One of the most important and fruitful branches of higher mathematics is group theory. And one of the most fruitful branches of group theory is the theory of Lie groups and the associated symmetries.
The group axioms are deceptively simple.
A group is a set of objects, A, and a binary operation, *, with the following properties:
- Closure: a * b is an element of A for every a and b in A.
- Associativity: For all a, b, and c in A, a * (b * c) = (a * b) * c.
- Identity: There is exactly one element in A, which we call the identity I, with the property that I*a = a*I = a for all a in A.
- Inverse: For every element a in A, there is exactly one element a-1 such that a * a-1 = a-1 * a = I.
From these four axioms we derive a vast number of theorems. For example, we can prove that no two elements a and b can have the property that a * c = b * c for every c in A; that is, the elements must be distinct. Otherwise we violate axiom 4.
Groups can have both a finite and an infinite number of members. Some groups have the property that a * b = b * a for all a and b in A, and we call these Abelian groups. But not all groups are Abelian.
The integers under the operation of addition are an infinite Abelian group, with 0 as the identity element. The following group is neither infinite nor Abelian:
Let the group members be rock, paper, scissors, Tom, Dick, Harry. Let their combination table be
… rock paper scissors Tom Dick Harry
rock | rock paper scissors Tom Dick Harry
paper | paper scissors rock Dick Harry Tom
scissors | scissors rock paper Harry Tom Dick
Tom | Tom Harry Dick rock scissors paper
Dick | Dick Tom Harry paper rock scissors
Harry | Harry Dick Tom scissors paper rock
You can show by brute force that this combination satisfies all the group axioms, with “rock” as the identity element. As it happens, this group is a somewhat unconventional representation of the smallest non-Abelian group, the dihedral group.
A special category of groups are the Lie groups, named for the Norwegian mathematician who first investigated them. These are groups whose members are parametrized by one or more real parameters. For example, rotation in the plane is a group, which we call U(1). Any two rotations applied successively yield a third rotation, and it can be shown that the laws for combining rotations satisfy the group axioms. The parameter is the angle of rotation. In this particular case, the group is Abelian, and U(1) is probably the simplest example of an Abelian Lie group.
Rotations in three dimensions also form a Lie group, parametrized by the three Euler angles for the rotation. This group is non-Abelian. Try this experiment when the wife and kids aren’t watching: Take a book. Hold it in front of you with the cover up and and the book oriented the way you’d normally read it. Rotate it 90 degrees around the vertical axis. Now rotate it 90 degrees around a horizontal axis pointing to your right. Note the final orientation of the book. Now return the book to its starting position, and reverse the operations. The final orientation is different when the order of rotations is reversed. (Really. Try it.)
Groups are closely associated with the notion of symmetry, and Lie groups are closely associated with continuous symmetries. The circle is symmetric under U(1); that is, you can rotate the plane as much as you like, but a circle in the plane will appear unchanged. Likewise, you can rotate space all you like, and a sphere in space will appear unchanged.
Some symmetries are bit more subtle than this.
Consider translational symmetry. Translations in space form a Lie group, parametrized by the three components of the translation vector. It turns out that the laws of physics are symmetric under a translation. You can see this by performing a thought experiment: If God took the entire universe and moved it three light-years to the right, how would we know?
Likewise, the laws of physics are symmetric under rotation. Again, the thought experiment: If God took the entire universe and rotated it thirty degrees around an axis pointing at Polaris from Earth, how would we know?
Likewise time symmetry. If God froze the entire universe for a thousand years, then had it pick up precisely where it had left off, how would we know?
There’s a further level of depth to this. Each such continuous symmetry has associated with it a conservation law. This is Noether’s Theorem, named after the existence proof that it is possible for women to be brilliant at mathematics. Translational symmetry gives rise to conservation of momentum. Rotational symmetry gives rise to conservation of angular momentum. Time symmetry gives rise to conservation of energy.
And the final twist is gauge symmetry.
To understand this one, pretend you are Captain Kirk in the Star Trek episode, “The Tholian Web.” You’ve been sucked into a parallel universe whose entire contents are … yourself.
Where are you in that universe? Does it matter? Which direction is up? Does it matter? Not when there is translational and rotational symmetry to the laws of physics in this parallel universe.
And does time pass at the same rate as in our universe? How would you know?
Well, you’d know because your watch would tick at a different rate. Except that your brain’s perception of time would also be at the new rate, so, no, you wouldn’t know. It’s even possible to have the rate at which time passes change continuously, but you still wouldn’t know, because all physical processes, including your brain’s perception of the passage of time, would change continuously as well. This is an example of a gauge symmetry.
Gauge symmetry was first recognized in connection with electromagnetism. The electromagnetic field is most naturally described by a scalar potential V plus a vector potential A. But you can add the divergence of any arbitrary field f to the vector potential A, and all physically measurable phenomenon will remain unchanged, so long as you subtract a counterterm from the scalar potential equal to the time derivative of f. Physicists first regarded this as a nuisance; then realized it was a reflection of a gauge symmetry, reflecting the arbitrariness with which we assign an internal phase to the electromagnetic field.
Put another way, the electromagnetic field is Nature’s way of communicating a choice of phase within an internal dimension of the universe through space and time. Other physical interactions are also gauge fields communicating a choice of various internal phases through space and time. And gravitation is the gauge field communicating a choice of gauge for space and time itself through space and time. This may explain why gravitation has proven impossible, so far, to quantize. Some physicists (not many) are even beginning to argue that gravitation is the one field that is not quantized in nature.
My point?
Think again of Captain Kirk trapped in his private universe where there is nothing but Kirk. Take it a step further: Suppose it is only Kirk’s mind that is trapped in this private universe; he has no body that might provide him some scale. Kirk’s size is now meaningless. His position is now meaningless. His orientation is meaningless. His location in time is meaningless. His velocity through space and his passage through time are likewise meaningless. He has no agency, because there is nothing to act upon. To me, this sounds like an excellent model of Hell. Niven and Pournelle would probably place this in the vestibule of Hell, but in some ways they depict the vestibule as worse than Hell itself.
Now suppose Spock is sucked into Kirk’s private Hell. It is now a bit less hellacious. We can suppose Kirk is aware that Spock’s mind is there, through some kind of gauge field. (Maybe he sees a bright point representing Spock: Light is electromagnetic radiation, a gauge field.) Well, that’s something. Perhaps they can even communicate — this implies a gauge field for time, since communication requires synchronization. So we have gravitation.
It’s still meaningless to talk about how far Spock is from Kirk, or in which direction. Compared to what? Why, compared to McCoy, whose mind we find has also now been sucked into this private Purgatory. McCoy is just a little closer than Spock and in the opposite direction. And now we find that Scotty is present; he’s at right angles to Spock and McCoy, and some distance further out.
Now Kirk has some power to act. If nothing else, he can communicate blessings or cursings on his fellow universe-dwellers. He now has meaningful agency.
I do not know if God was once the sole entity in a Hell that he turned into Heaven by populating it with His children. It seems unlikely that it worked quite like this. But I am inclined to believe that some of the principles were the same.
Who hath ears to hear, let him hear.
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