All primitive mathematical procedures are extended to be generic over
symbolic arguments. When given symbolic arguments, these procedures
construct a symbolic representation of the required answer. There are
primitive literal numbers. We can make a literal number that is
represented as an expression by the symbol “a” as follows:
(literal-number 'a) ==> (*number* (expression a))
The literal number is an object that has the type of a number, but its
representation as an expression is the symbol “a”.
(type (literal-number 'a)) ==> *number* (expression (literal-number 'a)) ==> a
— SCMUTILS Reference Manual
.
.
2019.08.17 Saturday ACHK
and 
, as described above, are groups.
(called the group law of G) that combines any two elements a and b to form another element, denoted 
or 
. To qualify as a group, the set and operation, 
, must satisfy four requirements known as the group axioms:
.
holds. Such an element is unique, and thus one speaks of the identity element.
(or 
, if the operation is denoted “+”), such that 
, where e is the identity element.
be a path-independent function and 
be a path-dependent function; then![\displaystyle{\delta_\eta h[q] = \left( DF \circ g[q] \right) \delta_\eta g[q]~~~~~\text{with}~~~~~h[q] = F \circ g[q].~~~~~(1.26)}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%7B%5Cdelta_%5Ceta+h%5Bq%5D+%3D+%5Cleft%28+DF+%5Ccirc+g%5Bq%5D+%5Cright%29+%5Cdelta_%5Ceta+g%5Bq%5D%7E%7E%7E%7E%7E%5Ctext%7Bwith%7D%7E%7E%7E%7E%7Eh%5Bq%5D+%3D+F+%5Ccirc+g%5Bq%5D.%7E%7E%7E%7E%7E%281.26%29%7D&bg=ffffff&fg=333333&s=0 "\displaystyle{\delta_\eta h[q] = \left( DF \circ g[q] \right) \delta_\eta g[q]~~~~~\text{with}~~~~~h[q] = F \circ g[q].~~~~~(1.26)}")
![\displaystyle{\delta_\eta F \circ g[q] = \left( DF \circ g[q] \right) \delta_\eta g[q]}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%7B%5Cdelta_%5Ceta+F+%5Ccirc+g%5Bq%5D+%3D+%5Cleft%28+DF+%5Ccirc+g%5Bq%5D+%5Cright%29+%5Cdelta_%5Ceta+g%5Bq%5D%7D&bg=ffffff&fg=333333&s=0 "\displaystyle{\delta_\eta F \circ g[q] = \left( DF \circ g[q] \right) \delta_\eta g[q]}")
 - F \circ g[q](t)}{\Delta t} \right) \lim_{\epsilon \to 0} \left( \frac{g[q+\epsilon \eta]-g[q]}{\epsilon} \right)}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%7BRHS+%3D+%5Clim_%7B%5CDelta+t+%5Cto+0%7D+%5Cleft%28+%5Cfrac%7BF+%5Ccirc+g%5Bq%5D%28t%2B%5CDelta+t%29+-+F+%5Ccirc+g%5Bq%5D%28t%29%7D%7B%5CDelta+t%7D+%5Cright%29+%5Clim_%7B%5Cepsilon+%5Cto+0%7D+%5Cleft%28+%5Cfrac%7Bg%5Bq%2B%5Cepsilon+%5Ceta%5D-g%5Bq%5D%7D%7B%5Cepsilon%7D+%5Cright%29%7D&bg=ffffff&fg=333333&s=0 "\displaystyle{RHS = \lim_{\Delta t \to 0} \left( \frac{F \circ g[q](t+\Delta t) - F \circ g[q](t)}{\Delta t} \right) \lim_{\epsilon \to 0} \left( \frac{g[q+\epsilon \eta]-g[q]}{\epsilon} \right)}")
![\displaystyle{ \begin{aligned} LHS &= \delta_\eta F \circ g[q] \\ &= \lim_{\epsilon \to 0} \left( \frac{F \circ g[q+\epsilon \eta]-F \circ g[q]}{\epsilon} \right) \\ &= \lim_{\epsilon \to 0} \left( \frac{F \left[ g[q+\epsilon \eta] \right] - F \left[ g[q] \right]}{\epsilon} \right) \\ \end{aligned}}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%7B+%5Cbegin%7Baligned%7D+LHS+%26%3D+%5Cdelta_%5Ceta+F+%5Ccirc+g%5Bq%5D+%5C%5C+++%26%3D++%5Clim_%7B%5Cepsilon+%5Cto+0%7D+%5Cleft%28+%5Cfrac%7BF+%5Ccirc+g%5Bq%2B%5Cepsilon+%5Ceta%5D-F+%5Ccirc+g%5Bq%5D%7D%7B%5Cepsilon%7D+%5Cright%29+%5C%5C++++%26%3D++%5Clim_%7B%5Cepsilon+%5Cto+0%7D+%5Cleft%28+%5Cfrac%7BF+%5Cleft%5B+g%5Bq%2B%5Cepsilon+%5Ceta%5D+%5Cright%5D+-+F+%5Cleft%5B+g%5Bq%5D+%5Cright%5D%7D%7B%5Cepsilon%7D+%5Cright%29+%5C%5C+++%5Cend%7Baligned%7D%7D&bg=ffffff&fg=333333&s=0 "\displaystyle{ \begin{aligned} LHS &= \delta_\eta F \circ g[q] \\ &= \lim_{\epsilon \to 0} \left( \frac{F \circ g[q+\epsilon \eta]-F \circ g[q]}{\epsilon} \right) \\ &= \lim_{\epsilon \to 0} \left( \frac{F \left[ g[q+\epsilon \eta] \right] - F \left[ g[q] \right]}{\epsilon} \right) \\ \end{aligned}}")
![\displaystyle{ \begin{aligned} LHS &= \lim_{\epsilon \to 0} \left( \frac{F \left( g[q+\epsilon \eta ] \right) - F \left( g[q] \right)}{\epsilon} \right) \\ \end{aligned}}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%7B+%5Cbegin%7Baligned%7D+LHS+++%26%3D+%5Clim_%7B%5Cepsilon+%5Cto+0%7D+%5Cleft%28++%5Cfrac%7BF+%5Cleft%28+g%5Bq%2B%5Cepsilon+%5Ceta+%5D+%5Cright%29+-+F+%5Cleft%28+g%5Bq%5D+%5Cright%29%7D%7B%5Cepsilon%7D+%5Cright%29+%5C%5C+++%5Cend%7Baligned%7D%7D&bg=ffffff&fg=333333&s=0 "\displaystyle{ \begin{aligned} LHS &= \lim_{\epsilon \to 0} \left( \frac{F \left( g[q+\epsilon \eta ] \right) - F \left( g[q] \right)}{\epsilon} \right) \\ \end{aligned}}")
![\displaystyle{ g[q+\epsilon \eta] = g + \Delta g}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%7B+g%5Bq%2B%5Cepsilon+%5Ceta%5D+%3D+g+%2B+%5CDelta+g%7D&bg=ffffff&fg=333333&s=0 "\displaystyle{ g[q+\epsilon \eta] = g + \Delta g}")
.![\displaystyle{ \begin{aligned} LHS &= \lim_{\epsilon \to 0} \left( \frac{F \left( g[q] + \Delta g[q]] \right) - F \left( g[q] \right)}{\epsilon} \right) \\ &= \lim_{\epsilon \to 0} \left( \frac{F \left( g[q] + \Delta g[q]] \right) - F \left( g[q] \right)}{\Delta g[q]}\frac{\Delta g[q]}{\epsilon} \right) \\ \end{aligned}}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%7B+%5Cbegin%7Baligned%7D+LHS+++%26%3D+%5Clim_%7B%5Cepsilon+%5Cto+0%7D+%5Cleft%28++%5Cfrac%7BF+%5Cleft%28+g%5Bq%5D+%2B+%5CDelta+g%5Bq%5D%5D+%5Cright%29+-+F+%5Cleft%28+g%5Bq%5D+%5Cright%29%7D%7B%5Cepsilon%7D+%5Cright%29+%5C%5C+++%26%3D+%5Clim_%7B%5Cepsilon+%5Cto+0%7D+%5Cleft%28++%5Cfrac%7BF+%5Cleft%28+g%5Bq%5D+%2B+%5CDelta+g%5Bq%5D%5D+%5Cright%29+-+F+%5Cleft%28+g%5Bq%5D+%5Cright%29%7D%7B%5CDelta+g%5Bq%5D%7D%5Cfrac%7B%5CDelta+g%5Bq%5D%7D%7B%5Cepsilon%7D+%5Cright%29+%5C%5C+++%5Cend%7Baligned%7D%7D&bg=ffffff&fg=333333&s=0 "\displaystyle{ \begin{aligned} LHS &= \lim_{\epsilon \to 0} \left( \frac{F \left( g[q] + \Delta g[q]] \right) - F \left( g[q] \right)}{\epsilon} \right) \\ &= \lim_{\epsilon \to 0} \left( \frac{F \left( g[q] + \Delta g[q]] \right) - F \left( g[q] \right)}{\Delta g[q]}\frac{\Delta g[q]}{\epsilon} \right) \\ \end{aligned}}")
, 
.![\displaystyle{ \begin{aligned} LHS &= \lim_{\substack{\epsilon \to 0 \\ \Delta g \to 0}} \left( \frac{F \left( g[q] + \Delta g[q]] \right) - F \left( g[q] \right)}{\Delta g[q]}\frac{\Delta g[q]}{\epsilon} \right) \\ &= \lim_{\Delta g \to 0} \left( \frac{F \left( g[q] + \Delta g[q]] \right) - F \left( g[q] \right)}{\Delta g[q]} \lim_{\epsilon \to 0} \frac{g[q + \epsilon \eta] - g[q]}{\epsilon} \right) \\ &= DF \left( g[q] \right) \delta_\eta g[q] \\ &= RHS \\ \end{aligned}}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%7B+%5Cbegin%7Baligned%7D+LHS+++%26%3D+%5Clim_%7B%5Csubstack%7B%5Cepsilon+%5Cto+0+%5C%5C+%5CDelta+g+%5Cto+0%7D%7D+%5Cleft%28++%5Cfrac%7BF+%5Cleft%28+g%5Bq%5D+%2B+%5CDelta+g%5Bq%5D%5D+%5Cright%29+-+F+%5Cleft%28+g%5Bq%5D+%5Cright%29%7D%7B%5CDelta+g%5Bq%5D%7D%5Cfrac%7B%5CDelta+g%5Bq%5D%7D%7B%5Cepsilon%7D+%5Cright%29+%5C%5C+++%26%3D+%5Clim_%7B%5CDelta+g+%5Cto+0%7D+%5Cleft%28++%5Cfrac%7BF+%5Cleft%28+g%5Bq%5D+%2B+%5CDelta+g%5Bq%5D%5D+%5Cright%29+-+F+%5Cleft%28+g%5Bq%5D+%5Cright%29%7D%7B%5CDelta+g%5Bq%5D%7D+%5Clim_%7B%5Cepsilon+%5Cto+0%7D+%5Cfrac%7Bg%5Bq+%2B+%5Cepsilon+%5Ceta%5D+-+g%5Bq%5D%7D%7B%5Cepsilon%7D+%5Cright%29+%5C%5C+++%26%3D+DF+%5Cleft%28+g%5Bq%5D+%5Cright%29+%5Cdelta_%5Ceta+g%5Bq%5D+%5C%5C+++%26%3D+RHS+%5C%5C++%5Cend%7Baligned%7D%7D&bg=ffffff&fg=333333&s=0 "\displaystyle{ \begin{aligned} LHS &= \lim_{\substack{\epsilon \to 0 \\ \Delta g \to 0}} \left( \frac{F \left( g[q] + \Delta g[q]] \right) - F \left( g[q] \right)}{\Delta g[q]}\frac{\Delta g[q]}{\epsilon} \right) \\ &= \lim_{\Delta g \to 0} \left( \frac{F \left( g[q] + \Delta g[q]] \right) - F \left( g[q] \right)}{\Delta g[q]} \lim_{\epsilon \to 0} \frac{g[q + \epsilon \eta] - g[q]}{\epsilon} \right) \\ &= DF \left( g[q] \right) \delta_\eta g[q] \\ &= RHS \\ \end{aligned}}")
of a real-valued observables, but, instead, a move from a commutative algebra to a partial commutative algebra of observables.
affect the size of its automorphisum group and, hence, of the allowable symmetries of the system (the weaker the assumed structure on 
unitary matrices clearly coincides with the circle group; the unitary condition is equivalent to the condition that its element have absolute value 1. Therefore, the circle group is canonically isomorphic to 
is unitary if its conjugate transpose 
is also its inverse—that is, if
is the identity matrix.
) and the equation above becomes
