Wave function is always in abstract space?

1. Feb 11, 2012

waterfall

Wave function is always in abstract space in any quantum interpretation be it Copenhagen or Bohmian or Many Worlds because wave function is in many dimensional abstract Hilbert Space. Correct?

Since the counterpart of Hilbert space in QM is Fock Space in QFT. Then the fields in QFT live in abstract space too. Correct?

This means electron field in QFT live in higher abstract space. But in QED, electromagnetic field live in Fock space too in abstract space, but how come we can detect electromagnetic field? Or is the answer we can't really detect electromagnetic field directly but just the photons when they couple to the electrons in an antenna and vibrate them up and down?

If the answer is we can detect electromagnetic field directly. But electromagnetic field is supposed to live in Fock space in abstract space. So how can we detect or touch things in Fock space when it is supposed to be located in higher abstract space?

Hope someone can clear this up as it sometimes still confused me. Thanks.

2. Feb 11, 2012

Staff: Mentor

You pose a lot of hard questions here so lets just start with the first, and I wont attempt an answer to them all. Basically QM is the way it is with complex numbers because it is the only way to have continuous transformations between what are known as pure states - check out:
http://arxiv.org/pdf/quant-ph/0111068v1.pdf

Your other ones require quite a bit of discussion also but just as a start since QFT is a relativistic theory you don't really have an electric field its an electromagnetic field and the quantum of that field is a photon. Electric fields are defined by the force exerted on test particles and that force is the result of an exchange of photons between particles. In this way fields have been reduced to something a little different.

The mathematical spaces they reside in are not the real key - its the physical ideas behind them.

Thanks
Bill

3. Feb 12, 2012

martinbn

The wave functions in QFT are not the quantum fields, they still describe the states. The fields are operator valued functions (actually distributions), which are defined on spacetime. So in QFT the fields may be represented by more complicated mathematical obejcts, but they do have as their domain spacetime.

Also it seems that in your questions you make a Freudian distinction between the Hilbert space in QM and Fock space in QFT. They are isomorphic as Hilbert spaces. All the Hilbert spaces, in quantum physics, are isomorphic as abstract Hilbert spaces.

4. Feb 12, 2012

waterfall

I see. I was not clear on the distinctions between the wave functions and quantum fields and Fock space. So Fock space is synonym to wave functions when these are put in Hilbert space. And quantum fields are not these.

I heard that electron field are not observable while electromagnetic field are observable.

Just to be clear on something. Although the electron field is not observable, I heard it has components of "grassman numbers". Now if these "grassman numbers" were altered by say the components of real numbers in the electromagnetic field, then there would be corresponding change in the electron particle even though the electron field is non-observable? Something similar to the Aharonov-Effect?

5. Feb 12, 2012

To talk of "observability", you need to have a theory of observations. This is not a part of the standard quantum theory exposition.

There are several different philosophies here and competing schemes. You must decide in advance what you consider as an "observation" - it must produce a fact, not a "probability distribution of different facts". And this fact should have an appropriate representation in the formal scheme.

By the way: Fock space, or, more generally, a representation space for the algebra of fields, is also a Hilbert space.

Last edited: Feb 12, 2012
6. Feb 12, 2012

waterfall

What you talking about. We are all familiar only with interpretations of quantum mechanics, but not interpretations of QFT. We can't imagine what is it like to have a BM or Many Worlds version of QFT. Anyway. When I mentioned wave functions in this thread. I mean the wave functions in quantum fields theory, not nonrelativistic quantum theory of particles (the normal QM everyone knows).

7. Feb 12, 2012

"We?" Who? And why are you restricting yourself to BM or Many Worlds?

8. Feb 13, 2012

waterfall

I'm not referring to interpretations in QM. But to how are the fields related to the wave functions. But you mentioned it has all to do with interpretations? Anyway. How do you embed Many Worlds in QFT? How do worlds in QFT split?

9. Feb 13, 2012

.

You already got the answer: In QFT "wave functions" represent quantum states of the field. Field itself is represented by a space-time net of local algebras (generated by field operators). When the field is a "charged field", then in the field algebra you have a subalgebra of "observables" (very bad name, but used). This subalgebra consists of those field operators that preserve "charge superselection rule" - very important concept.

Ask those who split the world. There are other approaches to the measurement problem in QFT that do not rely on scholastic reasonings such as "how many universes can simultaneously dance on the head of a pin?"

10. Feb 14, 2012

waterfall

I heard that electron field are not observable while electromagnetic field are observable.

Just to be clear on something. Although the electron field is not observable, I heard it has components of "grassman numbers". Now if these "grassman numbers" were altered by say the components of real numbers in the electromagnetic field, then there would be corresponding change in the electron particle even though the electron field is non-observable? Something similar to the Aharonov-Effect?

11. Feb 14, 2012

Staff: Mentor

Scratching my head about what you even mean - real numbers altering Grassman Numbers? In QFT everything is a quantum field.

Thanks
Bill

12. Feb 14, 2012

waterfall

Let's take an aquarium as analogy, the fishes and bubbles are the quanta, the water is the electron field. Now if walk toward the aquarium or equivalent to photon field (or electromagnetic field) travelling nearby that touch the aquarium water or electron field, would it affect the fishes or quanta or electrons?

13. Feb 14, 2012

Polyrhythmic

Where did you hear that? Electrons can be observed just as well as photons.

14. Feb 14, 2012

waterfall

In QFT, the electrons are quanta of the electron field. While the photons are quanta of the electromagnetic field. The electron field is unobservable. We can only observe the electrons. In the case of the EM field, we can detect both photons and electromagnetic field.

15. Feb 14, 2012

Staff: Mentor

Again I have zero idea what you are talking about - quanta are not like bubbles and the water is not like an an electron field. These things are are not really describable by classical analogs.

If you are trying to say do photons and electrons interact - yes they do - but the proper description of it is very abstract and mathematical - for a discussion that does not involve it I suggest Feynmans QED Lectures:
http://vega.org.uk/video/subseries/8

Thanks
Bill

Last edited: Feb 14, 2012
16. Feb 14, 2012

Electrons are observable. They leave tracks. These track exist whether you look at them or not. Observations of electron positions can be described using quantum field.

17. Feb 14, 2012

waterfall

Ah... I was thinking that in the electromagnetic field, the antennae can feel its magnetic and electric field but like what Hobba said, in QFT, there is no magnetic or electric fields but just virtual particles that transfer those fields. I was thinking of antennae to receive the electron fields... but they are just electrons... confusing sometimes.. so what's the equivalent of magnetic and electric fields in electron fields and what are the corresponding virtual particles that transfer the corresponding fields? ..

18. Feb 14, 2012

Staff: Mentor

Not quite - for the EM field all we can detect is photons. An electric field by definition is the force on a test particle exerted by the field, but in QED that is the result of virtual photon exchange from the source of the electric field with the test particle.

Thanks
Bill

19. Feb 14, 2012