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--- formulas:canonical_commutation_relations [2018/04/08 13:24]
jakobadmin
+++ formulas:canonical_commutation_relations [2018/05/13 09:18] (current)
jakobadmin ↷ Page moved from equations:canonical_commutation_relations to formulas:canonical_commutation_relations
@@ Line 1: / Line 1: @@
-<WRAP lag>$[\hat{p}_i,\hat{x}_j] =  -i \delta_{ij}$</WRAP>
+<WRAP lag>$  \color{black}{[}\color{royalblue}{\hat{p}_i},\color{firebrick}{\hat{x}_j}\color{black}{]} \color{black}{ =  -i} \color{olive}{\hbar} \color{black}{\delta_{ij}}, \ \color{black}{[}\color{firebrick}{\hat{x}_i} , \color{firebrick}{\hat{x}_j}\color{black}{]} \color{black}{=} \color{darksalmon}{0}, \  \color{black}{[} \color{royalblue}{\hat{p}_i} , \color{royalblue}{\hat{p}_j} \color{black}{]} \color{black}{=} \color{darksalmon}{0}$</WRAP>
 ====== Canonical Commutation Relations ======
@@ Line 5: / Line 5: @@
 <tabbox Intuitive>
-The canonical commutation relations tell us that we can't measure the momentum and the location of a particle at the same time with arbitrary precision.
+The canonical commutation relations tell us that we can't measure the <color royalblue>momentum</color> and the <color firebrick>location</color> of a particle at the same time with <color olive>arbitrary precision</color>.
+However, can measure the <color firebrick>location</color> on different axes - e.g. the location on the x-axis and the location on the y-axis - with <color darksalmon>arbitrary precision</color>. Equally, we can measure the <color royalblue>momentum</color> in the direction of different axes with <color darksalmon>arbitrary precision</color>.
+<blockquote>The action, in the form h/2π, also occurs in the Heisenberg Uncertainty Relationship, and here it sets the limit of precision to which the conjugate coordinates (momentum and position; or energy and time) can be determined simultaneously for one particle. It also occurs in the Dirac commutator relations (discovered by the British theoretical physicist, Paul Dirac (1902–84)) which show that the order in which measurements of these conjugate coordinates are carried out does make a difference - a tiny difference of magnitude h/2π .
+<cite>The Lazy Universe by Coopersmith</cite></blockquote>
 <tabbox Concrete>
@@ Line 11: / Line 20: @@
 **Quantum Mechanics**
-\begin{equation}  \label{eq:commquantummech} [\hat{p}_i,\hat{x}_j] =  -i \delta_{ij} .\end{equation}
+\begin{equation}  \label{eq:commquantummech} [\hat{p}_i,\hat{x}_j] =  -i \hbar \delta_{ij} .\end{equation}
@@ Line 50: / Line 59: @@
 \begin{equation} \label{qftcomm}  [\Phi(x), \pi(y)]=\Phi(x) \pi(y) - \pi(y) \Phi(x) = i \delta(x-y) \end{equation} where $\delta(x-y)$ is the Dirac delta distribution and $\pi(y) = \frac{\partial \mathscr{L}}{\partial(\partial_0\Phi)}$ is the conjugate momentum.
-It tells us that the fields in [[theories:quantum_field_theory|quantum field theory]] $\Phi(x)$ can‘t be simply a function, but must be __operators__. In contrast, ordinary functions and numbers commute:
+It tells us that the fields in [[theories:quantum_field_theory:canonical|quantum field theory]] $\Phi(x)$ can‘t be simply a function, but must be __operators__. In contrast, ordinary functions and numbers commute:
 For example $f(x)=3x$ and $g(y)= 7y^2 +3$ clearly commute $$ [f(x) , g(x)]= f(x)g(x) - g(x) f(x) = 3x (7y^2 +3) -(7y^2 +3) 3x =0. $$
@@ Line 83: / Line 92: @@
 <tabbox Abstract>
+The mathematically rigorous theory of these commutation relations can be found in
-<note tip>
+  * Jørgensen, P.E.T.; Moore, R.T.: Operator commutation relations. Dordrecht: Reidel, 1984
-The motto in this section is: //the higher the level of abstraction, the better//.
+  * Schmüdgen, K.: Unbounded operator algebras and representation theory. Birkhäuser Verlag, Basel, 1990
-</note>
+However, take note that the canonical commutation relations are mathematically problematic nevertheless.
+One problem is that the operators involved are unbounded. If we want that they represent physical observables they have to be self-adjoint; but on their respective domains of self-adjointness the commutator on the left-hand side is undefined.
+Another problem is that the canonical commutation relations rely on the possibility of choosing global coordinates on $\mathbb{R}^3$. Since this is, in general not possible on arbitrary configuration spaces it is unclear how they can then be generally valid.
+The problem of finding an appropriate mathematical interpretation of the canonical commutation
+relations is the subject of [[advanced_tools:quantization|quantization theory]].
+----
+  * A good discussion can be found [[https://mathoverflow.net/questions/55988/quantum-mechanics-formalism-and-c-algebras/56003|here]].
 <tabbox Why is it interesting?>
@@ Line 96: / Line 117: @@
 In quantum field theory it tells us that our fields are operators.
-<tabbox History>
-<blockquote>“I went back to Cambridge at the beginning of October 1925, and resumed my previous style of life, intense thinking about these problems during the week and relaxing on Sunday, going for a long walk in the country alone. The main purpose of these long walks was to have a rest so that I would start refreshed on the following Monday. It was during one of the Sunday walks in October 1925, when I was thinking about this (uv vu), in spite of my intention to relax, that I thought about Poisson brackets. I remembered something which I had read up previously, and from what I could remember, there seemed to be a close similarity between a Poisson bracket of two quantities and the commutator. The idea came in a flash, I suppose, and provided of course some excitement, and then came the reaction “No, this is probably wrong”. I did not remember very well the precise formula for a Poisson bracket, and only had some vague recollections. But there were exciting possibilities there, and I thought that I might be getting to some big idea. It was really a very disturbing situation, and it became imperative for me to brush up on – 12 my knowledge of Poisson brackets. Of course, I could not do that when I was right out in the countryside. I just had to hurry home and see what I could find about Poisson brackets. I looked through my lecture notes, the notes that I had taken at various lectures, and there was no reference there anywhere to Poisson brackets. The textbooks which I had at home were all too elementary to mention them. There was nothing I could do, because it was Sunday evening then and the libraries were all closed. I just had to wait impatiently through that night without knowing whether this idea was really any good or not, but I still think that my confidence gradually grew during the course of the night. The next morning I hurried along to one of the libraries as soon as it was open, and then I looked up Poisson brackets in Whitackers Analytical Dynamics, and I found that they were just what I needed.”<cite> [[http://www.damtp.cam.ac.uk/user/tong/dynamics/four.pdf|Dirac]]</cite></blockquote>
 </tabbox>

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