Fields in quantum field theory

[evk]

Hi, I'm a little confused about the nature of fields in quantum field theory. I sometimes see people make reference to an "electron field" or other matter field of some sort, and in my understanding, in quantum field theory, ALL the different fundamental particles can be represented as quantized fields.

My questions:

(1) What exactly does it mean for a field to be "quantized"?
(2) Wherever there is an electron (or other particle) in space, in QFT is it described by a field that somehow exists around that area of space? How is this field related to the Schrödinger wave equation? Or is there one electron "field" that surrounds all of space, and all electrons are just disturbances in this field (so wherever there is no disturbance in the field, there is no electron)?

Sorry if I sound confused. Please correct me if I have some fundamental misunderstandings. Thanks,

--Erik

 

[vanesch]

(1) What exactly does it mean for a field to be "quantized"?

It means that the field (originally a function of spacetime) became an operator over Hilbert space instead of a real number or a set of real numbers (like a vector field).

Or is there one electron "field" that surrounds all of space, and all electrons are just disturbances in this field (so wherever there is no disturbance in the field, there is no electron)?

This last view is pretty good. Electrons are to be seen as excitations of the "electron field". Different types of excitations correspond to different dynamical states of electrons (fast ones, slow ones...).

 

[Norman]

It means that the field (originally a function of spacetime) became an operator over Hilbert space instead of a real number or a set of real numbers (like a vector field).

An alternative, but just as correct view is:
In NRQM, what happens to our measurables (position and momentum for instance)? They obey a commutation relation. This is usually how quantization is first introduced. Well in QFT, not only do the measurables obey commutation relations, but the fields also do.

This last view is pretty good. Electrons are to be seen as excitations of the "electron field". Different types of excitations correspond to different dynamical states of electrons (fast ones, slow ones...).

The problem with the SE is that it is an non-relativistic equation. The search for a relativistic qm equation is well documented and typically in Relativistic QM and QFT one looks at the Dirac Equation and the Klein-Gordon equation to start with. Take a look at the end of most graduate level QM books and they will usually give and introduction to RQM. For RQM, Greiner has a GREAT book called "Relativistic Quantum Mechanics- Wave Equations." Check if your library has it. From what I can tell your understanding seems on the right track. Good luck with your studies.
Cheers,
Ryan

 

[Juan R.]

Hi, I'm a little confused about the nature of fields in quantum field theory. I sometimes see people make reference to an "electron field" or other matter field of some sort, and in my understanding, in quantum field theory, ALL the different fundamental particles can be represented as quantized fields.
My questions:
(1) What exactly does it mean for a field to be "quantized"?
(2) Wherever there is an electron (or other particle) in space, in QFT is it described by a field that somehow exists around that area of space? How is this field related to the Schrödinger wave equation? Or is there one electron "field" that surrounds all of space, and all electrons are just disturbances in this field (so wherever there is no disturbance in the field, there is no electron)?
Sorry if I sound confused. Please correct me if I have some fundamental misunderstandings. Thanks,
--Erik

First part was adressed by Vanesch.

On (2) One cannot ask questions like where one electron 'was' because position is not an observable. In fact, wavefunctions of QFT are not function of position. Fields are related to the Schrödinger one via the Hamiltonian density. The fields are used for the construction of interactions.

In traditional QFT, particles are considered excitations of the field. For example, and electron is the first excitation of a fermion field. However this view is not very accurate. In fact, Weiberg begins his book on QFT in a inverse manner. He begins from particles and derives the fields.

 

[Juan R.]

The problem with the SE is that it is an non-relativistic equation. The search for a relativistic qm equation is well documented and typically in Relativistic QM and QFT one looks at the Dirac Equation and the Klein-Gordon equation to start with. Take a look at the end of most graduate level QM books and they will usually give and introduction to RQM. For RQM, Greiner has a GREAT book called "Relativistic Quantum Mechanics- Wave Equations." Check if your library has it. From what I can tell your understanding seems on the right track. Good luck with your studies.
Cheers,
Ryan

That is not true. The basic equation of relativistic QFT is a SE-like equation for the wavefunction funtional of the quantum field. Both the KG and Dirac equations are NOT wavefunction equations in R-QFT, they are identity equations for KG and Dirac fields obtained from RQFT Lagrangians.

Precisely the fact that both Dirac and KG equations are NON suitable relativistic generalization of the SE (even if one can use Dirac wave equation for computing first relativisitic corrections to hydrogen atom energy spectra, the equation is incorrect) what DID were abandoned in R-QFT. I remark again that are substituted by field theoretic identities.

The equation fundamental in R-QFT is Schrödinger one for the wavefunction of the field. See Weinberg manual (volume I) for details.

 

[Juan R.]

Hi, I'm a little confused about the nature of fields in quantum field theory.

Some people -i between them- think that fields are not fundamental items of nature. Just mathematical artifacts for the assitance in the construction of relativisitic interactions operator.

See thread about quantum fields in sci.physics.research

http://www.lns.cornell.edu/spr/2005-09/msg0071506.html

On my discusion, i argue that fields are not fundamental and in fact are only derived when one does the approximation of one-body states. In R-QFt this is doen via aplication of the cluster decomposition principle to scattering states.

Perhaps you could see Eugene book accesible online on quantum fields. I think that is not all correct, therefore read it with a bit of salt. However, would be useful for you until i publish my own views on the topic.

Some of my post (with relevant quotes of Steven Weinberg manual) on why quantum fields are non fundamental are

http://www.lns.cornell.edu/spr/2005-10/msg0071725.html
http://www.lns.cornell.edu/spr/2005-10/msg0071775.html
http://www.lns.cornell.edu/spr/2005-10/msg0071793.html
http://www.lns.cornell.edu/spr/2005-10/msg0071813.html
http://www.lns.cornell.edu/spr/2005-10/msg0071831.html
http://www.lns.cornell.edu/spr/2005-10/msg0071878.html
http://www.lns.cornell.edu/spr/2005-10/msg0071926.html

 

[QMrocks]

Hi, I'm a little confused about the nature of fields in quantum field theory. I sometimes see people make reference to an "electron field" or other matter field of some sort, and in my understanding, in quantum field theory, ALL the different fundamental particles can be represented as quantized fields.
My questions:
(1) What exactly does it mean for a field to be "quantized"?
(2) Wherever there is an electron (or other particle) in space, in QFT is it described by a field that somehow exists around that area of space? How is this field related to the Schrödinger wave equation? Or is there one electron "field" that surrounds all of space, and all electrons are just disturbances in this field (so wherever there is no disturbance in the field, there is no electron)?
Sorry if I sound confused. Please correct me if I have some fundamental misunderstandings. Thanks,
--Erik

(1) One quantized the field by imposing the commutation relation (anti-commutator for Fermions) on the canonical variable (position and momentum) of via second quantization on the creation and annihilation operator.

(2) i dun have much appreciation of this question now...

 

[Norman]

That is not true.

What is not true? I am offering an example (yes only an example) of how one typically goes from QM to QFT.

The basic equation of relativistic QFT is a SE-like equation for the wavefunction funtional of the quantum field.

Yes, but it is NOT the usual SE that one sees in NRQM. That is the point I was making and it appears I did it poorly- sorry for that.

Both the KG and Dirac equations are NOT wavefunction equations in R-QFT, they are identity equations for KG and Dirac fields obtained from RQFT Lagrangians.
Precisely the fact that both Dirac and KG equations are NON suitable relativistic generalization of the SE (even if one can use Dirac wave equation for computing first relativisitic corrections to hydrogen atom energy spectra, the equation is incorrect) what DID were abandoned in R-QFT. I remark again that are substituted by field theoretic identities.
The equation fundamental in R-QFT is Schrödinger one for the wavefunction of the field. See Weinberg manual (volume I) for details.

I really don't understand what point you are trying to make here... Yes the Dirac and KG are NOT relativistic generalizations of the SE. So what? The Dirac and KG equations provide a nice introductory way to work in QFT.

Your last statement about the Schrödinger equation in QFT makes absolutely no sense to me (I am not trying to be brash here, I truly don't know what you mean). What matters is connecting QFT to observable physics. Can I calculate a cross section (up to some order in perturbation theory) that agrees well with a collider experiment? How do I do that? Well I write the Lagrangian for my theory, calculate the generating functional and the feynman rules, then calculate the cross section (again up to some order in perturbation theory).

So I would contend that the fundamental equation in QFT (for a given theory) is the Lagrangian for that theory. This is where all the physics resides- maybe that comes out wrong- this is where all the information about the physics resides (probably a better way of saying it).