# QFT vs Wave Function

• I
Gold Member
My level is not sufficient enough to easily understand QFT yet there is some basic question I need to understand in it - what in QFT corresponds to a wave function in QM, for a single particle case and, say, for a more general case of multiparticle nonseparable state (suppose the particles are identical). I do understand Fock space idea as well as creation-annihilation operators idea, yet I cannot grasp what is that operator field considered in QFT and how it may correspond to the general multiparticle case of QM. If this field defines an operator at each space point, then on which vector space does it act and how it may be related to the multiple particle wave function? Or maybe it is not the operator field itself which corresponds in QFT to the wave function/particles state, but something else?
I tried to look through some discussions here and on other sites but did not succeed to see or grasp what is the solution.

Thank you.

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A. Neumaier
The quantum field ##\phi(x)## corresponds in meaning to the quantum mechanical ##q_j## (specifically, ##\phi\to q## and ##x\to j##, but with a continuum of indices). But its action on an ##N##-particle state gives (in the nonrelativistic case) a linear combination of ##(N-1)##-particle and ##(N+1)##-particle states.
See https://en.wikipedia.org/wiki/Second_quantization.

vanhees71
Gold Member
But how the quantum field and particle states are related? I was under an impression that quantum particles are an emergent property of a quantum field, that is particles are some discrete modes of oscillation of that quantum field. So I expected that a state of the quantum field itself may somehow be interpreted as particle state.

From what I can see and understand in second quantization, operator field introduced there cannot contain any state-specific information.

atyy
The creation operator acting on the vacuum state creates an excited state, which is a state with one particle.

When you read about non-relativistic second quantization, first keep in mind non-relativistic quantum mechanics of many identical particles (ie. with the standard interpretation of the wave function etc.) Second quantization is simply non-relativistic quantum mechanics of many identical particles, but formulated in a different language.

A. Neumaier
operator field introduced there cannot contain any state-specific information.
Just like in ordinary quantum mechanics, the position operator cannot contain any state-specific information. The latter is always in the state, not in the operators.

Demystifier
Gold Member
In QFT there are 3 different things which, in some sense, correspond to the wave function ##\psi({\bf x},t)## in QM.
1. Field operator ##\hat{\phi}({\bf x},t)##
2. Wave functional ##\Psi[\phi({\bf x}), t]##
3. Wave function itself, which in QFT can be calculated as ##\psi({\bf x},t)=\langle 0|\hat{\phi}({\bf x},t)|\Psi\rangle##

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Geofleur, MichPod and vanhees71
kith
From what I can see and understand in second quantization, operator field introduced there cannot contain any state-specific information.
Yes. The field operator ##\phi(\vec x)## is not the state but it acts on the state. One way to picture it is by its action on the vacuum. The vacuum state is the state with zero particles which is denoted by $|0\rangle$. If I act on it with the field operator, I get a particle localized at location $\vec x$ (compare this with Demystifier's point 3 in post #6 above):
$$\phi(\vec x) |0\rangle = |\vec x\rangle$$
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 I act with the field operator on the vacuum, it creates in effect a particle with definite location.

(What I have written here is by no means rigorous and there are quite a few difficulties. For example, we cannot define a position operator with the usual properties for massless particles like photons. So the very concept of localizing particles isn't always well-defined.)

kith
Your confusion about the "operator field" and the "state of a quantum field" may be related to the fact that the term "field" in these quotes refers to different things. I started a thread about this a couple of years ago.

MichPod
vanhees71
Gold Member
2021 Award
Yes. The field operator ##\phi(\vec x)## is not the state but it acts on the state. One way to picture it is by its action on the vacuum. The vacuum state is the state with zero particles which is denoted by $|0\rangle$. If I act on it with the field operator, I get a particle localized at location $\vec x$ (compare this with Demystifier's point 3 in post #6 above):
$$\phi(\vec x) |0\rangle = |\vec x\rangle$$
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 I act with the field operator on the vacuum, it creates in effect a particle with definite location.
This has to be taken with very great care. E.g., in the case of photons, this "state" (it's a generalized state in the sense of a distribution in any case), this cannot be interpreted as a "generalized position eigenstate" since a photon doesn't permit the definition of a position operator in the strict sense. A photon has not a position as an observable at all (within standard QED)!

bhobba
Mentor
But how the quantum field and particle states are related?

There are a number of equivalent formulations of QM:

QFT corresponds to formulation F - the second quantization formulation based on creation and annihilation operators. As operators they operate on states and that's how states come into it. The interesting thing about QFT is its based on operators with states being of secondary importance. They are there of course as they must be - but most of the time they take a back seat.

Thanks
Bill

Gold Member
In QFT there 3 different things which, in some sense, correspond to the wave function ##\psi({\bf x},t)## in QM.
1. Field operator ##\hat{\phi}({\bf x},t)##
2. Wave functional ##\Psi[\phi({\bf x}), t]##
3. Wave function itself, which in QFT can be calculated as ##\psi({\bf x},t)=\langle 0|\hat{\phi}({\bf x},t)|\Psi\rangle##

It'll take me some time to work on it. Some immediate questions to clarify:

Are the ##|0\rangle## and ##|\Psi\rangle## appearing in the item 3 the wave functionals of the same kind as brought in item 2?

Is the wave function brought in item 3 eqivalent to the single particle QM wave function or something else?

Most importantly, how still can we get, say, a nonseparable wave function of two particles from the wave functional brought in item 2?

bhobba
Mentor
Some immediate questions to clarify:

Read the link I gave on the second quantization formulation of QM.

Thanks
Bill

MichPod
DarMM
Gold Member
Ignoring distribution theoretic issues, the wave functions in QFT are:

##\Psi[\phi(\vec{x})]## in the Heisenberg picture. That is a function over the space of field configurations in ##\mathbb{R}^{3}##. Roughly speaking it is the amplitude to observe the field in that configuration.

If the field is a free field then it turns out that this space of field wave functionals has a basis where it takes the form of a Fock space over a single particle space Hilbert space. Hence one can view the Hilbert space in either a particle or field manner.

Technical details for others (I'll fill them in when I'm less tired!):
If the space of classical fields is ##\mathcal{S}## and ##d\nu## is a measure on this space obeying the axioms of QFT and the specific identities of a free field, then one can prove that:

##L^{2}(\mathcal{S},d\nu) = \mathcal{F}(\mathcal{S}^{*})##

(The single particle quantum Hilbert space is the dual of the classical field space)

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bhobba and MichPod
Gold Member
Thanks to everybody!
I think I've got some directions and stuff to work on.

The creation operator acting on the vacuum state creates an excited state, which is a state with one particle.

When you read about non-relativistic second quantization, first keep in mind non-relativistic quantum mechanics of many identical particles (ie. with the standard interpretation of the wave function etc.) Second quantization is simply non-relativistic quantum mechanics of many identical particles, but formulated in a different language.

What is the relativistic quantum mechanics of many identical particles? Or spacetime background independence theory of many particles.. must we come to this latter for completeness?

bhobba
Mentor
What is the relativistic quantum mechanics of many identical particles?

Its called the Quantum Field Theory of Bosons or Femions which are all identical. We either have Bosons or Fermions in QFT - this is the famous Spin Statistics Theorem:
http://www.worldscientific.com/worldscibooks/10.1142/3457

Or spacetime background independence theory of many particles.. must we come to this latter for completeness?

Cant see how background independence has anything to do with this.

Thanks
Bill

Its called the Quantum Field Theory of Bosons or Femions which are all identical. We either have Bosons or Fermions in QFT - this is the famous Spin Statistics Theorem:
http://www.worldscientific.com/worldscibooks/10.1142/3457

Cant see how background independence has anything to do with this.

Thanks
Bill

Simple. General relativity is not yet united with quantum field theory. When you have general relativistic quantum field theory.. the particles will no longer occurred against a backdrop of fixed spacetime.. but dynamical where it is more than the particles curving spacetime... but particles and spacetime becoming united into a symmetry just like space and time where they become mere shadows. Is this not the final goal of QFT?

bhobba
Mentor
Simple. General relativity is not yet united with quantum field theory.

Thats a very common misconception, but totally false:
http://arxiv.org/abs/1209.3511

We have a perfectly valid theory up to about the Plank scale. QED is in the same boat - its only valid to a cutoff before the electroweak theory takes over and none would say QED is not EM united with QFT.

Thanks
Bill

atyy
What is the relativistic quantum mechanics of many identical particles? Or spacetime background independence theory of many particles.. must we come to this latter for completeness?

At the non-rigourous level, the relativistic quantum mechanics of many identical particles is conceptually the same as the non-relativistic quantum mechanics of many identical particles, except that particles can be created and destroyed. However, if you use second quantization in non-relativistic quantum mechanics, there you will already see that there are "particles" that can be created and destroyed, such as phonons in non-relativistic quantum mechanics, eg. http://yclept.ucdavis.edu/course/242/2Q_Fradkin.pdf.

As DarMM points out above, the "particle" picture and Fock space are not rigourously correct when the field theory is interacting. So all the derivations in textbooks are only "physically" correct, although they are mathematically wrong.

Background independence is not required for quantum field theory (or any theory with known physical relevance, including quantum gravity at low energies).

Demystifier
Gold Member
Are the ##|0\rangle## and ##|\Psi\rangle## appearing in the item 3 the wave functionals of the same kind as brought in item 2?
Yes.

Is the wave function brought in item 3 eqivalent to the single particle QM wave function ...?
Yes.

Most importantly, how still can we get, say, a nonseparable wave function of two particles from the wave functional brought in item 2?
Let me explain it for the simplest case of hermitian field operator. It can be written as
$$\hat{\phi}(x)=\hat{\psi}(x)+\hat{\psi}^{\dagger}(x)$$
where ##\hat{\psi}^{\dagger}(x)## contains the creation operators and ##\hat{\psi}(x)## the destruction operators. Then the non-separable 2-particle wave function is
$$\psi(x_1,x_2)=\langle 0|\hat{\psi}(x_1)\hat{\psi}(x_2)|\Psi\rangle$$

Even though ##\hat{\psi}(x_1)## and ##\hat{\psi}(x_2)## are local operators, the product ##\hat{\psi}(x_1)\hat{\psi}(x_2)## is not a local operator. This is why "local" QFT can have non-local (or non-separable) effects.

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MichPod
Gold Member
the non-separable 2-particle wave function is
$$\psi(x_1,x_2)=\langle 0|\hat{\psi}(x_1)\hat{\psi}(x_2)|\Psi\rangle$$

Thank you! Yet what I basically do not understand in this is how we are able to get both single-particle wave function and 2-particle wave function from the same wave functional? I'd expect that for a given arbitrary field wave functional we can (in the best case) just determine the number of particles (if it is definite) and then determine a QM wave function for this number of particles only. I fail to see how can we extract a QM wave function for whatever number of particles we want from the same field wave functional. Or maybe I fail to see what is the meaning of such a wave function.

A. Neumaier
what is the meaning of such a wave function.
It is a superposition of wave functions with 0,1,2,... particles. The particle number is not definite.

bhobba
Gold Member
It is a superposition of wave functions with 0,1,2,... particles. The particle number is not definite.

Yes. But suppose some very specific case - that we have a state of the field where the number of particles is definite and it equals two (can we have a definite number of particles at least for non-relativistic QFT?), then we try to calculate a 3-particle wave function. What are we supposed to get for such a case?

Nugatory
Mentor
Yet what I basically do not understand in this is how we are able to get both single-particle wave function and 2-particle wave function from the same wave functional?
Even in ordinary QM, a multi-particle quantum system is a single quantum system with a single wave function. You can see that clearly when you look at a pair of particles in the singlet state - there's a single wave function and the two spins are two different observables on it.

So if you're already OK with the idea that a multi-particle system has a single wave function.... The new thing is that you're being asked to do is think of the number of particles itself as an observable with an associated Hermitian operator. The wave function may be eigenvector of that observable or it may he a superposition of eigenvectors with different eigenvalues; in the latter case the particle number will not be fixed.

Gold Member
The wave function may be eigenvector of that observable or it may he a superposition of eigenvectors with different eigenvalues; in the latter case the particle number will not be fixed.

And I meant the case where the wave functon(al) is the eigenvector of the number operator (with eigenvalue 2).
Then we apply $$\psi(x_1,x_2,x_3)=\langle 0|\hat{\psi}(x_1)\hat{\psi}(x_2)\hat{\psi}(x_3)|\Psi\rangle$$ transformation and my question was what can we get? I would be happy if the answer is zero, but I am not sure.

Demystifier
Gold Member
Yes. But suppose some very specific case - that we have a state of the field where the number of particles is definite and it equals two (can we have a definite number of particles at least for non-relativistic QFT?), then we try to calculate a 3-particle wave function. What are we supposed to get for such a case?
We get ##\psi(x_1,x_2,x_3)=0##

MichPod and vanhees71
bob012345
Gold Member
The creation operator acting on the vacuum state creates an excited state, which is a state with one particle.

When you read about non-relativistic second quantization, first keep in mind non-relativistic quantum mechanics of many identical particles (ie. with the standard interpretation of the wave function etc.) Second quantization is simply non-relativistic quantum mechanics of many identical particles, but formulated in a different language.

And what exactly and I mean physically, is a creation operator? Don't give me math, I want physics. Thanks.

A. Neumaier
what exactly and I mean physically, is a creation operator?
It is the negative frequency part of the Hermitian field operator, and the annihilation operator (its adjoint) is the positive frequency part. For classical fields (and in signal processing) one would call the latter the analytic signal of a real field (or signal).

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bhobba

One thing there are no particles in QFT. The electrons are not a particles, but excitations of the electron-field. Particle-like appearance is when the excitations form a packet (many wave lengths) that is localized.

In this sense the wave picture is more correct than the particle in QFT.

bhobba
Nugatory
Mentor
An attempted thread hijack has been successfully prevented, but unfortunately a number of informative and even humorous posts by some of the SAs were lost in the cleanup.