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Very often in the literature of research papers and other articles, and maybe text books, on topics of quantum field theory, a theory may be described as a 3+1 or 0+1, or maybe even 1+1 theory. I have never read any definition of this notation but I believe that the general form is (d-1,1) where d is the overall spacetime dimension. Thus, a 4-dimension space time theory might be described as 3+1; or, sometimes (3,1).

A few examples I know of are that the motion of a classical particle that is parameterized by time would simply be a 0+1 theory. Or, I have read that a particular String Theory is described as a 1+1 theory.

I have just exhausted any knowledge I might have gleaned on this question though. I have never seen any formal definition or motivation for its use.

My question, who and when was this notation developed and what was the motivation behind it. In the very simple examples I have seen, it does not seem to add much than what a few words could say more clearly (at least to a philistine such as I).

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    $\begingroup$ Isn't this just derived from the notation for inertia indices of quadratic forms and metric signatures? Lorentzian metric has (3,1) signature so Lorentizan 4-manifolds are labeled 3+1, etc. The substance goes back to Jacobi and Sylvester, I am not sure who introduced the signature notation. $\endgroup$ – Conifold Jun 19 '18 at 19:06
  • $\begingroup$ Normally a metric signature is represented by using either (+---) or (-+++) depending on the style or conventions of the author. Both are equivalent and negative of each other but signify the same physics. However, with the notation of something like 0+1 or 1+1 the user is saying something slightly different in that his theory is dependent on no space coordinates (0+1) and one time coordinate. That same theory actually could be extended (sometimes to a 3+1 theory but the author (it appears) may be limiting to the 0+1 case due to some kind of symmetry constraint or something. $\endgroup$ – K7PEH Jun 19 '18 at 20:55
  • $\begingroup$ Continuing -- I must admit that with respect to the rhyme or reasons of saying something like 3+1 verses 1+1 I am merely guessing as I do not know for a fact. I read one String Theory paper where the author declared his theory to be 1+1. I still do not know why that was the case because the math seemed to involved all of the typical generalized coordinates. $\endgroup$ – K7PEH Jun 19 '18 at 20:56

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