What is the rigorous mathematical underpinning of Schwinger's quantum action principle? I'm reading J. Schwinger's book "Quantum Kinematics and Dynamics" and I'm trying to make sense of his formulation of his famous quantum action principle. In essence, he starts from considering arbitrary variations of the the quantum amplitude $\langle a, t_1 |b, t_2 \rangle$, where $|a, t_1 \rangle$ is an eigenvector, corresponding to eigenvalue $a$, of some arbitrary but fixed Hermitian operator $A(t_1)$ (considered in the Heisenberg picture of QM), and $|b, t_2 \rangle$ is an eigenvector, corresponding to eigenvalue $b$, of another arbitrary but fixed Hermitian operator $B(t_2)$. An arbitrary variation of the amplitude reads, obviously, $$ \delta (\langle a, t_1 |b, t_2 \rangle)= (\delta\langle a, t_1 |)|b, t_2 \rangle + \langle a, t_1 |(\delta |b, t_2 \rangle) $$ where $\delta\langle a, t_1 |$ and $\delta |b, t_2 \rangle$ are arbitrary variations of the eigenbra and eigenket.
Now comes the puzzle. Schwinger writes $$\delta |b, t_2 \rangle = -iG_b(t_2)|b, t_2 \rangle$$$$\delta\langle a, t_1 | = iG_a(t_1)\langle a, t_1 |$$where he says $G_a(t_1)$ and $G_b(t_2)$ are infinitesimal Hermitian operators. I don't understand as to why an arbitrary infinitesimal variation $\delta |b, t_2 \rangle$ should be generated by a Hermitian operator, as if only unitary infinitesimal variations (instead of general ones) are considered, and not by a more general operator. 
Could you, please, help clarify as to why the generators must be Hermitian for the most general variation $\delta |b, t_2 \rangle$?
 A: It is natural to endow any time-evolution of isolated systems with the natural abelian group structure of reals, i.e. $E(t)E(s)=E(t+s)$, $E(t)^{-1}=E(-t)$. This is because we want our mathematical definition of evolution to behave accordingly like a reversible physical time evolution that can be applied step-by-step.
On the other hand, it is also natural to assume the quantum time-evolution to be a group of linear operators acting on the Hilbert space of the theory, and that at each time it preserves probabilities (because of our probabilistic physical interpretation of wavefunctions).
With these assumptions, it follows that the quantum evolution $(E(t))_{t\in\mathbb{R}}$ should be a group of unitary operators, i.e. satisfying:


*

*$(\forall t\in\mathbb{R})$ $E(t)$ is a unitary operator on the Hilbert space $\mathscr{H}$ of wavefunctions;

*$(\forall t,s\in\mathbb{R})$ $E(t)E(s)=E(t+s)$ (i.e. $E(t)^*=E(-t)$).
Finally, it is also desirable that such evolution is strongly continuous, i.e.


*

*$(\forall s,t\in\mathbb{R})(\forall\psi\in\mathscr{H})$ $\lim_{s\to t}E(s)\psi=E(t)\psi$ (where the limit holds in the strong topology of $\mathscr{H}$);


because we want our probability distribution to behave smoothly as time passes.
Combining these three properties, we obtain a strongly continuous group of unitary operators, and by Stone's theorem its infinitesimal generator must be a self-adjoint operator.
The above reasoning can be extended to any more general symmetry transformation that behaves like time-evolution (i.e. has an underlying abelian group structure, preserves quantum probabilities and is smooth).
A: A short answer would be that because states are transformed via unitary transformations, infinitesimal transformations would be given by i times a hermitian operator.
For a longer answer one can use some model for the states where one expresses an arbitrary state $|a,t\rangle$ as a corresponding unitary operator $U(a,t)$ that converts some fixed reference state $|\Omega\rangle$ into that state
$|a,t\rangle=U(a,t) |\Omega\rangle . $
The variation of the state then becomes the variation of the unitary operator
$\delta|a,t\rangle \rightarrow \delta U(a,t) . $
Any unitary operator can be expressed in terms of an exponential function that contains hermitian matrices in its argument
$U(a,t)=\exp(i{\bf k}(a,t)\cdot{\bf A}) , $
where ${\bf k}(a,t)$ is a vector of real-valued parameters and ${\bf A}$ is a vector of hermitian matrices (think generators).
So the variation would then take the form
$\delta U(a,t)=i(\delta{\bf k}(a,t)\cdot{\bf A}) U(a,t) . $
Allow this to operate on the reference state and we end up with 
$\delta|a,t\rangle = i G(a,t) |a,t\rangle , $
where
$G(a,t) = \delta{\bf k}(a,t)\cdot{\bf A} $
is hermitian.
