Multiparticle Relativistic Quantum Mechanics in an external potential

[curious_mind]

TL;DR Summary

It is often argued that Dirac Equation is not valid as relativistic quantum mechanics requires the creation of antiparticles. But, there are also some arguments that suggest otherwise. For example, I saw Arnold Neumaier's website on this that there are multiparticle relativistic quantum mechanics also exist.

It is often argued that Dirac Equation is not valid as relativistic quantum mechanics requires the creation of antiparticles. But, there are also some arguments that suggest otherwise. For example, I saw Arnold Neumaier's website on this that there are multiparticle relativistic quantum mechanics also exist.

I would like to list some references on this:

[1] https://arnold-neumaier.at/physfaq/topics/multi.html
[2] https://www.physicsforums.com/threads/qft-vs-relativistic-quantum-mechanics.493656/ (Not open for reply, this is why starting new thread to put up my question)
[3] https://www.physicsoverflow.org/42599/whats-the-role-of-the-dirac-vacuum-sea-in-quantum-field-theory
[4] https://www.physicsoverflow.org/27619

I read about Prof. Arnold Neumaier's stand for relativistic quantum mechanics. However, in [1] it is clearly written that it is still somewhat not clear how to incorporate external potential in relativistic quantum mechanics.
Does that mean that Klein Paradox is not resolved by Relativistic Quantum Mechanics ? Do Relativistic QFT handles external potentials consistently ?

Any suggestions would be helpful. Thanks.

 

[vanhees71]

Strictly speaking there's no consistent first-quantized version of relativistic QT, and indeed that's due to the possibility of creation and annihilation processes for interacting particles. To make relativistic QT consistent with relativistic causality the only working theories are local relativistic QFTs, where local means that the microcausality constraint on local observables is fulfilled, i.e., operators, which represent local observables, must commute with space-time arguments, that are space-like separated. This holds true particularly for all commutators of any local observable with the Hamiltonian density, which itself also must be a local observable. This ensures that there are no faster-than-light signals and thus no causal connections between space-like separated local observations.

The microcausality constraint leads to very fundamental conclusions:

-there are always particles and antiparticles, described by local Hamiltonians; there's the possibility that particles and antiparticles are the same (socalled strictly neutral particles, e.g., photons)
-particle numbers are not conserved in interacting theories but only conserved charges like the electric charge or net-baryon number
-spin-statistics relation: Particles with integer (half-integer) spin are described by bosonic (fermionic) QFTs
-CPT theorem: Any local relativistic QFT that obeys the covariance under the proper orthochronous Poincare group (which makes the theory consistent with the special-relativsitic spacetime model, i.e., Minkowski space) is also covariant under the CPT transformation. This means that for any possible reaction also the reaction, where all particles are substituted with their anti-particles, and space reflection and time-reversal is also a possible reaction. NB: All other of the "discrete" spacetime symmetries like P, T, C, CP, PT, CT, are shown to be violated by the weak interactions

For a detailed and thorough discussion of all these foundations, see

S. Weinberg, Quantum Theory of Fields, vol. 1

Particularly there's an entire chapter on the question, in which sense the "external-potential approximation" can be derived from relativistic local QFT.

 

[curious_mind]

Thank You sir for the detailed explanation, but I still have doubts.

I am aware of Weinberg's first volume and I know whatever I am looking for is there in probably the first five chapters - but as per the information I have about Weinberg's textbook, probably it is not giving an answer whether there can be the valid relativistic quantum mechanical formulation for fixed particles or not.

[Actually, first I started QFT from Peskin's book, then looked at Srednicki, and I think I understand stuff till renormalization in scalar fields - and compared to that when I looked at Weinberg I was really surprised at the fact that - what is the need for these many chapters just to introduce fields from particles !! That's why I searched for relevant stuffs on various sources. ]

As per information I read from various sources(mostly cited in question), the conditions you described are proven in Weinberg's text as sufficient conditions and not necessary ones. Please correct, if I am wrong. I think Neumaier's posts also suggest the same
https://arnold-neumaier.at/physfaq/topics/multi.html.

I have only information regarding these issues and have not studied weinberg properly therefore I am not very eligible to argue further. The reason for asking this question is, I want to know, whether is it possible to explain subtleties like Klein Paradox in relativistic quantum mechanics, or in general multiparticle relativistic quantum mechanics in presence of an external field, and yes, I am not talking about interactions for now.

Strictly speaking there's no consistent first-quantized version of relativistic QT,

Do you mean is it true for free multi-particle as well ? If so, again, is not it contrary to the opinion than in https://arnold-neumaier.at/physfaq/topics/multi.html ?

 

[vanhees71]

Thank You sir for the detailed explanation, but I still have doubts.

I am aware of Weinberg's first volume and I know whatever I am looking for is there in probably the first five chapters - but as per the information I have about Weinberg's textbook, probably it is not giving an answer whether there can be the valid relativistic quantum mechanical formulation for fixed particles or not.

It's an approximation, where the situation is not too far from the non-relativistic approximation.

[Actually, first I started QFT from Peskin's book, then looked at Srednicki, and I think I understand stuff till renormalization in scalar fields - and compared to that when I looked at Weinberg I was really surprised at the fact that - what is the need for these many chapters just to introduce fields from particles !! That's why I searched for relevant stuffs on various sources. ]

That's the disadvantage when you start from a particle picture. Nevertheless Weinberg's book is the most thorough and consistent treatment of the subject. At the same level and complementary is

A. Duncan, The conceptual framework of quantum field
theory, Oxford University Press, Oxford (2012).

As already in the classical realm a consistent relativistic dynamics of point particles is possible only in a very limited sense, indeed the natural starting point is field theory and thus also for quantum theory field quantization. Thus, for me the best introductory textbook is

S. Coleman, Lectures of Sidney Coleman on Quantum Field
Theory, World Scientific Publishing Co. Pte. Ltd., Hackensack
(2018), https://doi.org/10.1142/9371

As per information I read from various sources(mostly cited in question), the conditions you described are proven in Weinberg's text as sufficient conditions and not necessary ones. Please correct, if I am wrong. I think Neumaier's posts also suggest the same
https://arnold-neumaier.at/physfaq/topics/multi.html.

I have only information regarding these issues and have not studied weinberg properly therefore I am not very eligible to argue further. The reason for asking this question is, I want to know, whether is it possible to explain subtleties like Klein Paradox in relativistic quantum mechanics, or in general multiparticle relativistic quantum mechanics in presence of an external field, and yes, I am not talking about interactions for now.

The Klein paradox is the prime example that a first-quantization view is at best unnecessarily complicated. I'd not recommend to study socalled "relativistic quantum mechanics" a la the 1st volume of Bjorken Drell at all. It's a waste of time and more confusing than helpful. You end up with the old hole-theoretical formulation of QED a la Dirac, which is pretty much equivalent to modern field-theoretical QED, but much more complicated.

Do you mean is it true for free multi-particle as well ? If so, again, is not it contrary to the opinion than in https://arnold-neumaier.at/physfaq/topics/multi.html ?

Of course, there's relativistic many-body QFT. Introductory textbooks for the equilibrium case are

M. LeBellac, Thermal Field Theory, Cambridge University
Press, Cambridge, New York, Melbourne (1996).

J. I. Kapusta and C. Gale, Finite-Temperature Field Theory;
Principles and Applications, Cambridge University Press, 2
edn. (2006).

M. Laine and A. Vuorinen, Basics of Thermal Field Theory,
vol. 925 of Lecture Notes in Physics, Springer (2016),
https://doi.org/10.1007/978-3-319-31933-9

A. Schmitt, Introduction to Superfluidity: Field-theoretical
approach and applications, vol. 888, Springer, Cham,
Heidelberg, New York, Dordrecht, London (2015),
https://dx.doi.org/10.1007/978-3-319-07947-9
https://arxiv.org/abs/1404.1284

 

[A. Neumaier]

TL;DR Summary: It is often argued that Dirac Equation is not valid as relativistic quantum mechanics requires the creation of antiparticles. But, there are also some arguments that suggest otherwise. For example, I saw Arnold Neumaier's website on this that there are multiparticle relativistic quantum mechanics also exist.

it is still somewhat not clear how to incorporate external potential in relativistic quantum mechanics.

You should first read the survey article on relativistic multiparticle equations by Keister and Polyzou. You'll see that it is possible to get closed relativistic (and causal) equations for $N>1$ particles but that to incorporate cluster separability becomes more and more complicated as $N$ increases.

That there is no such complexity barrier in the relativistic QFT approach (where causality and cluster separability are implemented through locality) is the reason why the relativistic multiparticle approach has remained a minority point of view. It can be used efficiently for quark models where cluster separability is invalid anyway (because of confinement); see, e.g., the work by Klink and collaborators.

On the other hand, the relativistic QFT approach has difficulties with describing bound states, which is still a quite poorly understood subject.

 

[A. Neumaier]

Of course, there's relativistic many-body QFT.

But there is no relativistic few-particle QFT that would give a temporal dynamics for hadrons in terms of quarks, say. (One only has the S-matrix, which describes the scattering without temporal resolution.)

This is where relativistic multiparticle quantum mechanics currently has an advantage.

 

[vanhees71]

As if there were a relativistic quantum mechanics for hadrons in terms of quarks and gluons...

 

[A. Neumaier]

As if there were a relativistic quantum mechanics for hadrons in terms of quarks and gluons...

There are effective models that correlate experimental data.

 

[vanhees71]

There are quantum-field theoretical effective models for hadrons, based on chiral symmetry, but no relativistic QM models. This wouldn't make sense in the applications (of relativistic heavy-ion collisions) since one deals with creation and annihilation processes all the time.

 

[A. Neumaier]

There are quantum-field theoretical effective models for hadrons, based on chiral symmetry, but no relativistic QM models.

no relativistic QM models ???

There is a recent survey that tells quite a different story:

• Plessas, W. (2019). The Constituent-Quark Model. Slides (75 pages). https://indico.cern.ch/event/799284...tachments/1895384/3126852/Plessas_HiX2019.pdf

And there is a whole book about relativistic QM models

• Klink, W. H., & Schweiger, W. (2018). Relativity, Symmetry, and the Structure of Quantum Theory, Volume 2: Point form relativistic quantum mechanics. Morgan & Claypool Publishers. https://iopscience.iop.org/book/mono/978-1-6817-4891-7

''This book develops what is called relativistic point form quantum mechanics, which, unlike quantum field theory, deals with a fixed number of particles in a relativistically invariant way. A chapter is devoted to applications of point form quantum mechanics to nuclear physics.''

And of course many other papers, such as

• Faustov, R. N., Galkin, V. O., & Kang, X. W. (2020). Semileptonic decays of D and D_s mesons in the relativistic quark model. Physical Review D, 101(1), 013004. https://link.aps.org/pdf/10.1103/PhysRevD.101.013004
• Jung, J. H., Schweiger, W., & Biernat, E. P. (2019). Constituent-Quark Model with Pionic Contributions: Electromagnetic N→ Δ Transition. Few-Body Systems, 60(1), 13. https://link.springer.com/content/pdf/10.1007/s00601-018-1479-3.pdf https://arxiv.org/abs/1804.01750

 

[vanhees71]

Sure, there are approximations, which are "not too relativistic", where you can get along with "first-quantization" formulations. It's not a fully consistent relativistic QT as is relativistic QFT though.

 

[A. Neumaier]

Sure, there are approximations, which are "not too relativistic", where you can get along with "first-quantization" formulations. It's not a fully consistent relativistic QT as is relativistic QFT though.

It is fully relativistic (Poincare invariant) and causal, hence fully consistent. Compared to relativistic QFT, only the asymptotic cluster decomposition principle is violated. But for relativistic QFTs with confinement, this is anyway violated.

 

[vanhees71]

In your first quoted source it explicitly says that the relativistic constituent quark model is "not a fundamental dynamical theory".

For relativistic QFTs with confinement the asymptotic states are color-neutral bound states and not "quarks and gluons". Of course, we don't have a complete understanding of confinement from first principles, but for sure it doesn't imply that there's a consistent dynamical theory of interacting particles in the "1st-quantization formalism". Also, why should one look for such a theory? Particle annihilation and creation processes indeed are what's observed in relativistic collisions!

I don't have access to the book you quoted.

 

[A. Neumaier]

In your first quoted source it explicitly says that the relativistic constituent quark model is "not a fundamental dynamical theory".

I never claimed that it is a fundamental dynamical theory. The words 'consistent' and 'fundamental' have a different meaning. The first is a mathematical property of a theory, the second is a philosophical claim about a theory.

I only claim that relativistic multiparticle quantum mechanics gives consistent, covariant, and causal dynamical models that can accommodate particle decay and have been successfully matched with experimental data. The agreement in the case of the quark model is excellent.

In contrast, conventional QFT (supposedly more fundamental) only gives an asymptotic description of few particle processes at times $t=\pm\infty$.

 

[vanhees71]

That's definitely not true. Relativistic QFT is not only vacuum QFT and S-matrix elements but also in-medium theory. In equilibrium there's also lattice-QCD at finite temperature, which is even non-perturbative.

 

[A. Neumaier]

That's definitely not true. Relativistic QFT is not only vacuum QFT and S-matrix elements but also in-medium theory. In equilibrium there's also lattice-QCD at finite temperature, which is even non-perturbative.

I know. But this is the infinitely many body case, not the few-body case I was talking about. (And lattice QCD is not covariant but has only a few discrete symmetries.)

But how do you describe the finite time dynamics of a quantum system with baryon number $N=2$ (baryon scattering by meson exchange) with relativistic QFT? Please offer a reference where I can update my lack of knowledge.

 

[vanhees71]

I don't know, whether this has ever been done, not even for non-relativistic two-body scattering. Also there you usually calculate the asymptotic states only.

 

[A. Neumaier]

I don't know, whether this has ever been done,

Yes, that's what I thought. Therefore it seems that I am right that

conventional QFT (supposedly more fundamental) only gives an asymptotic description of few particle processes at times $t=\pm\infty$.

This is where relativistic quantum mechanics, though not fundamental, is superior to QFT, hence finds its niche.

not even for non-relativistic two-body scattering. Also there you usually calculate the asymptotic states only.

Not at all.

For a translation invariant interaction, non-relativistic two-body dynamics can be reduced through separation of variables to the case of a single particle in an external potential. There are many, many studies that look at the finite time evolution of such systems, and every student is expected to be able to compute it.

 

[vanhees71]

That's of course true, but I guess, I don't know, what you are after. It's for sure a very limited point of view to say that relativistic QFT only deals with the calculation of S-matrix elements. In my field of research, relativistic heavy-ion collisions, there are many applications of relativistic QFT to describe many-body systems, of course using many kinds of approximations.

At the fundamental level one uses the Schwinger-Keldysh real-time contour to describe off-equilibrium situations for the hot and dense medium, leading to the Kadanoff-Baym equations for the time evolution of the one-particle Green's function. Starting from this you can derive via coarse graining the relativistic Boltzmann-Uehling-Uhlenbeck equations, including "off-shell transport" for broad resonances (and nowadays extended from the usual 2->2 collision terms also to n <->m collision terms). The next step of simplification is to derive relativistic hydrodynamics for systems close to local thermal equilibrium, including a consistent treatment of dissipative effects (viscous hydro).

 

[A. Neumaier]

It's for sure a very limited point of view to say that relativistic QFT only deals with the calculation of S-matrix elements.

This very limited point of view was never my view.

I repeatedly mentioned in other threads that in the context of extended media, relativistic QFT is very powerful. For example, it can be used in equilibrium statistical mechanics to derive the thermodynamical properties of materials, and with the closed time path formalism to derive hydrodynamical and kinetic equations for bulk matter.

But this is independent of the fact that it also has shortcomings in the context of few-particle systems, where relativistic QFT currently only deals with the calculation of S-matrix elements. Moreover it has conceptual difficulties in the description of bound state properties (beyond their mass as poles of Green's functions).

 

[LittleSchwinger]

It would be extraordinarily difficult to treat QFT for finite times. To even say there are $N = 2$ particles at finite time would be difficult for an interacting theory where multiparticle states are only defined at asymptotic times. States at finite times would not strictly have a particle decomposition. Interesting conceptual problem.

 

[vanhees71]

For interacting fields there is no particle interpretation, only for asymptotic free fields.

 

[LittleSchwinger]

For interacting fields there is no particle interpretation, only for asymptotic free fields.

I really wish more QFT texts would make this explicit.