TITLE: $p$-adic numbers in physics
QUESTION [14 upvotes]: As far as I know, in modern physics we assume that the underlying field of work is the field of real numbers (or complex numbers). Imagine one second that we make a crazy assumption and suggest that the fundamental equations of physics can be expressed with $p$-adic numbers. What could be really rewritten formally? Does it make sense in the physical world to work over $p$-adic numbers. Is there such attempt in the history of mathematical physics ?  What is the most convincing justification (in physics) that we need to work over the field of real numbers ?

REPLY [23 votes]: 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.