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How to solve the Klein Gordon Complex Field?

  1. May 25, 2009 #1
    Hi

    I am teaching myself QFT, and this is not a homework problem. Right now, I am learning Classical Field Theory, and I am not sure how to proceed with the following problem. I would be grateful to receive inputs and suggestions.

    Problem Given the Klein Gordon Lagrangian with a complex field,

    [tex]\mathcal{L} = \partial_{\mu}\Phi^{\dagger}\partial^{\mu}\Phi - m^2 \Phi^{\dagger}\Phi[/tex]

    find the conserved current and charge.

    My problem is two fold:

    1. How do I differentiate the first term with respect to [itex]\partial_{\mu}\Phi[/itex] (the term with the conjugate)? I know this is probably a trivial question, but I am new to this notation.

    2. What is the significance of a complex field, as opposed to a real field?

    Thanks.
     
  2. jcsd
  3. May 25, 2009 #2
    consider the other term as constant

    the real field describes particles with no charge (neutral), and complex field describe particles with two opposite charges.
     
  4. May 25, 2009 #3

    Fredrik

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    1. The function you're taking a partial derivative of is just a (first order) polynomial in 10 variables, so you differentiate it just like any other polynomial. There's nothing fancy going on here. It's no different than finding the partial derivatives of the function f defined by [itex]f(u,v,w,x,y,z)=uv-wx-m^2 yz[/itex] where m is a constant. (This is the 1+1-dimensional version of your [itex]\mathcal L[/itex]).

    2. The particles that correspond to a real field are their own antiparticles. Not so for a complex field. You can think of [itex]\phi[/itex] and [itex]\phi^\dagger[/itex] as two independent fields. This can be justified by checking that you get the same results as if you write [itex]\phi=\phi_1+i\phi_2[/itex] with [itex]\phi_1[/itex] and [itex]\phi_2[/itex] both real, and treat them as two independent fields.
     
    Last edited: May 25, 2009
  5. May 25, 2009 #4

    malawi_glenn

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    There is nice section on the complex Klein-gordon field in the book my Mandl, chapter 3 section 5 if i remember correctly.
     
  6. May 25, 2009 #5
    Thanks, what I originally meant to ask was how to compute

    [tex]\frac{\partial}{\partial[\partial_{\mu}\Phi]}\partial^{\nu}\Phi^{\dagger}[/tex]
     
  7. May 25, 2009 #6

    Fredrik

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    That's just [tex]\frac{\partial}{\partial A}B[/tex] with A and B expressed in a way that makes the expression look much more complicated than it is.
     
  8. May 25, 2009 #7
    With B = conjugate of A, right? What does it evaluate to? I'm a bit confused here.
     
  9. May 25, 2009 #8

    malawi_glenn

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    but A and B are INDEPENDENT here, see Fredrik post #3, thus

    [tex]
    \frac{\partial}{\partial A}B = 0
    [/tex]
     
  10. May 25, 2009 #9
    Thank you malawi_glenn and everyone else. I get this now.
     
  11. May 25, 2009 #10

    malawi_glenn

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    Have you done your exercises concerning the REAL KG field yet? :-)
     
  12. May 25, 2009 #11
    If you mean questions about determining the field equations from the real field Lagrangian, conserved currents and charge, then yes I have. But I did not encounter the notion of independence of the field and its conjugate field anywhere when I was doing them. I just came across the complex field question and got me thinking -- evidently in the wrong way. I had no idea that the field and its conjugate are independent fields, hence my question.

    Once again thank you for your help :-)
     
  13. May 25, 2009 #12

    malawi_glenn

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    well I personally think the REAL KG is harder to evaluate things with hehe, good luck, and just post the questions you have. Tell us also if you need more practice material.
     
  14. May 25, 2009 #13
    Oh okay, I'm just a beginner. Taken two courses on Quantum Mechanics with no prior exposure to relativistic quantum mechanics, so am having to learn stuff on the way. I am working through Landau CTF and Peskin/Schroeder mainly, and keep reading McMahon. Will definitely keep posting questions here :-)
     
  15. May 25, 2009 #14

    malawi_glenn

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    Well you can go a quick course on RQM here, since most QFT texts assumes that one is quite famililar with KG eq and Dirac eq and gamma matricies etc.

    http://www.phys.ualberta.ca/~gingrich/phys512/latex2html/node1.html [Broken]
     
    Last edited by a moderator: May 4, 2017
  16. May 26, 2009 #15
    Even in classical physics, if you have a function f(z), then the derivative with respect to z-conjugate is zero, and if you have a function f(z-conjugate), then the derivative with respect to z is zero (at least this is what I recall).
     
  17. May 26, 2009 #16

    vanesch

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    Nevertheless, that *is* puzzling and in fact totally incomprehensible when you think of derivative as the thing given by its definition. I remember having had a hard time with that question too, as somehow you want to apply the chain rule of some kind:

    if you have A and B = F(A), and then you have a function G(A,F(A)), and you're supposed to find the derivative wrt A, then it's difficult to understand how you should consider B = F(A) as an independent variable.

    The trick is that in fact, your "true" fields are the real and the imaginary part of phi, and phi is just a construct to lump them nicely together: phi = R + i J say, with R and J real "true" fields. You want the field equations in fact for R and J.
    Now, in that view, phi-dagger is nothing else but R - i J. And now you can consider phi as ONE linear combination of R and J, and phi-dagger as another, linearly independent one. Transformation of variables gives you then phi as one independent variable, and phi-dagger as another one, and not to be seen (for the moment) as the conjugate of phi, but rather as a different linear combination of R and J.

    And then you can show, and it is fun to work it out for yourself, that if you were to do the algebra for the R and J fields, you'd find the same field equations for them than as if you were going to consider phi and phi-dagger as independent.

    But indeed, at first sight, this doesn't make any sense.
     
  18. May 26, 2009 #17

    Landau

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    (I had a similar question once: click)
     
  19. May 26, 2009 #18
    Well, the proof is simple I think:

    [tex]
    f(a,b)=f(\frac{z+z^*}{2},\frac{z-z^*}{2i}) [/tex]

    [tex]
    \frac{\partial f}{\partial z}=\frac{\partial f}{\partial a}\frac{\partial (z+z*)}{\partial z}\frac{1}{2}+\frac{\partial f}{\partial b}\frac{\partial (z-z*)}{\partial z}\frac{1}{2i}
    [/tex]

    and then setting f=z=a+ib (and noting that [tex]\frac{\partial a}{\partial b}=0[/tex] and vice versa) should give you that:

    [tex]
    \frac{\partial z*}{\partial z}=0
    [/tex]

    So it's not intuitive, but if you've seen the proof before, then it's easy to understand why the conjugate is independent.

    But I agree that it is worthwhile breaking it up into 2 fields: O(2)~U(1) is related to this complex variables thing?
     
  20. May 27, 2009 #19

    vanesch

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    Let's see:
    If we take f = z = a + i b, then [itex]\frac {\partial f}{\partial a} = 1[/itex] and
    [tex]\frac{\partial f}{\partial b} = i[/tex]

    From this we have:
    [tex]
    \frac{\partial f}{\partial z}=1 \frac{\partial (z+z*)}{\partial z}\frac{1}{2}+i\frac{\partial (z-z*)}{\partial z}\frac{1}{2i} = \frac{\partial{z}}{\partial z}/2 + \frac{\partial{z*}}{\partial z}/2 + \frac{\partial{z}}{\partial z}/2 - \frac{\partial{z*}}{\partial z}/2 = \frac{\partial{z}}{\partial z}
    [/tex]

    and [itex]\frac{\partial{z*}}{\partial z}[/itex] disappears from the equation, no ?
     
  21. May 27, 2009 #20
    Okay, so just to confirm

    for

    [tex]\mathcal{L} = (\partial_{\mu}\Phi)^{\dagger}(\partial^{\mu}\Phi})-m^2\Phi^{\dagger}\Phi-V(\Phi^{\dagger}\Phi)[/tex]

    [tex]\frac{\partial\mathcal{L}}{\partial[\partial_{\mu}\Phi]} = \frac{\partial}{\partial[\partial_{\mu}\Phi]}((\partial_{\nu}\Phi)^{\dagger}(\partial^{\nu}\Phi)})[/tex]
    [tex]= \partial_{\nu}\Phi^{\dagger}\frac{\partial}{\partial[\partial_{\mu}\Phi]}g^{\nu\rho}\partial_{\rho}\Phi = \partial_{\nu}\Phi^{\dagger}\delta_{\rho}^{\mu}g^{\nu\rho} = \partial_{\nu}\Phi^{\dagger}g^{\nu\mu} = \partial^{\mu}\Phi^{\dagger}[/tex]

    Is this correct?
     
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