User:UniversalExplanation/appendix1

Appendix 1: A Basic Implication of Mapping Freedom
The fact that our results cannot depend on the exact mapping chosen implies some subtle consequences. In order to understand the impact, consider the possibility that we have found the correct $$\vec\Psi((x,\tau)_1,(x,\tau)_2,\cdots,(x,\tau)_n,t)$$ for some set Bk. Suppose we create a second mapping by simply adding the number a to every mapped numeric label. This is entirely equivalent to moving the origin of the real x axis by a distance a. This change requires

$$\vec\Psi((x,\tau)_1,(x,\tau)_2,\cdots,(x,\tau)_n,t)\rightarrow\vec\Psi((x-a,\tau)_1,(x-a,\tau)_2,\cdots,(x-a,\tau)_n,t)$$

but cannot change the probability of the referenced set Bk: i.e.,

$$P((x,\tau)_1,(x,\tau)_2,\cdots,(x,\tau)_n,t) = P((x-a,\tau)_1,(x-a,\tau)_2,\cdots,(x-a,\tau)_n,t)$$

for any and all values of a. This implies, by the very definition of a derivative, that

$$\frac{d}{da}P((x-a,\tau)_1,(x-a,\tau)_2,\cdots,(x-a,\tau)_n,t) = 0$$.

Forgetting, for the moment, exactly where this expression came from, consider the change of variables (x-a)i into zi. The chain rule of calculus allows us to write $$\frac{d}{da} = \sum_i^n\frac{\partial z_i}{\partial a}\frac{\partial}{\partial z_i}$$. But $$\frac{\partial z_i}{\partial a} = -1$$, which implies

$$\frac{d}{da}P = -\sum_i^n\frac{\partial}{\partial z_i}P((z,\tau)_1,(z,\tau)_2,\cdots,(z,\tau)_n,t) = 0$$.

We cannot make that equation false simply by changing the name of the variable, thus we know that

$$\sum_i^n\frac{\partial}{\partial x_i}P((x,\tau)_1,(x,\tau)_2,\cdots,(x,\tau)_n,t) = 0$$.

This implies that a differential constraint exists on the set of algorithms available to solve our problem. It is easy to show by direct substitution that the proper constraint on $$\vec\Psi$$ is

$$\sum_i^n\frac{\partial}{\partial x_i}\vec\Psi = i\kappa\vec\Psi$$ ,

where $$\kappa$$ is a real number.

The complex conjugate of this equation is

$$\sum_i^n\frac{\partial}{\partial x_i}\vec\Psi^\dagger = -i\kappa\vec\Psi^\dagger$$.

It follows that, by simple application of the chain rule, one has:

$$\sum_i^n\frac{\partial}{\partial x_i}P = \sum_i^n\frac{\partial}{\partial x_i}(\vec\Psi^\dagger\cdot\vec\Psi) = -i\kappa\vec\Psi^\dagger\cdot\vec\Psi + i\kappa\vec\Psi^\dagger\cdot\vec\Psi = 0$$

Exactly the same arguments may be used with regard to both the $$\tau$$ dependence and the t dependence. Thus we may immediately write down the following additional constraints on $$\vec\Psi$$:

$$\sum_i^n\frac{\partial}{\partial \tau_i}\vec\Psi = i\kappa_\tau\vec\Psi\;\;\text{ and }\;\;\frac{\partial}{\partial t}\vec\Psi = im\vec\Psi$$

where $$\kappa_\tau$$ and m are real numbers.

Note that what has been presented is fundamentally the mathematical argument behind conservation of momentum in conventional quantum mechanics.

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