Can the Residue Theorem be Extended to Higher Dimensions?

[Janitor]

Due to the functional form of typical Lagrangian densities that arise in particle physics, field theorists run into integrals having integrands that are fractions with polynomial denominators when they calculate propagators and Green‘s functions. That is where talk of “poles” and “contour integrals” comes into the picture. Pi makes an entry via the Residue Theorem. Pi shows up in the Residue Theorem because a circular contour surrounding a pole in the complex plane is used as the path of integration of an analytic function.

In another thread, Matt Grime said:

pi is only special, and commented on because we hacve chosen our geometry in such a way that circles are special. To say that we live in a 3-d world, and let's for the sake of argument accept that, is it not surprising that we didn't pick the ratio between the volume of a sphere and the cube of its radius?

For some reason, that triggered this thought: Is there any motivation to consider higher dimensional versions of the Residue Theorem? Or would nothing further be gained by this? For instance, would it make sense to consider analytic functions over quaternion values instead of over complex values? Would a quaternion version of the Residue Theorem then result from considering an integral over four-dimensional quaternion space, with the path of integration somehow being a three-dimension volume within the four-dimensional space, analogous to the familiar one-dimensional closed curve within the two-dimensional complex plane? Or does it not even make sense to speak of path integrals where the path is more than one-dimensional?

Sorry if this seems kind of fuzzy and ill-defined. At the moment I am not familiar enough with the subject to sharpen my question any further.

 

[Hurkyl]

In one case, the corresponding result is trivial:

In Cⁿ, with n > 1, if a function is analytic on a deleted neighborhood of a point, then it is also analytic at that point!

In other words, analytic functions do not have isolated poles. More specifically, the points of discontinuity form an analytic hypersurface.

There's probably something you can do with that, like when integrating along a loop around a path of discontinuity in C².


You can certainly talk about analytic functions of quaternions, but the extent of knowledge on that is roughly how you would define such a thing.


One of our algebraic geometers can probably shed some light on the topic, but I don't know if that light would relate to pi at all.

 

[Janitor]

Thanks for the quick response, Hurkyl.

Just to make sure that I am sort of understanding you, when I spoke of a four-dimensional quaternion space, I was implying that it was a four-dimensional real space R^4. I think you are saying that it can also be looked at (is isomorphic to) a two-dimensional complex space C^2.

Is your statement "In C^n, with n > 1, if a function..." easy to prove? What sort of book would one look at to see the proof?

 

[Hurkyl]

Is your statement "In C^n, with n > 1, if a function..." easy to prove? What sort of book would one look at to see the proof?

I would guess any multivariable complex analysis course would have the theorem. I had borrowed a little thin book on the subject from a friend for a couple months, that's where I encountered the theorem.

I think you are saying that it can also be looked at (is isomorphic to) a two-dimensional complex space C^2.

I think that's false, but I haven't tried working out a proof. The quaternions sort of act like they have one real axis and three imaginary axes, rather than two real and two imaginary.


I brought up C^2 because I thought you would find it interesting, and might be relevant.

Or does it not even make sense to speak of path integrals where the path is more than one-dimensional?

Nope. You can still talk about path integrals, and they might have nifty properties. You could also talk about surface integrals, but those don't seem very pretty.

 

[Janitor]

... The quaternions sort of act like they have one real axis and three imaginary axes, rather than two real and two imaginary...

Right, I know that quaternions themselves have a noncommutative multiplication rule. I somehow thought that they could be considered as forming the basis (maybe that is bad terminology) for an R^4 vector space, such that a point like (1,2,3,4) corresponds to the quaternion 1+2i+3j+4k. As long as all I cared about was addition of vectors in R^4, it seems obvious that they model addition of quaternions. You could then come up with a certain outer product multiplication rule (bad terminology?) that would make an ordered pair of vectors in R^4 correspond to the multiplication of quaternions, no?

Or am I getting things hopelessly muddled?

 

[Janitor]

I would guess any multivariable complex analysis course would have the theorem...

I will keep an eye out for that sort of book. I have seen a book or two on complex analysis, but maybe just single-variabled. 'z' seems to be the letter of choice for a single complex variable.

 

[Janitor]

...

There's probably something you can do with that, like when integrating along a loop around a path...

I was hoping to make good use of the Jordan-Brouwer Separation Theorem by specifying a three-dimensional region of integration.