Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Taylor expansion for SSB

  1. Mar 27, 2013 #1
    I am reading about spontaneous symmtry breaking for superconductors and came a cross to this simple statement:

    Here is the potential for complex scalar field: [itex] V = 1/2 \lambda^2 (|\phi|^2 -\eta^2)^2 [/itex].
    Scalar field is small and we can expand its modulus around [itex] \eta [/itex]:

    [tex]

    \phi(x) = |\phi(x)| e^{i \alpha(x)} = (\eta + \frac{1}{\sqrt{2}} \phi(x)) e^{i \alpha(x)}
    [/tex]

    How did he do that expansion???
     
  2. jcsd
  3. Mar 27, 2013 #2

    Fredrik

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    If that's just an expansion, then where did that ##\eta## come from? What is the relationship between ##\eta## and ##\phi##?

    I might help to provide an exact reference. Link directly to the relevant page at google books if that's possible.
     
  4. Mar 27, 2013 #3

    fzero

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    This expression is using ##\phi## for two related, but different quantities. It would be better to rewrite this as


    [tex]

    \phi(x) = |\phi(x)| e^{i \alpha(x)} = (\eta + \frac{1}{\sqrt{2}} \rho(x)) e^{i \alpha(x)}.
    [/tex]

    This formula defines a real scalar field ##\rho##. You might try to rewrite the potential in terms of ##\rho## to get an idea of why one might want to do this field redefinition.
     
  5. Mar 27, 2013 #4
    You are right, I think it was wrong in the text, there was no mentioning of this newly defined real field, no notation change, so I got confused.
     
  6. Mar 28, 2013 #5
    here is the attachment with that page from the book.
     

    Attached Files:

    • mmm1.pdf
      mmm1.pdf
      File size:
      115.3 KB
      Views:
      84
  7. Mar 28, 2013 #6

    fzero

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    The book uses ##\varphi## (\varphi in TeX) to distinguish the newly defined field from ##\phi##. I almost used it too, but figured the redefinition would be clearer with a completely different symbol.
     
  8. Mar 30, 2013 #7
    Yes, I was confused. For me phi is phi. it's interested how brain doesn't notice the difference even with close inspection.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook