of observables
\[
F^{\phi_s}: \mathcal{C}^{\infty}(T^*Q) \longmapsto \mathcal{C}^{\infty}(T^*Q)
\]
by means of
\[
g \longmapsto F^{\phi_s}(g)=g \circ \phi_s
\]


I interpret observables as little machines that you can place where you want (in a broad sense of the meaning of "where", because this includes time and any other variable properties like temperature or energy, but we are treating them as "space") and then you can read a number in their screen. What this is telling to us is that given any of those machines, say \(f\), we can construct a modifier of machines that depends on one parameter and that yields a group. Observe that the definition of the modifier use the symplectic form, or the Poisson bracket, which is a bit annoying for me. I think that this would be removed if we go to the \(C^*-\)algebra formulation...


On the other hand, we have that

\[
\frac{d}{ds} F^{\phi_s}(g)=\frac{d}{ds}(g\circ \phi_s)=dg_{\phi_{s}(-)} \cdot X_f=X_f(g\circ \phi_s)=\{F^{\phi_s}(g),f\}
\]
and
\[
F^{\phi_0}(g)=g.
\]

And also, \(F^{\phi_s}\) respect the Poisson algebra structure.



Reciprocally, if we start with a function \(f\) and an associated family of transformations of \(\mathcal{C}^{\infty}(T^*Q)\) to \(\mathcal{C}^{\infty}(T^*Q)\) that preserve the structure of Poisson algebra and such that
  • \(\frac{d}{d t} F_{t}^{f}(g)=\left\{F_{t}^{f}(g),f \right\}\)
  • \(F_{0}^{f}(g)=g\)

we can define a flow \(\phi_t\) in \(T^*Q\) such that
\[
F_t^f(g)=F^{\phi_t}(g)
\]

We proceed by taking the vector field \(\{-,f\}\) and the associated flow \(\phi_t\), and using the two required properties. Since
\[
\frac{d}{ds} F^{\phi_s}(g)=\frac{d}{d s} F_{s}^{f}(g)
\]
and
\[
F^{\phi_0}(g)=F_{0}^{f}(g)
\]
we can show that \(F_s^f(g)=F^{\phi_s}(g)\) by fixing a point \(P\in T^*Q\).


We indeed have a duality, between transformations in \(T^* Q\) and transformation in \(\mathcal{C}^{\infty}(T^* Q)\). Why do we pass from the flow in the phase space \(T^*Q\) to the flow of observables? Because this way, I guess, we can formulate everything in terms of algebras, and we get a deeper insight in the connection with QM.

Comments

Post a Comment

Popular posts from this blog

About the definition of angle, from Geometric Algebra

Image
In GA, bivectors represent  2-dimensional directions. Any simple bivector can be written as \[ \theta e_1e_2 \] with \(\theta \in \mathbb{R}\) and \(\{e_i\}\) unitary and orthogonal vectors. The unitary vectors specify the 2- direction itself, and \(\theta\) is the size (analogous to the length of a vector, that is a 1-dimensional direction). Given two vectors \(a\) and \(b\) (that we take to be unitary, wlog) we say that the simple bivector \(\theta e_1e_2\) is the angle formed by them if \[ ab=e^{\theta e_1 e_2} \] where we take as a definition \[ e^A=1+A+A^2 /2+\cdots=\lim_N (1+\frac{A}{N})^N \] Real analysis let us assure that this expression is well defined when \(A\) is a 2-blade. In fact, since \(e_1e_2e_1e_2=-1\), we can write \[ e^{\theta e_1 e_2}=cos(\theta)+e_1 e_2sin(\theta) \] where \(cos\) and \(sin\) are defined by their power series. Oserve that \(b=a e^{\theta e_1 e_2}\) and then \[ b=a(cos(\theta)+e_1 e_2sin(\theta))=a\cdot cos(\theta)+ ae_1e_2 \cdot sin(\theta) \]
Read more

Spherical, hyperbolic and parabolic geometry

Image
Let's consider the space \(\mathbb{M}^4\), that is, the manifold \(\mathbb{R}^4\) with coordinates \((x,y,z,t)\) but with the pseudo-Riemannian metric given in each tangent space by \[ ds^2=dx^2+dy^2+dz^2-dt^2. \] This metric is left invariant by the group \(O(3,1)\) when acting on \(\mathbb{M}^4\). It also leaves invariant the cone \[ t^2-x^2-y^2-z^2=0 \] so is logical that we pay attention to it. Now, following TRTR from Penrose page 423, we can consider the family of hyperplanes inside \(\mathbb{M}^4\) given by \[ z+t+\lambda(t-z)=2 \] intersecting our cone in different 2-dimensional submanifolds \(P_{\lambda}\) of \(\mathbb{M}^4\). Here we have a picture from that book, where we consider only \((x,z,t)\) for purposes of drawability. If we analyse the case \(\lambda=0\), we observe that we obtain the submanifold \(P_0\), \(E\) in the picture with a shape of a parabola, given by the embedding \begin{align*}   \phi \colon \mathbb{R}^2 &\longrightarrow \mathbb{M}^4\\   (u_1,u_2
Read more

General covariance and contravariance (new version)

Image
Suppose that a group \(G\) acts transitively on a space \(X\). A transformation \(T\in G\) can be thought of indistinctly as either "moving" the objects in \(X\) or as a change of "point of view" in the following way: Suppose we have a comfortable, mathematically speaking, space with a distinguished point, \((S,s_0)\). We are going to think of \((\mathbb Z,0)\), for easy. Suppose also a bijection \(b_1:S\to X\) (in the line of reasoning of the notes homogeneous space#Intuitive approach and basis and change of basis ). It is better to think in terms of \(b_1^{-1}\). That is, for every \(x\in X\) we have a kind of "coordinates" \(s\) for \(x\), given by \(s=b_1^{-1}(x)\in S\). Think, for example, of a vector space \(V\) and the isomorphism \(b_1:\mathbb R^2 \to V\) (fixing a basis in \(V\)). We can think of this as our initial point of view , and we can imagine as if we were located at \(x_0=b_1(0)\in X\) with some devices to take measurements that let u
Read more