First and second quantization of relativistic mechanics Classical mechanics can be written in a lagrangian formalism. If one quantizes this theory, we get quantum mechanics. Let us continue this process:

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*Relativistic mechanics can also be written in a lagrangian formalism, first quantizing it should therefore give us relativistic quantum mechanics. Is this theory consistent? What are its problems?


*Classical relativistic field theory can also be written in lagrangian formalism and quantized, thus getting QFT. In this context, why do we say that QFT is the only coherent framework reuniting relativity and quantum mechanics? Is it equivalent, in some ways, the relativistic quantum mechanics?
 A: The Schrödinģer equation is a non-relativistic equation. Naively, we replace momentum and loacation in the Hamiltonian formulation of classical mechanics by operators acting on a Hilbert space. The linear shell of an element of that space is represented by a wavefunction.
We can do the same replacement for the energy momentum relation derived in special relativety. This gives the Klein-Gordon equation. It describes relativistic particles with zero spin (scalar-bosons).
The Dirac equation is a little bit more subtle to derive. It acts on spinors, which describe relaitivtic fermions.
Last and most importantly we have the Yang-Mills equation. It describes vector bosons like photons gluons and depends on a Lie-Group and describes Lie algebra-valued connection one-forms on a principal G bundle over space-time.
The important thing is that quantization of these theories gives a propagator describing the propability amplitude for a field to move from state A to state B. However, all the equations (except for Yang-Mills in the non-abelian setting) are linear equations and we are interested in the interaction of the various fields. The Feynman path integral is however only approximatable if the interaction is very weak and even then the math is not rigorous. So there is no mathematical rigorous formulation of an interacting QFT in four-dimensions. Altough the approximation in terms of power series in the coupling variable can be very precise for example in the case of QED.
Another viewpoint is that QFT is one-dimensional quantum-gravity because in perturbation theory we sum over one-dimensional topologies aka Feynman graphs. From this perspective is string theory two-dimensional quantum gravity because the point particles become one-dimensional and the Feynman graphs are two-dimensional topologies. In this sense is QFT a special case of string theory and much of the research there is dedicated to give a better understanding of QFT itself.
