QFT vs QM 101

waterfall

Thanks. I'm presently reading on Teller "Interpretative Introduction to QFT". The above will be very useful as we go to the heart of what really is QFT.

Fredrik

What I'm saying is that you need at least three lines meeting at a point to have an interaction. For example, the diagram representing two electrons exchanging a photon looks like an H. There are two points where three lines meet. If there are no points where three or more lines meet, then all your diagrams look like this: | Such diagrams are present in interacting theories too, but they're ignored because they don't contribute to anything observable, except the energy density of the vacuum.

I don't think that's a good way to think about quantum fields. Neither Schrödinger's theory nor any QFT says that particles have positions, so in my opinion, neither should we.

The_Duck

In noninteracting theories, there are no vertices in Feynman diagrams. As a result the only Feynman diagrams you can draw consists of a bunch of straight lines that don't touch each other, representing particles that simply travel along without interacting with each other. You can see why this is a simple but boring kind of theory.

waterfall

Why, doesn't Fock space involve this 3 lines meeting at a point or standard Feynmann Diagram with interaction? Are you (and The_Duck) saying Fock space just involves noninteracting vertical lines? Is this related to perturbation theory which is Fock space pretending to have interaction when it doesn't really?


Oh. So quantum fields don't have particles before measurement. If this is so. Then quantum field is just like the classical electromagnetic field with only photons appearing during measurement? Then what's the advantage of QFT? I thought it involves particles being created and annihilated as part of the quantum field.

Fredrik

A Fock space is constructed from the Hilbert space associated with the single-particle theory. You use the single-particle space to construct a space of 2-particle states, a space of 3-particle states, and so on, and then you combine them all into a Hilbert space that contains all the 1-particle states, all the 2-particle states, and so on. This Hilbert space is called a Fock space. So it's just an algebraic construction. You need nothing more than the Hilbert space from the single-particle theory to define it, and the single-particle theory can be defined using a Lagrangian with no products of more than two field components or derivatives of field components.

However, in non-rigorous QFT, I think the idea is just to ignore that the interacting Hilbert space is really a different Hilbert space, and just introduce operators that can take n-particle states to (n+1)-particle states for example. In this context, Fock space is, as you put it, "pretending to have interaction when it doesn't really". I really suck at QFT beyond the most basic stuff, so I can't explain it better, and I might even be wrong (about the stuff in this paragraph).

I didn't say that. I said that the theory doesn't give us any reason to think that particles have positions. A position operator can be defined for massive particles, but it's kind of weird. I suppose we could use it to define "approximately localized" states, in a way that's similar to how its done in Schrödinger's theory. But there's no position operator for massless particles.

Anyway, "particle" doesn't mean "classical particle", so you can't assume that something has properties like position just because a quantum theory calls it a "particle".

atyy

What do you think of this comment in Collins's notes? He does acknowledge that a point of view different from his is more common.
http://www.phys.psu.edu/~collins/563/LSZ.pdf: "Note that in both formulae, the vacuum state |0> is very definitely and strictly the true vacuum. This is just the same as in the definition of the coefficient c, (3), where the vacuum and one-particle states are definitely the true vacuum and one-particle states, i.e., the true physical states. In contrast, many textbook treatments appear to suggest that the state |0> should be the free-field unperturbed vacuum; if that approach is tried, very delicate limits involving adiabatic switching of the interaction are called for."

Edit: Here's another presentation by Srednicki that starts off with the more common point of view, but he goes on to discuss that it's wrong, and says that renormalization computes the corrections to having started the calculation with the wrong ground state. http://web.physics.ucsb.edu/~mark/ms-qft-DRAFT.pdf, p51: "However, our derivation of the LSZ formula relied on the supposition that the creation operators of free field theory would work comparably in the interacting theory. This is a rather suspect assumption, and so we must review it."

Fredrik

I don't know QFT well enough to answer that, so I'll leave it for someone who does.

waterfall

Are you saying not all physicists with Ph.D. are experts in QFT? I thought they all wer. But using Fock space noninteracting terms, how could they make the Large Hadron Collider function and successfully predict those scattering angles and interactions of the numerous particles. Is Fock space enough to analyze them including predicting the mass of the Higgs? Or do Large Hadron Collider physicists use purely rigorous QFT that normal physicists don't tackle?


So how does one imagine a quantum field? I thought it should have particles vibrating like harmonic oscillator.. but now saying particles don't have position.. then how does one picture it? Or is it possible space and time only occur during interaction with the quantum field, and without interaction, space and time doesn't really exist as we know it in the quantum field? And it is just a blob of untime and unspace?

Fredrik

They are not, but most of them have at least taken a QFT course. But that's not the point I was trying to make when I mentioned rigorous QFT. The point was questions like what the Hilbert space of the interacting theory is aren't answered in typical QFT courses, or typical QFT books. Actually, I don't think anyone even knows how to properly define the Hilbert space for QED in 3+1 dimensions. (Maybe they know that and are still struggling with other things, but they're struggling with something, because I know that no rigorous version of QED in 3+1 dimensions has been found).

I suspect that even some QFT experts don't know rigorous QFT. It's like an entirely different field of physics. A typical student at an "introduction to QFT" course would probably need two more years of math before he can really begin to learn rigorous QFT.

They do approximate calculations using the lowest orders of Feynman diagrams, and don't worry much about the behavior of the entire series.


I think most LHC physicists work on the hardware components and with data analysis. They are more likely to be good at programming than at QFT. But the theoretical particle physicists know QFTs of course. I don't know if they use rigorous methods much. I suspect that they don't. It would surprise me if they use them a lot.

I don't know if there is a way.

atyy

I think it's believed that QED is fundamentally unsound - it is inconsistent at high energies. Strictly speaking, there's no proof of that since it's only perturbatively unsound. Anyway, this belief of unsoundness is called the "Landau pole". At the same time, the renormalization flow having a low energy fixed point explains why such an inconsistent theory is still usable.

QCD is believed to be completely consistent. It's still a Clay problem, but you can see that they do make use of axiomatic field theory. For example, Gupta's notes (p23) say that QCD has Osterwalder-Schrader reflection positivity. This is a condition for the analytic continuation of a Euclidean theory to meet the Wightman axioms, which is constructive field theory.

waterfall

Wikipedia entry on QFT is wrong then, it depicts things as almost complete and rosy. For example the following words are not right:

http://en.wikipedia.org/wiki/Quantum_field_theory

Wiki:"Quantum field theory is thought by many[who?] to be the unique and correct outcome of combining the rules of quantum mechanics with special relativity."

Fact: it is not exactly correct as you emphasized.

Wiki:"In perturbative quantum field theory, the forces between particles are mediated by other particles. The electromagnetic force between two electrons is caused by an exchange of photons. Intermediate vector bosons mediate the weak force and gluons mediate the strong force. "

Fact: Fock space doesnt handle interactions so those pertubative approach are just temporary and is fundamentally invalid"

Wiki:"In QFT, photons are not thought of as "little billiard balls" but are rather viewed as field quanta – necessarily chunked ripples in a field, or "excitations", that "look like" particles."

Fact: Particles dont have positions so they are not really excitations of the field. One must not visualize it that way.

Agree with everything? Maybe its time to correct Wiki and state things are not that rosy and indeed bleak.

waterfall

I read the Mysearch shared site in http://www.quantumfieldtheory.info/Chap01.pdf and need to ask a critical question:

"1.8 Points to Keep in Mind 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. "

Why not call it Quantum Operator Theory instead of Quantum Field Theory as the above fact showed that the Field in QFT was not related to the classical field. I thought QFT was just about performing canonical quantization on the classical field. Or could be this true only to QED? Isnt QED about performing quantization on the electromagnetic field?

lugita15

Yes, in some sense we are dealing with harmonic oscillators, but these are quantum simple harmonic oscillators, not classical SHO. Just as a particle in nonrelativistic quantum mechanics, like a quantum harmonic oscillator, does not have a definite position but only a probability of being measured at different positions, in quantum field theory there is no definite position associated with field quanta AKA particles.

waterfall

When you mentioned "field quanta", are you referring to operator field quanta or actual field quanta? This is because as detailed in message 29, the field in QFT are field operator, not the usual field we understood as electromagnetic field for example.

Oudeis Eimi

Most physicists work in condensed matter, not particle or high-energy physics. They have some
knowledge of QFT (mostly the non-relativistic kind) as part of their training in QM, but needn't be
experts in the mathematical foundations of QM.

The particles are excitations, the most basic 'vibration states' of the fields. When we say they don't
necessarily have a position, what we mean, in layman's language, is that those 'basic vibrations'
aren't confined to a single point in space. Note however that they may (but don't NEED to be)
confined to a very tiny region from our macroscopic point of view. This is completely analogous to
the case of non-relativistic ordinary QM.

As to how to visualise a quantum field... well, quantum operators behave a lot like stochastic
variables. They have an expectation value and a complete set of moments which give you the
indeterminacy of said expectation value. So in principle, any such operator can be visualised as
a 'fuzzy' quantity, centered around the expectation value and with the fuzziness being proportional
to the indeterminacy. So for the case of a field, it's a 'fuzzy' field.

As a visualisation technique, this is probably only useful for bosonic fields in states such that
the indeterminacy is much smaller than the expectation value. This is the case for instance for
the electromagnetic field in most ordinary cases. Fermionic fields OTOH don't have a classic
limit and are thus much harder to visualise.

Oudeis Eimi

No, not agreed. Perturbative techniques work well within their range of applicability. They're not the
ideal solution, but are necessary for those cases where the full solution to the problem isn't available.
Note such techniques are /extensively/ used across both pure and applied physics (including
engineering). For instance, we don't have a general solution for the N-body problem, so we need
to resort to approximations like numerical and/or perturbative methods.

This doesn't follow. Your conclusion is invalid.

Oudeis Eimi

Waterfall, a quantum field is a quantum 'quantity'. In the formalism of quantum physics, these are
operators (or POVMs, which are a related but more complicated object). The 'actual' field IS the
'operator' field.

I'll give you two examples: the total momentum of a system, P, and the electromagnetic field, A.
In CLASSICAL physics, these, or their components in some reference frame, are numbers.
P = {Px, Py, Pz}; A = {phi, Ax, Ay, Az}.

In QUANTUM physics, these are operators. That's a more complicated kind of object. An important
difference with the above case is, operators don't have a value by themselves. This is where the
state comes in in the theory. Quantum states give operators their values (and their indeterminacy).

So, while in classical physics you have A=A(x,y,z,t) as a vector with a definite value assigned
to every point (x,y,z), in quantum physics you have A=A(x,y,z,t) as an operator field, that is,
an operator assigned to every point of space (and time). Once you're given a state you can
assign a value (actually, an expectation value and an indeterminacy) to those operators. If the
indeterminacy is sufficiently small, it can be ignored and you recover the classical field (this
can only happen for fields which do possess a classical limit, of course. The em field does.)