Sorry, but that doesn't make any sense. Can you just outline the path integral approach for the QHO? I.e, start by writing down the relevant Hamiltonian and work from there?
Your last post has confused me.
I am back, but not for long... so i will quicly surmize our misunderstandings. Perhaps from my view, or yours. Who knows?
Right, the above works i quickly gave are related to temperature equations and we relate temperature to the known $$\beta$$ symbol, as most of us will know here, are related to the thermodynamical law, where entropy is an increasing value from the past cone, and where systems do have wavelike structures, then the system is undefined, and takes out all possible paths it can take.
But we need to remember about the statistics here. A wave forward without a collapse, is like a wave backwards, so temperature is involved in the Hamiltonian workings. So, if we took the Schrodinger Equation as an example, and even the Heat equations, are in fact primary examples of a Wick Rotation.
In a heat equation, we find that even for macroscopic systems, like an ice cube in a glass, the ice cube will displace more as time passes. If you have three spatial dimensions, and one time, usually the heat equation yields a zero value.
In a single dimension, the derivation of such a single point like configuration, now yields the equation, $$q=-k \nabla u$$ so that the reduction of the equation using the Fourier Law yields $$q=-ku_{x}$$. This means that the k is an acting thermal variable of conductivity, so in only one dimension, the gradient produces what we have seen.
This means, that there is an inverse law in which the equations are held, when $$<\psi \psi>< \psi* \psi*>$$ are the functions of inverse conjugates. In other words, the process forward, is NEVER seen the same way, unless, mind the pun, it is seen from some observational point of view, where the wave duration of the system yields $$\psi \psi*= \pm 1$$, and we say $$\pm$$ because it is representing the temperature, $$\beta$$, or if you like, the wave vector as being non-biased about any eigentstate it can have, when considered over a sum of its histories.
I cannot make it any more simpler than that I am afraid. You’ll just need to accept that these functions operate together in this way, and with the understanding of Generalized Absorber Theory, can one see a particle in imaginary time, taking all of it’s actions and non-actions into consideration, so the wave fields have no problem in defining a specific path, so long as relativity takes hold of it, and allows it to manifest as some kind of Least Action at very small values.
We link the time variable as an imaginary vector, and this creates rotations in spacetime where the beginning and end play the same roles. Of course, this statistically is very improbable, when taking a large clump of defined matter, so it’s not easy to see it from that point of view. Instead, we have resolved the work down to simple deduction.
$$x- \Delta x \le \xi \le x + \Delta x$$
because the time variable exists as a mathematical discipline that is a conjecture made by Einsteins teacher Minkowski… given as $$–ds^2+ds^2+ds^2=ds^2$$ where the $$-ds^2$$ is the acting time variable, whilst the other dimensions remain as spatial dimensions, but they switch roles all the time, according to relativity in strange gravitational singular area’s of spacetime.