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Hi,
I'm trying to learn some QFT at the moment, and I'm trying to understand how interactions/nonlinearities are handled with perturbation theory. I started by constructing a classical mechanical analogue, where I have a set of three coupled oscillators with a small nonlinearity added. The eqns. of motion of that system are:
##m\frac{d^2 x_1(t)}{dt^2}=-kx_1(t)+k'(x_2(t)-x_1(t))-\lambda [x_1(t)]^3##
##m\frac{d^2 x_2(t)}{dt^2}=-kx_2(t)+k'(x_1(t)-x_2(t))+k'(x_3(t)-x_2(t))-\lambda [x_2(t)]^3##
##m\frac{d^2 x_3(t)}{dt^2}=-kx_3(t)+k'(x_2(t)-x_3(t))-\lambda [x_3(t)]^3##
Now, if ##\lambda = 0##, this system is easy to solve by finding out the linear combinations of ##x_1,x_2,x_3## that are the normal modes of this system and behave like independent oscillators.
In the perturbed case, where ##\lambda > 0##, I can still plot how the system oscillates in the normal modes of the unperturbed system, but now because of the nonlinear term, energy (kinetic+potential) is transferred between those normal modes (some transitions are symmetry-disallowed, though).
The QFT equivalent of this would be a one-dimensional Klein-Gordan field with the phi-fourth perturbation, which is obviously a system of infinite number of oscillators (normal modes of the KG field), unless I arbitrarily assign a cut-off at some value of momentum and discretize the k-space.
Now, suppose I have an initial state that describes a particle with definite value of momentum:
##\left|p\right>=a_p^\dagger\left|0\right>##
Now I'd like to calculate the temporary rate at which there happen transitions to states with some other values of momentum. From the Schrödinger equation, I have:
##\frac{\partial\left|\psi (t)\right>}{\partial t} = -iH\left|\psi (t)\right>## ,
where H is the hamiltonian operator. The initial transition amplitude from momentum state ##p_1## to momentum state ##p_2## is:
##-i< 0 | a_{p_2} H a_{p_1}^{\dagger} |0>## .
Now, the question I run into, is how do I simplify the term ##-i< 0 | a_{p_2} [\hat{\psi} (x)]^4 a_{p_1}^{\dagger} |0>##, where the psi-hat is the field operator? I should somehow be able to write that fourth power of ##\hat{\psi}(x)## in terms of the creation and annihilation operators. All textbooks I've read seem to do these things in the Heisenberg or interaction pictures, but I'd personally understand this easier if I saw it done in the Schrödinger picture.
EDIT: I understand that a particle of one momentum can't simply turn into a particle with some other momentum, because momentum must be conserved... It seems, however, that an excitation with some value of p could be converted to two excitations with different momenta.
I'm trying to learn some QFT at the moment, and I'm trying to understand how interactions/nonlinearities are handled with perturbation theory. I started by constructing a classical mechanical analogue, where I have a set of three coupled oscillators with a small nonlinearity added. The eqns. of motion of that system are:
##m\frac{d^2 x_1(t)}{dt^2}=-kx_1(t)+k'(x_2(t)-x_1(t))-\lambda [x_1(t)]^3##
##m\frac{d^2 x_2(t)}{dt^2}=-kx_2(t)+k'(x_1(t)-x_2(t))+k'(x_3(t)-x_2(t))-\lambda [x_2(t)]^3##
##m\frac{d^2 x_3(t)}{dt^2}=-kx_3(t)+k'(x_2(t)-x_3(t))-\lambda [x_3(t)]^3##
Now, if ##\lambda = 0##, this system is easy to solve by finding out the linear combinations of ##x_1,x_2,x_3## that are the normal modes of this system and behave like independent oscillators.
In the perturbed case, where ##\lambda > 0##, I can still plot how the system oscillates in the normal modes of the unperturbed system, but now because of the nonlinear term, energy (kinetic+potential) is transferred between those normal modes (some transitions are symmetry-disallowed, though).
The QFT equivalent of this would be a one-dimensional Klein-Gordan field with the phi-fourth perturbation, which is obviously a system of infinite number of oscillators (normal modes of the KG field), unless I arbitrarily assign a cut-off at some value of momentum and discretize the k-space.
Now, suppose I have an initial state that describes a particle with definite value of momentum:
##\left|p\right>=a_p^\dagger\left|0\right>##
Now I'd like to calculate the temporary rate at which there happen transitions to states with some other values of momentum. From the Schrödinger equation, I have:
##\frac{\partial\left|\psi (t)\right>}{\partial t} = -iH\left|\psi (t)\right>## ,
where H is the hamiltonian operator. The initial transition amplitude from momentum state ##p_1## to momentum state ##p_2## is:
##-i< 0 | a_{p_2} H a_{p_1}^{\dagger} |0>## .
Now, the question I run into, is how do I simplify the term ##-i< 0 | a_{p_2} [\hat{\psi} (x)]^4 a_{p_1}^{\dagger} |0>##, where the psi-hat is the field operator? I should somehow be able to write that fourth power of ##\hat{\psi}(x)## in terms of the creation and annihilation operators. All textbooks I've read seem to do these things in the Heisenberg or interaction pictures, but I'd personally understand this easier if I saw it done in the Schrödinger picture.
EDIT: I understand that a particle of one momentum can't simply turn into a particle with some other momentum, because momentum must be conserved... It seems, however, that an excitation with some value of p could be converted to two excitations with different momenta.
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