It’s easy, relatively speaking, to develop an intuition for higher spatial dimensions, usually by induction on familiar lower-dimensional spaces. But I’m having difficulty envisioning a universe with multiple dimensions of time. Even if such a thing may not be real or possible, it seems like a good intellectual exercise. Can anyone offer an illustrative example?
Several authors (in particular Itzhak Bars) have written papers about two-time-physics that should help build intuition for the topic.
Infinitely many 'times' appear in integrable systems.
Theories can be formulated in any number of space-time dimensions. Examples of multidimensional theories:
F-theory, 12d theory with two times.
S-theory, 13d theory with three times.
Kalitzin's relativity with r-times or even infinite-dimensional times (the latter much less developed) was studied.
Indeed, I think to remember some people tried to understand quantum mechanics in a multitemporal set-up.
In philosophy, some people studied three-dimensional time theories. These times are called time, hyparxis and eternity.
If you are happy to focus on the intuitive without worrying about too much maths, the classic work on multiple time dimensions in physics must surely be J W Dunne's The Serial Universe - the second edition, published in 1942, is shorter and (comparatively) more readable. It was the sequel to his bestseller An Experiment with Time and elaborated on the role of the observer in modern physics. His regress of multiple time dimensions was inhabited by a similar regress of observers, thus (in retrospect) proffering a solution to the Schroedinger's Cat + Wigner's Friend regress of discrete real observers.
A multiverse interpretation of additional time dimensions accounts for the expected stochastic process variability, empirically captured by repeated sampling. Some experimentalists and frequentists do not consider multiple time dimensions as realistic, as the solutions of various (operator) equations, derived model predictions, and data-driven inference may generally not be deterministic, but are rather holistic, i.e., they are probabilistic in nature or may imply non-local constraints.
See for example the rigorous mathematical formulation of complex time (kime), which addresses some problems of time, represents classical temporal dynamics as kime-surface manifolds, and offers unique ways to model time dynamics of natural phenomena by observing repeated measurements from controlled longitudinal experiments.
I have been a self-guided student of theoretical physics for over 30 years. I don't have any formal training. I have been working on a model of our universe ever since I purchased and read Dr dr Michio Kaku book "hyperspace" in high school 25+ years ago. My model utilizes multiple dimensions of space time. It is still a work in progress. The assumption that my model is built of is this: Because spacetime is an inseparable entity it is assumed that space and time are proportionate to each other. What that means is, as space expands in dimensions, so too does time. So rather than with 5 spatial dimensions there is just one dimension of time there ate two axis of time. 6 dimensions of space 3 of time, [7D 4t], [8D 5t]. Lets just say time gets really really weird when you get to 4t and above. at one point you end up having a prime observer in two separate timelines within the same 3D space. Nuts! Another implication that is poking its head out is that energy density is relative to spacetime/matter-energy. That is, energy density determines the number of space time dimensions. There might be some explanation to the hyper expansion of our univers right after the big bang. This is still just a work in progress and no ware near complete. But some interesting hints at the possible nature of our univers.