Joy Christian wrote:Yablon wrote:FrediFizzx wrote:Go ahead and put it this function.
 := \lim_{{\bf s_A} \rightarrow\, -\text{sgn}({\bf a}\cdot{\bf s_A}){\bf a} }\Big[\langle\phi_{\bf n \text R}|\{ ({\boldsymbol\sigma}\cdot{\bf a}) \,(\lambda \,{\boldsymbol\sigma}\cdot (-{\bf s_A}))\}|\phi_{\bf n \text R}\rangle \Big] = \mp 1)
Let's see what it looks like.
Change “lim” to “set,” then go ahead and set things to what you are saying to set them to. And with bra and ket normalized, you will get
just as you intend.
The difference between "set" and "lim" is that the former is a discrete process whereas the latter is a continuous process.
In the GA model, which is based on a classical, continuous geometry, "lim" is more appropriate. "lim" is also more appropriate physically because there are no mysterious discrete jumps in nature. Ultimately the choice between "set" and "lim" is a matter of preference for underlying physics.
"lim" is not a process. It's an operator, and it's definition involves topology. There are discrete topologies and there are continuous topologies. If we write "limit as x tends to a of f(x) equals y" there is implied a space X in which x and a both live and a topology on that space, there is a space Y in which y lives, there is a topology on Y, and there is a function f from X to Y.
A "topology" T on X is just a set T of what are called the "T-open subsets" G of X. T must be closed under arbitrary unions and finite intersections and it must contain the whole space X.
Of course, a "topology" T' on Y is just a set T' of what are called the "T'-open subsets" G' of Y. T' must be closed under arbitrary unions and finite intersections and it must contain the whole space Y.
The definition of "limit as x tends to a of f(x) equals y" is that for every open subset G' of Y which contains a, there exists an open subset G of X containing a such that f(G) is contained in G'.
There is no "process". You may imagine a process, but you are letting your physics imagination run ahead of your formal logical/mathematical foundation, on the top of which you "do" physics.
Maybe that everything would appear to be stated differently nowadays in the language of category theory, but as I understand it, that is just a new and more powerful language, it has not altered the mathematical content - at least, not once one starts applying category theory to "sets". Category theory is about "categories", not about "sets".
Maybe Joy should explore the use of category theory in order to expand the domain of application of his work.
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