Where exactly does QFT differ from QM? (in their formalisms)

[kith]

How can both that operator and the creation operator create a particle out of the vacuum state? They seem to be different things. Does the first creates a particle with a definite position and the second a particle with a definite momentum?

Yes. In ordinary QM, the position wavefunction can be written as a superposition of momentum eigenstates. Analogously in QFT, the field operator can be written as a sum of operators which create or annihilate a particle with definite momentum. If we act with the field operator on the vacuum, it creates in effect a particle with definite location.

 

[Jilang]

Perhaps like many others, I have been under the (false?) impression that these fields are real In some sense? I am thinking of the buckyball field.

 

[A. Neumaier]

Perhaps like many others, I have been under the (false?) impression that these fields are real In some sense? I am thinking of the buckyball field.

Why do you mention this? There is no conflict. For every bound state (and hence for buckyballs) there is a corresponding asymptotic field. It is real since one can measure the buckyball density everywhere, though it will be nonzero only where one can actually find the buckyballs. But even a buckyball field operator can be written as the sum of a corresponding creation and annihilation operator.

 

[Jilang]

I think what I am struggling with is that an infinite number of different fields are permeating the universe at all times. Are they just mathematical constructs (like virtual particles)?

 

[A. Neumaier]

I think what I am struggling with is that an infinite number of different fields are permeating the universe at all times. Are they just mathematical constructs (like virtual particles)?

There are a number of basic local fields (those in the standad model and gravity). All other fields are composite fields. In any field theory one can create lots of local composite fields (technically these form the Borchers class of local fields of a theory). In a free QFT, the most general composite local field is given by a linear combination of normally ordered products of local field operators at the same space-time position $x$. A few of these appear in the Lagrangian density defining a QFT.

In general, composite fields are just mathematical constructs. But some of them have a physical interpretation since they are measurable in an operational sense; the most important ones are the composite fields corresponding to bound states and the associated currents.

A fully defined quantum field theory assigns in each state expectation values of all nonlocal products of the basic fields (technically correlation functions), from which the composite local field expectations (which in the cases mentioned are in principle measurable) are obtained by a limiting procedure (technically through Haag-Ruelle theory). An effective theory concentrates on the few fields and currents relevant at a particular description level for a particular purpose.

So people studying entanglement of buckyballs ignore everything except for the buckyballs. They even dispense with the fields (needed in case the number of buckyballs is not certain) and just look at 1- and 2-particle states where the particle is a buckyball. It would be overkill (and distract from the real physics in buckyball experiments) to represent the buckyballs in terms of quarks and leptons.

For the same reason, engineers concerned with everyday physics ignore the (far too detailed) description of flowing water, say, in terms of quantum fields and represent water instead by a few classical fields, for example energy density, momentum density, and temperature. Each applications therefore has its own effective description, but from a fundamental point of view all these are expectations of certain composite fields.

 

[RockyMarciano]

The mathematical model is quite different so it is not easy to explain how QM and QFT differ in a post. I'll try giving the general change of picture in a few strokes.
The main difference is that one goes from a Hilbert space of functions acted by operators(observables) that is suited for imagining the wavefunction somewhat as belonging to a quantum particle, to the concept of quantum field as a distribution whose values are operators (functionals) acting on states in a Fock space. These operators can be thought of (if one continues with the analogy of wavefunctions of particles in QM) as creating particles(in the case of creation operators) from the vacuum state they act on. So one can see the mathematical model changes quite a bit, from an infinite-dimensional Hilbert space with finite degrees of freedom to an infinite dimensional Fock space with infinite degrees of freedom and that allows changing numbers of particles.
All this works fine mathematically as long as you only consider free fields. With interacting fields there is no longer well defined Fock space (as mentioned in another post) due to certain theorem and QFT is just an extremely useful heuristic and predictive calculational tool.