Deriving equations of motion of abelian gauge field coupled to scalar

In summary, the equations of motion are three sets of equations. One for each of the fields ##\phi,\phi^\dagger, A_\mu##.
  • #1
Geigercounter
8
2
Homework Statement
Consider the following theory in three dimensions (1 time and 2 space)
$$\mathcal{L} = -(D^\mu\phi)^\dagger(D_\mu\phi)-\frac{1}{4}\lambda( \phi^\dagger\phi - v^2)^2-\frac{1}{4}F_{\mu\nu}F^{\mu\nu},$$ i.e. an abelian gauge field coupled to a complex scalar. Here $$F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu,$$ ##\lambda## and ##v## are real numbers and $$D_\mu \phi = \partial_\mu \phi -iq A_\mu \phi.$$ Now we parametrize the spatial plane with polar coordinates ##\vec{x} = (r\cos \phi, r \sin \phi)## and take the solutions to be of the form (this is an assumption) $$\phi(r,\psi) = vf(r)e^{in\psi} \qquad \qquad \qquad \vec{A}(r,\psi) = \frac{i}{q}a(r)e^{in\psi} \nabla e^{-in\psi}, A_0 = 0.$$ Here ##n \in \mathbb{Z}##, ##a## and ##f## both go to ##1## as ##r \rightarrow \infty## and to zero at the origin.
Relevant Equations
See above
I want to compute the equations of motion for this theory in terms of the functions ##f## and ##a##. My plan was to apply the Euler-Lagrange equations, but it got confusing very quickly.

Am I right that we'll have 3 sets of equations? One for each of the fields ##\phi,\phi^\dagger, A_\mu## ?

So for example the equation of motion when differentiating to ##\phi##becomes $$-\frac{1}{2} \lambda(\phi^\dagger \phi - v^2) \phi^\dagger + \partial^\mu\phi^\dagger (iqA_\mu) - q^2A^\mu A_\mu\phi^\dagger + \partial_\mu\partial^\mu \phi^\dagger + iq \partial_\mu(A^\mu \phi^\dagger) = 0$$ Is this correct?

Then I'm also confused on the ##\partial_\mu## since we are now in spherical coordinates. Are these still derivatives with respect to ##x^1## and ##x^2## or with respect to ##r## and ##\psi##? I'm having some brain lag on this part.

EDIT: I've worked some more on the problem and obtained these equations of motion:
$$-\frac{1}{2}\lambda\phi^\dagger(\phi^\dagger \phi - v^2) +iqA_\mu\left(\partial^\mu + iqA^\mu \right) \phi^\dagger = - \partial_\alpha\left(\partial^\alpha + iqA^\alpha\right)\phi^\dagger$$

$$-\frac{1}{2}\lambda\phi(\phi^\dagger \phi - v^2) -iqA_\mu\left(\partial^\mu - iqA^\mu \right) \phi = - \partial_\alpha\left(\partial^\alpha - iqA^\alpha\right)\phi$$

$$\partial_\alpha F^{\alpha\mu} = iq\left(\phi(\partial^\mu + iq A^\mu )\phi^\dagger - \phi^\dagger(\partial^\mu - iqA^\mu)\phi \right)$$

Now plugging in our ansatz for the first one gives a large second order differential equation: $$-\frac{1}{2}\lambda v^3f(r)(f^2(r) - 1) + n^2a(r)\frac{1}{r^2}vf(r)\left(1 - a(r)\right) = vf''(r) -n^2vf(r)$$ This looks very messy to me...
 
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  • #2
Should your polar coordinates read ## \vec x = (r \cos \psi , r \sin \psi)##?

Have you tried rewriting your scalar fields as ##\phi = \rho (x) + \text{i} \eta (x)##, ##\phi^\dagger = \rho (x) - \text{i}\eta (x)## and rewritten the lagrangian in terms of the real scalar fields ##\rho## and ##\eta## instead?
 
  • #3
Yes ##\psi## indeed. I'd edit the post but seems like I can't do that anymore. I don't think I'd need to rewrite in a real and imaginary part though, since I have an explicit solution (ansatz)...
 
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  • #4
You can always do the same trick with the ansatz.
Have you noticed what happens if you take the complex conjugate of the eq of motion for ##\phi##?
What about the time parameter of the field?
 
  • #5
Well the fields are time-independent. The hermitian conjugate of the EOM for ##\phi## is precisely the EOM for ##\phi^\dagger##.

I don't see ow doing your trick in the ansatz will help further. I wrote it out but that makes everything more complicated in my opinion. What's wrong with the method I'm applying now?
 
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  • #6
How can you have eq of motion with time independent fields?
Where is this problem from?
 
  • #7
This ansatz is in the form of a soliton. In easiest case it is time independent.
 
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  • #8
Geigercounter said:
This ansatz is in the form of a soliton. In easiest case it is time independent.
And this is a textbook problem? Can you provide the original problem statement?
 
  • #9
The problem statement is as I've written it here. I got it from lecture notes I studied a while back when looking at solitons.
 
  • #10
Where is the actual statement?
1684868332122.png

Is the problem to "find a closed solution to ##f(r)## and ##a(r)##"?
 
  • #11
Not exactly. From the equations of motion I want to find two second order differential equations in ##a(r)## and ##f(r)##. That statement is just below the relevant equations section.

My apologies for the bad formatting.
 

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