>>12831089In physics, particularly QFT, dimensional regularisation assumes that the dimension of spacetime is a complex number. In particular, you assume that , with .
However, they don't really use a theory of geometry where the dimension is complex, really they write down equations that depend on d, which for is the dimension of spacetime, and then analytically continue to new equations where d is complex, but is no longer be interpreted as the dimension of spacetime.
I'll give you a comparison. Say you have a theory of a particle of mass m, and in your equations somewhere you have a term involving m^2, and we'll define a new variable M = m^2 = 4, so the particle has mass m = 2. Now we analytically continue to M = -4, and we can no longer interpret M as the mass squared of a particle (what would imaginary mass even mean?), but we can (possibly) still use those equations -- it's (potentially) still a valid theory -- its just M is a variable that for M => 0 is interpreted as the mass squared of the particle, and for M < 0 is just some negative number appearing in our equations (see: Higgs mechanism, symmetry breaking). Dimensional regularisation is just the same thing but instead of working with the mass of a particle, we're doing the same thing with the dimension of spacetime.
