Introduction to relativistic quantum mechanics and maybe QFT

[samalkhaiat]

Indeed, in the standard treatment, you have
$$S^{-1}(\Lambda)\gamma^{\mu} S(\Lambda)={\Lambda^{\mu}}_{\nu} \gamma^{\nu},$$
where $S(\Lambda)$ is the bi-spinor representation of the Lorentz group and $\Lambda$ a Lorentz-transformation matrix. Written the 6 parameters of the Lorentz group in the usual way as $\omega_{\mu \nu}=-\omega_{\nu \mu}$ it turns out that
$$S(\Lambda)=\exp \left (-\frac{1}{4} \omega_{\mu \nu} \sigma^{\mu \nu} \right) \quad \text{with} \quad$$
$$\sigma^{\mu \nu}=\frac{\mathrm{i}}{2} [\gamma^{\mu},\gamma^{\nu}].$$
All this should be consistent with what samalkhaiat wroten in #41, maybe in a somewhat different convention, but I haven't checked this carefully.

Yes, I used 2-spinor notation. The group-theoretic treatment is generally clearer in the 2-spinor notation. This is because $\chi_{A} \in (1/2 , 0)$ and $(\varphi_{A})^{*} \equiv \bar{\varphi}_{\dot{A}} \in (0 , 1/2)$ are the irreducible representations of the Lorentz group, while Dirac bispinors are reducible $$\Psi = \chi_{A} \oplus \ \epsilon^{\dot{A}\dot{B}}\bar{\varphi}_{\dot{B}} .$$ The transformation laws for the fundamental spinor and it’s conjugate are given by $$\chi_{A} \to S_{A}{}^{B} \ \chi_{B} = \left( e^{- \frac{i}{2}\omega_{\mu\nu}s^{\mu\nu}} \right)_{A}{}^{B} \ \chi_{B} ,$$
$$\bar{\varphi}^{\dot{A}} \to \bar{S}^{\dot{A}}{}_{\dot{B}} \ \bar{\varphi}^{\dot{B}} = \left( e^{- \frac{i}{2}\omega_{\mu\nu}\bar{s}^{\mu\nu}} \right)^{\dot{A}}{}_{\dot{B}}\ \bar{\varphi}^{\dot{B}} ,$$ where $$(s^{\mu\nu})_{A}{}^{B} = \frac{i}{4} ( \sigma^{\mu}\bar{\sigma}^{\nu} - \sigma^{\nu} \bar{\sigma}^{\mu})_{A}{}^{B} ,$$ and $$(\bar{s}^{\mu\nu})^{\dot{A}}{}_{\dot{B}} = \frac{i}{4}(\bar{\sigma}^{\mu}\sigma^{\nu} - \bar{\sigma}^{\nu}\sigma^{\mu})^{\dot{A}}{}_{\dot{B}} ,$$ are the corresponding Lorentz generators. From these transformations, and with no reference to Weyl or Dirac equations, we obtain the transformation law for the Dirac bispinor $$\Psi \to ( S \otimes \bar{S} ) \Psi = (e^{-\frac{i}{2}\omega_{\mu\nu}(s^{\mu\nu} \oplus \bar{s}^{\mu\nu})}) \Psi ,$$ with the generators in block diagonal form
$$\sigma^{\mu\nu} = \begin{pmatrix} s^{\mu\nu} & 0 \\ 0 & \bar{s}^{\mu\nu} \end{pmatrix} = \frac{i}{4}[\gamma^{\mu},\gamma^{\nu}] ,$$ where $$\gamma^{\mu} \equiv \begin{pmatrix} 0 & \sigma^{\mu} \\ \bar{\sigma}^{\mu} & 0 \end{pmatrix} .$$

Sam

 

[Demystifier]

I can mathematically prove all the statements I made including the one you quoted.

I am sure you can. But to prove them, you must use some mathematical axioms, that's how mathematical proofs work. But axioms, as you certainly know, cannot be proved. So one is free to choose different axioms (as long as they are not logically inconsistent), and then prove different statements. And that's what I do; I take different axioms (which I call a different "picture") and obtain different theorems. With different axioms I obtain that new gamma matrices do depend on the Lorentz frame.

From the physical point of view I show that the new axioms lead to the same physics because the new axioms do not affect the physical currents such as the Dirac current.

 

[microsansfil]

But axioms, as you certainly know, cannot be proved.

Yes we can. A an axiom of the Theory T, then A |-- A ; prove A (A is a theorem). I agree this tautology isn't useful to physics.

Patrick

 

[Demystifier]

Yes we can. A an axiom of the Theory T, then A |-- A ; prove A (A is a theorem). I agree this tautology isn't useful to physics.

It's not useful even in mathematics (except in pure logic, which some mathematicians (such as Poincare) do not even consider to be a part of "real" mathematics).

 

[Demystifier]

Already classically, a multi-particle relativistic picture is inconsistent.

I took a look at the paper by Currie, Jordan and Sudarshan that you suggested, and also at the book
https://www.amazon.com/dp/0521143624/?tag=pfamazon01-20
(Sec. 16.4 The no-interaction theorem in classical mechanics).

What appears to be inconsistent is not to have relativistic interacting particles, but to have only relativistic interacting particles. It is still possible to have a classical relativistic interaction that involves particles and fields. This, indeed, is what we have in classical electrodynamics, but also, in a certain sense, in Bohmian mechanics.

 

[vanhees71]

Yes, I used 2-spinor notation. The group-theoretic treatment is generally clearer in the 2-spinor notation. This is because $\chi_{A} \in (1/2 , 0)$ and $(\varphi_{A})^{*} \equiv \bar{\varphi}_{\dot{A}} \in (0 , 1/2)$ are the irreducible representations of the Lorentz group, while Dirac bispinors are reducible $$\Psi = \chi_{A} \oplus \ \epsilon^{\dot{A}\dot{B}}\bar{\varphi}_{\dot{B}} .$$
Sam

Sure, this makes the Lie algebra/group structure clearer. If I remember right, the Dirac representation is a reducible representation for $\mathrm{SO}(1,3)^{\uparrow}$, but irreducible, if you take space reflections (parity) in addition, i.e., as a representation for $\mathrm{O}(1,3)^{\uparrow}$. Thus you have Dirac spinors in QED and QCD, because the electromagnetic and the strong interaction respect parity symmetry. The approximate chiral symmetry in QCD in the light-quark sector is an "accidental gift" for hadron-model builders :-), and parity is not broken.

Of course for the weak interaction, parity is broken, and the left- and right-handed quark transform differently under the gauge group (as a WISO doublet and singulet, respectively).

 

[Demystifier]

The textbooks treatments are correct and consistent with every thing I said.

Please answer each of the following questions by either "yes", "no", or "I don't know"!
1. Do textbooks (which consider spinors in curved spacetime) say that there is gamma matrix which transforms as a vector under general coordinate transformations? (yes/know/I don't know)
2. Is Lorentz transformation a special case of a general coordinate transformation? (yes/know/I don't know)
3. If both answers above are "yes", does it logically imply that (according to those textbooks) there is gamma matrix which transforms as a vector under Lorentz transformations? (yes/know/I don't know)

 

[Demystifier]

Another question is the issue about QFT in a GRT background space-time. That's much more complicated, even for free fields/particles. I'd not consider this a standard topic for the introductory QFT lecture.

By the same kind of argument, one might say that QFT is much more complicated than relativistic QM, even for free fields/particles.
Just as you point out that there are still good reasons to teach QFT before relativistic QM, similarly one may point out that there are also good reasons to teach QFT in curved spacetime before QFT in flat spacetime.

In particular, to learn QFT in curved spacetime you must unlearn some overemphasized concepts that looked so fundamental in flat spacetime. For instance, Poincare invariance is no longer fundamental for definition of particles, the number of particles is no longer observer independent, the gamma matrix is no longer invariant under coordinate transformations (including the Lorentz ones), and so on...

My point is - there is no one correct way to teach physics. There are many complementary ways to do that, each with its advantages and disadvantages.

 

[A. Neumaier]

It is still possible to have a classical relativistic interaction that involves particles and fields.

Yes, but once you admit classical fields as fundamental, it is more parsimonious to reard only these as fundamental. You get classical particles for free as the idealization where a field is supported on worldlines only. These are usually not exact solutions of the field equations, whence the particle concept is approxiamte only. Indeed, nothing hat we know about real particles suggests that they should be moving points.

 

[Jano L.]

Yes, but once you admit classical fields as fundamental, it is more parsimonious to reard only these as fundamental. You get classical particles for free as the idealization where a field is supported on worldlines only. These are usually not exact solutions of the field equations, whence the particle concept is approxiamte only. Indeed, nothing hat we know about real particles suggests that they should be moving points.

The description of interaction in the particle theory via fields does not necessarily mean fields are "fundamental", although I am not sure what you meant by this. One possible view is that charged particles are material points and their *interaction* has advantageous description in terms of fields, or less advantageously and less generally, in terms of particular solution to Maxwell equations.

The view that all particles are just continuous fields concentrated to small region of space is common and deserves exploration, but it is mathematically problematic.

For example, in classical physics, having one or few fields only without reference to discrete particles more often than not leads to UV catastrophe. Even if we assume there is some acceptable configuration of the field with finite energy, there does not seem to be a way to convincingly choose a model of relativistic continuum out of endless possibilities available that would faithfully describe what we know as structureless electron.

On the other hand, theories of particles with interaction are free of this problem (Newtonian gravity, EM theory of charged particles considered by Frenkel, Tetrode, Fokker where there is no self-interaction). Also Wheeler-Feynman, although their absorber condition is unrealistic and has questionable benefits.

 

[A. Neumaier]

there does not seem to be a way to convincingly choose a model of relativistic continuum out of endless possibilities available that would faithfully describe what we know as structureless electron.

We don't know of a ''structureless electron'' - it is a theoretical idealization of the real electron, which has nontrivial form factors, hence a nontrivial structure. It is precisely this difficulty that you point out that makes point particles unrealistic and problematic. Field theory itself leads classically to no catastrophe, only the assumption of truly pointlike (and hence singular) sources does.

 

[Jano L.]

We don't know of a ''structureless electron'' - it is a theoretical idealization of the real electron, which has nontrivial form factors, hence a nontrivial structure.

All mathematical concepts of theoretical physics are theoretical idealizations of real things, including quantum field and scattering form-factor. Structure-less electron is not a particularly deficient concept in this respect.

Much of the knowledge we have on electricity can be explained with taking basis in the idea of structure-less point particle. This concept has been incredibly fruitful (classical mechanics, Schroedinger's equation) and has its place in physics. It also has certain mathematical advantages when compared to field model of matter. In particular, it has finite number of degrees of freedom, while the matter field model leads to infinite number of degrees of freedom, inevitably leading to hard-to-resolve issues with mathematical consistency and consistency with the rest of physics.

Form-factors also have their place in the world of idealized concepts, especially for fitting approximate many-parameter models to results of complicated experiments. It would be incorrect to claim everything can be currently explained with point electrons, but it is also unwise to take the success of the form-factor scattering calculations as a basis for what the "real electrons are". There are experiments that were used to estimate the size of the charge distribution of the electron and they have consistently lowered down the upper limit, currently to $10\text{E}-18$ m or so I recall, certainly less than the classical electron radius or the Compton wave length. In such a situation it is not unreasonable to consider models of an electron where it is a point, structure-less particle.

It is precisely this difficulty that you point out that makes point particles unrealistic and problematic. Field theory itself leads classically to no catastrophe, only the assumption of truly point-like (and hence singular) sources does.

Indeed, field theory by itself does not lead to catastrophe. One is lead to UV catastrophe in a specific but important case. Let's ignore this part for a while though.

The difficulty with matter field as a model for electron I am pointing out is that once there is infinity of degrees of freedom, there isn't really any convincing way to formulate a definite model of internal state evolution of such a continuum. I believe it is possible to find such models, but I do not see a way to select one that would be any better that the others.

 

[vanhees71]

The classical point-particle concept is only simple with pretty crude approximations, and it works quite well thanks to the weakness of the electromagnetic interaction. A "classical electron" can in some cases be treated as a classical point particle, but it's not structureless, because it has electric charge and a magnetic moment. It's inevitable that (in the standard picture of local interactions in relativity theory) it has both an electric (Coulomb) field and a magnetic (dipole) field, defined in an inertial frame, where the electron is at rest (or in motion with constant velocity). That "lonely" electron is the only case, where we can make sense of it in a fully self-consistent way, i.e., you have a full solution of the dynamical system consisting of the single electron and its electromagnetic field.

If you now consider the motion of an electron in an external electromagnetic field (which is due to charge distributions around different from the single electron) it gets accelerated and the electromagnetic field of the electron becomes one consisting of the above described self-field of the electron plus a radiation field due to the acceleration of the charge and magnetic moment carried by it. This radiation in turn acts back on the electrons accelerated motion, and at this point all hell breaks loose. In my opinion, there's not a satisfactory exact solution to this problem but only approximate ones with the Landau-Lifshitz modification of the old Abraham-Lorentz treatment, avoiding artificial effects like self-acceleration.

The reason for this is that the point-particle "idealization" cannot be made rigorous in classical physics but introduces problems. From a microscopic point of view that's understandable, because classical physics can only deal with these phenomena in a "coarse-grained" way, and a point charge of 0 extent must be seen as a small body of finite extent, where the scale of its size is negligible against the resolution of the classical observables like its position, and indeed one can find a better description of radiation reaction, introducing appropriate stresses a la Poincare and von Laue, for a compact charged object of finite extent. The resulting equation of motion, however, is far from a simple relativistic point-particle equation (as the one one gets in the usual text-book treatment of "point charges" moving in external electromagnetic fields when neglicting all the radiation reaction). It's a differential-difference equation which is non-local in time, but one can show that it is well-behaved (at least for the most simple "point-particle models" like a charge sphere or spherical shell).

Quantum field theory is in somewhat better shape than the classical point-particle model, because with ("soft-photon resummed" renormalized) perturbation theory you can at least define a clear scheme to find approximate solutions. As far as I know also here we still do not have a fully selfconsistent non-perturbative solution nor a mathematical proof that one exists.

 

[Jano L.]

... A "classical electron" can in some cases be treated as a classical point particle, but it's not structureless, because it has electric charge and a magnetic moment.

Classical point electron does not have magnetic moment, I believe. Magnetic moment was introduced for extended models of electron (charged rotating sphere) and in quantum theory.

This radiation in turn acts back on the electrons accelerated motion, and at this point all hell breaks loose. In my opinion, there's not a satisfactory exact solution to this problem but only approximate ones with the Landau-Lifshitz modification of the old Abraham-Lorentz treatment, avoiding artificial effects like self-acceleration.

The only satisfactory solution I know is modifying the premise; radiation of point particle does not act back on the point particle. Assuming it does has lead to contradictions nobody was able to resolve.

The reason for this is that the point-particle "idealization" cannot be made rigorous in classical physics but introduces problems.

The idea of limiting charged sphere to point introduces problems. If we begin with point particles right from the start and do not allow self-interaction in the first place, none of those problems arise. Self-interaction is not necessary to explain known experiments and it only brings problems.
Consistent theories of charged point particles were described many times in the past. The first case I know of is the paper by Frenkel:

J. Frenkel, Zur Elektrodynamik punktfoermiger Elektronen, Zeits. f. Phys., 32, (1925), p. 518-534.http://dx.doi.org/10.1007/BF01331692

In English, this article also explains it concisely:

R. C. Stabler, A Possible Modification of Classical Electrodynamics, Physics Let- ters, 8, 3, (1964), p. 185-187. http://dx.doi.org/10.1016/S0031-9163(64)91989-4

Quantum field theory is in somewhat better shape than the classical point-particle model, because with ("soft-photon resummed" renormalized) perturbation theory you can at least define a clear scheme to find approximate solutions. As far as I know also here we still do not have a fully selfconsistent non-perturbative solution nor a mathematical proof that one exists.

Could you please give some references which discuss these issues in quantum field theory?

 

[vanhees71]

You can take everything concerning the magnetic moment of a classical electron out of my privious posting. The point I wanted to make becomes also clear for a fictitious charged point particle without magnetic moment (although there's no such thing in nature since all (pseudo-)scalar bosons like the pion are not pointlike to begin with). The problems remain.

Of course, radiation reaction is very real and must be taken into account, at least when constructing electron accelerators. Synchrotron radiation is either unwanted, if you want to accelerate electrons to very high energies, and thus one better uses linear colliders rather than accelerator rings for electrons, or it is what you are after to get specialized light sources for materials and biological research (like the Free Electron X-Ray Laser at the Helmholtz center in Hamburg (DESY), which turned from a HEP lab to one applying highly coherent X-rays).

 

[Jano L.]

The point I wanted to make becomes also clear for a fictitious charged point particle without magnetic moment (although there's no such thing in nature since all (pseudo-)scalar bosons like the pion are not pointlike to begin with). The problems remain.

If your point is there is no consistent self-interaction theory for point particles , I agree with you. If the point is there is no way a useful theory with point particles can be constructed, I don't think that is a necessary viewpoint.

Of course, radiation reaction is very real and must be taken into account, at least when constructing electron accelerators.

Indeed. In electron accelerators, billions and more particles move together in so-called bunches and their retarded fields are highly correlated. The bunch radiates and loses energy and it would slow down and dissipate if the acceleration cavities were not actively managing their motion state.

This does not provide any evidence that individual particles experience force due to their own field or that they would lose energy if accelerated on their own. In a bunch, billions of particles interact via their individual EM fields and it is easy to see retarded interaction of the particles has the result of an effective drag.

 

[vanhees71]

I'm far from claiming this. Classical "electron theory" is, e.g., successfully used to construct high-energy acclerators like the LHC! I only say that there is no fully consistent classical theory of interacting point charges (and even not a fully consistent relativistic quantum theory of interacting quanta in 4 space-time dimensions).

 

[Jano L.]

I'm far from claiming this. Classical "electron theory" is, e.g., successfully used to construct high-energy acclerators like the LHC! I only say that there is no fully consistent classical theory of interacting point charges (and even not a fully consistent relativistic quantum theory of interacting quanta in 4 space-time dimensions).

There is self-consistent theory of point particles! Check out the links I gave above - the papers of Frenkel and Stabler.

 

[Dale]

This thread has seriously drifted. It is closed now.