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In quantum mechanics we work using wave functions, which represent (if we take the module squared) the probability of finding a particle in a certain position or momenta. On the other hand in the case of QFT we work with fields. Now, I fail to understand the physical meaning of the field in QFT: in the classical case, for example, a field is associated with a physical quantity that we can measure, like the Electric field, and at each point in space the field associates the value of the quantity. Now the same interpretation doesn't seem be true for the quantum field and I don't really understand what its value at a certain spacetime point represents. It looks like the quantum field is much similar to a quantum mechanical wave function, in the sense that in order to extract physical information you have to perform some operations on it (like taking the module square for the wave function). I'm also wondering if the quantum field and the quantum mechanical wave function are related in some way or if, when doing QFT, we just throw away the concept of wave function.

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    $\begingroup$ Related/possible duplicates: physics.stackexchange.com/q/54603/50583, physics.stackexchange.com/q/13157/50583 $\endgroup$
    – ACuriousMind
    Dec 11, 2021 at 17:32
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    $\begingroup$ IMHO the best answer to your question is given in Chapters 2 to 5 of Weinberg's The Quantum Theory of Fields, Vol. 1. Fields are introduced because they are convenient building blocks to construct relativistic interactions obeying cluster decomposition. Personally Weinberg's perspective is the one that makes most sense for me. $\endgroup$
    – Gold
    Dec 11, 2021 at 18:29
  • $\begingroup$ @Gold Does cluster decomposition (CDP) even apply in the non-relativistic case with any potential? W motivates fields (start of ch. 5) as the solution to a problem arising from wanting CDP and wanting creation/ann.. operators to implement it, but this looks irrelevant non-relativistically (NR), we can set up NRQF's for any NR multi-particle problem in about 3 steps from the HUP (See Landau vol. 3 Ch. 1 & 9, it's actually a completely general argument) - it doesn't seem like W's approach makes sense as anything but one possible plausibility argument that misses e.g. NR stuff, or does it? $\endgroup$
    – bolbteppa
    Dec 11, 2021 at 18:48

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Quantum field theory is emergent from the basic quantum mechanical postulates and principles first used in bound state problems that led to the wave functions and the probability distributions for the observations of bound states.

As scattering experiments became dominant in the study of particle interactions it was necessary to formulate a theory for more than two particles in the interaction. This led to the use of Feynman diagrams for calculating crossections with the development of Quantum Field Theory.

The quantum field theory for calculating with Feynman diagrams, is based on differential operators operating on the plane wave solutions of the relevant quantum mechanical equations, i.e. with zero potential. Dirac for fermions, Klein Gordon for bosons, quantized Maxwell for photons,( for particle physics, for QFT of other subjects the appropriate equations).

Creation and annihilation operators operate on the plane wave solutions of the corresponding particles , which represent the fields in QFT, covering all space : electron field, photon field etc.

There are other formulations isomorphic to this , that lead to theoretical studies, but when coming down to basics, it is the success of Feynman diagram calculations that makes QFT a physics theory .

What does a field represent in QFT?

The fields of QFT, in this representation, are the plain wave solutions on which the differential creation and annihilation operators operate. These functions, representing the fields of electrons, muons, .... are like a mathematical coordinate system on which the creation and annihilation of the corresponding particles can model the behavior of particles that are observed in experiments. Similar to the real field of the (x,y,z) coordinate system that allows to model the trajectory of particles in classical mechanics.

when doing QFT, we just throw away the concept of wave function.

No, it is in the underlying level of which QFT is a meta level, in the mathematics of the plane wave wavefunctions that the creation and annihilation operators work on. It is what makes the results of QFT calculations related to probability distributions.

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It's not very well known that QFT as well as classical field theory can be constructed geometrically, just as is done for GR.

The relevant geometry here is the theory of principal and vector bundles and these are both incarnations of fibre bundles.

Here, a field, in its most general incaranation is a section of a fibre bundles and they assemble into a sheaf of fields.

Physically, we should see a quantum field as a quantisation of a field of harmonic oscillators. This is how the creation and annihilation operators make their appearance.

There are other formalisms, for example Kevin Costellos axiomatisation of perturbative QFT. This uses his notion of a factorisation algebra and which is indebted to Segals axiomatic notion of a conformal field theory.

I am also wondering whether the quantum field and quantum mechanical wave function is related in some way.

They are. QFT in zero dimensions is essentially QM. It's an exercise in the first chapter of Srednicki's book on QFT.

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(tl;dr) A field represents an integration of every possible wave function associated with its every possible creation and annihilation states.

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Quantization can happen in two levels. First is, obviously, quantization of classical mechanics: you introduce wave function to treat an observable of a particle as the eigenvalues of its operator acting on that wave function. A formalism that governs the wave function of a system.

But wave functions have problem when you try to formalize many-body problem. Why? Because, from statistical mechanics, we know that quantum particles of the same sort are indistinguishable. So, trying to extract some state of, say, particle #1 and that of particle #2 doesn't mean anything, because they are exactly identical. In other words, applying operators $\hat{S}_1$ and $\hat{S}_2$ which are for particle #1 and #2, respectively, to $\psi$ is simply nonsense, because you cannot label the particles.

What you want to do instead is trying to count how many particles are there occupying each possible state. This basically means you need new formalism that governs the occupation numbers of a system. You can say that this is also a quantization, a quantization of occupation numbers (often called the second quantization). We know that wave functions live in the Hilbert space. Analogously, occupation numbers live in the Fock space.

In a Fock space, operators acting on a Fock state is just annihilation and creation operators, which is straightforward because all you need to change an occupation number is addition or subtraction. For example, say a particle in a many-particle system has three possible eigenstates $\psi_1$, $\psi_2$, and $\psi_3$. Then the general form of a Fock state of this system can be expressed by $|n_1,n_2,n_3\rangle$. If the system is fermionic, then each occupation number is either $0$ or $1$, whereas for bosonic, any non-negative integer. Consider the Fock state $|0,1,0\rangle$ which evolves to the state $|2,0,0\rangle$. All you need to do is simply applying creation operator $\hat{a}^\dagger_1$ for eigenstate $\psi_1$ twice and annihilation operator $\hat{a}_2$ for $\psi_2$ once to the initial state so that $\hat{a}^\dagger_1\hat{a}_2|0,1,0\rangle=|2,0,0\rangle$ which is just $\psi=1/\sqrt{2}\left(\psi_1\otimes\psi_1+\psi_1\otimes\psi_1\right)$ in terms of eigenstate. Notice that the annihilation and creation operators commute each other, because this is a bosonic system.

When you turn all these quantum mechanics into field-theoretic formalism, i.e., making occupation numbers more or less continuous, then any field can be simply expressed in terms of an integration over the conjugate momentum space of the general linear combination of annihilation and creation operators acting on the eigenstates associated with a conjugate momentum. For scalar field $\phi$, which is often used to model the Higgs boson, pions, etc,

$$\phi(x)=\int\frac{\mathrm{d}^3p}{\sqrt{(2\pi)^3 2E}}\left[\hat{a}_p\mathrm{e}^{-ip\cdot x}+\hat{a}^\dagger_p\mathrm{e}^{+ip\cdot x}\right]$$

where $\mathrm{e}^{\pm ipx}$ are simply the eigenstates (or eigenwavefunctions if you like) whose conjugate momentum is $p$.

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