I Translational Symmetry: Effects on Laws & World

[Robin04]

What would the world look like if there was no translational symmetry? How would certain laws be different and what wouldn't change?

 

[Dale]

What would the world look like if there was no translational symmetry? How would certain laws be different and what wouldn't change?

Momentum would not be conserved. Other than that, I don’t think more can be said without specific details about how the symmetry was broken

 

[Robin04]

Momentum would not be conserved. Other than that, I don’t think more can be said without specific details about how the symmetry was broken

And in what ways could the symmetry be broken? I know that this is not about "real" physics, rather about fantasy, but I just saw a lecture online where the teacher talked about symmetries and he said that we could imagine a world where certain symmetries could be broken, and this really made me curious. I suspect there is an infinite number of ways these alternative worlds could be imagined, but I'm just wondering how our univserse would look like if some laws of physics were different (without being exhaustive).

For example, if I understood well, the translational symmetry is the same thing as space being homogeneous. So if space did have a center, how would that affect the conservation of momentum and the rest of physics?

 

[Dale]

And in what ways could the symmetry be broken?

Basically, translational symmetry means that the Lagrangian is not a function of position. Any Lagrangian which is a function of position would break translational symmetry. So the ways that the symmetry could be broken are every way you could make the Lagrangian a function of position.

 

[Dale]

So if space did have a center, how would that affect the conservation of momentum and the rest of physics?

Oh, I missed responding to this earlier. If there were a center and if the universe were isotropic then you would have some sort of central potential. Instead of force-free objects traveling in a straight line at constant speed they would spontaneously accelerate towards or away from the central point.

 

[Robin04]

Oh, I missed responding to this earlier. If there were a center and if the universe were isotropic then you would have some sort of central potential. Instead of force-free objects traveling in a straight line at constant speed they would spontaneously accelerate towards or away from the central point.

Would it be a good example if gravity depended on the position? Like let's say the gravitational constant is actually not a constant. It would be fun to calculate this.

 

[Dale]

Like let's say the gravitational constant is actually not a constant.

Generally to have a physical effect what must change is a dimensionless constant. If only dimensionful constants change then all you are doing is changing your units.

 

[Robin04]

Generally to have a physical effect what must change is a dimensionless constant. If only dimensionful constants change then all you are doing is changing your units.

I don't really understand the difference in physical effect between the two options. Could you explain it more precisely, please?

 

[Dale]

I don't really understand the difference in physical effect between the two options. Could you explain it more precisely, please?

Here is a good article by John Baez describing the difference:

http://math.ucr.edu/home/baez/constants.html