Editing Canonical transformation (section)

== One parameter subgroup of Canonical transformations ==
Allowing the values of <math>\alpha </math> to take continuous range of values in:

<math display="block">\begin{align}
& 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{align}  </math>

which can be expressed as <math>\epsilon^\mu(\eta,t;\alpha ) </math> where <math>\epsilon^\mu(\eta,t;0)=\eta^\mu </math>.


One parameter subgroup of Canonical transformations are those where the generator of the transformation can be used to generate coordinates where<math>\epsilon^\mu(\epsilon(\eta,t;\alpha_1);\alpha_2)=\epsilon^\mu(\eta,t;\alpha_1+\alpha_2) </math> is satisfied, i.e. composition of two canonical transformations of parameter <math>\alpha_1 </math> and <math>\alpha_2 </math> are the same as that of a single canonical transformation of parameter <math>\alpha_1+\alpha_2 </math>.


The condition on the transformations of the one parameter subgroup kind can be expressed equivalently as a differential equation:

<math display="block">\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) \implies \frac{d\epsilon^\mu(\eta,t;\alpha)}{d \alpha}= J^{\mu\nu}\frac{\partial G}{\partial \epsilon^\nu}(\epsilon(\eta,t;\alpha ),t)   </math>

for all <math>\eta </math> given that the generator has no explicit dependance on <math>\alpha </math>. The conditions <math>\epsilon^\mu(\epsilon(\eta,t;\alpha_1);\alpha_2)=\epsilon^\mu(\eta,t;\alpha_1+\alpha_2) </math> can be recovered since this equation is trivially satisfied when <math>\alpha_2=0 </math> which is considered initial values and the differential equations of both sides are of the same form implying the relation due to uniqueness of solutions with given initial values. Hence one parameter subgroups of canonical transformations are extension of infinitesimal canonical transformations to finite values of <math>\alpha </math> by using the same functional form of its generator independent of parameter <math>\alpha </math>.<ref name=":2" />


As a consequence of the generator having no explicit dependance on <math>\alpha </math>, the generator is also implicitly independent of <math>\alpha </math>.

<math display="block">\frac{d G(\epsilon(\eta;\alpha),t)}{d \alpha}=\{G,G\}=0,\,\forall \alpha  \implies G(\epsilon(\eta;\alpha),t)=G(\eta,t)  </math>

This can be used to express the differential equation as:

<math display="block">\frac{d\epsilon^\mu(\eta,t;\alpha)}{d \alpha}= \{\epsilon^\mu(\eta,t;\alpha),G(\eta,t)\}_\eta=:-\tilde G \epsilon^\mu </math> 

where the linear differential operator is defined as <math>\tilde G:= (\nabla_\eta G)^T J \nabla_\eta </math>.
=== Active view of transformation ===


Upon iteratively solving the differential equation, the solution of the differential equation follows as:<ref name=":2">{{Harvnb|Sudarshan|Mukunda|2010|p=50-57}}</ref>

<math display="block">\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</math>


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

<math display="block">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)  </math>

=== Passive view of transformation ===
Change in a function can be invoked by preserving its values on the same physical states in phase space as <math>f(\epsilon,t)=f(\epsilon(\eta;\alpha),t)=f'(\epsilon(\eta;\alpha+\delta\alpha),t)= f'(\epsilon',t) </math> can be expressed as upto first order as:

<math display="block">\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\}  </math>

Including the change in the function as some explicit dependance on parameter of transformation <math>\alpha </math>, it can be expressed as <math>f(\epsilon,t;\alpha) </math> where it is explicitly dependant on <math>\alpha </math> such that <math>\frac{\partial f(\epsilon,t;\alpha)}{\partial \alpha} =-\{f,G\} </math> which indicates that the function transforms oppositely to that due to the coordinates to preserve well defined mapping from a physical point in phase space to its scalar values. It is also possible that functions transform without needing to preserve its values on the same physical states in phase space. Such as, for example, the Hamiltonian whose explicit dependance on the canonical transformation can be different from the above form, restated from its previous derivation as

<math display="block">\frac{\partial H(\epsilon,t;\alpha)}{\partial \alpha} =\frac{dG}{dt} </math>

which is similar to previous relation but also accounts for any explicit time dependence of the generator. Hence, if the Hamiltonian is invariant in passive view for infinitesimal canonical transformations, its generator is a constant of motion.<ref name=":2" />