QFT vs QM 101

mysearch

Is the word ‘actual’ in the statement above based on mathematical consistency or any level of physical verification?

I do not want to be accused of blatant scepticism, although it is said that a certain amount is healthy. Equally, I do not want be accused of just cherry-picking comments by other people out of context just because they might appear to question some aspect of QFT. However, from the perspective of somebody simply interested in the subject, I am beginning to wonder just how many years of maths is now required to even come close to understanding QFT, let alone questioning any of its fundamental premises. As such, it seems that QFT may now extend beyond the reach of most people to quantify for themselves and therefore they must “stand on the shoulders of giants” or, at least, on the shoulders of somebody taller than themselves. However, it seems that any conclusions drawn will still depend on whose shoulders you decide to pick, e.g. see article “The search for a quantum field theory” for a somewhat pessimistic, and possibly outdated, take on the current state of play. Of course, this author, although apparently well qualified, may have simply lost his way and been left behind by leading edge thinking. Therefore, I am assuming that his concerns can now be dismissed?

waterfall

Electromagnetic field and fermionic (or matter) fields are not directly the gauge fields which are unobservable. We can observe the electromagnetic field. Is it possible we just haven't yet invented the technology to detect matter fields? We detect electrons by scattering events and hits in detector. But the more subtle matter fields may need other methods of detection. What would it take to detect them?

Oudeis Eimi

It was sloppy language on my part. What I meant is, the operator-valued fields are
the mathematical models that correspond in the quantum theory to the number-valued
fields of the classical theory.

StevieTNZ

Is this why formulating a Quantum Gravity theory is being problematic?

lugita15

No, as far as we know quantum gravity may not even have a Landau pole, which is the big issue with QED.

It's hard to do exact calculations in any quantum field theory, so in order to get approximate answers we use perturbation theory to get infinite series. But it turns out that most of these series are divergent, so we apply a procedure known as renormalization to get finite results. Renormalization requires knowing the values of so-called "running constants", parameters which must be determined by experiment. Most theories like QED and QCD just require the determination of a few such constants, but quantum gravity requires infinitely many constants, and it's not very practical to do infinite experiments. The hope with theories like string theory is that perhaps there are undiscovered symmetries (e.g. supersymmetry) which would provide relations between these constants so that only finitely many experiments need to be done. Another idea is to somehow to quantum gravity calculations nonperturbatively, and thus avoid the need for renormalization altogether.

BTW, the Landau pole problem with QED is that at very high energies, renormalization fails to give sensible answers. Again, a possible solution to this would be to find a nonperturbative method of calculation.

waterfall

After reading many books on Quantum Field Theory (each in one sitting). I got the feeling that somehow QFT is only an approach for calculational purposes. This means it is not something permanent. Meaning Quantum Field Theory can someday be replaced by others which doesn't involve the quantum field especially the matter field by second quantization. Do you agree with this analysis? Therefore when MySearch asked in the other thread "what is the field in QFT?". Well. I think the fields are just for certain calculational approach and is not something definite like spin and can be replaced someday. Do you agree?

The latest I'm reading is M.Y. Han's book "A Story Of Light: A Short Introduction To Quantum Field Theory Of Quarks And Leptons"

http://www.amazon.com/Story-Light-In...8438503&sr=8-1

Which part of the following do you think is inaccurate and why?

Fredrik

I'm not sure what that would even mean. You're suggesting that quantum fields can be replaced by something else. How? Like when "action at a distance" (of the gravitational force) was replaced by the view that spacetime is a manifold with a metric to be determined from an equation. That gave us a completely different theory. Is that the sort of thing you're talking about? Or are you suggesting that there's a better way to state theories like QED, that may not involve fields? I very much doubt that there is, and even if there is, the fields would still be present in the theory.

Here's a quote from Steven Weinberg:

...it is very likely that any quantum theory that at sufficiently low energy and large distances looks Lorentz invariant and satisfies the cluster decomposition principle will also at sufficiently low energy look like a quantum field theory.

It's from this transcript of one of his talks, but he's also mentioning this idea in his QFT book.

waterfall

So quantum field will be with us forever. I heard QFT could be low energy limit of superstrings or something. So you mean there may be a larger theory but QFT will just be the classical limit of it?

I wonder what else besides Superstrings or M-Theory that can comprise the larger theory....

juanrga

1.
Fock Space is based in Hilbert space.

2.
QM has position as observable because has operator.
QFT has not position operator and position is not obdservable but a dummy unphysical parameter.

Momentum and spin are observables given by operators in QFT

3.
(Non-relativistic) QM uses no relativity.
(relativistic) QFT uses relativity in the sense of using a dummy version of special relativity, where x and t are not measurable.

4.
QED, QCD, and QWD are examples of QFT.

The equivalent of the electromagnetic field in QED for electrons (fermions) is the fermion field.

waterfall

After days of discussions. I know all of them already. But I have new questions.

I heard it said that an electron around a proton or even a travelling single electron can be modelled by QFT. So how does one start to do that? I want to imagine the matter field of electron and proton and how they behave and also the matter field of the single travelling electron. I know QFT is appropriate for an "infinite numbers of particles". But again I heard it can be done for a single or two particles. How?

mysearch

Waterfall, my apologises for posting a general reply to one of your earlier posts, as I can see you are anxious to try to get specific answers to the following questions.
Whether this is entirely possible is unclear to me, if QFT is not a final theory, but I will continue track the thread with interest. However, one of the confusing things about QFT, at least to me, is that at one level it underwrites the standard particle model, which has amassed a huge amount of observational data suggesting that the assumptions of QFT must reflect some tangible aspect of quantum reality, but which I assume we ultimately describe in terms of a classical approximation, e.g. a particle. However, this said, the ‘reality’ of the developing quantum description still seems to be surrounded by much ambiguity, given the level of scientific, mathematical and philosophical conjecture, e.g.

Why Quantum Theory?
“The usual formulation of quantum theory is very obscure employing complex Hilbert spaces, Hermitean operators and so on. While many of us, as professional quantum theorists, have become very familiar with the theory, we should not mistake this familiarity for a sense that the formulation is physically reasonable. Quantum theory, when stripped of all its incidental structure, is simply a new type of probability theory.”

So What are the Fields in QFT? Here is a summary of some suggested descriptions from the thread referenced above.
How do other references define/quantify fields? The first essentially appears to align with Juanrga’s breakdown of the ‘types’ of fields referenced in QFT.

Quote taken from p.41 summarised:
Particles with zero spin, such as pions and the famous Higgs boson, are known as scalars, and are governed by the Klein-Gordon equation. Particles with ½ spin, such as electrons, neutrinos, and quarks, and known as spinors, defined by the Dirac equation. And particles with spin 1, such as photons and the W’s and Z’s that carry the weak charge, and known as vectors discovered by Alexandru Proça. The Proça equation reduces, in the massless (photon) case, to Maxwell’s equations.

However, the tangibility of these fields then seems to recede in the following (1.8) paragraph on p.9
When the word “field” is used classically, it refers to an entity, like fluid wave amplitude, E, or B, that is spread out in space, i.e., has different values at different places. By that definition, the wave function of ordinary QM, or even the particle state in QFT, is a field. But, it is important to realize that in quantum terminology, the word “field” means an operator field, which is the solution to the wave equations, and which creates and destroys particle states. States (= particles = wave functions = kets) are not considered fields in that context.

So at one level, the idea of scalar, spinor and vector fields seems rooted in a mathematical description, although at another level the quantization of the EM field into photon particles almost seems tangible. Of course, one might still have to question the physicality of a photon in spacetime. For example, here are some further clarifications of the idea of a field in QFT taken from this thread:
As a somewhat off-the-cuff thought, has anybody ever attempted to quantify the nature of energy as a scalar quantity and its apparent ability to move in spacetime in terms of some fundamental ability of spacetime to be distorted? In part, it seems that that general relativity alludes to the idea of curved spacetime, such that we might re-interpret John Wheeler’s original quote:

“Matter tells spacetime how to curve, and spacetime tells matter how to move!”. ->
“Energy tells spacetime how to curve, and spacetime tells energy how to move?”

Please accept this as a question, not as a proposal, but could spacetime itself be the basis of the field?

juanrga

Only some aspects of the single electron or of the electron around a proton can be studied in QFT. There is not such thing as «the matter field of electron and proton» but a field for the electron and other for the proton.

It is not right that QFT is appropriate for an "infinite numbers of particles", because those systems are plagued with infinities, which have to be regularized and renormalized. Such techniques are not general.

waterfall

mysearch, in M.Y. Han book. It is mentioned that the gauge symmetry craze in the 1970s have physicists hooked on QFT because of the electromagnetism U(1) which clued them to electroweak U(1)xSU(2) and strong force is SU(3), this third phase is called the (Lagrangian) gauge field theory. This is what made them forgive QFT having non-interacting fields.. because they think gauge theory can somehow save the day. But I wonder if gauge theory can also be hold on without the path of QFT exactly (does anyone know the answer?). The M.Y. Han book can give you a bird eye view of QFT. If you have other interesting QFT book recommendation which you have read or encountered, let me know. Thanks.

waterfall

To people who only participate in this quantum forum. I learnt from M.Y. Han book that there are 3 phases of development of quantum field theory and how they deal with non-interacting fields. I'll summarize it.

First phase (Early 1950s) - Langrangian Field Theory - based on canonical quantization, success in QED followed by non-expandability in the case of strong nuclear force and by non-renomalizability in the case of weak nuclear force.

Second phase (1950s-1960s) - Axiomatic QFT - for example S-Matrix theories and other axiomatic approaches, however they did not bring solutions to quantum field theories any closer than the Lagrangian field theories.

Third phase (1970s) - (Lagrangian) gauge field theory - ongoing

My question is. Can you make use of Gauge Theory without using Quantum Field Theory? Or the two completely related? But noether theorem can be applied to newtonian physics so can the gauge symmetry concept of electromagnetism U(1), electroweak U(1)xSU(2), Strong SU(3) can be developed without using the concept of quantum field theory?

Fredrik

I don't think the idea of gauge fields is useful in classical field theory. Take electrodynamics for example. How do you modify the classical theory of an electromagnetic field in Minkowski spacetime to make it gauge invariant? By introducing the electron/positron field, which is a spin-1/2 field. I think the gauge fields are always fermionic (half-integer spin) and that this is what makes them useless in a classical context.

Edit: This is obviously wrong. I realized that after seeing atyy's post. See my correction in post #54.

waterfall

Classical electrodynamics is already automatically lorentz invavariant. In fact Einstein built the SR from following it.

So I guess gauge invariance is another issue. Are you sure spin 0 and spin 2 can't be properties of gauge invariance but only spin 1/2? How come?

Btw.. in QED.. do they analyze the electric field as coulomb potential or only as virtual particles... like every analysis in QED involves perturbation of particles?

Fredrik

Yes. A gauge transformation isn't a coordinate transformation.

No, I'm not sure because the only gauge theory I've studied is QED, and it was a long time ago. Now that you mention it, gravitons have spin 2. Not sure what that means though. There are a few threads here where the question of whether gravity is a gauge theory is debated. I think the conclusion was that it's not a gauge theory in the traditional sense, but the answer still depends on what exactly you mean by a gauge theory. I still think that what I said is correct, but if someone tells you that I'm not and they sound like they know what they're talking about, they're probably right.

I think it can be treated as a potential in approximate calculations, but as I said, it was a long time ago.