The basis of Quantum Mechanics is contained in the postulates which tell us how to describe quantum systems (below I disconsider possibly degenerate spectra just for simplicity):
To describe a quantum system there is a Hilbert Space $\mathcal{H}$ whose elements represent the possible states of the system.
For each physical quantity associated to a system there is one hermitian operator defined on $\mathcal{H}$. We call these operators observables.
The only possible values to be measured of an observable are the elements of its spectrum.
If $A$ is an observable with discrete spectrum $\sigma(A)=\{a_i : i \in \Bbb N\}$, then the probability of measuring $a_i$ on the state $|\psi\rangle$ is $P(a_i)=|\langle \varphi_i|\psi\rangle|^2$, where $|\varphi_i\rangle$ is the eigenstate corresponding to the eigenvalue $a_i$. Analogously, if $A$ is an observable with continuous spectrum $\sigma(A)$ then the probability density for the values of $A$ on the state $|\psi\rangle$ is $\rho(x)=|\langle x|\psi \rangle|^2$ where $x\in \sigma(A)$ and $|x\rangle$ is the generalized eigenvector corresponding to $x$.
When one performs a measurement of the observable $A$ the state colapses to the eigenstate corresponding to the eigenvalue measured.
The time evolution is governed by the requirement that the observable corresponding to the energy is the generator of time translations. That is, the time evolution equation is $i\hbar \frac{d|\psi\rangle}{dt}=H|\psi\rangle$.
Everything else follows from this. If we want to describe also spin we include Pauli's postulates and again, everything works just fine.
Now, apart from Quantum Mechanics there is Quantum Field Theory. I'm just starting to study it and there is something I still didn't get: Quantum Field Theory is just one application of Quantum Mechanics or it modifies Quantum Mechanics?
In other words, is QFT just one application of these postulates I've stated, or it modifies these postulates somehow? And if it does, how does QFT modifies Quantum Mechanics and its postulates?
