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Quantum field theory basic concepts

  1. Nov 21, 2013 #1
    Would it be right to say that QFT tries to bring together the many-particles(many-body) discrete systems of quantum mechanics and the relativistic fields that are basically continuous systems?
    Of course the discrete particle of classical mechanics that when found in big numbers must be dealt with many-body statistical mechanics has always a certain conceptual and methodological clash with classical continuous fields and with continuum mechanics that has traditionally required convinient mathematical objects like dirac delta generalized functions.
    It is IMO a significative parallelism that just like when approximating matter many-body systems thru continuous models like it is done in fluid and solid continuum mechanics one makes the unrealistic assumption that at ech point there are infinite molecules, in QFT there are also the appearance of infinities that have to be treated with regularization-renormalization.
    Can we say that QFT solves rigorously the continuous-discrete controversy that seems to always chase physical models?
  2. jcsd
  3. Nov 21, 2013 #2


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    Many-body theory in condensed matter is non-relativistic QFT.

    Because of the landau pole in QED and electroweak theory, it is often said that even relativistic QFT fundamentally assumes a lattice. For example, 't Hooft writes "Often, authors forget to mention the first, very important, step in this logical procedure: replace the classical field theory one wishes to quantize by a strictly finite theory. Assuming that physical structures smaller than a certain size will not be important for our considerations, we replace the continuum of three-dimensional space by a discrete but dense lattice of points." (http://www.staff.science.uu.nl/~hooft101/lectures/basisqft.pdf (p12))

    The continuum emerges only at low energies, because we use long wave length probes that cannot resolve the lattice (Wilson's renormalization group).

    However, one problem with assuming that relativistic QFT fundamentally assumes a lattice is that there appears to be some problem with getting chiral interactions on the lattice. Kaplan's review http://arxiv.org/abs/0912.2560 says "there is currently no practical way to regulate general nonabelian chiral gauge theories on the lattice." Poppitz and Shang http://arxiv.org/abs/1003.5896 say "we do not yet have a method of approximating an arbitrary chiral gauge theory by latticizing and then simulating it on a computer even in principle."

    Some QFTs like QCD are thought to have a continuum limit even at arbitrarily high energies (asymptotic freedom or asymptotic safety).
    Last edited: Nov 21, 2013
  4. Nov 21, 2013 #3
    Yes, Landau poles and latticing problems seem to hint that QFT hasn't completely solved the tension between discreteness and continuity in physics.
    We have in classical field theory the dichotomy between matter point sources and vaccum free fields, that we try to improve in quantum field theory by quantizing the field only to get as obvious interpretation of this quantisation a many-particle lattice picture that is still in conflict with the continuous field and that demands the addition of a complex and arbitrrary machinery(even accepting Wilson's explanatory tools) to get rid of the infinities.
  5. Nov 21, 2013 #4


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    UV infinities arise in the context of perturbation theory -- attempting to approximate the full interacting theory in terms of solutions of the free theory.

    Unphysical infinities also arise in classical dynamics -- e.g., when approximating the anharmonic oscillator with quartic potential by solutions of the ordinary harmonic oscillator (iirc -- see Jose & Saletan). The resolution there is to change the constants of the theory in such a way that the infinites are absorbed. This is the Poincare-Lindstedt method: http://en.wikipedia.org/wiki/Poincaré–Lindstedt_method
    Renormalization in QFT is essentially this method on steroids. :biggrin:

    My point is that one must distinguish between nonperturbative features of a theory, and unphysical inconveniences arising by perturbation of another theory (the free theory) which isn't entirely suitable for the task.
  6. Nov 21, 2013 #5


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    But isn't it ok for the standard model to have landau poles (not so sure about the chiral lattice problems) since we don't expect the standard model to be the final theory?
  7. Nov 22, 2013 #6
    Sure, I'm aware that all the possible mathematical inconsistencies related to Landau poles and quantum triviality in general are moreless ignored on the grounds of practical convenience and the implicit idea that after all the standard model is not expected to be the final theory, this is similarly approached in GR with say singularities by acknowledging it is not the final theory and the hope that they'll disappear in a future quantum gravity theory. I guess pretty much most physicists may agree on this.
    I'm more interested here in getting a glimpse of what might be the conceptual blocks that keep us not quite getting there (the next theory free from this nagging problems). And in that vein I find evident the need to address the ever present difficulty to reconcile what we observe as discrete in nature(the more quantum side) with the continuous models that pervade physics without having to "shack up" with possible mathematical inconsistencies, quantum triviality, singularities,... and actually more importantly the apparent impossibility to unify in a model what we observe as coexisting in nature at low energies like gravity with the rest of the interactions without the scapegoat of recurring to infinite or near infinite energy scales of renormaization theoretical apparatus.
    Is this discrete vs continuous dichotomy even perceived as a problem by theoretical physicists?, probably not, but I'm interested in what anyone might think about this.
  8. Nov 22, 2013 #7
    Yes, as I said the discrete-continuous dichotomy affects classical physics models as well as quantum field ones, it is just that it is more prominent or visible in the quantum case.
    The very need to talk about interacting vs free fields and theories in QFT, or point sources vs free fields in classical field theory is the dichotomy I'm talking about. I guess it's the old particle-field(wave) duality that keeps coming up, meaning it is not really solved.

    I think this is something that surpasses (but it's related to or it manifests itself also as) the purely mathematical issue of the limitations of perturbative methods.
  9. Nov 22, 2013 #8


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    If we avoid gravity, there are relativistic QFTs that are complete.

    I don't understand these, but at the mathematical level, these are discussed in http://www.claymath.org/millennium/Yang-Mills_Theory/yangmills.pdf (section 6.2) and http://www.rivasseau.com/resources/book.pdf . Although they are constructed as Euclidean field theories, the Osterwalder-Schrader axioms allow them to be rigourously "rotated" into Minkowski space. Other references for rigourous constructions are Glimm and Jaffe's https://www.amazon.com/Quantum-Physics-Functional-Integral-Point/dp/0387964770 and Barry Simon's https://www.amazon.com/The-Euclidean-Quantum-field-theory/dp/0691081441 . You can also look at Haag's book, which is used even though it does not construct explicit examples.

    Although not satisfactory to mathematicians, physicists also believe that Yang Mills and QCD are complete because of "asymptotic freedom" or "asymptotic safety". One possibility being researched is that gravity is asymptotically safe, and that the difficulties with Einstein gravity as a quantum field theory are only perturbative, but disappear non-perturbatively. If gravity is not asymptotically safe, then it must be "emergent" from a more fundamental theory.

    The Weinberg-Witten theorem rules out a wide range of ways in which quantum gravity in 3+1 d may emerge from a relativistic field theory in 3+1 d. (The Weinberg-Witten theorem does not rule out certain types of emergence such as Sakharov's induced gravity. However, Sakharov's induced gravity seems to need a high energy cutoff.)

    The gauge/gravity duality conjecture, if correct, circumvents the Weinberg-Witten theorem by having quantum gravity in d+2 dimensions emerge from a d+1 relativistic quantum field theory. Since supersymmetric Yang-Mills is thought to be complete, and because supersymmetric Yang-Mills is one of the theories from which gravity emerges according to this conjecture, the gauge/gravity duality suggests that we do have a complete theory of quantum gravity for some universes. Although our universe does not appear to be one of these, it is hoped that by studying the duality, generalizations can be found.
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