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What are the Fields in QFT?

  1. Jan 28, 2012 #1


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    I have just started to try to understand some ‘basic’ quantum field theory (QFT), if this is even possible, but not sure that I have any real understanding of the scope of the fields implied within the QFT model. As such, the following description may be completely wrong, but may serve as a description of my confusion.

    As I understand it, QFT is the combination of quantum mechanics and special relativity, which has its roots in Dirac’s equation, but which later developed in terms of Quantum Electro-Dynamics (QED), Quantum Chromo-Dynamics (QCD) and the Electro-Weak Theory (EWT). In combination these theories seem to underpin the standard model of particle physics, although the semantics of the word ‘particle’ should not necessarily be taken literally as the terms may be more reflective of an underlying field theory.

    However, I am struggling to understand whether the ‘field’ in question has any fundamental characteristics. For example is the quantum field of an electron different to the quantum field of a proton or photon? Is the idea of a quantum wave just an undulation of the quantum field?

    By way of example, how might QFT describe a hydrogen atom, consisting of a proton and electron is floating in space, free of any obvious ‘classical’ fields. Under the particle model, the proton is a baryon made of quarks bound by a strong force described in terms of boson gluons. In contrast, electron is a lepton that presumably sits in a ground state orbital, but I am not sure how the stability of this orbital is described in QFT, e.g. is the electromagnetic ‘force’ linked to photons as a force carrier within the atomic structure?

    Classically, a stable atom is said not to radiate EM, even though it appears to contain an accelerating charge, e.g. electron. In QFT, does the photon force carrier exist as quantized EM energy within the atom?

    Equally, how does QFT describe a ‘free’ photon travelling through space. Is it a quantization of the EM field or is there a more fundamental description in terms of some quantum field description. In contrast, I have seen the suggestion that boson force carriers are described as scalar fields. Are there both quantum scalar fields and quantum vector fields? As such, I have become confused as to the scope and meaning of the fields in QFT and would very much appreciate any general insights before getting too lost, again, in the all the associated maths. Thanks
    Last edited: Jan 28, 2012
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  3. Jan 28, 2012 #2

    QFT is different from quantum mechanics (Mandl and Shaw QFT textbook).

    The term particle is correct, because what one detects in experiments are particles (Weinberg QFT textbook). Fields are unobservable, by definition.

    Yes, the quantum field of an electron is different to the quantum field of a photon.

    There are not quantum waves in QFT.

    QFT cannot completely describe a hydrogen atom, because QFT does not deal with bound states, but with free states and with scattering of free states.

    In QFT position is not observable; therefore, QFT cannot physically describe a "photon travelling through space".
  4. Jan 28, 2012 #3


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    You need to read about the Lamb shift.
  5. Jan 28, 2012 #4


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    It can!

    Nobody would use QED to describe the hydrogen atom "from scratch", but one used QED corrections to calculate the lamb shift.
    In QCD one describes bound states (hadrons) using the lattice gauge formalism and one can derive mass spectrum, magnetic moment etc.

    The fact that in many QFT textbooks one does not find descriptions of bound states is a problem of these books, not a problem of QFT
  6. Jan 29, 2012 #5


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    Appreciate the comments and will read up on the issue of Lamb shift and bound states starting with the initial links, as cross reference. However, I would like to follow up on Juanrga's comments because they seem to address some of the fundamental issues I am interested in:
    I have been trying to follow the historical developments leading to QFT. As I have understood things, Schrodinger’s wave equation did not account for special relativity (SR) and was limited to modelling just a few ‘particles’ at most. The Klein_Gordon equation was the first attempt to include SR, but did not account for spin. For this reason, Dirac’s equation seems to be the accepted starting point of QFT as a merger of quantum mechanics (QM) and SR. However, post-war development of QFT seems to have split into various branches, e.g. QED, QCD & EWT as mentioned in post #1. What I am not too sure about is whether QFT research continues as a subject in its own right?
    While I agree that the term ‘particle’ is commonly used and certainly the focus of the particle model, I am not sure that the semantics of its use in QFT can be equated to any classical concept of a particle having tangible substance within the context of a field theory? In the spirit of the title of this thread, ‘What are the Fields in QFT?’, I would also like to try to clarify the comment about fields being unobservable. I am not questioning the comment, but the implication of what level of physical existence is attached to these fields. Do they have ‘any’ physical existence outside their mathematical description. From my initial review of the Euler-Lagrange equation for fields, it would seem that an aspect of the fields can be assigned both energy and momentum density. If so, are quantum fields described in terms of both scalar or vector fields? What physical concepts, if any, underpins these fields? See further comment below.
    Is there any fundamental description of the nature of quantum fields, i.e. can I assume these fields are some sort of quantized energy field? Is a photon a quanta of EM energy that conforms to the general description of an EM field? So how is the make-up of the quantum field of an electron different? Is this field fundamentally different again when describing the quantum field of a proton?
    If energy is a scalar quantity, how does energy get transported between two points in spacetime without any concept of a basic wave mechanism?
    While I am not sure that I necessarily understand the detailed implications of the points raised by Bill_K and Tom.Stoer, I thought one of the reasons for developing QFT was due to the limitation of QM, i.e. Schrodinger’s equation, to describe ‘N’ particle systems. As far as I am aware, I thought QED was limited to electron-photon interactions, while QCD was more focused on quark interactions in the nucleus. Therefore, I thought the example of a hydrogen atom, given in post #1, would be relatively simple as a model to describe the various quantum fields at work. What is the state of play in quantum theory in the sense of providing a physical description versus just a probability outcome?
    By that argument, can it describe an electron travelling through space? I realise that the idea of a photon can be problematic, but regarding the comment ‘there are not quantum waves in QFT’ – how does the energy of the photon move through spacetime according to QFT?

    As always, would appreciate any help on offer. Thanks
    Last edited: Jan 29, 2012
  7. Jan 29, 2012 #6
    Whilst you're trying to grasp the physical nature of quantum fields, it's important not to forget that they need to be considered in conjunction with states in order to describe a system. This thread contains some useful insights. It's a long thread and there are a few disagreements, but there is some good stuff there !

    For a not-too obscure discussion of what it all really means, Paul Teller's book is quite useful.
  8. Jan 29, 2012 #7


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    Appreciate both references. However, there seems to be some mixed reviews on the Teller book.

    Therefore, started to review the thread suggested, but finding most posts apparently focused on escalating a purely mathematically description of evermore abstracted concepts hard going. Couldn’t help thinking that Einstein’s quote “if you can't explain it simply, you don't understand it well enough” might have some relevance here. While I realise that QFT requires a given level of maths to understand the details in depth, I was hoping to get a few basic concepts sorted out before going straight to the 10 metre board to use a diving analogy.

    I think you are being quite diplomatic, when you say that there are a ‘few disagreements’, e.g. Post #37 makes a possibly insightful general comment, which is then immediately refuted in post #38. I don’t think there is a single page in which one post doesn’t appear to be completely contradicted by another.

    Having spent some time reviewing QM, I was left with doubts as to whether a quantum wave could be anchored to any sort of physical description due to the implications of wave dispersion required by a deBroglie wave packet. Therefore, posts #1 and #5 were simply trying to ascertain whether there was any consensus of the physical nature of quantum fields. Thanks
  9. Jan 29, 2012 #8
    You would pay attention to the word "completely"
  10. Jan 29, 2012 #9
    First I said "completely" a word that another poster also missed.

    Second, even if we restrict ourselves to working in an approximation where QED or QCD; i.e. where QFT is valid, there is not consistent and complete description of bound states.


    And Arnold has only covered the surface of this topic.
  11. Jan 29, 2012 #10


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    mysearch, are you familiar with operators in QM by now? In QFT, the most important dynamical equations like the Dirac equation are statements about operators, not about wavefunctions. So I think it is absolutely necessarry to learn how operators are used in simple QM and why they are needed, in order to understand how QFT works.

    Note that there is a wavefunction interpretation of the Dirac equation for a single particle. This is sometimes called Realtivistic Quantum Mechanics. It's important to distinguish this from QFT. QFT is absolutely needed, if particle creation and annihilation is possible, that is in every collider experiment for example.
  12. Jan 29, 2012 #11


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    The problem is not the word "completely" but the statement "because QFT does not deal with bound states, but with free states and with scattering of free states."

    I agree that may problems are not yet solved rigorously, but if you look a t lattice QCD you will find bound states of quarks (hadrons) in some approximation (e.g. quenched QCD) with e.g. 5% accuracy of mass spectra. So I would say that QFT is not able to fully describe all aspects of bound states rigorsously, but the statement that it "does not deal with bound states" is not correct; there are other QFT models of bound states, e.g. effective field theories, which are not so bad.

    What do you mean by that? In order to apply QCD you don't need any approximation in principle; the only thing you need is an appropriate method or maybe approximation depending on the regime you may want to study.

    What is the regime where QCD is not valid but where you have bound states?
  13. Jan 29, 2012 #12


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    Regarding your link:

    The second assumption is wrong b/c it is by no means clear that 'first expand then resume' is mathematically well defined. So yes, bound states would show up in the S-matrix, but a) it is not clear if this is the best way to calculate them and b) things get worse by a perturbative treatment of the S-matrix.



    Does that mean that QFT is incomplete? No, not necessarily. It simply means that a perturbative treatment of QFT is limited to scattering states and that bound states have to be treated differently (and there are other problems which you don't see in any perturbation expansion: color confinement, chiral symmetry breaking, instantons, ...)

    When all you own is a hammer, every problem starts looking like a nail.
  14. Jan 29, 2012 #13
    If you are going to read those, you would at least read the next as well


    There is not rigorous and complete treatment of bound states in QFT. In rigor, QFT only deals with free states although, using some tricks, QFT can deal with some energetic aspects of bound states as Lamb shift.

    The Schrodinger equation is valid for systems of N particles, where N can be relatively large.

    There are particle without spin. The problem of the Klein & Gordon equation is that is inconsistent. Dirac equation was an attempt to add spin but this equation is also inconsistent.

    Both equations are the basis of RQM (relativistic quantum mechanics) but are completely inconsistent and were abandoned.

    Then QFT born, rejecting RQM, and developing an different theory. For instance Dirac wavefunction equation of RQM is replaced by an identity for an operator named dirac operator, which is quantized and which describes a field: fermion field.

    QED and QCD are subfields of QFT. QED is the quantum field theory of electromagnetic and fermion fields.

    The concept of particle used in quantum theory is not a classical concept. E.g. the electron in the Schrödinger equation of QM is not the electron in Lorentz equation of the classical theory.

    A field is unobservable, therefore in rigour (scientific method) you cannot attach "physical existence outside their mathematical description", although some physicists do (but well, you can find physicists believing in aliens and in parallel universes also).

    Effectively fields have associated such concepts as energy and momentum density, and you cannot measure globally energy and momentum densities but only locally where there are particles. As Weinberg emphasizes in CERN experiments we detect particles, not fields. The field is a mathematical abstraction.

    There are scalar, vector, and tensor quantum fields.

    You are asking in essence, can I assume these fields are some sort of field?

    Quantum fields have energy and momentum, but are not "energy fields", but fermion fields, boson fields...

    The photon is the quanta of the EM quantum field.

    Each field and its quanta has different properties as charge, spin, mass...

    An electron with energy E travelling is transporting energy. A photon transports energy as well, but a photon is a quantum particle. The macroscopic wave phenomena is the result of a large collection of photons. Wave optics follow from QED as an approximation.

    QED also deals with interactions electron-electron, electron-positron, positron-photon...

    Even limiting ourselves to study an hydrogen atom within the framework of QFT, you cannot describe it completely using only QED and QCD.

    I do not know why you think that "physical description" and "probability outcome" are antonyms.

    QFT cannot describe arbitrary motion. It only can describe simple cases as scattering, where spacetime coordinates are not measured. In QED the velocity of an electron is not even defined, the different components of the velocity do not commute and you cannot build a vector velocity
  15. Jan 29, 2012 #14


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    I think you still don't get the difference between QFT and perturbative QFT.
  16. Jan 29, 2012 #15
    The meaning of the word "completely" in my original posts must be understood as close to your "I would say that QFT is not able to fully describe all aspects of bound states rigorously".

    QFT is not the final theory of the universe, but based in many approximations.
  17. Jan 29, 2012 #16
    The link starts by merely reporting what is "supposed" in the literature. In fact the author think something different:

    As said before Arnold has only considered the surface of this topic.

    Effectively, QFT is incomplete and cannot deal completely with bound states.

    The impossibility to describe all the aspects of a bound state has nothing to see with perturbative vs non-perturbative QFT, although for some strange reason you believe the contrary.
  18. Jan 29, 2012 #17


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    Tell me one single approximation that applies to QFT in general.

    You can repeat it many times - w/o any profound reason it's implausible.

    What are the aspects of bound states you miss in QFT? which bound states? and in which formalism?
    Last edited: Jan 29, 2012
  19. Jan 30, 2012 #18


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    Kith, thanks for the pointers and clarification regarding RQM versus QFT. I have only just started to work my way into post-war developments, which seem to underpin the standard particle model, i.e. QFT, QED, QCD, EWT. However, I am still only at the stage of working through the update of the Euler-Lagrange equation for fields, which then raised a number of questions in my mind regarding the nature of quantum fields, hence this thread.

    As regard to operators, while I think I understand the basic concept I am not sure I fully appreciate the full scope within QFT. I am posting the following summary of my understanding so that somebody might be able correct any misconceptions on my part.

    My Understanding
    Historically, the Klein-Gordon equation was interpreted in terms of particles. Subsequently it was re-interpreted in terms of a field [Φ], which was considered as an operator that could describe the creation and annihilation of particles. Again, in term of historical developments, QM was initial based on what is called the ‘first quantization’ that was re-interpreted in terms of fields via a ‘second quantization’. Originally, the first quantization defined position and momentum as operators, but in QFT the fields are quantized and become operators, while position, momentum and time are all now treated as parameters. However, the scope of the field operators is to act on the ‘state’ of the field. I think ‘Sheaf’ was trying to point me in this direction in post #6, although the idea of the field as an operator and the state of the field are still somewhat abstract concepts in my mind at this stage -see reply to Juanrga for examples:confused:. This said, the field operators are assumed to act on the field state to create or destroy particles, which is a key idea in QFT such that the number of particles is not fixed.

    However, I will be honest and say that I not sure whether this approach is simply the preference of the mathematicians, which might still be described in some other way so that the physics of these interactions seem more tangible. :wink:
    Last edited: Jan 30, 2012
  20. Jan 30, 2012 #19


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    Juanrga, thanks for taking the trouble to respond to my questions in post #5. The following reply is intended primarily as comments rather than further questions.
    Thanks for the reference.
    Thanks, understanding the chronology is useful.
    I am not a physicist and do not necessary rule out the possibility of aliens or parallel universes, at least, at this stage. Therefore, it may come as no surprise that I struggle with physics that seems to be predicated on only mathematical concepts. It’s a limitation I know:rolleyes:
    I know I am going to run into trouble with the following comments, but based on a ‘need’ to try to describe these concepts in some form of lowest common denominator, I think in terms of energy, fields and waves. The idea of a ‘particle’ which has no definable ‘substance’ seems to have no meaning other than in terms of a descriptive convenience at larger scales. In this context, a field seems to align to the idea of potential energy, where waves or perturbations in the field would appear analogous to kinetic energy of motion. Please understand I am not promoting this idea, especially in this forum, simply trying to explain my current difficulties in understanding QFT.:redface:
    Fair comment and a good word. In part, I have explained some of 'my' issues above, where basically I can’t reconcile the ability to predict the probability of a certain outcome or state with an actual physical description of why that outcome occurred. I accept that many will see this a complete failure to understand the probabilistic nature of quantum physics. However, I am still prepared to work on these issues:tongue2:
  21. Jan 31, 2012 #20
    Some precision is needed here I am not objecting to people believing in many-worlds or aliens, but to people who pretends that such issues have the same scientific status that the law of gravity of the laws of thermodynamics.

    I do not know what you mean by «substance» (it seems some kind of metaphysical idea), but the concept of elementary particles as building blocks of nature seems particularly elegant and powerful for me.

    Regarding fields they are modelled as a collection of harmonic oscillators. And if you ask what is oscillating? Then either you avoid to answer or you return to a particle concept. Moreover, the concept of field is only approximate. It is now generally accepted that QFT is only an effective theory that breaks down to higher energies. Field theory also breaks in other situations, and alternatives are under active research.
  22. Jan 31, 2012 #21
    Everybody says that in QFT there is no position operator, time and postion are considered as labels, but...
    what is phi(x)?
    what represents the state phi(x)|0>? (I thought that it represented the state in which there is 1 particle in position "x")
    what represents the state phi(x)|phi(y)|0>? (I thought that it represented the state in which there is 1 particle in position "x" and another in position "y")
    Thanks for everything you can do!
  23. Jan 31, 2012 #22
    Here's a fun little calculation to try that demonstrates (to me at least) just how little "substance" there is in elementary particles. If you divide the radius of the Earth by the classical radius of an electron, then multiply that by the mass of an electron you will have effectively grown the mass of the electron to what it would be if you grew the size of the electron to be the size of the Earth.

    The surprise is that the electron would weigh about 1/10 of an ounce (if memory serves correctly)
  24. Jan 31, 2012 #23


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    My apologises, I knew what you meant, my comment was somewhat tongue-in-cheek.:tongue2:
    While I don’t want to waste your time on semantics, I meant the statement about ‘substance’ quite literally, not metaphysical. While, I agree with your comment in post #13 that “the concept of particle used in quantum theory is not a classical concept” it is not always clear just how people are defining a particle in QM versus QFT. Therefore, all I was implying is that an electron has no substance by virtue of there being no definition of material substance at this level other than in terms of E=mc^2. If so, is a particle just a concentration of energy in a small volume of space, ignoring spin and charge for now? Equally, I am not sure if the QM time evolution of a matter wave, i.e. dispersion, prior to the wave function collapse still fits into the QFT description. Sorry to belabour the point.
    I have just found what appears to be a useful site, especially Chapter-1. Any thoughts: http://www.quantumfieldtheory.info/
    Last edited: Jan 31, 2012
  25. Jan 31, 2012 #24


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    Here is the calculation as I have understood it, but I not sure whether this is what you meant.
    Earth Radius/Electron Radius * Electron Mass
    = (6378.1*10^3)/(2.817*10^-15)*(9.1*10^-31) = 2.01*10^-9 kg
    This seems a lot less than 1/10 of an ounce, if the figures are correct.
    The calculations were also a bit rushed as it is now getting late in my time zone.

    However, wouldn’t the density of the electron be more relevant here?
    Electron Density=Electron Mass/Electron Volume

    Earth Density=Earth Mass/Earth Volume

    On this basis it is the Earth that has no substance, while an electron would appear to be full of it. However, the question of ‘substance’ was more to do with what we think the (classical) Earth is made of versus what we think a (quantum) electron is made of. See issue raised by Juanrga in previous post.
  26. Jan 31, 2012 #25


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    A field in QFT has just as much substance as a field in classical physics.

    Take the electromagnetic field. If you think it has substance or no substance in classical physics, then it is the same in QFT. The only difference in "substance" between classical and quantized EM fields is that classically we may work with the electromagnetic fields, whereas quantum mechanically we have to work with the electromagnetic potentials (which we also call the electromagnetic field).

    A photon is a quantized excitation of the electromagnetic field.
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