Anuncio

Colapsar
No hay ningún anuncio todavía.

ejercicio de thermal field theory

Colapsar
X
 
  • Filtro
  • Hora
  • Mostrar
Borrar todo
nuevos mensajes

  • 2o ciclo ejercicio de thermal field theory

    Hola
    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.

Contenido relacionado

Colapsar

Trabajando...
X