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Defeated by quantum field theory yet again

  1. Nov 25, 2012 #1
    I haven't taken a course in qft yet, just looking ahead to see what's to come, and so far things are not looking good, I read the firet few chapters of qft in a nutshell, and jesus christ what is this stuff, where are the postulates? The equations of motion? How do I even do these crazy path integrals??What is this business about everything being harmonic oscillators??
    *crawls back into my nonrelavistic cave where things are nice and cozy
     
  2. jcsd
  3. Nov 25, 2012 #2
    What is your background in quantum mechanics?
     
  4. Nov 25, 2012 #3

    dextercioby

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    We have axiomatical quantum field theory, don't worry. But one should first start with the basics and have a firm grip on non-relativistic quantum mechanics, including the path integral formulation by Feynman.
     
  5. Nov 25, 2012 #4
    Check out chapter two in Sakurai's quantum book "modern quantum mechanics." This outlines dynamics, and has a good part on the path integral in non-relativistic quantum mechanics. Also, DON'T GIVE UP! YOU CAN DO IT! :) Start over in Zee's book! and do it again until it works.
     
  6. Nov 26, 2012 #5

    tom.stoer

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    One can do quantum field theory in the canonical i.e. Hamiltonian approach (it's not popular in most text books b/c all scattering and Feynman stuff becomes rather complex, but there are applications where this is advantageous). In this approach one can make QFT look like QM with (countable) infinitly many d.o.f. and one does not need to introduce any path integral.
     
  7. Nov 26, 2012 #6

    atyy

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    There's a way to make non-relativistic QM for many identical particles into QFT called second quantization.
    http://www.mit.edu/~levitov/8514/#lecturenotes (Lecture 3)
    http://research.physics.illinois.edu/ElectronicStructure/598SCM-F04/lecture_notes/lect18a-2ndQ.pdf [Broken]
    http://arxiv.org/abs/hep-th/0409035 (p13-15).

    Postulates for reloativistic QFT are in section 3.5 of http://uqu.edu.sa/files2/tiny_mce/plugins/filemanager/files/4282179/non11.pdf
     
    Last edited by a moderator: May 6, 2017
  8. Nov 26, 2012 #7
    Could I get some help tho, because I have no where Zee is getting this stuff from:
    1. Why is the klein gordon equation relavistic??
    2.Why do the number of indicies on the stress energy tensor tell us that the spin of the gravitons is 2?
    3. How do I interperet particle fields? Are they still probability amplitudes?
     
  9. Nov 26, 2012 #8

    tom.stoer

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    1.: b/c it is formulated in terms of scalars and 4-vectors
    2.: it's the metric tensor which tell's us that it's spin 2

    The metric is symmetric, so it has 4+3+2+1 = 10 indep. components; one can identify a gauge invariance allowing us to gauge away 8 components, so we are left with 2 d.o.f. which indicates that we are talking about a massless spin 1,2,... field. The reason why it's spin 2 (not spin 1) can be seen by looking at its multipole expansion: it starts with the quadrupole instead of a dipole
     
  10. Nov 26, 2012 #9
    "How do I interperet particle fields? Are they still probability amplitudes?"
    That's my favorite question ;-) (talked about that a lot in the past few weeks)
    The problem you have with QFT may be at least in part due to the fact that nobody ever tells you what exactly a QFT field and its state is.
    In QFT, the fields themselves are (in path integral formulation) classical fields, not prob. amplitudes. Consider an elastic membrane - the field is the displacement at each point.
    In QFT, you now have a probability amplitude for each possible field configuration - the state of a quantum field is a superposition of all possible field configurations, each with its own probability amplitude.
    Since it is very difficult to calculate with this kind of object (its a wave functional - a wave function with functions as arguments), people usually don't use it, but conceptionally I think it is important to understand this.

    If you do a fourier analysis, each fourier component of your field behaves like a harmonic oscillator - so in the ground state the probability of finding a value a_k of the k'th fourier mode is given by a gaussian centered at zero. (And that is why people will tell you that the expectation value of the field is zero for a vacuum state - but there is still a probability of measuring a non-zero state, exactly as for the position of a particle in a QHO).

    If you have a 1-particle state in mode k, this means that the prob. amplitude for the fourier coefficient of a_k looks like the first excited function of the QHO. It still has a zero expectation value, but now it has a different prob. amplitude. (And this is why you can read that a state with a definite particle number has a vanishing expectation value of the corresponding classical field.)

    Hope this helps.
     
  11. Nov 26, 2012 #10
    Yet, I am still not told what these particle fields represent physically
    also I have no idea how to calculate these runctional integrals, no where have I ever seen them explained, how is it even defined? What does the infitesimal represent now?
     
  12. Nov 26, 2012 #11
    "what these particle fields represent physically"
    To make it concrete, think of the electromagnetic field (slightly simplified): The classical field is nothing but the four-potential. This is quantised, so in a quantum state, you have different amplitudes for different values of the four potential.

    If the functional integral (I assume you mean the path integral?) seems complicated, just discretise it in your mind. Think of a space-time grid with a discrete number of points. Then the iegral becomes a sum over all possible field configurations, i.e., for each field configuration, you calculate exp(iS/hbar) and then you sum up all these to get the total value of the functional integral. It is completely analoguous to the path integral formulation of quantum mechanics (which is explained by Zee):
    Instead of summing over all possible paths, you sum over all possible field configurations.

    As so often in physics, QFT may seem so difficult because you have to learn new physics and new mathematical tricks, and many books don't make it clear whether something is just a mathematical trick or "true" physics...

    Hope this helps a bit more - if not, it would be nice if you could make your questions more specific.
     
  13. Nov 26, 2012 #12

    dextercioby

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    The energy-momentum 4 tensor has 2 spacetime indices irrespective of the spin content of the field. The linearized metric tensor has spin content 0 + 2. Removing the spin 0 content is done for example by requiring it to be traceless.
     
  14. Nov 26, 2012 #13
    Well I never had a firm grasp on the path integral formalism even in quantum mechanics, and most of the texts I've read are difficult to understand for me, I don't understand how the integral is defined. It clearly is no longer some multiple Reinman sum, so what is it?
     
  15. Nov 26, 2012 #14

    George Jones

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    It doesn't exist; it is very useful. :biggrin:

    Form "Quantum Field Theory: A Tourist Guide for Mathematicians" by Gerald Folland
     
  16. Nov 26, 2012 #15
    I really dislike this kind of ill-defined procedures...what is rhe best text on path integrals?
     
  17. Nov 26, 2012 #16

    tom.stoer

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    I would recommend either Feynman's lectures or Sakurai. The PIs can be defined rigorously in QM, but not in QFT. Anyway, I think it's better to get an idea what they are about, what they mean and how they are related to ordinary quantum mechanics.
     
  18. Nov 26, 2012 #17
    so..
    1. what do particle fields represent
    2. why don't we just use good old kets and bras for QFT?
    3. why can't we just solve for a bunch of eigen fields from the sources and then just superimpose them
     
  19. Nov 26, 2012 #18

    tom.stoer

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    as I said in post #5: we can, but you don't find it in many text books (unfortunately)
    b/c it's too complicated
     
  20. Nov 27, 2012 #19
    "what do particle fields represent"
    As I told you - for the photon, the field A is the vector potential.

    "why don't we just use good old kets and bras for QFT?"
    because - again as I told you - the state in QFT is a wave functional (a wave function of functions), making dealing with it awkward. You can do it - I explained in my post before how for a vacuum and 1-particle state. See the book by Hatfield "QFT oof point particles and strings" (I seem to recommend that a lot...)

    "why can't we just solve for a bunch of eigen fields from the sources and then just superimpose them"
    Because in the end what we are most interested in are scattering processes and for those that would be difficult to do.
     
  21. Nov 27, 2012 #20

    tom.stoer

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    Sorry to say that but scattering processes are only one sector of QFT; non-perturbative effects like, chiral symmetry breaking, quark condensates, color confinement, QCD bound states, their masses, their form factors etc. are likewise interesting - and there you will a lot of work based on the canonical formalism not using path integrals.
     
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