samples/texts/1749790/page_10.md

Form operators

$$X_{\lambda }:=1\mathrm{/}2i[Z^2,X^{\mathrm{\prime }}]+\lambda X^{\mathrm{\prime }}X\_\{\backslash lambda\}\; :=\; 1/2\; i\; [Z^2,\; X\text{'}]\; +\; \backslash lambda\; X\text{'}$$

$$Y_{\lambda }=1\mathrm{/}2i[Z^2,Y^{\mathrm{\prime }}]+\lambda YY\_\{\backslash lambda\}\; =\; 1/2\; i\; [Z^2,\; Y\text{'}]\; +\; \backslash lambda\; Y$$

$$Z_{\lambda }=Z\mathrm{.}Z\_\{\backslash lambda\}\; =\; Z.$$

Then, as was shown in [4],

$$[X_{\lambda },Y_{\lambda }]=-Z[X\_\{\backslash lambda\},\; Y\_\{\backslash lambda\}]\; =\; -Z$$

$$[Z,X_{\lambda }]=Y_{\lambda }[Z,\; X\_\{\backslash lambda\}]\; =\; Y\_\{\backslash lambda\}$$

$$[Z,Y_{\lambda }]=-X_{\lambda }[Z,\; Y\_\{\backslash lambda\}]\; =\; -X\_\{\backslash lambda\}$$

these operators ($X_λ, Y_λ, Z$) form a representation of the Lie algebra of $SL(2, R)$.

We now want to investigate more precisely what happens as $\lambda \to \infty$. Let us set $\varepsilon = 1/\gamma$. Define

$${\phi }_e(X^{\mathrm{\prime }})=1\mathrm{/}2\epsilon i[Z^2,X^{\mathrm{\prime }}]+X^{\mathrm{\prime }}=\epsilon X_{\lambda }\backslash varphi\_e(X\text{'})\; =\; 1/2\; \backslash varepsilon\; i\; [Z^2,\; X\text{'}]\; +\; X\text{'}\; =\; \backslash varepsilon\; X\_\lambda$$

$${\phi }_e(Y^{\mathrm{\prime }})=\epsilon Y_{\lambda }(5.2)\backslash varphi\_e(Y\text{'})\; =\; \backslash varepsilon\; Y\_\lambda \; \backslash quad\; (5.2)$$

$${\phi }_e(Z^{\mathrm{\prime }})=Z\mathrm{.}\backslash varphi\_e(Z\text{'})\; =\; Z.$$

Then,

$$[{\phi }_e(X^{\mathrm{\prime }}),{\phi }_e(Y^{\mathrm{\prime }})]=-{\epsilon }^{2}Z[\backslash varphi\_e(X\text{'}),\; \backslash varphi\_e(Y\text{'})]\; =\; -\backslash varepsilon^2\; Z$$

$$[Z,{\phi }_e(X^{\mathrm{\prime }})]=\epsilon Y_{\lambda }={\phi }_e(Y^{\mathrm{\prime }})[Z,\; \backslash varphi\_e(X\text{'})]\; =\; \backslash varepsilon\; Y\_\lambda \; =\; \backslash varphi\_e(Y\text{'})$$

$$[Z,{\phi }_e(Y^{\mathrm{\prime }})]=-{\phi }_e(X^{\mathrm{\prime }})\mathrm{.}[Z,\; \backslash varphi\_e(Y\text{'})]\; =\; -\backslash varphi\_e(X\text{'}).$$

These formulas can be interpreted as follows:

Let $\mathfrak{G}$ be the vector space spanned by the elements $X', Y', Z'$. For each $\epsilon$, define a Lie algebra structure as $[\cdot, ]_\epsilon$ on $\mathfrak{G}$ by the following formulas:

$$[X^{\mathrm{\prime }},Y^{\mathrm{\prime }}{]}_{ϵ}=-{ϵ}^{2}Z(5.3)[X\text{'},\; Y\text{'}]\_\backslash epsilon\; =\; -\backslash epsilon^2\; Z\; \backslash quad\; (5.3)$$

$$[Z,X^{\mathrm{\prime }}{]}_{ϵ}=Y^{\mathrm{\prime }},[Z,Y^{\mathrm{\prime }}{]}_{ϵ}=X^{\mathrm{\prime }}\mathrm{.}[Z,\; X\text{'}]\_\backslash epsilon\; =\; Y\text{'},\; [Z,\; Y\text{'}]\_\backslash epsilon\; =\; X\text{'}.$$

Define $\varphi_\epsilon$ as above. Then, for each $\epsilon$, the above formulas define $\varphi_\epsilon$ as a linear representation of the $[\cdot, ]_\epsilon$ Lie algebra. There is no longer any singularity at $\epsilon = 0$ or $\lambda = \infty$. Thus, passing from the “Inonu-Wigner” picture with which we began (where the Lie algebra structure remains fixed, and the representation is continued and the basis of the algebra is changed simultaneously) to the “Kodaira-Spencer” picture (where the Lie algebra and representation are continued simultaneously) is an enormous aid to a proper mathematical understanding of the situation.

Thus, we can look at the Gell-Mann formula (5.1) in the following way: start off with the Lie algebra defined by (5.1), which is the Lie algebra of the group of rigid motions of the plane. Define an analytic continuation of the Lie algebra structure by the formulas (5.3). This continuation is nonrigid in the Kodaira-Spencer sense, since for $\epsilon > 0$