The Beginning of the World from the Point of View of Quantum Theory.

G. LEMAÎTRE

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SIR ARTHUR EDDINGTON1 states that, philosophically, the notion of a beginning of the present order of Nature is repugnant to him. I would rather be inclined to think that the present state of quantum theory suggests a beginning of the world very different from the present order of Nature. Thermodynamical principles from the point of view of quantum theory may be stated as follows : (1) Energy of constant total amount is distributed in discrete quanta. (2) The number of distinct quanta is e…

$M$ - smooth manifold, $G$ - Lie group, $\ell:G\times M\to M$ - smooth left action. We write $\ell_x:=\ell(\cdot,x):G\to M, g\mapsto gx$.

Let $X\in\mathfrak g=\Lie(G)$. Then the \emph{fundamental vector field} associated to $X$ is defined by $\bar X_x=\dd \ell_x(X)$. We write $\sigma:\mathfrak g\to\mathfrak{X}(M)$ for the map $X\mapsto \bar X$.

*Proposition.*
For $X,Y\in\mathfrak g$, we have $[\bar X,\bar Y]=-\overline{[X,Y]}.$ In other words, $\sigma$ is a Lie algebra antihomomorphism.

*Proof.*