On Tuesday, February 3, 2015 at 8:18:58 PM UTC+1, Aaron Meurer wrote:

> Do physics applications actually use summation notation with noninteger 
> limits? If so, what convention do they use?
>

Summing over the states of fermions. But that's a convention.
 

> @asmeurer. I only know they are used in summing the "half-integer" spins. 
> But i have no idea regarding the rules used.
>

Spins are usually half-integers in spacetime, but I know of some special 
cases, like propagation on waves on 2D surfaces, where one may build a 
theory with any rational spin quasi-particles, or something like that.

Sometimes people with a background in mathematics get really scared when 
they see how mathematics is used in physics. The funniest I can think of is 
that multiple summations and integrals are often silently assumed to 
commute, even if they sum over to infinity.

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