Unintuitive equivalency involving multiple dual operators

Obligatory apologies if this subject is too "homework-y" for the mainline math sub.

• I tend to get exceedingly long-winded when I type out reddit posts so I will do my best to keep this short. Skip to the quoted material if you don't eff with mostly-unnecessary context.

• I am an ex math major who had my higher-education cut short for medical reasons. I have always been fond of recreational maths. Dual numbers in particular have always tickled my brain in a particular way. This question specifically I have been coming back to for over a year with little-to-no forward progress. I have done my best to read the appropriate reference material, but (as most technical writing online) it is rather dense. Here's hoping the answer to this leans-trivial or at least can be condensed to a handful of wikipedia articles rather than the 30 I've been going back to.

• I am less familiar with all of this now than I once was; its been a while I've been back to this. Sorry. Hopefully I haven't retreaded some of my previous mistakes.

• Also I've never used latex and cannot find a reliable plugin/script for firefox and thus have no way of knowing if this will display properly. At the very least it is formatted poorly. Sorry. I will be including a link to an image of the rendered text in case I wiff the formatting and/or for mobile users

• I also realize I include a significant amount of math that's likely unnecessary. I get I could've condensed this to like three lines; most papers I read on grassman numbers didn't bother to touch this algebraic stuff. Or if they did they wrote it in lanquage i couldn't understand. It is included here to give context to my thought process.

[; \varepsiloni * \varepsilon_j = -\varepsilon_j * \varepsilon_i :, \ Let: \varepsilon_i = \varepsilon_j:; \varepsilon_i * \varepsilon_i = 0 = -\varepsilon_i * \varepsilon_i \ \dot{.\hspace{.095in}.}\hspace{.5in} \varepsilon_i² = 0\ \forall\ \varepsilon \ \dot{.\hspace{.095in}.}\hspace{.5in} \varepsilon : \textrm{are } \emph{nilpotent} \ \ (\varepsilon_i + \varepsilon_j)² = \varepsilon_i² + \varepsilon_i \varepsilon_j + \varepsilon_j \varepsilon_i + \varepsilon_j² = 0 + \varepsilon_i \varepsilon_j - \varepsilon_i \varepsilon_j + 0 = 0 \ (\sum{k=0}^{n} \varepsilon{i_k})² = \sum{k=0}^{n} \varepsilon{i_k}² + \sum{k=0}^{{n}(\varepsilon{i_k}\sum{j=k+1}{n}\varepsilon_{i_j})} - \sum{k=0}^{{n}(\varepsilon}{ik}\sum{j=k+1}^{{n}\varepsilon_{i_j})} = 0 \ \dot{.\hspace{.095in}.}\hspace{.5in} \textrm{sums of } \varepsilon: \textrm{are } \emph{nilpotent} \ \ e^x = 1 + x + \frac{x^2}{2!} + ... + \frac{x^n}{n!} + ... = \sum{i=0}^{n} \frac{x^n}{n!} \ e^\varepsilon = 1 + \varepsilon :, : (\textrm{for } \emph{all} \textrm{ nilpotents of degree 2}) \ e^{{\varepsilon_i+\varepsilon_j}} = 1 + \varepsilon_i + \varepsilon_j : (\textrm{as } \varepsilon_i + \varepsilon_j \textrm{ is nilpotent}) \ \dot{.\hspace{.095in}.}\hspace{.5in} e^{\sum{k=0}^{n} \varepsilon{i_k}} = 1 + \sum{k=0}^{n} \varepsilon{i_k} \ \ \prod{k=0}^{n}(1 + \varepsilon{i_k}) \ = 1 + \varepsilon{i0} + \varepsilon{i1} + ... + \varepsilon{in} \ + \varepsilon{i0}\varepsilon{i1} + \varepsilon{i0} \varepsilon{i2} + ... + \varepsilon{i0} \varepsilon{in} \ + \varepsilon{i1}\varepsilon{i2} + \varepsilon{i1}\varepsilon{i3} + ... + \varepsilon{i1}\varepsilon{in} \ + ... \ + \varepsilon{i{n-1}}\varepsilon{in} \ + \varepsilon{i0}(\varepsilon{i1}\varepsilon{i2} + \varepsilon{i1}\varepsilon{i3} + ... + \varepsilon{i1}\varepsilon{in}) \ + \varepsilon{i1}(\varepsilon{i2}\varepsilon{i3} + \varepsilon{i2}\varepsilon{i4} + ... + \varepsilon{i1}\varepsilon{in} + ... + \varepsilon{i{n-1}}\varepsilon{in}) \ + ... \ + \varepsilon{i{n-2}}\varepsilon{i{n-1}}\varepsilon{i{n}} \ + ... \ + \varepsilon{i{0}}\varepsilon{i{1}}...:\varepsilon{i{n-1}}\varepsilon{i{n}} \ \ =\prod{k=0}^{{n}e{\varepsilon_{i_k}}} \ \ = e^{{\sum{k=0}{n}\varepsilon{i_k}}} \ \ = 1 + \sum{k=0}^{{n}\varepsilon}{ik} = 1 + \varepsilon{i0}+\varepsilon{i1}+ ... + \varepsilon{i{n-1}} + \varepsilon{i_n} ;]

SUMMARY AND WHY IT MATTERS:

• These epsilon operators are defined to anti-commute with one another. As a result each operator is nilpotent (epsilon² = 0)
• Sums and products of these operators are also nilpotent. ( i think? maybe this is as simple as me messing up the non-commutative multiplication.)
• e^nilpotent = 1 + nilpotent
• Every combination of n coefficient-less operators can be generated with the provided product formula
• the product can be rewritten as the product of exponential functions which can be condensed into a single exponential function
• this exponential can be rewritten as a simple sum of single operators

My problem is that this sum is included in the original product (which it is ostensibly equal to), but the product goes on to include more items. A few questions arise:

• Where did I mess up this math?
• when using one of these numbers is it simply identical to leave out the remainder of the original product? I can see intuitively why this may be the case but do not know how to "prove" or otherwise explore this exhaustively.
• If there is nothing wrong this result, does it continue to hold with an infinite amount of operators? (i am interested in exploring the properties of these for myself)
• Can a similar result be found when considering the "full" sum where each variable has unique coefficients? (i am interested in exploring the properties of these variations myself; this is what got me looking into this in the first place)
• if not, do there exist coefficients where the equality still holds true?
• why are these "supernumbers" always formatted even differently than this? example (it is particularly strange to me that, in the case of the infinite, I am hardpressed to find an example that does not include the converging factorials) *in Dewitt's paper on the subject (afaik the premier source on the topic) why does he say "the soul need not be nilpotant"? is this related? incomplete google archive. (Found directly under 1.1.4)

I would appreciate any insight on this topic. I find it hard to maintain my interest in mathematics for very long when I inevitably run into large roadblocks like these. I am not opposed to doing the research; the roadblock for me is knowing what exactly to research. I endeavor to get to a point where I can stay out of the hospital for long enough to continue taking classes; i can practically feel my synapses rerouting around the rudimentary base I had built up five years ago.

Please do not feel the need to speculate on my knowledge-base or use that to temper your response. I have more than enough time to actually look into this--the right way. I finished a significant amount of my undergraduate mathematics in high school and have a handful of semesters under my belt filling out that base. I suspect I am a course or two away from this being a "well duh"-type problem.

EDIT: sorry in advance if i do not reply to your comments. I promise they are no less appreciated. I cannot use my hands for long periods of time and I am afraid i may have already overdid it with all this. My access to people who are generous to notate for me is limited and i am working on a more permanent accessibility solution

submitted by /u/SigmaGod
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from math https://ift.tt/3ab6zs2