
#37
Feb412, 07:18 PM

P: 66

the mathematical models that correspond in the quantum theory to the numbervalued fields of the classical theory. 



#38
Feb412, 10:05 PM

PF Gold
P: 777





#39
Feb412, 11:26 PM

P: 1,583

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 socalled "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. 



#40
Feb512, 06:52 AM

P: 381

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/StoryLightIn...8438503&sr=81 Which part of the following do you think is inaccurate and why? 



#41
Feb512, 07:33 AM

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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. 



#42
Feb512, 07:44 AM

P: 381

I wonder what else besides Superstrings or MTheory that can comprise the larger theory.... 



#43
Feb512, 08:06 AM

P: 476

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. (Nonrelativistic) 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. 



#44
Feb512, 08:12 AM

P: 381

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? 



#45
Feb512, 09:36 AM

PF Gold
P: 513

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. 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 KleinGordon 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: “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? 



#46
Feb512, 04:34 PM

P: 476

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. 



#47
Feb512, 04:36 PM

P: 381

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 noninteracting 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.




#48
Feb512, 05:52 PM

P: 381

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 noninteracting fields. I'll summarize it.
First phase (Early 1950s)  Langrangian Field Theory  based on canonical quantization, success in QED followed by nonexpandability in the case of strong nuclear force and by nonrenomalizability in the case of weak nuclear force. Second phase (1950s1960s)  Axiomatic QFT  for example SMatrix 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? 



#49
Feb512, 07:11 PM

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PF Gold
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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 spin1/2 field. I think the gauge fields are always fermionic (halfinteger 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. 



#50
Feb512, 07:24 PM

P: 381

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? 



#51
Feb512, 08:01 PM

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PF Gold
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#52
Feb512, 08:05 PM

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#53
Feb512, 08:40 PM

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P: 8,006

The electromagnetic field is a gauge field when potentials are used, as they are QFT.
The more common definition of a gauge field just means that several different ways of naming the field are physically equivalent. So electric potential in circuit theory has a gauge invariance in this sense  it is only potential difference that is physical, the potential itself can be shifted arbitrarily. In the same sense, the diffeomorphism invariance is a gauge invariance  metrics that are related by diffeomorphisms are physically equivalent. This is why you will see the term "de Donder gauge" with reference to classical general relativity. There is a second different definition of a gauge field as the connection on a bundle, and gravity is not a gauge field in this sense. 



#54
Feb512, 08:51 PM

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Ah, I was confused about the most important detail. I probably shouldn't be posting this late at night. I remembered that QED is found by taking one theory and adding another field to make the theory gauge invariant. But I was thinking that this process adds the Dirac field to electromagnetism, when in fact it's the other way round. You start with the Lagrangian for a noninteracting Dirac field, note that it's not gauge invariant, and add a vector (spin1) field with special properties to get a theory that is gauge invariant. This vector field is the electromagnetic 4potential.



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