How is the conjugate momentum defined in the Schrödinger picture of QFT?

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SUMMARY

The discussion clarifies the definition of conjugate momentum in the Schrödinger picture of Quantum Field Theory (QFT). It emphasizes that while operators do not depend on time in this picture, the concept of conjugate momentum can still be understood by drawing parallels with single-particle Quantum Mechanics (QM) and classical Hamiltonian mechanics. The key takeaway is that the relationship between the Lagrangian and the time derivative of the field, d(L)/d(d(phi)/dt), remains valid and can be analyzed through these frameworks.

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alphaone
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Hi,
I have been wondering why we can consider d(phi)/dt when we are in Schrödinger picture (phi is just the usual scalar field here). Isn't this 0 as operators do not depend on time in this picture? However then how does it make sense to talk about the conjugate momentum in this picture which is
d(L)/d(d(phi)/dt)?
 
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To answer it, I would suggest to make it simpler.
First, replace QFT by single-particle QM, the problem is essentially the same.
Second, instead of quantum physics in the Schrödinger picture, study classical physics in the Hamiltonian formulation. The problem is still essentially the same.
Now I hope that you will be able to answer your question by yourself.
 

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