Many books have described the path integral for non-relativistic quantum. For example, how to get the Schrödinger equation from the path integral. But no one told us the relativistic version. In fact, the relativistic version is impossible to be perfect, it must be replaced by quantum field theory. But why?
The answer I want is not that the quantum mechanics will give us a negative energy or negative probability. We need a answer to explain why non-relativistic Lagrangian $$L=\frac{p^2}{2m}$$ can lead a correct Schrödinger equation? Why if we replace it by relativistic Lagrangian $$L=-mc^2\sqrt{1-v^2/c^2},$$ we can not obtain any useful information? So how can we think of the correctness of the non-relativistic path integral?
And how a quantum field can give a relativistic quantum theory?
Here give us a simple answer Special relativity version of Feynman's "Space-Time Approach to Non-Relativistic Quantum Mechanics" but it is useless. It doesn't tell us how to develop it step by step in mathematical frame and physical frame...
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