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One parameter subgroup of Canonical transformations

Allowing the values of ${\displaystyle \alpha }$  to take continuous range of values in:$${\displaystyle {\begin{aligned}&Q(q,p,t;\alpha )\quad \quad \quad &Q(q,p,t;0)=q\\&P(q,p,t;\alpha )\quad \quad {\text{with}}\quad &P(q,p,t;0)=p\\\end{aligned}}}$$ which can be expressed as ${\displaystyle \epsilon ^{\mu }(\eta ,t;\alpha )}$  where ${\displaystyle \epsilon ^{\mu }(\eta ,t;0)=\eta ^{\mu }}$ .

One parameter subgroup of canonical transformations concerns when the infinitesimal generator function is of the same functional form regardless of the coordinates, that is when

${\displaystyle \delta \epsilon ^{\mu }(\eta ,t;\alpha )=\delta \alpha \{\epsilon ^{\nu },G\}=\delta \alpha J^{\mu \nu }{\frac {\partial G}{\partial \epsilon ^{\nu }}}(\epsilon (\eta ,t;\alpha ),t)}$ 

for all ${\displaystyle \eta }$  such that the generator has no explicit dependance on ${\displaystyle \eta }$  or ${\displaystyle \alpha }$ . This forms a differential equation as follows

${\displaystyle {\frac {d\epsilon ^{\mu }(\eta ,t;\alpha )}{d\alpha }}=J^{\mu \nu }{\frac {\partial G}{\partial \epsilon ^{\nu }}}(\epsilon (\eta ,t;\alpha ),t)}$ 

It follows that ${\displaystyle \epsilon ^{\mu }(\epsilon (\eta ,t;\alpha _{1});\alpha _{2})=\epsilon ^{\mu }(\eta ,t;\alpha _{1}+\alpha _{2})}$  since this equation is trivially satisfied when ${\displaystyle \alpha _{2}=0}$  which is considered initial values and the differential equations are of the same form implying the relation due to uniqueness of solutions with initial values given. Hence it also follows that such canonical transformations form a one parameter subgroup.

Another conclusion can follow from the nature of the generator

${\displaystyle {\frac {dG(\epsilon (\eta ;\alpha ),t)}{d\alpha }}=\{G,G\}=0,\,\forall \alpha \implies G(\epsilon (\eta ;\alpha ),t)=G(\eta ,t)}$ 

which implies that ${\displaystyle {\frac {d\epsilon ^{\mu }(\eta ,t;\alpha )}{d\alpha }}=\{\epsilon ^{\mu }(\eta ,t;\alpha ),G(\eta ,t)\}_{\eta }=:-{\tilde {G}}\epsilon ^{\mu }}$  upon taking repeatedly in steps we get the solution as:

${\displaystyle \epsilon (\eta ,t;\alpha )=\eta +\alpha \{\eta ,G(\eta ,t)\}+{\frac {1}{2!}}\alpha ^{2}\{\{\eta ,G(\eta ,t)\},G(\eta ,t)\}+\cdots =e^{-\alpha {\tilde {G}}}\eta }$ 

where the differential operator is defined as ${\displaystyle {\tilde {G}}:=(\nabla _{\eta }G)^{T}J\nabla _{\eta }}$ .

Change in function values ${\displaystyle {\frac {df(\epsilon (\eta ;\alpha ),t)}{d\alpha }}=\{f(\epsilon (\eta ;\alpha ),t),G(\eta ,t)\}_{\eta }=:-{\tilde {G}}f(\epsilon (\eta ;\alpha ),t)}$  upon taking repeatedly in steps and using ${\displaystyle \epsilon (\eta ,t;0)=\eta }$  we get similarly

${\displaystyle f(e^{-\alpha {\tilde {G}}}\eta ,t)=f(\epsilon (\eta ;\alpha ),t)=f(\eta ,t)+\alpha \{f(\eta ,t),G(\eta ,t)\}+{\frac {1}{2!}}\alpha ^{2}\{\{f(\eta ,t),G(\eta ,t)\},G(\eta ,t)\}+\cdots =e^{-\alpha {\tilde {G}}}f(\eta ,t)}$ 

Change in a function that preserves values on physical states in phase space as ${\displaystyle f(\epsilon ,t)=f(\epsilon (\eta ;\alpha ),t)=f'(\epsilon (\eta ;\alpha +\delta \alpha ),t)=f'(\epsilon ',t)}$  can be expressed as upto first order as:

${\displaystyle \delta 'f=f'(\epsilon )-f(\epsilon )=f'(\epsilon )-f'(\epsilon ')\approx f(\epsilon (\eta ;\alpha -\delta \alpha ))-f(\epsilon (\eta ;\alpha ))=-\delta \alpha \{f,G\}}$ 

Including the change in the function, it can be expressed as ${\displaystyle f(\epsilon ,t;\alpha )}$  where it is explicitly dependant on ${\displaystyle \alpha }$  such that ${\displaystyle {\frac {\partial f(\epsilon ,t;\alpha )}{\partial \alpha }}=-\{f,G\}}$  which indicates that the function transforms oppositely to that of the coordinates to preserve well defined mapping from a physical point in phase space to scalar values.

Realization of Lie algebra by canonical transformations

Let the set of generators of canonical transformations of the one parameter kind be closed under Poisson brackets i.e. ${\displaystyle \{\lambda _{i},\lambda _{j}\}=c_{ij}^{k}\lambda _{k}+d_{ij}}$  then it also follows that${\displaystyle {\widetilde {\{\lambda _{i},\lambda _{j}\}}}=c_{ij}^{k}{\tilde {\lambda }}_{k}}$ . Using Jacobi identity of Poisson bracket, it can be shown that the exponential map of the group elements formed by ${\displaystyle {\tilde {\lambda }}}$ 's forms a Lie group.

Using Baker-Campbell-Hausdroff formula on the exponential map, since ${\displaystyle {\tilde {\lambda }}}$  are linear operators, closure of the exponential set is given if the commutator, ${\displaystyle {\tilde {\lambda }}_{i}{\tilde {\lambda }}_{j}-{\tilde {\lambda }}_{j}{\tilde {\lambda }}_{i}}$  and hence all other terms arising in the BCH formula can be given as element of the generator set.

${\displaystyle ({\tilde {\lambda }}_{i}{\tilde {\lambda }}_{j}-{\tilde {\lambda }}_{j}{\tilde {\lambda }}_{i})f=\{\lambda _{i},\{\lambda _{j},f\}\}-\{\lambda _{j},\{\lambda _{i},f\}\}=\{\lambda _{i},\{\lambda _{j},f\}\}+\{\lambda _{j},\{f,\lambda _{i}\}\}=\{\{\lambda _{i},\lambda _{j}\},f\}={\widetilde {\{\lambda _{i},\lambda _{j}\}}}f=c_{ij}^{k}{\tilde {\lambda }}_{k}f}$ 

Since the commutator is closed it follows that ${\displaystyle e^{a_{i}{\tilde {\lambda }}_{i}}e^{a_{j}{\tilde {\lambda }}_{j}}=e^{a_{ij}^{k}{\tilde {\lambda }}_{k}}}$ .