# Concept of wavefunction and particle within Quantum Field Theory

• A
• Jufa
In summary: There is no demystification without prior mystification. I just want to understand this stuff better and I feel that the currently existing literature does not explain this stuff clearly. I am not (yet) in a phase to demystify this stuff, I'm just trying to think of it from different angles. Eventually I hope that I will be able to write something about it in a demystifying way, but first I have to look at it from different angles, including the wrong... angles.
Jufa
TL;DR Summary
I am struggling to properly understand this two concepts.
-1st: Could someone give me some insight on what a ket-state refers to when dealing with a field? To my understand it tells us the probability amplitude of having each excitation at any spacetime point, but I don't know if this is accurate. Also, we solve the free field equation not for this wavefunction but for the field itself. The latter sounds rather strange to me, since the field is indeed an operator. To me it looks as if we solved the Schrodinger equation not for the wavefunction but for the operator X, which is just an observable.

Once you solve the equation for a free field you see that applying the creation operator (p) to a ket-state creates a particle with momentum p.
-2nd: As far as I know the only thing we know is that applying the creation operation increases the momentum of the system by a quantity p. It is fair then to think that this operation is analogous to creating a particle, but what do we know about this particle? For instance, where do we have created it?

A ket ##|\psi\rangle## is not a probability amplitude of anything. To get a probability amplitude you must consider a quantity of the form ##\langle a|\psi\rangle##, which is the probability amplitude that the measurement of observable ##A## will give the value ##a##. The observable ##A## may be almost anything you like, e.g. energy, momentum, electric field at a point, or particle position (in relativistic QFT the particle position is somewhat tricky and requires additional clarifications).

The fact that we solve the equation for the field operator ##\phi({\bf x},t)## just means that we work in the Heisenberg picture. In nonrelativistic QM it is analogous to solving the harmonic oscillator for the position operator as ##x(t)=ae^{-i\omega t} +a^{\dagger}e^{i\omega t}##. Alternatively, in QFT one can also work in the Schrodinger picture in which the field operator ##\phi({\bf x})## does not depend on ##t##, but it is rarely used in practice.

If you create particle with a definite momentum, then you cannot know its position. There is an operator that creates a particle at definite position (as I said it's tricky in relativistic QFT, but it's not problem at all in nonrelativistic QFT used e.g. in condensed matter physics), but in this case you know nothing about its momentum.

For more details see also my https://arxiv.org/abs/quant-ph/0609163 Secs. 8 and 9.

topsquark, CuriousLearner8 and vanhees71
Many thanks for your answer, it helped me a lot! Also I checked your paper and found it really interesting.

Demystifier
Demystifier said:
A ket ##|\psi\rangle## is not a probability amplitude of anything. To get a probability amplitude you must consider a quantity of the form ##\langle a|\psi\rangle##, which is the probability amplitude that the measurement of observable ##A## will give the value ##a##. The observable ##A## may be almost anything you like, e.g. energy, momentum, electric field at a point, or particle position (in relativistic QFT the particle position is somewhat tricky and requires additional clarifications).

The fact that we solve the equation for the field operator ##\phi({\bf x},t)## just means that we work in the Heisenberg picture. In nonrelativistic QM it is analogous to solving the harmonic oscillator for the position operator as ##x(t)=ae^{-i\omega t} +a^{\dagger}e^{i\omega t}##. Alternatively, in QFT one can also work in the Schrodinger picture in which the field operator ##\phi({\bf x})## does not depend on ##t##, but it is rarely used in practice.

If you create particle with a definite momentum, then you cannot know its position. There is an operator that creates a particle at definite position (as I said it's tricky in relativistic QFT, but it's not problem at all in nonrelativistic QFT used e.g. in condensed matter physics), but in this case you know nothing about its momentum.

For more details see also my https://arxiv.org/abs/quant-ph/0609163 Secs. 8 and 9.
Nice paper. Thank you for sharing.

vanhees71, Demystifier and gentzen
CuriousLearner8 said:
Nice paper. Thank you for sharing.
It is indeed a nice paper. I really like it. That's why he is the Demystifier!

vanhees71, Demystifier and CuriousLearner8
gentzen said:
It is indeed a nice paper. I really like it. That's why he is the Demystifier!
Usually, although he's currently mystifying us with his free quark at rest!

A. Neumaier, topsquark, vanhees71 and 2 others
PeroK said:
Usually, although he's currently mystifying us with his free quark at rest!
There is no demystification without prior mystification. I just want to understand this stuff better and I feel that the currently existing literature does not explain this stuff clearly. I am not (yet) in a phase to demystify this stuff, I'm just trying to think of it from different angles. Eventually I hope that I will be able to write something about it in a demystifying way, but first I have to look at it from different angles, including the wrong ones.

CuriousLearner8, gentzen, topsquark and 2 others
PeroK said:
Usually, although he's currently mystifying us with his free quark at rest!
But here he makes us think about very subtle issues with local gauge symmetries, and I think there are many myths in the standard literature (and not only textbooks!) which have to be resolved!

Demystifier, gentzen and topsquark

## 1. What is a wavefunction in Quantum Field Theory?

A wavefunction in Quantum Field Theory is a mathematical description of a quantum system, which contains information about the probability of finding a particle at a certain location or with a certain energy. It is represented by a complex-valued function that evolves over time according to the Schrödinger equation.

## 2. How does the concept of wavefunction differ from the classical particle model?

In the classical particle model, particles are described as having definite positions and momenta at all times. However, in Quantum Field Theory, particles are described as excitations of quantum fields and their behavior is governed by probabilities rather than definite values. The wavefunction represents the probability amplitude of finding a particle at a certain location, rather than its exact position.

## 3. What is the role of the wavefunction in the measurement process?

The wavefunction plays a crucial role in the measurement process in Quantum Field Theory. When a measurement is made, the wavefunction collapses to a specific state, which determines the outcome of the measurement. This collapse is known as the "measurement problem" and is still a topic of debate among scientists.

## 4. Can the wavefunction be observed or measured directly?

No, the wavefunction itself cannot be observed or measured directly. It is a mathematical concept used to describe the behavior of particles in Quantum Field Theory. However, the effects of the wavefunction can be observed through experiments and measurements, such as the double-slit experiment.

## 5. How does the concept of a particle fit into Quantum Field Theory?

In Quantum Field Theory, particles are described as excitations of quantum fields. These fields permeate all of space and time, and particles are created and destroyed through interactions with these fields. The concept of a particle is essential in understanding the behavior of matter and energy at the quantum level.

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