I am new to the concept of weak and strong equalities, and I have a doubt trying to derive an expression. In section $1.2.1$ of Henneaux and Teitelboim's Quantization of Gauge Systems, there is a strong equation in the context of gauge transformations

$$\delta F=\delta v^a[F,\phi_a]\tag{1.35},$$

where $\delta v^a=\delta t~(v^a-\tilde v ^a)$, and $\delta F=F(t_2)-\tilde{F}(t_2)$ is the difference between the values of a dynamical variable $F$ at time $t_2=t_1+\delta t$ corresponding to two different choices of $v^a$, $\tilde{v}^a$ of the arbitrary functions at time $t_1$.

Using equation $(1.26)$, which gives the evolution of a dynamical variable


I get

$$F(t_2)=F(t_1+\delta t)=F(t_1)+\delta t ~\dot{F}(t_1)\approx F(t_1)+\delta t~[F,H'+v^a\phi_a]=\\= F(t_1)+\delta t~[F,H']+\delta t~[F,v^a]\phi_a+\delta t~v^a[F,\phi_ a]\approx F(t_1)+\delta t~[F,H']+\delta t~v^a[F,\phi_ a], \tag{1}$$ where I used the second "$\approx$" because $\phi_a$ vanishes on the constaint surface, so

$$F(t_2)\approx F(t_1)+\delta t~[F,H']+\delta t~v^a[F,\phi_ a].\tag{2}$$

Doing the same but using $\tilde{v}^a$,

$$\tilde F(t_2)\approx F(t_1)+\delta t~[F,H']+\delta t~\tilde{v}^a[F,\phi_ a]\tag{3},$$

and subtracting these expressions I get $(1.35)$ but with a "$\approx$" symbol instead of a "$=$". I can't see how this strong equality arises. Perhaps my procedure is not correct, but in that case I wouldn't know why.


You are correct that, if one conceives of the $\delta F$ in eq. (1.35) as $F(t_2) - \bar{F}(t_2)$, where the time dependence in both $F(t)$ and $\bar{F}(t)$ is obtained from the equations of motions, the equality would be weak.

However, what we are interested in here is not the difference $F(t_2) - \bar{F}(t_2)$, but defining a transformation on phase space - namely the gauge transformations. So eq. (1.35) is a definition of $\delta F$ - of the change of an observable $F$ under the gauge transformations generated by the primary first-class constraints $\phi_a$. Symmetry transformations should always be considered as acting off-shell, so this is an off-shell definition.

Precisely because this is equal to $F(t_2) - \bar{F}(t_2)$ on-shell, this does not alter physical states, and is hence exactly what we mean by a transformation being gauge.

  • $\begingroup$ I thought that, due to the form $(1.35)$ takes, it is identified as a (gauge) transformation generated by the first-class constraits, not the other way. I mean, $(1.35)$ is presented as "the difference between the values of a dynamical variable $F$ at time $t_2$ corresponding to two different choices of the arbitrary functions $v^a$". So, if that equation is a definition, what motivates that form? $\endgroup$ – AFG Apr 30 at 8:28
  • $\begingroup$ @AFG It is motivated by $F-\bar{F}$ being of that form! But eq. (1.35) is a definition of an arbitrary change $\delta F$ of an observable $F$ at a fixed time, not merely the difference of two on-shell evolutions of observables. $\endgroup$ – ACuriousMind Apr 30 at 9:09
  • $\begingroup$ Let's see if I got it right. In eq. $(1.35)$ we define a transformation generated by $\phi_a$ with parameter $\delta v^a=(v^a-\tilde{v}^a)~\delta t$. We then calculate $F(t_2)-\tilde{F}(t_2)$ on shell, which we know should be physically irrelevant, and the result coincides with the definition $(1.35)$. We conclude that the transformation is gauge. Am I right? $\endgroup$ – AFG Apr 30 at 11:36
  • $\begingroup$ @AFG that's what I meant to argue in my answer, yes. If you have suggestions how to make it clearer, feel free to comment them here or suggest an edit. $\endgroup$ – ACuriousMind Apr 30 at 11:50
  • $\begingroup$ It's okay, I was just confused by the way the book presents it. Thank you very much! $\endgroup$ – AFG Apr 30 at 11:56

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