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Groups as symmetry

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* The more structural conditions we impose, i.e. the more things we demand remain invariant, the less the symmetry: If you settled for a less-than-perfect notion of ''invariance'', you'd get more symmetry elements.
* The larger the universe of transformations we look in, the more the symmetry
==Groups and the big advances in physics==
===The role of symmetry in the physical world===
A lot of serious work in physics is devoted to finding out the ''group of symmetries'' for physical laws. For instance, should physical laws provide a natural way of distinguishing between left and right, or do they possess mirror-symmetry? Do physical laws provide us with natural units of length, mass and time, or do they possess a scaling-invariance?
Symmetry may have been the realm of philosophers, who believed that the symmetries of the universe should be things that we can derive by thought. For instance, if we do not see any way of distinguishing one direction from the other in space, then all spatial directions should be equivalent. If we do not see a natural way of distinguishing one moment of time from another, then all moments of time must be equal. These philosophical arguments led to the initial study of symmetry in science and art and led to the belief that certain groups of symmetry described the symmetry of the physical world.
However, our thinking is limited by the specific world we live in, and our intuition may not explain what happens at the microscopic (atomic) scales and the macroscopic (galactic) scales. It may not help understand events that happen in extremely short periods of time or be any help in comprehending cosmic timescales.
===The special theory of relativity===
Intuitively, we expect that, sitting in space, there is no way to distinguish one direction from the other, and no way to distinguish one origin from another, but the notion of distance remains invariant under symmetries. Thus, the symmetry group the physicists believed to capture spatial symmetry, was the group of those transformations of <math>\R^3</math> that preserve distance. Thus, these symmetries preserve everything geometric but do not give a distinguished origin or direction (in technical parlance, we say that space is ''homogeneous'' for having no distinguished origin, and ''isotropic'' for having no distinguished direction). In other words, a physical law should have the same prediction if we translate or rotate.
Independently, there is a symmetry group for time: we expect that time has no natural origin, but an interval of time is an intrinsic quantity. Thus, the group of symmetries of time was the group of time translations. In other words, a physical law should have the same predictions regardless of the point in time when they are applied.
The special theory of relativity challenged the assumption that we have two independent symmetry groups for space and time. Rather, in the special theory of relativity, a single symmetry group was discovered that governed both space and time, ''together''. This symmetry group roughly treats time as a ''fourth'' dimension along with space, but as a fourth dimension that is not directly interchangeable with the other three dimensions. In other words, the four-dimensional spacetime doesn't look the same in all four dimensions: the three spatial dimensions look similar, but the time dimension behaves differently. The group of symmetries proposed by Einstein contains the earlier groups of symmetries, but offers new symmetries that allow for ''exchanging'' space and time, albeit in a convoluted way.
In summary, the special theory of relativity proposed a new group of symmetries of the physical universe.
===Quantum mechanics===
Quantum mechanics makes significant use of groups, particularly, the non-Abelian nature of group operations. Moreover, it does so in an unexpected way.
One of the key insights of quantum mechanics is to view quantities that were previously considered ''measurements'' (like where an object is, and how fast it is moving) into ''operators'': things ''acting'' on something. Further, these operators can be composed with each other, and become part of a group, where they don't commute. Thus, instead of adding two momentum vectors, we end up composing the corresponding operators.
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