Going over some of my old posts I came across this one:
2 points in 4-space - Factual Questions - Straight Dope Message Board
Now, whether or not Time is or is not imaginary is incidental…but can this four dimensional matrix
image-20-1024x431.jpg (1024×431) (profoundphysics.com)
compute something…interesting…if one (or more) of the real dimensions are replaced by an imaginary number?
(such as i=sqrt-1)
For anyone else interested, the “four dimensional matrix” linked here is the stress-energy tensor (a matrix is a kind of tensor, but not all tensors are matrices). In GR, the stress-energy tensor is the mathematical description of, well, stuff. In ordinary experience, it’s mostly matter, though in some extraordinary circumstances, pressure can also be relevant, and in principle so can shear stresses or momentum flux, though I don’t know of any specific cases where those are non-negligible.
Anyway, I won’t definitively say no, but I can’t think of any interpretation where a non-real value would make any sense for any of that. Why do you ask?
Something that has stayed with me from a casual conversation many years ago with a physics mate of mine. Conversation had drifted onto the reality of numbers. And imaginary numbers. (He is a renegade engineer turned lattice gauge QCD physicist.) He opined that he didn’t think there was any physical reality to imaginary or complex numbers, but that they were a great way of handling phase. (And Quantum Dynamics is full of that.)
If you can work out how to make GR compatible with QD there is a free trip to Stockholm waiting for you.
This question has a solid factual answer? I thought this was still in the realm of theoretical physics.
And theoretical physics doesn’t have any facts?
Sure there is. i is a left turn. No, really, that’s all. When we say that i^2 = -1, that’s just another way of saying that making two left turns points you in the opposite direction from where you started.
That said, not all sorts of numbers are applicable to all sorts of phenomena, and it’s unclear, at best, what it would mean for something’s mass, say, to be rotated by 90º.
Geometrically speaking, if you want to introduce imaginary numbers, what you have is some form of complex manifold. These certainly do show up in physics, e.g. to formulate string theory. In any case, if you had some sort of gravitation(?) on your complex manifold then it might be more clear what geometric quantities on your complex manifold occur when formulating your conservation laws or other physical laws.
There may be some philosophical questions there for philosophers, but I don’t think you are going to pass too many physics courses without using complex numbers, so I would not worry too much about such provocative “renegade” statements.
As for this lattice gauge theory stuff, if your space-time is discrete that seems a natural setting to try to introduce relevant forms of quantum gravity but that is by its nature already not general relativity (except perhaps in some limit)
That’s not what I said. I asked if the OP’s stated question had one factual answer.
Not a mathematician or a physicist, but I think you could attack the physicality of real numbers on the same grounds. “Sure you get the right answer but are numbers really real?”
Isn’t “factual” the problem word? It’s true that the field is normally divided into theoretical physics and experimental physics, but they are two sides of the same coin.
Physics is written in math. In the best case, that math describes known physical reality as well as models and predicts aspects of physical reality that are yet either unobserved or not solidly tied to a known set of equations.
Poking around in the equations is the way that progress is made on the theoretical side of the coin. Experimenters then have to test the predictions that fall out of the changes. Sometimes the changes lead to major discoveries; sometimes they fail because they produce impossible or meaningless results, sometimes the answer isn’t yet clear. The Higgs boson was theoretical until it was found. Saying that those equations weren’t factual until they became factual is more a misuse of language than an understanding of physics.
Applying common English terms to science is always fraught. Philosophers have battled over the “reality” of math for hundreds of years. Do humans create math or “find” it? Are numbers themselves real? That common language term “real” is a different sense of “real” than the precisely-defined scientific term “real numbers” which are parallel to “imaginary numbers” and when put together form the “complex numbers.” Mathematicians create new forms of math every day, and those new forms often find usefulness in physics, answering problems that have long been unsettled. Is all math “factual”? In the sense that it gives answers, maybe, but the infinite summation of positive integers 1+ 2+ 3+ 4+ … equals -1/12. And yes, this profoundly nonintuitive fact has a place in theoretical physics.
tl;dr Theoretical and factual are not mutually exclusive.
Perpendicular.
I get that! But I was using common English. So a perfect capsulation of the problem.
In the Journal of Mathematical Physics, Vol. 12 No.3 there is a short note “The Einstein Tensor and its Generalizations” where Lovelock investigates the problem of finding a tensor A^{ij} which is symmetric, divergence-free, and a “concomitant” function of the metric tensor and its first two derivatives. He comes up with an expression that works in a space with an arbitrary number of dimensions.
There is also “Gravity Theories in more than Four Dimensions” by Zumino (Physics Reports 137, No. 1) with some Lagrangians like R_{ab}R_{cd}\cdots e_fe_g\cdots \epsilon^{abcd\cdots fg\cdots} (interpreted in terms of topological invariants) generalizing the Einstein one, which may provide an answer to the OP at least in terms of what happens when you increase the number of dimensions.
Having a foot in both the practical and theoretic (degrees in both engineering and mathematics), I’d almost say the opposite for complex numbers.
Rather than complex numbers having no ‘reality’ but useful as a way of handling phase, I’d say phase is a way for engineers (among others) to develop a better intuition/understanding of complex numbers based on something familiar from the physical realm.
Complex numbers aren’t intuitively easy for most people. So we will latch onto anything that helps us develop a mental model for them.
Personally I would agree. I found the viewpoint of my mate surprising. But in the context of the OP I though it an interesting one. The whole question of “what is mathematics” is amusing, but eventually doesn’t really lead anywhere. Given almost all real numbers cannot be constructed or expressed, it all gets a bit silly worrying.
I was aware of 4+n dimensional supergravity theories, such as Kaluza-Klein 5D theory, and, ultimately, M Theory
I was wondering if the introduction of imaginary numbers could provide an explanation, a mathematical basis, for such physical manifestations as the omnidirectional nature of time, or why higher dimensions are “curled up” really tiny on the Planck Scale.
Hey, if you can come up with a model that does it, great. But it takes a lot more brain sweat than just saying “Hey everyone, what if imaginary numbers?”.
Still generally speaking, all the geometry introduced, whether involving extra dimensions, various symmetries, holonomy, noncommutativity, etc. is supposed to be heavily constrained by having to output a consistent or realistic physical theory. It is going to be more complicated than is there or isn’t there a complex structure.
BTW there are a lot of experimental constraints on extra dimensions and how big or small they might have to be.
The same applies to negative numbers, too. Darn useful, but you can’t hand me a negative apple any more than you can handke me i apples. Phyaical reality is for predictions of physical effects, not intermediate results like complex probabilities or negative apples
The biggest problem with negative apples is that any room they occupy rapidly fills up with doctors.
