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Tengo un par de ejercicios que resolver de thermal field theory. No he conseguido avanzar casi nada por diversos motivos y la entrega es en breve. Desearía saber si alguien tiene el tiempo y las ganas para resolver alguno de ellos. Cuelgo el código en latex.
Gracias de antemano y un saludo a todos.
First Homework: consider scalar $\phi^4$ theory with Lagrangian density
$$
L = - \partial_\mu \phi^* \partial^\mu \phi - m^2 \phi^* \phi
- \lambda (\phi^* \phi)^2 /24
$$
where $\phi$ is a complex field $\phi = ( \phi_r + i \phi_i)/\sqrt{2}$.
1) Find the conserved Noether current $J_\mu$ associated with the
symmetry $\phi \rightarrow e^{i\theta} \phi$, phase rotation of the
complex field.
2) Write down an expression for the Fourier transform of the correlation
function of two of these current operators,
$$
G^>(k^0,\vec{k}) = \int dt e^{-i k^0 t}
\int d^3 x e^{i \vec{k}\cdot \vec{x}} \times
\langle J_\mu(0) J_\nu(t,\vec{x}) \rangle
$$
where the $\langle \rangle$ mean (quantum) expectation value in the
thermal ensemble. Work only to the lowest order (zero-order) in
$\lambda$ and set the mass to zero, that is, you can feel free to treat
$m^2 = 0 = \lambda$ and compute in the massless free theory. Do not try
to evaluate your expression just yet.
3) Evaluate your expression for the contraction
$\langle J_\mu(0) J^\mu(t,\vec{x}) \rangle$ (that is, $g^{\mu \nu}$ of
the expression you found in part 2 above) in TWO limits:
A) $\vec{k} = 0$ strictly, $k^0$ nonzero but small (specifically,
much smaller than the temperature $T$)
B) $k^0 = 0$ strictly but $|\vec{k}|$ nonzero but small (much
smaller than the temperature)
Show that these limits do not commute, that is, the limit as
$k^\mu \rightarrow 0$ is different depending on whether $k^0$ is
first set to zero or if $\vec{k}$ is first set to zero.
4) [Other students did not get this far, but to be fair I am including
steps analogous to what they were supposed to do in your homework as
well.] Extra Credit::::
Interactions mean that the propagator will not be strictly an
on-shell delta function; replace
$$ \Delta^>(k) \equiv
\int d^4 x e^{ik\cdot x} \langle \phi(0) \phi^*(x) \rangle
= 2\pi (n_b(k^0)+1) \delta(k^2) $$
with $2 (n_b(k^0)+1) \Gamma/((k^2)^2+\Gamma^2)$ a Lorentzian of
the same area. This simulates the effects of scattering.
Using this expression show that the behavior of the correlator found
above is regular, but show that current conservation
$$ \partial_\mu \langle J_\nu(0) J^\mu(x) \rangle = 0 $$
is NOT obeyed. Therefore there is something wrong with making this
substitution of the propagator.
==============
Assignment 2
1)
Consider the sum encountered in class:
$$
F(y) = \sum_{n \in {\cal Z}} \frac{1}{y^2 + n^2}
$$
that is, the sum over $n$ a member of the integers of $1/(n^2 + y^2)$.
Show that in the limit of small $y$, the sum approaches
$F(y) = 1/y^2 + C + ...$ and (if possible) evaluate $C$.
Then show that in the limit of large $y$, the sum approaches
$F(y) = \pi/y$.
Extra credit: show that the corrections to the last expression are
exponentially small, $\sim \exp(-y)$. Use the series expansion, do not
make reference to the explicit expression we found for the sum.
2)
We know that in vacuum the defining equation for the Green function
$$
\partial_\mu \partial^\mu G(x) = \delta^4(x)
$$
is solved by
$$
G(x) = \frac{1}{4\pi^2} \frac{1}{x^2} \,.
$$
(Here $G(x) = \langle \phi(0) \phi(x) \rangle$ is the Wightman
function.)
Argue that, for QED in Feynman gauge, the correlation function
$$
G_{\mu\nu}(x-y) \equiv \langle A_\nu(y) A_\mu(x) \rangle
$$
is
$$
G_{\mu\nu}(x-y) = \frac{1}{4\pi^2} \frac{1}{|x-y|^2} \,.
$$
Use this expression to determine the correlation function of two field
strengths,
$$
G_{\mu\alpha,\nu\beta}(x-y) =
\langle F_{\mu\alpha}(y) F_{\nu\beta}(x) \rangle
$$
directly in real-space by taking derivatives with respect to $x,y$ of
the expression for $G_{\mu\nu}(x-y)$ you found above.
3)
At finite temperature, find an expression for the
equal-time, magnetic field-magnetic field correlation function
$ G_{ij,kl}(\vec{x}-\vec{y}) $.
Your expression should have an infinite summation. If possible, perform
the infinite summation to get an explicit expression. It may be easiest
to first get an explicit expression for $G_{ik}(x-y)$ and then take
spatial derivatives.
Tengo un par de ejercicios que resolver de thermal field theory. No he conseguido avanzar casi nada por diversos motivos y la entrega es en breve. Desearía saber si alguien tiene el tiempo y las ganas para resolver alguno de ellos. Cuelgo el código en latex.
Gracias de antemano y un saludo a todos.
First Homework: consider scalar $\phi^4$ theory with Lagrangian density
$$
L = - \partial_\mu \phi^* \partial^\mu \phi - m^2 \phi^* \phi
- \lambda (\phi^* \phi)^2 /24
$$
where $\phi$ is a complex field $\phi = ( \phi_r + i \phi_i)/\sqrt{2}$.
1) Find the conserved Noether current $J_\mu$ associated with the
symmetry $\phi \rightarrow e^{i\theta} \phi$, phase rotation of the
complex field.
2) Write down an expression for the Fourier transform of the correlation
function of two of these current operators,
$$
G^>(k^0,\vec{k}) = \int dt e^{-i k^0 t}
\int d^3 x e^{i \vec{k}\cdot \vec{x}} \times
\langle J_\mu(0) J_\nu(t,\vec{x}) \rangle
$$
where the $\langle \rangle$ mean (quantum) expectation value in the
thermal ensemble. Work only to the lowest order (zero-order) in
$\lambda$ and set the mass to zero, that is, you can feel free to treat
$m^2 = 0 = \lambda$ and compute in the massless free theory. Do not try
to evaluate your expression just yet.
3) Evaluate your expression for the contraction
$\langle J_\mu(0) J^\mu(t,\vec{x}) \rangle$ (that is, $g^{\mu \nu}$ of
the expression you found in part 2 above) in TWO limits:
A) $\vec{k} = 0$ strictly, $k^0$ nonzero but small (specifically,
much smaller than the temperature $T$)
B) $k^0 = 0$ strictly but $|\vec{k}|$ nonzero but small (much
smaller than the temperature)
Show that these limits do not commute, that is, the limit as
$k^\mu \rightarrow 0$ is different depending on whether $k^0$ is
first set to zero or if $\vec{k}$ is first set to zero.
4) [Other students did not get this far, but to be fair I am including
steps analogous to what they were supposed to do in your homework as
well.] Extra Credit::::
Interactions mean that the propagator will not be strictly an
on-shell delta function; replace
$$ \Delta^>(k) \equiv
\int d^4 x e^{ik\cdot x} \langle \phi(0) \phi^*(x) \rangle
= 2\pi (n_b(k^0)+1) \delta(k^2) $$
with $2 (n_b(k^0)+1) \Gamma/((k^2)^2+\Gamma^2)$ a Lorentzian of
the same area. This simulates the effects of scattering.
Using this expression show that the behavior of the correlator found
above is regular, but show that current conservation
$$ \partial_\mu \langle J_\nu(0) J^\mu(x) \rangle = 0 $$
is NOT obeyed. Therefore there is something wrong with making this
substitution of the propagator.
==============
Assignment 2
1)
Consider the sum encountered in class:
$$
F(y) = \sum_{n \in {\cal Z}} \frac{1}{y^2 + n^2}
$$
that is, the sum over $n$ a member of the integers of $1/(n^2 + y^2)$.
Show that in the limit of small $y$, the sum approaches
$F(y) = 1/y^2 + C + ...$ and (if possible) evaluate $C$.
Then show that in the limit of large $y$, the sum approaches
$F(y) = \pi/y$.
Extra credit: show that the corrections to the last expression are
exponentially small, $\sim \exp(-y)$. Use the series expansion, do not
make reference to the explicit expression we found for the sum.
2)
We know that in vacuum the defining equation for the Green function
$$
\partial_\mu \partial^\mu G(x) = \delta^4(x)
$$
is solved by
$$
G(x) = \frac{1}{4\pi^2} \frac{1}{x^2} \,.
$$
(Here $G(x) = \langle \phi(0) \phi(x) \rangle$ is the Wightman
function.)
Argue that, for QED in Feynman gauge, the correlation function
$$
G_{\mu\nu}(x-y) \equiv \langle A_\nu(y) A_\mu(x) \rangle
$$
is
$$
G_{\mu\nu}(x-y) = \frac{1}{4\pi^2} \frac{1}{|x-y|^2} \,.
$$
Use this expression to determine the correlation function of two field
strengths,
$$
G_{\mu\alpha,\nu\beta}(x-y) =
\langle F_{\mu\alpha}(y) F_{\nu\beta}(x) \rangle
$$
directly in real-space by taking derivatives with respect to $x,y$ of
the expression for $G_{\mu\nu}(x-y)$ you found above.
3)
At finite temperature, find an expression for the
equal-time, magnetic field-magnetic field correlation function
$ G_{ij,kl}(\vec{x}-\vec{y}) $.
Your expression should have an infinite summation. If possible, perform
the infinite summation to get an explicit expression. It may be easiest
to first get an explicit expression for $G_{ik}(x-y)$ and then take
spatial derivatives.