Introduction to relativistic quantum mechanics and maybe QFT

[dsatkas]

I'm aware that most modern textbooks gloss over relativistic qm and jump to qft. Since I'm not that brilliant of a student I'm thinking that i should firstly familiarize myself with relativistic and then go to qft. So my first question:
Is it worth it to study relativistic qm or should i jump straight to qft?
If it is worth what textbooks are you suggesting? Do you recommend a book full on relativistic qm or one that has a fair introduction and then goes to qft?

Have bachelor in physics and I'm familiar with qm (up to perturbation methods and a little bit of scattering),em and the math stuff

 

[julian]

I do like the series of books by W. Greiner - he has many worked exercises. He has written a book called "Relativistic Quantum Mechanics - Wave Equations". I found familiarity with chapters of this book helped a lot when moving to his next book "Quantum Electrodynamics" - this second book is based on Feynman's approach, so not the more formal approaches (knowing some perturbation theory and scattering theory helps - even though Greiner goes through all that in detail)...these books guide you right to Feynman's rules for QED and are fun (I thought).

 

[atyy]

I'm an amateur, and at the introductory level it is fine to jump from QM to QFT. Relativistic QM is a very problematic theory, so it is ok to skip it although it is useful in practice. One way to make the jump from non-relativistic QM to relativistic QFT is to see that non-relativistic QM for many identical particles can be formulated as non-relativistic QFT (this is called "second quantization", a name that makes no sense but is used for historical reasons).

The reformulation of non-relativistic QM as QFT is used in condensed matter theory.

http://hitoshi.berkeley.edu/221b/QFT.pdf
http://www.phys.ens.fr/~mora/lecture-second-quanti.pdf
http://www.colorado.edu/physics/phys7450/phys7450_sp10/notes/2nd_quantization.pdf

 

[bhobba]

Here is THE book to get at the beginning level of QFT:
https://www.amazon.com/dp/019969933X/?tag=pfamazon01-20

It carefully explains why relativity and QM inevitably leads to QFT.

Normally QFT is left for graduate school, but IMHO that book can be done at senior undergraduate level after a serious first course in QM.

Thanks
Bill

 

[julian]

Here is THE book to get at the beginning level of QFT:
https://www.amazon.com/dp/019969933X/?tag=pfamazon01-20

It carefully explains why relativity and QM inevitably leads to QFT.

Normally QFT is left for graduate school, but IMHO that book can be done at senior undergraduate level after a serious first course in QM.

Thanks
Bill

As someone who first did QED Feynman's way (as undergrad - self taught from Greiner books - later on his books on non-abelian gauge theory), an undergrad +phd condensed matter QFT (operator and functional integral techniques), RG equations in quantum and statistical field theory, and knows something about topological field theory (being interested in condensed mater and QG) - this book gives a very good sketch of the wide facets and applications of QFT.

 

[vanhees71]

I'm aware that most modern textbooks gloss over relativistic qm and jump to qft. Since I'm not that brilliant of a student I'm thinking that i should firstly familiarize myself with relativistic and then go to qft. So my first question:
Is it worth it to study relativistic qm or should i jump straight to qft?
If it is worth what textbooks are you suggesting? Do you recommend a book full on relativistic qm or one that has a fair introduction and then goes to qft?

Have bachelor in physics and I'm familiar with qm (up to perturbation methods and a little bit of scattering),em and the math stuff

This is a misunderstanding! You think what's called "relativistic quantum mechanics" is simpler than "relativistic quantum field theory". This is plain wrong! The problem is that what's called "relativistic quantum mechanics" (the content of Bjorken&Drell volume 1, a book which I clearly NOT recommend to touch at all, and volume 2 is unfortunately outdated, particularly when it comes to renormalization of QED) is inconsistent in itself. The reason is that a naive first-quantization wave-function picture doesn't work in the relativistic realm, the reason being that at relativistic collision energies between particles you easily create new particles or destroy particles and transform them to something else, and for such processes the most natural way to describe them is quantum field theory, and the starting is not that difficult. The "Gifted Amateur" book, recommended by Bhobba, is very nice.

 

[Demystifier]

The problem is that what's called "relativistic quantum mechanics" (the content of Bjorken&Drell volume 1, a book which I clearly NOT recommend to touch at all,

I disagree. For example, in relativistic QM you can see how to solve the Dirac equation for the hydrogen atom, which gives you atom energies which are in better agreement with experiments than those from the non-relativistic Schrodinger equation. It's good to know that something useful and important can be obtained from relativistic QM without QFT.

Relativistic QM without QFT is also interesting from the point of view of string theory (for those who are not strictly against string theory). Namely, in Bjorken&Drell 1 one can see how to construct Feynman rules without "second quantization". In string theory one also constructs Feynman rules without "second quantization", because string-field theory is a rather poorly understood topic. If one thinks of perturbative string theory as "the fundamental" theory, then, from that point of view, relativistic QM looks more fundamental than QFT. Of course, it may not look so in non-perturbative string theory (which may be a string-field theory, or M-theory, or something else), but at the moment nobody really knows what non-perturbative string theory really is.

The point is that it is useful to think about physics from different (complementary) points of view. Relativistic QM certainly offers a point of view which differs from that of QFT, so it is not a good idea to completely neglect the relativistic-QM point of view.

For my own contribution to the relativistic-QM-potentially-better-than-QFT perspective, in the spirit above, see
http://lanl.arxiv.org/abs/0705.3542 [Europhys. Lett.85:20003, 2009]

 

[ddd123]

I think relativistic quantum mechanics is important for the hydrogen atom spectroscopy, which is usually glossed over by QFT texts. The theory is inconsistent but it gives a successful heuristics for such phenomena. See Sakurai - Advanced Quantum Mechanics.

 

[Demystifier]

I'm aware that most modern textbooks gloss over relativistic qm and jump to qft. Since I'm not that brilliant of a student I'm thinking that i should firstly familiarize myself with relativistic and then go to qft. So my first question:
Is it worth it to study relativistic qm or should i jump straight to qft?
If it is worth what textbooks are you suggesting? Do you recommend a book full on relativistic qm or one that has a fair introduction and then goes to qft?

Have bachelor in physics and I'm familiar with qm (up to perturbation methods and a little bit of scattering),em and the math stuff

Here is the list of all introductory textbooks I am aware specialized for teaching relativistic QM without QFT:
https://www.amazon.com/dp/0072320028/?tag=pfamazon01-20
https://www.amazon.com/dp/3540674578/?tag=pfamazon01-20
https://www.amazon.com/dp/904813644X/?tag=pfamazon01-20
https://www.amazon.com/dp/3540255028/?tag=pfamazon01-20
(The last one contains also some elements of QFT, but the emphasis is on relativistic QM.)

For advanced relativistic QM see also
https://www.amazon.com/dp/9401073260/?tag=pfamazon01-20
https://www.amazon.com/dp/3110262924/?tag=pfamazon01-20
https://www.amazon.com/dp/3642081347/?tag=pfamazon01-20

 

[vanhees71]

I disagree. For example, in relativistic QM you can see how to solve the Dirac equation for the hydrogen atom, which gives you atom energies which are in better agreement with experiments than those from the non-relativistic Schrodinger equation. It's good to know that something useful and important can be obtained from relativistic QM without QFT.

It's a question, whether you want to understand relativistic QT correctly from its principles in the beginning and not to learn old-fashioned stuff (an approach that I prefer; there's a reason why we don't require our students to learn Aristotelian physics before we teach them Newtonian mechanics) or you want to teach the history of quantum theory (which is a very interesting subject but it's not very relevant for physics).

Now the question arises, why the naive wave-mechanics approach to the Dirac equation works for the hydrogen atom (and also positronium for that matter) works so well. The answer is simply that it is a good approximation in this cases, and this can be derived from QFT (in this case QED in fact). The addional benefit of this additional effort is that you can also set up the radiative corrections systematically, and this is done up to 4 or 5 loops today, and is a triumph for QED, making it to one of the best checked theory ever.

Relativistic QM without QFT is also interesting from the point of view of string theory (for those who are not strictly against string theory). Namely, in Bjorken&Drell 1 one can see how to construct Feynman rules without "second quantization". In string theory one also constructs Feynman rules without "second quantization", because string-field theory is a rather poorly understood topic. If one thinks of perturbative string theory as "the fundamental" theory, then, from that point of view, relativistic QM looks more fundamental than QFT. Of course, it may not look so in non-perturbative string theory (which may be a string-field theory, or M-theory, or something else), but at the moment nobody really knows what non-perturbative string theory really is.

The point is that it is useful to think about physics from different (complementary) points of view. Relativistic QM certainly offers a point of view which differs from that of QFT, so it is not a good idea to completely neglect the relativistic-QM point of view.

For my own contribution to the relativistic-QM-potentially-better-than-QFT perspective, in the spirit above, see
http://lanl.arxiv.org/abs/0705.3542 [Europhys. Lett.85:20003, 2009]

I'll have a look at it.

 

[Demystifier]

It's a question, whether you want to understand relativistic QT correctly from its principles in the beginning

Suppose that I want to understand QT correctly from its principles. Does it mean that I should start immediately from relativistic QFT, even before learning non-relativistic QM?

Another related question:
Can one really derive non-relativistic QM as an approximation of relativistic QFT? If you think one can, do you know a reference where such a derivation is done systematically? In particular, how non-relativistic position operator and its physical interpretation is derived from the principles of relativistic QFT which lacks position operator?

 

[Fredrik]

Seems like what everyone here means by "relativistic quantum mechanics" is how to deal with wavefunctions that satisfy relativistic wave equations. I don't think it would be worth the effort to go deeply into this subject, but it doesn't hurt to take a look at the most basic aspects of it.

To me, the term "relativistic quantum mechanics" is not synonymous with relativistic wave equations. The relativistic wave equations define theories that I would consider examples of relativistic quantum theories. What I like to call "relativistic quantum mechanics" is a framework in which such theories can be defined. I'm thinking in particular of the idea that a relativistic quantum theory involves a Hilbert space and a projective representation of the connected part of the Poincaré group. These things can unfortunately be extremely mathematical. (Take a look at "Geometry of quantum theory" by Varadarajan if you don't mind having your head explode). Chapter 2 of Weinberg's QFT book cover these things at a reasonable level. I think it's worth the effort to study this chapter in some detail.

I should probably warn you that studying these things won't help you pass your QFT exam, unless they're part of the course.

 

[vanhees71]

Suppose that I want to understand QT correctly from its principles. Does it mean that I should start immediately from relativistic QFT, even before learning non-relativistic QM?

Another related question:
Can one really derive non-relativistic QM as an approximation of relativistic QFT? If you think one can, do you know a reference where such a derivation is done systematically? In particular, how non-relativistic position operator and its physical interpretation is derived from the principles of relativistic QFT which lacks position operator?

No, first you learn non-relativistic QM to learn the concepts. You also learn classical mechanics and classical field theory before you learn QT, but one should learn all these subjects from a modern perspective and you should not be forced to learn concepts which you should better forget afterwards again!

In relativistic QFT massive particles have a position operator (for massless particles only those with spin <=1/2), and then you can systematically do an expansion in powers of 1/c to get non-relativistic approximations. Non-relativistic physics doesn't make sense for massless particles. The reason is deep in the group-theoretical structure of the underlying space-time models. It's a very nice subject :-).

 

[Demystifier]

In relativistic QFT massive particles have a position operator (for massless particles only those with spin <=1/2)

I guess these position operators are not Lorentz covariant, am I right?
In any case, can you give references for those two statements?

 

[vanhees71]

http://arnold-neumaier.at/physfaq/topics/position.html

and references cited therein.

 

[Demystifier]

http://arnold-neumaier.at/physfaq/topics/position.html

This is a great summary, but let me add some comments.
Those position operators are not Lorentz covariant. It is OK as long as one accepts that a particle does not have a position independent of observation. It is, of course, in agreement with orthodox instrumental interpretation of quantum theory, according to which quantum theory is nothing but a tool to predict the results of measurements. But then it is disappointing that the most fundamental theory of Nature we have can tell us nothing about Nature per se, about properties which Nature possesses even when we don't observe it.

By contrast, my paper mentioned a few posts above offers a possible particle/string picture of the world meaningful even when those particles/strings are not observed. Such a picture is necessarily more speculative, but potentially also more fundamental.

 

[atyy]

Suppose that I want to understand QT correctly from its principles. Does it mean that I should start immediately from relativistic QFT, even before learning non-relativistic QM?

Actually, I don't disagree with what you say that it is valuable to learn relativistic QM, especially for the hydrogen atom. However, since vanhees71 is arguing that one should only learn coherent theories, then yes, that argument does not rule out first learning non-relativistic QM. Probably a non-perturbative definition of the standard model, certainly of QED comes from lattice theory, which is non-relativistic QM. I'm not sure what to do about the chiral fermions - I guess if we follow vanhees71's argument, one would need to learn string theory (but that still has no established non-perturbative definition).

Another related question:
Can one really derive non-relativistic QM as an approximation of relativistic QFT? If you think one can, do you know a reference where such a derivation is done systematically? In particular, how non-relativistic position operator and its physical interpretation is derived from the principles of relativistic QFT which lacks position operator?

No, but we also don't have a systematic derivation of say the BCS Hamiltonian from the Schroedinger equation. I don't think we have the effective Lagrangian of chiral perturbation theory from QED either. But it is an interesting question. I think the main argument is that under certain circumstances, we can get bound states in external fields, say as in Weinberg Chapter 10. Then we assume we are in some regime where we can ignore particle creation. Then working in the path integral formulation, we can get the Galilean symmetry at slow speeds. Although relativistic QFT has no exact position operator, there should be an inexact position operator in relativistic QFT which if the Wilsonian program can really be carried out, should yield the non-relativistic poisition operator. In principle, if we start from a relativistic QFT path Lagrangian and get to a non-relativistic path integral Lagrangian, since the Lagrangians allow formulations of quantum mechanical theories, we can get the position operator that way. But it is indirect (and assumes that going from Hilbert space to path integral and back commutes with renormalization, which is doubtful). It would be better if the renormalization could be carried out in the Hilbert space and operators (like http://arxiv.org/abs/1412.0732).

 

[A. Neumaier]

It is OK as long as one accepts that a particle does not have a position independent of observation.

Already the notion of a particle depends on the observer, as shown by the Unruh effect. Thus it is no surprise that the position of something observer-dependent is also observer-dependent. It explains naturally why position operators are necessarily noninvariant under Lorentz boosts.

Quantum fields are covariant and exist everywhere, so they need neither observers nor a particular position operator. That this is not the case for particles is - in view of the fact that physical objects existed long before observers came into existence - sufficient reasons why particles cannot be fundamental.

 

[Demystifier]

Quantum fields are covariant and exist everywhere, so they need neither observers ...

What do you mean by "quantum fields exist"? Do non-hermitian fermionic fields exist? Do the field and its conjugate momentum simultaneously exist? In an orthodox interpretation of QFT, the answers to both questions should be - no. So replacing particle ontology (which, as you correctly noted, has its problems) with field ontology does not really solve the problem.

 

[A. Neumaier]

What do you mean by "quantum fields exist"? Do non-hermitian fermionic fields exist? Do the field and its conjugate momentum simultaneously exist?

I mean with ''exist everywhere'' that they have values when integrated over arbitrary open and bounded subsets of space-time. This is independent of any ontological interpretation.

What exists in a real sense are the smeared expectation values of quantum fields and their (renormalized) products, since these carry measurable implications, even when nobody measures them.

 

[dextercioby]

All in all, a correct course on quantum mechanics should touch on special relativity: the Klein-Gordon equation + solutions for free particle + solutions for the H-atom + shortcomings with respect to the Born rule + the Dirac equation + solutions for free particle + solutions for the H-atom + shortcomings with respect to the Born rule.

That's the only natural way to impose the necessity of QFT. You can't end a QM course with the Born approximation in scattering theory.

 

[vanhees71]

Well, scattering theory is way more important than a bad introduction to relativistic QT in terms of a first quantized wave-function theory. A good argument to motivate a theory are experimental facts, and for me thus the proper motivation for the use of QFT is the fact that at relativistic energies particle number is not conserved, but you easily create new particles, and that's most elegantly described interms of QFT. Relativistic bound-state problems should come much later in the QFT course, when you can justify the approximations that allow you to use the wave-function treatment as the 0th order approximation for situations that are not "too relativistic", i.e., where the relativistic effects are small corrections to the non-relativistic approximation (hydrogen atom, positronium; if you like to treat also the scalar-boson hydrogen-like system, you may motivate it by the study of pionic atoms).

 

[Demystifier]

A good argument to motivate a theory are experimental facts, and for me thus the proper motivation for the use of QFT is the fact that at relativistic energies particle number is not conserved, but you easily create new particles, and that's most elegantly described interms of QFT.

I agree with that. The true physical motivation to introduce QFT should be to describe the experimentally observed particle creation and destruction, not to resolve the problems with relativity.

QFT may describe creation and destruction of particles even in the non-relativistic limit. For example, in condensed matter physics non-relativistic QFT describes creation and destruction of phonons.

In relativistic processes in which particles are not created and destructed, relativistic QM offers a self-consistent picture. Even a conserved probability in the position space can be introduced, provided that one considers wave packets made of either only positive or only negative frequencies. Of course, the resulting position operator is not Lorentz covariant, but as already stressed several times, that's OK as long as position is interpreted merely as a result of measurement, not as an intrinsic property of the quantum object existing without measurements.

In situations in which one must consider superpositions of positive and negative frequencies, in this regime there is particle creation involved, in which case relativistic QM should be replaced by relativistic QFT.

 

[vanhees71]

Of course, a good non-relativistic QM lecture also introduces QFT, and in many-body QFT (condensed-matter physics) there are quasiparticles and destruction and creation processes.

I still disagree that one should teach relativistic QM, because even for free particles, you have trouble with causality. For this argument, see, e.g., Peskin/Schroeder. QFT solves both, the problem of non-conserved particle numbers in relativistic scattering processes and the problems with causality.

 

[A. Neumaier]

For example, in condensed matter physics non-relativistic QFT describes creation and destruction of phonons.

Do you think phonons have an exact position? If not, why should other particles have one?

 

[vanhees71]

That's a very good point. The phonons are quantized collective vibrations of the chrystal lattice describing a solid body. It doesn't make much sense to give them a particle interpretation at all. Formally you can do so, and depending on the dispersion relations of the quasi particles it might be possible to formally define a position operator for them. Only, what's the physical interpretation of such a position operator is not clear to me.

 

[atyy]

Do you think phonons have an exact position? If not, why should other particles have one?

If I understand Demystifier correctly, his question is not so much that there has to be a position operator. The question is if relativistic QFT is more fundamental, then non-relativistic QM should be considered a slow speed and "coarse-grained" effective theory that is derived from relativistic QFT. The effective theory clearly has a position operator, so how can we derive that from relativistic QFT?

I think other similar tricky questions aobut the emergence of an effective theory are:
- Can we derive the solid state from Schroedinger's equation?
- Can we derive chiral perturbation theory from the standard model Lagrangian?

 

[A. Neumaier]

If I understand Demystifier correctly, his question is not so much that there has to be a position operator.

He was complaining about the position operator (which no one claimed not to be derivable from relativistic QFT) being observer-dependent. So I was questioning whether it is reasonable to require that there should be a position operator at all.

I think other similar tricky questions aobut the emergence of an effective theory are:
- Can we derive the solid state from Schroedinger's equation?
- Can we derive chiral perturbation theory from the standard model Lagrangian?

It depends on what one demands of a derivation whether such questions can be answered. What satisfies one person may be very unsatisfying or another.

 

[Demystifier]

Do you think phonons have an exact position? If not, why should other particles have one?

I don't think phonons have an exact position. But in the QFT of phonons, even the phonon field is not fundamental. What is fundamental in condensed matter physics are atoms, and it is not so unreasonable to imagine that atoms do have positions.

Of course, a particle physicist knows that atoms are also not fundamental, because they consist of more fundamental particles, which are states of standard-model fields. Perhaps these fields are also not fundamental, but describe fluctuations of some more fundamental particles? Or strings? And perhaps the ultimate fundamental objects do have positions?

 

[atyy]

Of course, a particle physicist knows that atoms are also not fundamental, because they consist of more fundamental particles, which are states of standard-model fields. Perhaps these fields are also not fundamental, but describe fluctuations of some more fundamental particles? Or strings? And perhaps the ultimate fundamental objects do have positions?

I will guess they don't. AdS/CFT probably doesn't model our universe, but in the universes it does model, bulk position is presumably emergent, since bulk spacetime is emergent, and the boundary theory is fundamental.

 

[A. Neumaier]

Those position operators are not Lorentz covariant.

Nobody ever constructed commuting Lorentz invariant position operators consistently. (The work of Hawton that claims the contrary is flawed.)

On the other hand, Hardy’s theorem states that any dynamical theory of measurement, in which the results of the measurements agree with those of ordinary quantum theory, must have a preferred Lorentz frame. See

L. Hardy. Quantum mechanics, local realistic theories and Lorentz-invariant realistic theories. Phys. Rev. Lett., 68:2981–2984, 1992.

and the sharpened version (making fewer assumptions) in

I.C. Percival, Quantum measurement breaks Lorentz symmetry." arXiv preprint quant-ph/9906005 (1999).

Thus on the level of observation, where the question of the existence of operators measuring something arise, it is unreasonable to expect Lorentz invariance.

 

[Demystifier]

Nobody ever constructed commuting Lorentz invariant position operators consistently.

Then, I guess, you will say that my own attempt is inconsistent too:
http://arxiv.org/abs/0811.1905 [Int.J.Quant.Inf.7:595-602,2009]

On the other hand, Hardy’s theorem states that any dynamical theory of measurement, in which the results of the measurements agree with those of ordinary quantum theory, must have a preferred Lorentz frame. See

L. Hardy. Quantum mechanics, local realistic theories and Lorentz-invariant realistic theories. Phys. Rev. Lett., 68:2981–2984, 1992.

I don't think that the Hardy theorem proves a preferred Lorentz frame, at least not in the sense of violation of Lorentz invariance. It proves non-locality without inequalities, but violation of Lorentz invariance is an interpretation of the theorem. Such an interpretation is not really proved in the strict sense. For my own critique of such an interpretation of the Hardy theorem see
http://arxiv.org/abs/1309.0400 [Appendix A.1.1]

 

[Demystifier]

AdS/CFT probably doesn't model our universe

What about its generalizations, such as gauge/gravity or bulk/boundary correspondence?

 

[A. Neumaier]

Then, I guess, you will say that my own attempt is inconsistent too:
http://arxiv.org/abs/0811.1905 [Int.J.Quant.Inf.7:595-602,2009]

Yes, your approach is inconsistent, too. https://www.physicsforums.com/threads/the-refutation-of-bohmian-mechanics.490095/reply?quote=3281129

Moreover, you break the Lorentz symmetry yourself when you discuss the dynamics in Section 3, by treating time differently from space. In this way you select from your covariant state space with the 4D inner product in an ad hoc fashion a subspace that gives the physically correct state space with the 3D inner product. The same argument can be used to argue (by singling out x_i rather than time) that the eigenstates of x_i are not physical, for any i=0,1,2, 3, and nothing physical is left...

Thus your own paper confirms what I had said, that sharp position requires breaking Lorentz invariance.

 

[ddd123]

I don't see a problem in teaching relativistic QM right before QFT. You can do the Dirac equation, bilinear covariants, charge conjugation, chirality and helicity with the neutrino parity violation example, the hydrogen atom, Klein's paradox, all stuff that is used later or is instructive anyway and they're basically all exercises. The student won't think that the theory is consistent because of Klein's paradox and won't really waste time in the process.