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Diffeomorphisms, Differential Structure, ETC.

 Quote by friend The integral, $$\int_{ - \infty }^{ + \infty } {\int_{ - \infty }^{ + \infty } { \cdot \cdot \cdot \int_{ - \infty }^{ + \infty } {{\rm{\delta (x - }}{{\rm{x}}_n}){\rm{\delta (}}{{\rm{x}}_n}{\rm{ - }}{{\rm{x}}_{n - 1}}) \cdot \cdot \cdot {\rm{\delta (}}{{\rm{x}}_1}{\rm{ - }}{{\rm{x}}_0})} } } d{x_n}d{x_{n - 1}} \cdot \cdot \cdot d{x_1} = {\rm{\delta (x - }}{{\rm{x}}_0})$$ ... if we use the gaussian form of the Dirac delta function, $${\rm{\delta (}}{{\rm{x}}_1}{\rm{ - }}{{\rm{x}}_0}) = \mathop {\lim }\limits_{\Delta \to 0} \frac{1}{{{{(\pi {\Delta ^2})}^{1/2}}}}{e^{ - {{({x_1} - {x_0})}^2}/{\Delta ^2}}}$$ with $${\Delta ^2} = \frac{{2i\hbar }}{m}({t_1} - {t_0})$$ the dirac deltas become $$\delta ({x_1} - {x_0}) = \mathop {\lim }\limits_{{t_1} \to {t_0}} {\left[ {\frac{m}{{2\pi i\hbar ({t_1} - {t_0})}}} \right]^{1/2}}\exp \left[ {\frac{{-im{{({x_1} - {x_0})}^2}}}{{2\hbar ({t_1} - {t_0})}}} \right]$$ ... the above (integral) becomes $$\int_{ - \infty }^{ + \infty } {\int_{ - \infty }^{ + \infty } { \cdot \cdot \cdot \int_{ - \infty }^{ + \infty } {{{(\frac{m}{{2\pi i\hbar \Delta t}})}^{n/2}}{e^{\,\,{\textstyle{-i \over \hbar }}\int_0^t {\frac{m}{2}{{(\dot x)}^2}dt} }}} } } d{x_n}d{x_{n - 1}} \cdot \cdot \cdot d{x_1}$$ Which is Feynman's path integral for a free particle. (See previous post for details)
But I haven't justified the complex nature in $${\Delta ^2} = \frac{{2i\hbar }}{m}({t_1} - {t_0})$$
And it seems to me that if I could find a good reason for complex numbers here, then I could justify the use of complex numbers in quantum mechanics, since the Dirac delta function can be used to derive the Feynman Path Integral.

I suppose the only way to justify the use of complex numbers within any formulism is to show that this formulism has the same algebraic properties as the complex numbers. As I understand it, the complex numbers lose the ability to determine magnitude (is i > -i ?), just as the quaternions also lose commutativity, and the octonions then also lose associativity.

Now since it is not possible to say which Dirac delta function is less than or greater than any other Dirac delta function, the Dirac delta function loses the property of magnitude (when compared to other Dirac deltas) but retains commutativity and asscociativity, just like the complex numbers do. So the complex numbers are a fair representation of the Dirac delta function since they share the same algebra. And we see here how the complex numbers enter quantum mechanics.

This brings me to wonder whether the properties of diffeomorphisms and differential structures change if we are talking about structures on a complex manifold instead of a real valued manifold.