Canonical partner of time in QFT and string theory In analytical mechanics, the Hamiltonian or total energy becomes the conjugate momentum of the time in the symmetric form of the equations. This seems very strange and interesting to me. Does it have an analogue in quantum field theory and string theory?
 A: The point is that if, on the Hilbert space $\cal H$ of the theory, there were a self-adjoint operator $T$ verifying $[T,H]= -i\hbar I$ on a dense domain, with some other technical hypotheses, for a celebrated theorem (Stone-von Neumann's theorem) one would be able to construct a unitary operator $U : {\cal H} \to \oplus_n L^2(\mathbb R)$ ($^*$) such that $UHU^\dagger= \oplus_n X$ and $UTU^\dagger= \oplus_n P$. Since unitary transformations preserve the spectrum of an operator, $H$ would have the same spectrum as $P$, that is the whole $\mathbb R$. Instead, the spectrum of $H$ is known a priori and there are good physical reasons to assume it is always bounded from below (to guarantee stability of the physical system).
So it is very difficult to suppose that time is represented by a self-adjoint operator conjugated with $H$. Some other ways are nevertheless  possible (using generalized observables relying on the notion of POVM, for instance), but dropping the idea that $T$ and $H$ are conjugated observables in a strict sense.
Addendum (after  Isidore Seville's comment). The above illustrated no-go result is known as Pauli's theorem in the literature.

Footnote ($^*$) We have to use many copies of $L^2(\mathbb R)$ when ${\cal H}$ is not irreducible under the unitary  representation of Weyl-Heisenberg group generated by $I,T,H$.
