samples/texts/1749790/page_18.md

that of the Galilean group. $\frac{X}{c}$ is analytic in $\frac{1}{c}$, and converges to the operator $\frac{d}{dv}$, which is just the operator of constant acceleration with respect to the Galilean group. However, the operator $E$ is not analytic in $\frac{1}{c}$. The physical interpretation suggests a way to proceed. Let us "re-normalize" $E$ by subtracting off a constant that "becomes infinite" at $c = \infty$. We interpret this in the following group-theoretic way: enlarge the Lie algebra defined by (7.5) by adding an element $p$ that commutes with all the other operators, i.e., the enlarged algebra is the direct sum with a one-dimensional Abelian subalgebra. Define:

$$E^{\mathrm{\prime }}=E-m{c}^{2}l\mathrm{.}E\text{'}\; =\; E\; -\; mc^2\; l.$$

$E'$ is now analytic in $\frac{1}{c}$ at $c = \infty$. In terms of the basis $(X/c, E', p, l)$ this algebra becomes:

$$[\frac{X}{c},E^{\mathrm{\prime }}]=p\backslash left[\; \backslash frac\{X\}\{c\},\; E\text{'}\; \backslash right]\; =\; p$$

$$[\frac{X}{c},p]=\frac{1}{c}2E=\frac{1}{c^2}(E^{\mathrm{\prime }}+m{c}^2)=-\frac{E^{\mathrm{\prime }}}{c^2}+m\mathrm{.}(7.6)\backslash left[\; \backslash frac\{X\}\{c\},\; p\; \backslash right]\; =\; \backslash frac\{1\}\{c\}\; 2E\; =\; \backslash frac\{1\}\{c^2\}\; (E\text{'}\; +\; mc^2)\; =\; -\backslash frac\{E\text{'}\}\{c^2\}\; +\; m.\; \backslash quad\; (7.6)$$

The limiting algebra as $c \to \infty$ now exists, and is by its construction, the representation depending on $c$ is analytic in $\frac{1}{c}$ at $c = 0$. However, the limiting algebra at $c = 0$ is not that of the Galilean group, but a central extension of it. This explains why the "interesting" physical representations of the Galilean group are not true representations but representations only up to a factor.

8. The Gell-Mann formula for contraction of the Poincaré to the Galilean group

Now, let us ask whether there is a formula representing the Lie algebra of the Poincaré group as functions of the generators of the Lie algebra of the central extension of the Galilean group constructed in the last section. (For simplicity, we continue to work with the groups corresponding to one-space dimension.) We suppose then that $X''$, $E''$, $p''$ and $l$ are the generators of a Lie algebra, with the structure relations.

$$[X^{\mathrm{\prime }\mathrm{\prime }},E^{\mathrm{\prime }\mathrm{\prime }}]=p^{\mathrm{\prime }\mathrm{\prime }},[X^{\mathrm{\prime }\mathrm{\prime }},p^{\mathrm{\prime }\mathrm{\prime }}]=1[X\text{'}\text{'},\; E\text{'}\text{'}]\; =\; p\text{'}\text{'},\; [X\text{'}\text{'},\; p\text{'}\text{'}]\; =\; 1$$

$$[1,X^{\mathrm{\prime }\mathrm{\prime }}]=0=[E^{\mathrm{\prime }\mathrm{\prime }},p^{\mathrm{\prime }\mathrm{\prime }}]=[1,E^{\mathrm{\prime }\mathrm{\prime }}]=[1,p^{\mathrm{\prime }\mathrm{\prime }}]\mathrm{.}(8.1)[1,\; X\text{'}\text{'}]\; =\; 0\; =\; [E\text{'}\text{'},\; p\text{'}\text{'}]\; =\; [1,\; E\text{'}\text{'}]\; =\; [1,\; p\text{'}\text{'}]\; .\; \backslash quad\; (8.1)$$

(For simplicity, we also suppose $m=1$). Define

$$X=c{X}^{\mathrm{\prime }\mathrm{\prime }}X\; =\; cX\text{'}\text{'}$$

$$E=\frac{c^2}{\sqrt{c^2-2E^{\mathrm{\prime }\mathrm{\prime }}}}(8.2)E\; =\; \backslash frac\{c^2\}\{\backslash sqrt\{c^2\; -\; 2E\text{'}\text{'}\}\}\; \backslash quad\; (8.2)$$

$$p=\frac{c{p}^{\mathrm{\prime }\mathrm{\prime }}}{\sqrt{c^2-2E^{\mathrm{\prime }\mathrm{\prime }}}}\mathrm{.}p\; =\; \backslash frac\{cp\text{'}\text{'}\}\{\backslash sqrt\{c^2\; -\; 2E\text{'}\text{'}\}\}\; .$$