Groups as symmetry
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Groups are everywhere. And it doesn't take a lot of effort to spot them, if you look. This article describes how one can locate groups easily. No knowledge except the definition of group, subgroup, trivial group and Abelian group is required.
Contents
Symmetry
Symmetry from a geometric perspective
What do we mean by symmetry?
Symmetry, arguably, is an aesthetic; it measures how nice something looks. But there is a more quantitative aspect to it. Namely, something symmetric is something that looks the same from different angle. For instance, if your face possesses left-right symmetry, then that means that it looks exactly the same as the face you see in the mirror. If, on the other hand, it does not possess left-right symmetry (say, if you had a mole on your left cheek) then your mirror image looks different from your own image. In the mirror image, the mole is on the right cheek.
A circle is an example of a lot of symmetry:
- Rotational symmetry: You can rotate the paper any amount around the center of the circle, and you still get a circle.
- Mirror symmetry: You can also reflect the circle about any of its diameters and get back the same circle: the circle enjoys a mirror symmetry about each diameter.
In other words, symmetry is the fact that if you make some change (either in the object itself or in your perspective) the object looks exactly the same. The extent of symmetry can now be described by the number of such different perspectives you can use. For instance, an equilateral triangle possesses some symmetry: if you rotate by certain angles, it doesn't change. But rotating by an arbitrary angle does not send the equilateral triangle to itself. So the equilateral triangle isn't quite as symmetric as the circle.
We can put it this way:
The extent of symmetry of an "object" is measured by the set of transformations that map the object to itself.
The "object" in the above sentence need no always be a concrete object. It could be a structural rule or principle or equation or law. We're used to thinking of symmetries of concrete objects (like mice, clocks, and historical monuments). But symmetry is a deeper concept.
Symmetry from a fairness perspective
Let's go a little further with the idea of symmetry in laws, but this time, from the point of view of law in the sense of laws made by humans. We'd ideally like our laws to enjoy a certain kind of symmetry: a fairness. For instance:
- Non-discrimination policy: The quality of service received by a person should be independent of his or her name. In other words, if we permute the names of the people, the quality of service received should not change.
- Who wins an election should depend only on the votes that the candidates receive. In other words, if we permute the names of the candidates, while preserving the votes they received, the outcome of the election should not be affected.
- Justice is blind: The outcome of a court case should also be independent of the names of the people involved. If we permute the names of the people involved in the court case, the outcome should not be affected.
Thus, the ideals of equality and fairness amount to an indifference or blindness to irrelevant factors. This translates in group theory to invariance under a certain group action. The group action is the one that permutes the irrelevant factors.
Symmetry from an indistinguishability perspective
Another name for fairness is indistinguishability. 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.
Here are some examples:
- Indistinguishability of points in time: If you believe that time has no natural origin, then any point in time looks like any other. You cannot give any a priori criterion that is satisfied by one point in time and not by the other. But given two points in time, you can certainly compare them: you can, for instance, say which one came earlier.
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. 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 , 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.
- Indistinguishability of lengths in space: You may believe that there is no a priori mathematical way of defining a particular unit of length. In other words, there is no criterion that is satisfied by one unit of length, and not by the other. However, given two units of length, you can certainly ask which is bigger.
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.
Where groups come in
We saw above that to measure the extent of a symmetry, we should specify two things:
- What thing we want to be invariant: It could be the region in space occupied by an object (like, a triangle or circle). It could be the effect of applying a law. It could be a differential equation.
- What transformations we are allowing
Given these two, we can define the group of symmetries as those transformations that preserve whatever we are trying to keep invariant.
Why is the group of symmetries a group? Firstly, if leaves something invariant, and leaves something invariant, then so does the composite . In other words, composing two things that don't move it, doesn't move it. So we can define a multiplication on the group of symmetries by composition.
The multiplication is associative
This is the fact that function composition is associative. Another way of thinking of it is: if you have instructions (1) and (2) on the first page and instruction (3) on the second page, that has the same net effect as having instruction (1) on the first page and instruction (2) and (3) on the second page. (The instructions just tell us what transformations to do).
The identity map is in
Certainly! The identity map preserves everything. So, the identity map is actually inside any group of symmetries, and it is the identity element of the group.
There may be a tendency to dismiss the identity element as unimportant or uninteresting. However, this is akin to thinking that vacuum is uninteresting in physics. The identity element is right in the middle of the group, and though it doesn't have much character of its own, it is needed for practically everything.
Inverses
The existence of inverses is a debatable point, but the rough idea is that if your transformation actually is a symmetry, you should be able to undo it. Symmetries that aren't reversible aren't genuine symmetries. The reverse transformation is precisely the inverse element in the group.
There are certain kinds of situations where we want to relax the assumption about invertibility. Groups without inverses (i.e. sets with associative binary operation having an identity element) are termed monoids. Monoids also act, just like groups do, but the actions aren't often dubbed symmetries. You can't call a one-way street symmetric.
Measuring symmetry by groups
The bigger the symmetry group, the more the symmetry. However, the number of elements in the symmetry group isn't always the best measure of how symmetry-rich the structure is. Rather, various measures of complexity of the group are better.
For example the straight line has quite a lot of symmetry, but not too many people would marvel at its symmetry. But a beautiful lattice picture with a huge symmetry group is worth noticing.
The relation between structure and symmetry
In general, the more structure we insist should remain invariant, the smaller the symmetry group. For instance:
- The group of symmetries of the plane that preserve collinearity is a huge group. It is called the general affine group of the plane. It includes translations, rotations, reflections, as well as transformations that compress the different axes in different ratios.
- The group of symmetries of the plane that preserve shape is a smaller group. This is termed the group of similarity transformations, and is also termed the affine orthogonal similitude group. This group is generated by translations, rotations, reflections, and dilations. It does not include transformations that compress the different axes in different ratios.
- The group of symmetries of the plane that preserve shape and size is an even smaller group. This is the affine orthogonal group, and is generated by translations, rotations and reflections. It does not include dilations except by .
- The group of symmetries that preserve shape, size and orientation is even smaller.
Thus, the more structure we demand must be preserved, the fewer the symmetries. The extreme case of this is when the only transformation that preserves everything is the identity map, in which case the symmetry group is the trivial group.
The relation between symmetry and the ambient universe of symmetries
An equivalent formulation of the point raised in the previous subsection is: the larger the overall group of transformations we are looking in, the larger the group of symmetries is likely to be. For instance, consider the square.
- If our universe of transformations is the group of all rotations about the center of the square, there are only four symmetries: the rotations by multiples of .
- If our universe of transformations includes both rotations about the center and reflections about lines through the center, there are eight symmetries: the four rotations, the two reflections about the diagonals, and the two reflections about the lines joining midpoints of opposite sides.
- If our universe of transformations includes all homeomorphisms from the plane to itself, then the group of symmetries of the square is huge and unwieldy.
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 dilation-invariance?
The philosophical view: 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.
The flipside to the philosophical view: 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, physicists believed that the symmetry group capturing spatial symmetry was the group of those transformations of 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.
Summary of the classical view: Physical laws are invariant under the group of all isometries of space (i.e., transformations of space that preserve distance: these include both space translations and rotations). They are also invariant under all time translations.
The special theory of relativity challenges the assumption that we have two independent symmetry groups for space and time. Rather, in the special theory of relativity, a single symmetry group governs 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 all the earlier symmetries: it allows for all spatial isometries and also allows for all time translations. However, it also introduces new symmetries that exchange space and time, albeit in an indirect and complicated way -- the symmetries must preserve a more complicated notion of distance.
Summary of the relativistic view: Physical laws are invariant under the group of isometries of spacetime with respect to a new, and more complicated, notion of distance. These isometries include the purely spatial isometries (that act on the space component without affecting time) as well as the time translations, but also include new isometries that mix space and time.
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.