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Groups as symmetry

436 bytes added, 23:18, 15 March 2009
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===Symmetry from an indistinguishability perspective===
Another name for fairness is indistiguishabilityindistinguishability. You shouldn't be able to, ''a priori'', give any criterion that one thing satisfies, that the other doesn't.
''A priori'' distinguishability is much weaker than ''a posteriori'' distinguishability. The former is like saying: ''I cannot give beforehand a criterion that is satisfied by one and not by the other''. The latter is saying: ''Given the two, I cannot tell whether they are the same''.
The ''group'' that describes this indistinguishability is the group of ''time translations''. A time translation is an operation that takes everything and moves it a certain amount forward or backward by a certain interval in time. The group of time translations is isomorphic to the group of real numbers under addition (the sum of two time translations is simply the time translation corresponding to the sum of the intervals). The fact that there's no ''a priori'' way of defining a point in time is related to the assumption that physical laws are invariant under time translation. In other words, a physical law that holds now should also hold ten days from now.
* ''Indistinguishability of points in space'': If you believe that space has no natural origin, then any point in space looks like any other. Youcannot You cannot give any ''a priori'' criterion that distinguishes a point here from a point there.
The ''group'' that describes this indistinguishability is the group of ''space translations''. A space translation moves everything by a certain vector in space -- i.e., by a certain length in a certain direction. The group of space translations is isomorphic to the group <math>\R^3</math>, i.e., to the group of three-dimensional real vectors (the sum of two space translations is the sum of the corresponding vectors). The fact that there's no ''a priori'' way of defining a point in space is related to the assumption that physical laws are invariant under space translation. In other words, a physical law that holds here holds just as well anywhere else.
The ''group'' that describes this is the dilation group -- this is a group that expands all lengths by a certain factor (The dilation factor). The fact that there's no ''a priori'' way of defining a length is related to the assumption that physical laws are invariant under dilation. In other words, a physical law that holds for a system holds equally well for a dilated version of the same system.
 
* ''Indistinguishability of directions in space'': You may believe that there is no ''a priori'' mathematical way of defining a direction. In other words, there is no criterion that is satisfied by one direction, and not by another.
 
The ''group'' that describes this is termed the orthogonal group -- it is a group of rotations. For instance, rotation of the flat plane shows that all directions in the plane are indistinguishable.
All these examples are, from a physical perspective, likely to be incorrect. Physical laws may not enjoy time-invariance, space-invariance, and dilation-invariance. However, these laws are useful models of our understanding and expectations of physical laws. They describe the symmetries we ''expect'' to see from physical laws, even if those symmetries are not actually achieved.
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