Difference between symmetries of a theory and symmetry of a system

Question

I am trying to motivate the role of symmetries in physics. In doing so, I would like to distinguish between a theory's symmetries and the symmetry of a system. The ideas are similar but I am not able to think about them in the proper manner. In the end, a system has to obey the symmetries that the theory exhibits but it can have additional symmetries. Also, a system is usually a specific example in a theory. How to think about this?

For example, a theory like GR described by a Lagrangian has symmetries, but a system like a black hole can have symmetries like rotational symmetry or time translation symmetry. How do we think about these two kinds of symmetries?

A system has to obey the symmetries that the theory exhibits. Note that symmetry breaking exists. — Ghoster, Oct 10 '22 at 04:10
Are you asking about the difference between e.g. space being isotropic but not all objects in space being spherical? — FlatterMann, Oct 10 '22 at 04:10
Related: Do an action and its Euler-Lagrange equations have the same symmetries? — Qmechanic, Oct 10 '22 at 04:51
The system doesn't have to do anything the theory demands of it, although for an effective theory it often does. When the system does something else, the theory needs revision. — John Doty, Oct 10 '22 at 11:30
I think you should clarify what you mean by Theory. Following your example of GR, I would not say the theory has symmetry. It is just a framework able to accommodate many possible space-times (flat space-time has not the same symmetries as Schwarzschild's spacetime or those of a space-time with torsion. — GiorgioP-DoomsdayClockIsAt-90, Oct 11 '22 at 07:04

score 0 · Answer 1 · answered Oct 10 '22 at 09:19

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When we consider the evolution of a system there are usually two inputs - the laws which determine the dynamics of the system (in physics, these are often differential equations) and the initial conditions. Even if the laws of the system have a particular symmetry, its evolution does not necessarily share that symmetry because the initial conditions may be asymmetric.

For example, the rules of chess have left-right symmetry, but game positions are asymmetric because the starting position does not share this left-right symmetry.

answered Oct 10 '22 at 09:19

gandalf61

52,505

Can you elaborate on this? – Khushal Oct 11 '22 at 02:47
@Khushal You will have to be more specific. Which part of my answer do you not understand ? – gandalf61 Oct 11 '22 at 08:34
I understand that the physical laws have symmetries, but how are symmetries present in the initial conditions? Laws have symmetries that the Lagrangian has but is there an origin of these other type of symmetries? – Khushal Dec 18 '22 at 05:00

Difference between symmetries of a theory and symmetry of a system

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