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I followed a basic course on QFT and know some basic concepts but I focussed mainly on calculations and understanding the Feynman rules e.g.. However, sometimes I have the idea that there a quite some fundamental concepts I might be missing, especially regarding the interpretation of QFT and "what's happening".

One of the things I wonder is how does wave collapsing of QM fit into QFT? In QM there was a lot of focus on measurements, probabilities, and (the collapsing) of the wave function.

Of course, particles in QM and QFT are fundamentally different. In QM they are probability waves and in QFT they are (excitations of) fields. And if I'm not mistaken I would say that for the field of a particle, the excitations of the field are governed by the rules of Quantum Mechanics. However, is the wave function really completely gone, or is there some way to link fields to wave functions again and effects of measurements etc.

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The fundamental principles of quantum mechanics are exactly the same for QFT as for nonrelativistic quantum mechanics. There are wavefunctions that live in a Hilbert space. Observables are represented by Hermitian operators. The wavefunction evolves in a unitary way, as specified by the Schrodinger equation.

These principles are very general. They apply to things like qubits that don't even have to exist in a background of space.

What is different about QFT is that the Hilbert space and observables are just different from the ones in nonrelativistic quantum mechanics. For instance, although there are wavefunctions, there is no wavefunction $\Psi(x,t)$ that we can interpret as "the wavefunction of a photon," and there is no observable that is "the position of the photon."

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    $\begingroup$ What is the interpretation of wave functions in this case? $\endgroup$ Dec 15, 2020 at 1:10
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    $\begingroup$ @user3397129, there are wave functionals rather than wave functions. See for example, this answer $\endgroup$ Dec 15, 2020 at 1:38
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    $\begingroup$ @AlfredCentauri not all wavefunctionals belong to the Hilbert space of the QFT. With normal wavefunctions we have the square-integrability condition to choose the ones that belong to the Hilbert space, bit with wavefunctionals the problem is a lot harder. It is unsolved in general, rigorous solutions exist only for some models in 2 and 3 space-time dimensions. $\endgroup$ Dec 15, 2020 at 21:41
  • $\begingroup$ The question was about collapse. It has no place in QFT. $\endgroup$ Dec 18, 2020 at 11:15
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    $\begingroup$ @ArnoldNeumaier I wouldn't express the measurement principles using the "collapse" idea either in QFT or in nonrelativistic QM, but that might be beside the point of the question. This answer emphasizes that the general measurement principles (expressed in terms of observables as operators on a Hilbert space) work the same way in any nonperturbatively well-defined QFT as they do in nonrelativistic QM. That might be what the question is really trying to ask, even though it uses the regrettably-popular "collapse" language. $\endgroup$ Dec 19, 2020 at 5:56
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I think you are distracted by dubious interpretational remarks that particles are probability waves or that they are excitations of the field. I would advise to ignore such remarks, and focus on mathematical structure. Particles are what they are, physical somethings. Since such somethings are physical, they are distinct from the mathematics which we use in physics. Mathematics (at least in its pure form) is a product of the human mind and quite distinct from physical reality.

In quantum mechanics we do not actually describe particles. We describe the probabilities for measurement results. This is done using states in Hilbert space. The wave function is simply the components of the state in a particular basis (the basis of position states). Since the state is essentially a description of probability, it changes when information changes, for example when a measurement is performed. Collapse is just a change of information resulting from measurement. To describe states of many particles we construct Fock space from the Hilbert space of states of a single particle.

Quantum fields are operators defined (albeit not rigorously in usual treatments) on Fock space, and describe the creation and annihilation of particles in interaction. In qft we are usually most concerned with describing the effects of interaction, but to bring it back to experiment, we still have to form S-matrix elements and calculate cross-sections using initial and final states. These are still states in Hilbert space, and collapse still takes place for the states when a measurement is performed.

There are important generalisations, however. For example, in elementary treatments of quantum mechanics, one may take the von Neumann projection postulate as axiomatic for the treatment of observable quantities, meaning that collapse always takes place in measurement. This is not generally true in quantum field theory. For example we cannot actually measure the position of a photon, but only the position at which the annihilation of a photon took place. Consequently, the projection postulate does not apply. Also we can regard the expectation of the photon field as the classical electromagnetic $A$ field. The $A$ field is strictly not observable, but the derivatives, the $E$ and $B$ fields are generally considered as observable quantities. There is no collapse when we observation is of the expectation of an observable, as distinct from elementary qm in which observable quantities are treated as eigenstates of observable operators.

For this reason the Dirac–von Neumann axioms (as given in Wikipedia) make no mention of eigenstates and eigenvalues (implicit in collapse) but only use expectations of observables, from which it is possible to show that eigenstates and eigenvalues are a special case.

I have given a rigorous treatment in A Construction of Full QED Using Finite Dimensional Hilbert Space, and a fuller account in my books (see profile).

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The distinction between a probability wave and an excitation of the zero-point field has two characteristics:

In part it is just a mathematical sleight of hand, representing the same wave function via two different formalisms. QFT is the "deeper" formalism from which QM may be derived.

And it is also a conceptual interpretation, one of those metaphysical debates that the mainstream "shut up and calculate" or Copenhagen school chooses to ignore.

The collapse of the wave function remains an enigma in both formalisms. In simple terms it represents an unexplained discontinuity where, in order to describe what we observe, we jump from one theoretical model (the wave) to another (the particle).

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