- #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...
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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