What is the quantized form of the strings lagrangian?
What you are doing isn't a proper way of learning science. Its obvious that you're not familiar with quantization of classical systems. So you're trying to learn string theory before learning QM and QFT which isn't a good idea. Also even if we put aside that, you should first read a textbook and when you have problem, come and ask.
I suggest you first read some textbooks about QM and QFT before going to string theory. But if you're insistent on learning string theory now, I can only suggest you read Barton Zwiebach's "A first course in string theory".
Quantization is the replacement of classical quantities (in the Hamiltonian) like momentum by their respective quantum operators. Thus quantization transforms a classical system into a quantum system equivalently a classical Hamiltonian is transformed into a quantum Hamiltonian. Quantum mechanics is the physics of the small scale. Each familiar classical observable is correlated to a quantum operator.
One uses the operator to operate on the wave function to find the required eigenstate. Now the normalizability condition limit the number of possible eigenstates and wave functions from ensuring the total probability is one across all of Hilbert space
Raising and lowering operators, give one the immediately higher or lower eigenstate.
The schrodinger time dependent equation is the eigenequation for the Hamiltonian.
Time independent equation governs the evolution of the wave function through time.
Hilbert space is the space of states.
Quantum field theory is the quantization of classical field theories like maxwells theory of the electromagnetic force
I hope you meant time dependent equation otherwise you are wrong.
Further you can look at the evolution from a different perspective.
For example the Heisenberg picture with wave functions that are constant w.r.t. time. An example where this picture is useful is the zitterbewegung of a particle
In QFT yet another picture is very useful, the interaction picture.
Next you would need some basics of understanding symmetry arguments, Lie groups/algebras are everywhere in ST/modern physics.
Familiarity with gauge choices is a necessity as well. Finally you need GR.
Not to mention we usually work with actions instead of Lagrangians.
And lastly, posting random sentences which for all we know are copy-pasted from wikipedia or whatever won't convince us that you know that stuff. In fact I'm quite certain you haven't undertaken a formal investigation of QFT. If you would have just posted a polite reply stating that you know the prerequisites would have been sufficient for most.
Be warned, there is a reason string theory isn't part of the curriculum in most schools (maybe in the near future with the good books available these days). Also it is definitely graduate level, there is a reason for that.
Ya sorry I hate typos I meant time dependent.
And thank you for the advice I have studied QM. Although I have not studied QFT
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