There are many question about life and physics in higher dimensions. but is there any physical thing (i mean things like force, momentum, speed,...) that cannot exist in 2D world? could 2D world have life forms(not necessarily the life form we have) or there is something used by life that needs at least 3 spatial dimensions?

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    $\begingroup$ While not by itself a physics book I would recommend reading "The Planiverse" which is a fairly well researched attempt at describing physics in a 2D world. $\endgroup$ – Slereah Jun 9 '17 at 16:26
  • $\begingroup$ Possible duplicate: physics.stackexchange.com/q/77883/2451 $\endgroup$ – Qmechanic Jun 2 '18 at 17:26

Let us start with Newtonian mechanics. For any point particle, position, speed, acceleration, momentum and force would still make sense, with two components instead of three. But the question is then how those forces would be modified.

Let us look at gravity for example. If we were to assume the same form for the equation the gravitational field $g$ is solution of,

$$\DeclareMathOperator{\div}{div} \iint_S g.dS = -4\pi M$$

where $S$ is a closed surface and $M$ is the mass inside $S$, then instead of getting a force in $1/r^2$, we would get a force in $1/r$, or equivalently a gravitational potential in $\log r$, in the classic two-body problem. This leads to very different a phenomenology, as such a force field does not result in close trajectories (Bertrand's theorem). So the Earth, for example, would keep "precessing" around the Sun, never coming back to the same point. The Ephemerides of the Solar system would be a lot of fun!

The equivalent Quantum mechanical problem of the Hydrogen atom, would also have a potential in $\log r$ if we assume the same form of Maxwell equations. This result in unbounded energy level: they go to $+\infty$ as the quantum number $n$ goes to $+\infty$, which completely changes the phenomenology of ionisation and conduction. I will find some interesting references when I get back to my desk!

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  • $\begingroup$ WOW! nice! really looking forward to more physical things in 2D $\endgroup$ – Nemexia Jun 10 '17 at 17:48

Yes, life can exist in a 2D world. This was done by a long simulation experiment called Conway's Game of Life where signs of life were shown by tiles on a computer screen(search it up for details).

It is true that life would be vastly different and in a 2D world the laws of physics would have to adapt to allow formation of molecules etc.( e.g. In a 2D space, if laws did not modify, elliptical orbits would be impossible) Remember this, just because a mathematical framework for this exists does not mean it is truly something physical. The models we create should fit what we observe to create a satisfactory model in physics.

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  • $\begingroup$ I think you're misunderstanding Conway's Game of Life. It doesn't prove life can exist in 2D, and it doesn't "show signs of life." $\endgroup$ – Jold Jun 9 '17 at 18:55
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    $\begingroup$ @Jold Fans of Life have constructed self-reproducing structures that can evolve. $\endgroup$ – Mitchell Porter Jun 9 '17 at 20:42

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