If we have the wavefunctional $\Phi[\psi]$ which tells us the probability density for finding $\psi$. Let's say we know the exact field state at $t=0;$ $\psi(x,0)$. Can we use the wavefunctional $\Phi[\psi]$ to evolve the wavefunction to $\psi(x,t)$. Using something similar to Bohmian mechanics for fields because if we know the initial position and momentum for a particle in bohmian mechanics we can calculate its new position and momentum.
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3$\begingroup$ I'm not aware of a Bohmian version of field theory, but there is the so-called "functional Schrodinger approach" which generalizes the Schrodinger equation to work for quantum fields. Just like in ordinary quantum mechanics, if you know the initial state of the field you can use unitary time evolution to compute the state at later times. $\endgroup$– AndrewCommented Nov 15, 2020 at 20:05
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Since Bohmian mechanics is a deterministic theory, if you now the initial state and the probability density of the wavefunction, you will be able to predict the final state. This is not true if you add noise to the system(for example collision between atoms), then you have to consider stochastic processes. This means that even if you know the probability density of the wavefunction and the initial state, you will be able to predict the final one. You will know it only when you get the time T that corresponds to that state, given that you know all random variables of the system until that moment.
