I wondered the other day if you could brutally calculate the histories of all the systems associated to an origin point. I know of Feynmans integral which sums over the histories of all quantum systems which originates at some starting point in history: Recent developments in further study of this area suggests that you must even sum over all the histories of a universe! Of course to do this, you need to model the universe like a quantum system with it's own path integral and so you get back all the information about the universe as though it followed superpositioning rules... and yes. It has many different histories due to the wave function, assuming nothing is there to initially disturb it. The main good thing which comes out of a path integral is the fact it calculates the probability amplitude of the system. It is at the heart of it all, an equation of motion which was extended to Langrangian mechanics but it's complete quantized form was given by Feynman. To make such an equation, you need to ''slice'' up either time or positions which are functions of time. I'll consider a really simple model, a scattering of a particle against something like an electron. We will call the initial time of scattering \(t_1\) and the resultant motion is calculated at \(t_2\). We can slice this ''action'' into very small time intervals anything down satisfying Planck constraints. The idea is to create a model which equivalently accounts for all the motions of the system after the scattering processes. So the origin of the systems history will be calculated as a deviation from the physical mediator. We will assume also that it's wave function is calculating all the possible trajectories from the point source. (Particles never properly interact, it is in fact the local fields of the particles which interacts). To do this, I wanted to find a simpler solution to calculating these possible trajectories and to do this, I simply use some calculus. \(a(t) = \frac{dv}{dt}\) We first integrate this \(\int_{t_0}^{t_1} a(t) dt = dv\) We now multiply \(dp = F\ dt\) on both sides to obtain \(\int_{t_0}^{t_1} a(t) dt Ft = Ft \ dv = dE(v)\) And one can noticed that this can be rewritten as \(d \dot{E}(v) = \int_{t_0}^{t_1} a(t) F dt\) This has units of power. Using the fundamental theorem of calculus, this is the same as saying \((v(t_2) - v(t_1))F = P\) We now consider a form which can be derived from this: \(\int (v(t_2) - v(t_1)) M\ dv = dE(v)\) ∴ \((v(t_2) - v(t_1)) M = \frac{dE(v)}{dv} = p\) Using the dot product, the squared part of the formula arises as \(p^{2}_{e'} = p_{e'} \cdot p_{e'} = (p_{\gamma} - p_{\gamma'}) \cdot (p_{\gamma} - p_{\gamma'})\) \( = p^{2}_{\gamma} + p^{2}_{\gamma'} - 2p_{\gamma} p_{\gamma '}\ cos \theta\) So we have to describe the single history of a system, between the origin time and the final calculation at \(t_2\) as \((v(t_1) - v(t_2))^2 M^2 = p^{2}_{\gamma} + p^{2}_{\gamma'} - 2p_{\gamma} p_{\gamma '}\ cos \theta\) ....... now to my question. What is the easiest way to calculate the sums of all the systems? I imagined something as straight forward as \(\sum_i M_i\) but I am not sure.
