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Why the topological term F\til{F} is scale independent?

  1. Jun 20, 2014 #1
    why the topological term in gauge theory, [itex]ε_{\mu\nu ρσ}F^{\mu \nu} F^{ρσ} [/itex] ,is scale-independent?
     
  2. jcsd
  3. Jun 20, 2014 #2

    samalkhaiat

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    Do you know how the field tensor transforms under scale transformation?
     
  4. Jun 22, 2014 #3
    if the spacetime coordinates scales as [itex]x_{0}→S^{a}\bar{x_0}, x_{i}→S^{b}\bar{x_i}[/itex],then the potential vector scales as follow: [itex]A_{0}→S^{a+d}\bar{A_0}, A_{i}→S^{b+d}\bar{A_i}[/itex]
     
  5. Jun 22, 2014 #4

    samalkhaiat

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    Why do space and time scale differently? Can you tell me your background in physics?
     
  6. Jun 22, 2014 #5
    We are considering the most general rescaling, so space and time scale differently. This is especially true for nonrelativistic case.
     
  7. Jun 23, 2014 #6

    samalkhaiat

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    No, not “especially”. Time and space scale differently ONLY in non-relativistic theory. But, your original question is meaningless in the non-relativistic domain. This is why I asked you about your background in physics.
    Any way, in relativistic field theories, the coordinates scale according to
    [tex]x^{ \mu } \rightarrow \bar{ x }^{ \mu } = e^{ - \lambda } x^{ \mu } ,[/tex]
    and the field transforms as
    [tex]F ( x ) \rightarrow \bar{ F } ( \bar{ x } ) = e^{ \lambda \Delta } F ( x )[/tex]
    where [itex]\Delta[/itex] is the scaling dimension of the field. In D-dimensional space-time:
    [tex]\Delta = \frac{ D - 2 }{ 2 }, \ \ \mbox{ for } \ \ A_{ \mu } (x) ,[/tex]
    and
    [tex]\Delta = \frac{ D }{ 2 } , \ \ \mbox{ for } \ \ F_{ \mu \nu } ( x ) .[/tex]

    See (for more detailed description) the link below

    www.physicsforums.com/showthread.php?t=172461
     
  8. Jun 23, 2014 #7
    Thank you for your reply. The link given by you is so long that I need some time to follow. However, I donn't agree with your point that
    "Time and space scale differently ONLY in non-relativistic theory". It is the textbook's convention that space and time scale the same in relativistic region, but it is not the principle law. Anyway, I don't want to get invoked into this aspect. There is quick question I want to ask you: How do you determine the scaling dimension of a field? By dimension analysis and keep kinetic term dimension-D?
    This textbook's answer doesn't make sense...
     
  9. Jun 24, 2014 #8

    samalkhaiat

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    I’m afraid you have to agree with me. This is not an opinion, it is a “principle law”. Scaling has to preserve the light-cone structure. You violate relativity principles if you scale space and time differently.

    Yes, you get it either from the action integral or by demanding scale-invariance of the equal-time commutation relations. This is explained in the thread I linked above. See equations (10.13) to (10.17).
     
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