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 ? 



$p$-adic numbers in physics
For an overview of applications of p-adic numbers in physics I would refer to the [Wikipedia](https://en.wikipedia.org/wiki/P-adic_quantum_mechanics) and [Physics.stackexchange](http://physics.stackexchange.com/questions/15252/p-adic-quantum-mechanic) links, and to this [nLab entry.](https://ncatlab.org/nlab/show/p-adic+physics) 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:](http://www.ams.org/notices/200902/rtx090200212p.pdf)



> 
> 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.
> 
> 
> 


