2 Added clarification about the language
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The question is a mixture of a few different questions. This answer tries to help sort them out.

(First, here's a note about language. The name "quantum mechanics" is sometimes used as a synonym for the general principles of quantum theory, and it is sometimes used to refer to a non-relativistic special case. Both languages are fine, as long as it's clear which language is being used. The following answer uses the name "quantum theory" for the general principles and "non-relativistic quantum mechanics" for claritythe special case.)

Do the various interpretations of quantum mechanics have any impact on how we interpret quantum field theory? Or can QFT be considered as one of the interpretations of QM ...

Quantum field theory (QFT) and non-relativistic quantum mechanics (QM) both respect the general principles of quantum theory. QFT might help clarify exactly what the "measurement problem" is, simply because QFT is more accurate and comprehensive, but it certainly doesn't resolve the measurement problem at all, and it isn't a new interpretation. The issue of interpreting quantum theory is the same in QFT as it is in non-relativistic QM. Those mysteries don't go away.

Can Bohm's interpretation survive the transition to QFT?

If by "Bohm's interpretation" you mean Bohmian mechanics (also called pilot-wave theory), then it is not an interpretation of quantum theory at all. It is an entirely different theory, one that tries to work as well as quantum theory does. But, although I have seen occasional claims, I have not seen any extension of that idea to anything that works as well as QFT.

Now from what I have learned from popular physics, QFT says space always has fields and when you put energy in them, the particles pop up as excitations. This seems very much independent of any interpretation of quantum mechanics.

Like you suggested, the occurrence of particles in QFT is independent of any interpretation of quantum theory. In contrast to non-relativistic QM, QFT is formulated in terms of fields, not particles. In QFT, particles are predicted rather than assumed, even though the most common computational methods (involving Feynman diagrams, etc) can make it look like particles are being assumed. The subject of particles in QFT is an interesting one, but it is independent of any interpretation of quantum theory.

The question is a mixture of a few different questions. This answer tries to help sort them out.

(First, here's a note about language. The name "quantum mechanics" is sometimes used as a synonym for the general principles of quantum theory, and it is sometimes used to refer to a non-relativistic special case. Both languages are fine, as long as it's clear which language is being used. The following answer uses the name "non-relativistic quantum mechanics" for clarity.)

Do the various interpretations of quantum mechanics have any impact on how we interpret quantum field theory? Or can QFT be considered as one of the interpretations of QM ...

Quantum field theory (QFT) and non-relativistic quantum mechanics (QM) both respect the general principles of quantum theory. QFT might help clarify exactly what the "measurement problem" is, simply because QFT is more accurate and comprehensive, but it certainly doesn't resolve the measurement problem at all, and it isn't a new interpretation. The issue of interpreting quantum theory is the same in QFT as it is in non-relativistic QM. Those mysteries don't go away.

Can Bohm's interpretation survive the transition to QFT?

If by "Bohm's interpretation" you mean Bohmian mechanics (also called pilot-wave theory), then it is not an interpretation of quantum theory at all. It is an entirely different theory, one that tries to work as well as quantum theory does. But, although I have seen occasional claims, I have not seen any extension of that idea to anything that works as well as QFT.

Now from what I have learned from popular physics, QFT says space always has fields and when you put energy in them, the particles pop up as excitations. This seems very much independent of any interpretation of quantum mechanics.

Like you suggested, the occurrence of particles in QFT is independent of any interpretation of quantum theory. In contrast to non-relativistic QM, QFT is formulated in terms of fields, not particles. In QFT, particles are predicted rather than assumed, even though the most common computational methods (involving Feynman diagrams, etc) can make it look like particles are being assumed. The subject of particles in QFT is an interesting one, but it is independent of any interpretation of quantum theory.

The question is a mixture of a few different questions. This answer tries to help sort them out.

(First, here's a note about language. The name "quantum mechanics" is sometimes used as a synonym for the general principles of quantum theory, and it is sometimes used to refer to a non-relativistic special case. Both languages are fine, as long as it's clear which language is being used. The following answer uses the name "quantum theory" for the general principles and "non-relativistic quantum mechanics" for the special case.)

Do the various interpretations of quantum mechanics have any impact on how we interpret quantum field theory? Or can QFT be considered as one of the interpretations of QM ...

Quantum field theory (QFT) and non-relativistic quantum mechanics (QM) both respect the general principles of quantum theory. QFT might help clarify exactly what the "measurement problem" is, simply because QFT is more accurate and comprehensive, but it certainly doesn't resolve the measurement problem at all, and it isn't a new interpretation. The issue of interpreting quantum theory is the same in QFT as it is in non-relativistic QM. Those mysteries don't go away.

Can Bohm's interpretation survive the transition to QFT?

If by "Bohm's interpretation" you mean Bohmian mechanics (also called pilot-wave theory), then it is not an interpretation of quantum theory at all. It is an entirely different theory, one that tries to work as well as quantum theory does. But, although I have seen occasional claims, I have not seen any extension of that idea to anything that works as well as QFT.

Now from what I have learned from popular physics, QFT says space always has fields and when you put energy in them, the particles pop up as excitations. This seems very much independent of any interpretation of quantum mechanics.

Like you suggested, the occurrence of particles in QFT is independent of any interpretation of quantum theory. In contrast to non-relativistic QM, QFT is formulated in terms of fields, not particles. In QFT, particles are predicted rather than assumed, even though the most common computational methods (involving Feynman diagrams, etc) can make it look like particles are being assumed. The subject of particles in QFT is an interesting one, but it is independent of any interpretation of quantum theory.

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The question is a mixture of a few different questions. This answer tries to help sort them out.

(First, here's a note about language. The name "quantum mechanics" is sometimes used as a synonym for the general principles of quantum theory, and it is sometimes used to refer to a non-relativistic special case. Both languages are fine, as long as it's clear which language is being used. The following answer uses the name "non-relativistic quantum mechanics" for clarity.)

Do the various interpretations of quantum mechanics have any impact on how we interpret quantum field theory? Or can QFT be considered as one of the interpretations of QM ...

Quantum field theory (QFT) and non-relativistic quantum mechanics (QM) both respect the general principles of quantum theory. QFT might help clarify exactly what the "measurement problem" is, simply because QFT is more accurate and comprehensive, but it certainly doesn't resolve the measurement problem at all, and it isn't a new interpretation. The issue of interpreting quantum theory is the same in QFT as it is in non-relativistic QM. Those mysteries don't go away.

Can Bohm's interpretation survive the transition to QFT?

If by "Bohm's interpretation" you mean Bohmian mechanics (also called pilot-wave theory), then it is not an interpretation of quantum theory at all. It is an entirely different theory, one that tries to work as well as quantum theory does. But, although I have seen occasional claims, I have not seen any extension of that idea to anything that works as well as QFT.

Now from what I have learned from popular physics, QFT says space always has fields and when you put energy in them, the particles pop up as excitations. This seems very much independent of any interpretation of quantum mechanics.

Like you suggested, the occurrence of particles in QFT is independent of any interpretation of quantum theory. In contrast to non-relativistic QM, QFT is formulated in terms of fields, not particles. In QFT, particles are predicted rather than assumed, even though the most common computational methods (involving Feynman diagrams, etc) can make it look like particles are being assumed. The subject of particles in QFT is an interesting one, but it is independent of any interpretation of quantum theory.