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Does spherical geometry govern physics at the quantum scale?

My motivation for this question came from studying non-Euclidean geometry. When we go down from general relativistic length scales to everyday length scales, geometry changes from non-Euclidean to Euclidean. Does this process continue as we move further down to quantum length scales? Does the shape of space continue to change? From a saddle to a flat sheet to a sphere, so to speak.

An example would be the 3 parallel lines that meet at the edge of the universe forming a triangle with sum of angles zero. As we shrink this triangle down the angle sum increases until at our scale it is a classical triangle. As this triangle shrinks down to a point does it's angle sum increase? Does it appear convex from our viewpoint and might this explain apparent faster than light travel from our viewpoint?

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  • $\begingroup$ Spacetime is non-Euclidean at all scales. It does not become Euclidean at our scale. However I wonder if you are thinking that space (just space not spacetime) looks flat at everyday scales and therefore you're wondering if we treat space as flat at quantum scales. Is this what you mean? $\endgroup$
    – John Rennie
    Commented Apr 26 at 15:26
  • $\begingroup$ That's certainly a better way of putting it? $\endgroup$
    – tony schofield
    Commented Apr 26 at 15:33

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Does spherical geometry govern Physics at the quantum scale?

No.

Our leading theory to describe quantum particles is quantum field theory and this is formulated in a flat spacetime - specifically in Minkowski spacetime. The space part of the spacetime is Euclidean in the sense that Pythagoras's theorem holds true, but the spacetime as a whole is Lorentzian not Euclidean. It is the same as our everyday spacetime except that in everyday life we are not generally dealing with velocities near the speed of light. Particle colliders like the Large Hadron Collider accelerate the particles to 99.9999991% of the speed of light, so there relativistic effects are very important.

Working in a curved spacetime presents some technical difficulties for quantum field theory, though it can be done, and indeed this is how Hawking discovered Hawking radiation. However the vast majority of quantum field theory calculations are done in a flat spacetime not a curved one.

It is generally true that space looks flatter as you zoom in. For example the Earth looks flat to me, though I know perfectly well that it's a sphere. A ping pong ball would look flat to an amoeba living on its surface. So we expect that space flattens as we zoom in, but there is no point at which it inverts and becomes curved again. It just gets flatter and flatter the more we zoom in.

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  • $\begingroup$ Is it necessary that in order to quantize space-time as in LQG, we need to have flattest space as possible. I have heard that area and volume are also quantised in LQG. And I feel that a curved space is more favourable for quantisation of area and volume rather than a flat one. Which could be correct? $\endgroup$
    – Proscionexium
    Commented Apr 26 at 16:29
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    $\begingroup$ @Proscionexium In LQG the loops are not quanta of spacetime. They are a mathematical structure from which area and volume can be calculated using the corresponding operators. The curvature does not arise from LQG in any simple way, and indeed last I heard it had still to be demonstrated that LQG yields general relativity as a classical limit. $\endgroup$
    – John Rennie
    Commented Apr 26 at 16:54

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