Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

I Introduction to relativistic quantum mechanics and maybe QFT

  1. Jul 21, 2015 #1
    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
     
  2. jcsd
  3. Jul 21, 2015 #2

    julian

    User Avatar
    Gold Member

    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).
     
    Last edited: Jul 21, 2015
  4. Jul 21, 2015 #3

    atyy

    User Avatar
    Science Advisor

    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
     
    Last edited: Jul 21, 2015
  5. Jul 22, 2015 #4

    bhobba

    User Avatar
    Science Advisor
    Gold Member

    Last edited by a moderator: May 7, 2017
  6. Jul 22, 2015 #4

    julian

    User Avatar
    Gold Member

    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.
     
    Last edited by a moderator: May 7, 2017
  7. Jul 23, 2015 #5

    vanhees71

    User Avatar
    Science Advisor
    2016 Award

    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.
     
  8. Jul 23, 2015 #6

    Demystifier

    User Avatar
    Science Advisor

    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]
     
    Last edited: Jul 23, 2015
  9. Jul 23, 2015 #7
    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.
     
  10. Jul 24, 2015 #8

    Demystifier

    User Avatar
    Science Advisor

    Here is the list of all introductory textbooks I am aware specialized for teaching relativistic QM without QFT:
    https://www.amazon.com/Relativistic-Quantum-Mechanics-James-Bjorken/dp/0072320028
    https://www.amazon.com/Relativistic...37726448&sr=1-2&keywords=greiner+relativistic
    https://www.amazon.com/Relativistic...37726410&sr=1-1&keywords=wachter+relativistic
    https://www.amazon.com/Relativistic...37726492&sr=1-1&keywords=pilkuhn+relativistic
    (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/Solutions-Re...&qid=1437726690&sr=1-3&keywords=bagrov+gitman
    https://www.amazon.com/Equation-Solutions-Gruyter-Studies-Mathematical/dp/3110262924
    https://www.amazon.com/Dirac-Equati..._sbs_14_1?ie=UTF8&refRID=1Z0RTYSMD0HWEZW3DFCB
     
    Last edited by a moderator: May 7, 2017
  11. Jul 24, 2015 #9

    vanhees71

    User Avatar
    Science Advisor
    2016 Award

    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.

    I'll have a look at it.
     
  12. Jul 24, 2015 #10

    Demystifier

    User Avatar
    Science Advisor

    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?
     
  13. Jul 24, 2015 #11

    Fredrik

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    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.
     
  14. Jul 24, 2015 #12

    vanhees71

    User Avatar
    Science Advisor
    2016 Award

    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 :-).
     
  15. Jul 24, 2015 #13

    Demystifier

    User Avatar
    Science Advisor

    I guess these position operators are not Lorentz covariant, am I right?
    In any case, can you give references for those two statements?
     
  16. Jul 24, 2015 #14

    vanhees71

    User Avatar
    Science Advisor
    2016 Award

  17. Jul 24, 2015 #15

    Demystifier

    User Avatar
    Science Advisor

    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.
     
  18. Jul 24, 2015 #16

    atyy

    User Avatar
    Science Advisor

    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).

    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).
     
    Last edited: Jul 24, 2015
  19. Jul 25, 2015 #17

    A. Neumaier

    User Avatar
    Science Advisor
    2016 Award

    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.
     
  20. Jul 25, 2015 #18

    Demystifier

    User Avatar
    Science Advisor

    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.
     
  21. Jul 25, 2015 #19

    A. Neumaier

    User Avatar
    Science Advisor
    2016 Award

    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.
     
    Last edited: Jul 25, 2015
  22. Jul 25, 2015 #20

    dextercioby

    User Avatar
    Science Advisor
    Homework Helper

    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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Introduction to relativistic quantum mechanics and maybe QFT
Loading...