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R6

From IFPA wiki
  1. a) Let [math]\vec l_1=(dx,0)[/math] and [math]\vec l_2=(0,dy)[/math] be two infinitesimal vectors along the sides of a parallelogram of angle [math]\theta[/math], and [math]g_{ij}[/math] the corresponding metric tensor. What are the lengths [math]l_1[/math] and [math]l_2[/math]? What is the relation between [math]\theta[/math] and [math]g_{12}[/math]? Show that the infinitesimal area is given by [math]\sqrt{det(g)}dx dy[/math]. b) Show that [math]\sqrt{-g} d^4 x[/math] is invariant under changes of variables.
  2. Consider a Bogolyubov transformation for fermionic fields [math]a^-_k=\alpha_k b_k^-+\beta_k b_{-k}^+[/math], [math]a^+_k=\gamma_k b_k^+ +\delta_k b_{-k}^-[/math]. What are the conditions on the coefficients?
  3. Find the relation between the bosonic two vacua [math]|0;a\gt [/math] and [math]|O;b\gt [/math] that are related by the Bogolyubov transformation [math]a^-_k=\alpha^*_k b_k^-+\beta_k b_{-k}^+[/math], [math]a^+_k=\alpha_k b_k^+ +\beta_k b_{-k}^-[/math].
  4. Show that [math] v_k(\eta) [/math] must have a Wronskian equal to [math]2i[/math] for the commutation relations of the fields to lead to a Hilbert space structure.
  5. Write the expression of the hamiltonian of a scalar field in an expanding universe as a function of creation and annihilation operators.

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