Thanks for the link! I'm a huge fan of Peter Woit - his blog especially.
So my understanding is that while the way Lagrangians are written in classical theory doesn't extend directly to quantum mechanics, the concept of a Lagrangian is still useful since the Lagrangian can be fed into the path integral formulation (as opposed to being used as an input to the Euler-Lagrange equations). Also, in quantum field theory the starting point for canonical second quantization is typically a Lagrangian, where the fields are changed to operator fields.
So my understanding is that while the way Lagrangians are written in classical theory doesn't extend directly to quantum mechanics, the concept of a Lagrangian is still useful since the Lagrangian can be fed into the path integral formulation (as opposed to being used as an input to the Euler-Lagrange equations). Also, in quantum field theory the starting point for canonical second quantization is typically a Lagrangian, where the fields are changed to operator fields.
Also, something interesting I came across - the Euler-Lagrange equations do have a quantum analogue as well: https://en.wikipedia.org/wiki/Schwinger%E2%80%93Dyson_equati...