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Kinetic lagrangian for non-abelian gauge theories

  1. Aug 6, 2008 #1
    hello everyone!

    i've got a very special question on the gauge invariance (gauge group: SU(N), non-abelian) of the kinetic term in the lagrangian. is it invariant for any representation? i only know, that for the fundamental rep. this is true.

    to be more specific:

    the usual kinetic term is [tex]-\frac{1}{4} F_{\mu \nu}^{a}F^{\mu \nu a}[/tex]

    arbitrary rep.: [tex]F_{\mu \nu} =F_{\mu \nu}^a T^a[/tex]

    the field tensor transforms like this: [tex]F_{\mu \nu}'=U F_{\mu \nu} U^{-1}[/tex], where [tex]U(x)=\exp(i \theta^a (x) T^a)[/tex]. (in an arbitrary representation with generators [tex]T^a[/tex])

    in the fundamental representation of SU(N), i.e. [tex]T^a=\frac{\lambda^a}{2}[/tex], the lagrangian is invariant under gauge transformations, which can easily be proved.

    and now for the adjoint representation of SU(N): [tex](T^a)_{ij}=-if^{aij}[/tex]
    (note: the generators are here [tex](N^2 -1)\times (N^2-1)[/tex]-matrices.) does anyone know, or is anyone smart enough :-) to proof, that this also holds for this representation?

    thank you, i'm looking forward to any answer!

    patrice
     
  2. jcsd
  3. Aug 6, 2008 #2

    samalkhaiat

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  4. Aug 7, 2008 #3
    hi sam!

    thank you for your answer! it was a great help for me. i remember my professor saying that the gauge fields transform in the adjoint representation. up to now it's not yet clear to me, how the gauge transformation "prescription" [tex]A_{\mu}'=UA_\mu U^{-1}+\frac{i}{g} U (\partial _\mu U^{-1}) [/tex] is connected with the generators of the adjoint representation of Lie group SU(N), namely the structure constants of the corrensponding Lie Algebra.

    Does [tex]A_\mu= A_\mu ^a T^a[/tex] not mean (which one follows from the prescription above), that [tex]A_\mu[/tex] is Lie Algebra valued for the representation of our choice for the matter multiplett [tex]\psi[/tex], even the fundamental rep?

    by the way: before i read your post i wanted to say, that the problem is solved, because [tex]F_{\mu \nu}^a[/tex] is independent from the representation and so the Lagrangian is too. but now i'm unsure...

    i found a maybe good article on adjoint endomorphisms at wikipedia which i will try to understand later on. but now for the moment my head is full :) i'm going to tell you, if i could get wise from that arcticle..

    best regards

    patrice
     
  5. Aug 7, 2008 #4

    samalkhaiat

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  6. Aug 15, 2008 #5
    hello

    sorry for not posting that long. yes, indeed this wikipedia article is a little short and cannot help me with my problem.
    neither could the textbooks that i tried: peskin & schroeder (introduction to quantum field theory), ryder (qft) and bjorken drell (relativistic quantum field theory)

    do you know or does anyone know a textbook which gives a more detailed discussion on representations of gauge potentials? that would be really nice.

    thanks a lot

    patrice
     
  7. Aug 15, 2008 #6

    Avodyne

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    In an arbitrary rep, the kinetic term is [itex]{\rm Tr}\,F^{\mu\nu}F_{\mu\nu}[/itex] (up to an overall constant).

    Try Srednicki's book (google to find a free draft version online).
     
  8. Aug 21, 2008 #7
    thank you, avodyne! i'll try this book. it really seems to be well structured and quite instructive.

    patrice
     
  9. Aug 21, 2008 #8
    Hi, you seem to be mostly on the right track....

    Gauge theory posits a symmetry described by the transformation rule [tex] \psi \rightarrow U \psi [/tex], where U belongs to a gauge group G, and [tex] \psi [/tex] belongs to a vector space which is referred to as the fundamental representation of G. The covariant derivative [tex] D_{\mu} \psi [/tex] transforms in the same way (in other words, it also belongs to the fundamental representation), and this can be used to show gauge invariance of the kinetic term for [tex] \psi [/tex] in the Lagrangian, as you seemed to refer to in your original post. The adjoint representation is the vector space of quantities which transform as [tex] Z \rightarrow UZU^{-1} [/tex]; the connection [tex]A_\mu[/tex] is of this type for *global* gauge transformations, while the field strength [tex] F_{\mu\nu} [/tex] is an even better example, since it transforms this way even under local gauge transformations. [tex]A_\mu[/tex] belongs to this space as a consequence of its definition in the covariant derivative. For your question, yes, both [tex]A_\mu[/tex] and [tex] F_{\mu\nu} [/tex] take values in the Lie algebra of G.

    I'm not entirely sure, but perhaps your question in your original post asks what changes if we substitute G for one of its other representations? If so, the answer is nothing, because G can be whatever group you like; the fundamental representation is still the space which holds \psi, and A and F will be in the adjoint representation of that space.

    Good luck with the studies!
    Dave
     
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