$p$-adic numbers in physics
For an overview of applications of p-adic numbers in physics I would refer to the Wikipedia and Physics.stackexchange links, and to this nLab entry. Regarding the second question "What is the most convincing justification in physics that we need to work over the field of real or complex numbers" I would like to quote Freeman Dyson in Birds and Frogs:
When I look at the history of mathematics, I see a succession of illogical jumps, improbable coincidences, jokes of nature. One of the most profound jokes of nature is the square root of minus one that the physicist Erwin Schrödinger put into his wave equation when he invented wave mechanics in 1926. Starting from wave optics as a model, he wrote down a differential equation for a mechanical particle, but the equation made no sense. The equation looked like the equation of conduction of heat in a continuous medium. Heat conduction has no visible relevance to particle mechanics. Schrödinger’s idea seemed to be going nowhere. But then came the surprise. Schrödinger put the square root of minus one into the equation, and suddenly it made sense. Suddenly it became a wave equation instead of a heat conduction equation. And Schrödinger found to his delight that the equation has solutions corresponding to the quantized orbits in the Bohr model of the atom.
It turns out that the Schrödinger equation describes correctly everything we know about the behavior of atoms. It is the basis of all of chemistry and most of physics. And that square root of minus one means that nature works with complex numbers and not with real numbers.
All through the nineteenth century, mathematicians from Abel to Riemann and Weierstrass had been creating a magnificent theory of functions of complex variables. They had discovered that the theory of functions became far deeper and more powerful when it was extended from real to complex numbers. But they always thought of complex numbers as an artificial construction, invented by human mathematicians as a useful and elegant abstraction from real life. It never entered their heads that this artificial number system that they had invented was in fact the ground on which atoms move. They never imagined that nature had got there first.
Since the question on physics.stackexchange that was referred to in the comments above has been closed, let me essentially repeat my answer here together with some updates.
I think there are mainly two reasons for motivating the introduction of $p$-adic models in physics.
- They could exist in nature.
- They provide insightful toy models for physical phenomena.
There is a vast physical literature on the subject which cite Reason 1 as justification, this is sometimes called the Vladimirov Hypothesis. Namely, we do not know the texture of spacetime at the Planck scale therefore it is possible that it might look more like $\mathbb{Q}_p^d$ than $\mathbb{R}^d$. It is a seductive idea, but there is no evidence for it. Moreover, this would beg the question of why would Nature choose a particular prime number. A perhaps better hypothesis is the Adelic one advocated by Manin, i.e., all primes should occur democratically. In any case, this is quite speculative at present.
The way I see it, $p$-adic numbers are useful in physics mainly because of Reason 2. By exaggerating only a little, one could argue that $p$-adic toy models is what guided Kenneth Wilson when making his great discoveries in the theory of the renormalization group which entirely revolutionized physics in the early seventies.
When studying complex multiscale phenomena it is often important to decompose functions into time-frequency atoms which live on a tree, e.g., when using a wavelet decomposition. Unfortunately for most questions of interest the metric which governs how these atoms interact with each other is not the natural (from the tree point of view) ultrametric distance, but the Euclidean metric of the underlying continuum. Hierarchical models in physics amount to changing the model so it is the ultrametric distance which defines atomic interactions. The same idea also appears in mathematics where such toy models are often called "dyadic models". See this wonderful post by Tao for a nice discussion of this circle of ideas. Given a problem in Euclidean space, there are lots of ways of setting up a simplified hierarchical model for it. The $p$-adics, in some sense provide the most canonical, structured and principled way of doing this.
To go back to Wilson and the RG, he also used a hierarchical model and gave it yet another name "the approximate recursion". The importance of this toy model in Wilson's path to discovery is clear from his quote:
"Then, at Michael's urging, I work out what happens near four dimensions for the approximate recursion formula, and find that d-4 acts as a small parameter. Knowing this it is then trivial, given my field theoretic training, to construct the beginning of the epsilon expansion for critical exponents."
which can be found here (about one third from the bottom of the page).
For example, the use of $p$-adics gives a toy model for rigorously understanding conformal field theories. This is explained in my article "Towards three-dimensional conformal probability" (these slides are perhaps easier to understand since they have lots of pictures). More recently, the following articles explored similar issues:
- "$p$-adic AdS/CFT" by Gubser, Knaute, Parikh, Samberg and Witaszczyk,
- "Tensor networks, $p$-adic fields, and algebraic curves: arithmetic and the ${\rm AdS}_3$/${\rm CFT}_2$ correspondence" by Heydeman, Marcolli, Saberi and Stoica,
- "Edge length dynamics on graphs with applications to $p$-adic AdS/CFT" by Gubser, Heydeman, Jepsen, Marcolli, Parikh, Saberi, Stoica and Trundy.
As another example, in the article "A second-quantized Kolmogorov-Chentsov theorem via the operator product expansion" I discovered a general result for making sense of products of random Schwartz distributions starting from Wilson's operator product expansion. The statement of the theorem and the method of proof was done over $\mathbb{Q}_p$ first.
Another remark is that exotic numbers like $p$-adics can appear in two different conceptual roles. Suppose one is interested in a "scalar field" $\phi:X\rightarrow Y$ where $X$ is space-time. The toy models I mentioned make $X$ $p$-adic but keep $Y$ real or complex. As in Dyson's quote from Carlo's answer, things should still be Archimedean-valued. This kind of distinction is well known to researchers in the area called the Langlands Program which splits in two separate arenas depending on whether one's favorite $L$ or zeta functions take values in $\mathbb{C}$ or $\mathbb{Q}_p$. There are works in $p$-adic physics (e.g. by Khrennikov) where $Y$ is taken $p$-adic, but I am not familiar with the motivation behind this.
Finally, and in relation to Laie's comment above, one possible use of $p$-adics is to try to exploit an Adelic product formula in order to compute a real quantity of interest as the inverse of the product of similar quantitites over $\mathbb{Q}_p$. This was observed for tree level Veneziano amplitudes and was one of the first motivations for developing $p$-adic physics. However, these adelic formulas break down for higher loop amplitudes. See this Physics Report article for more on this story.