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Say I have a multicomponent field $X_a(x,t)$ such that I know it Fourier modes satisfy the following equation of motion,

$(\delta_{ab} \partial_t + \Omega_{ab}(t))X_b(k,t) = e^t \int \frac{d^3p d^3q}{(2\pi)^3} \gamma_{abc} (k,-p,-q) X_b (p,t) X_c(q,t)$

where $\gamma_{abc}$ is a function of the 3 momenta and $\Omega_{ab}$ is a time dependent matrix.

  • Now is it legitimate to say that in real space the equation is,

    $(\delta_{ab} \partial_t + \Omega_{ab}(t))X_b(x,t) = e^t \int \frac{d^3k d^3p d^3q}{(2\pi)^{3+1.5}} e^{ik.x}\gamma_{abc} (k,-p,-q) X_b (p,t) X_c(q,t) $

  • Now if one wants to write a Lagrangian such that this would be its equation of motion then one is faced with the "Dirac equation" kind of problem where by an equation of motion which is first order in space-time derivatives can't be gotten from Bosonic varaibles and hence one has to think of the field as Grassmannian. But if I want to keep my $X$ Bosonic then one possible way to do it is to introduce an auxiliary field say $A_a(x,t)$ and write the action as,

$S = \int dt d^3k \{A_a(k,t)[(\delta_{ab} \partial_t + \Omega_{ab}(t))X_b(x,t) - e^t \int \frac{d^3k d^3p d^3q}{(2\pi)^{3+1.5}} e^{ik.x}\gamma_{abc} (k,-p,-q) X_b (p,t) X_c(q,t)]\}$

Then it seems that the equation of motion w.r.t variation of $A$ gives the equation of motion for $X$

  • But does this make sense? (isn't the auxilliary field now becoming interacting with X?)
  • Now can one just go ahead and do the usual field theory of calculating n-point functins etc by adding to this $S$ above a source field for both $X$ and $A$ as say $S'= \int dt d^3x [K_a (x,t) A_a (x,t) + J_a (x,t) X_a (x,t) ]$

Or is this all a bit too naive and something subtle is being missed?

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