Physical principles are almost always phrased as (or equivalent to) symmetries, and symmetries are often phrased as the invariance of some special inner product or bilinear form. For example, the statement that physics should be rotationally invariant is the statement that the Euclidean dot product, \(\mathbf x\cdot\mathbf y\), should be remain invariant under transformations which are symmetries of our physics.

However, there is a subtle point in going between the invariance of an inner product and a symmetry group which I know was never explicitly pointed out to me. While specifying a symmetry group is enough to specify the collection of invariant quantities, specifying an invariant quantity is not quite enough to specify the symmetry.

Suppose we take the invariance of \(\mathbf x\cdot\mathbf y\) as a first example. Suppose that \(R\) is a suggestively named prospective transformation which acts on \(\mathbb{R}^n\) linearly as the matrix multiplication \(R\mathbf x\). Then if \(R\) is to be a transformation which leaves the inner product invariant, we must have \(\mathbf x^T(R^TR)\mathbf y=\mathbf x^T\mathbf y\) so \(R^TR=1\). If we had two such transformations, \(R,R^\prime\), then we could apply both successively. The composite transformation can be easily checked to also leave the inner product invariant, so the collection of good transformations is closed under composition. If we are to eventually call these transformations symmetries, then it would also be reasonable to require them to be invertible — this is essentially the requirement that if \(\mathbf x\) is to be equivalent to some \(\mathbf x^\prime\) under the symmetry, then there must exist some \(R\) leaving the inner product invariant such that \(\mathbf x^\prime=R\mathbf x\), but for this relation to be symmetric, and hence a true equivalence, then there must exist \(R^\prime\) such that \(\mathbf x=R^\prime\mathbf x^\prime\). But we want to say \(\mathbf x\sim R\mathbf x\) for all \(\mathbf x\), and so \(R\) must be invertible. All of this is to say, we are interested only in those linear transformations which leave \(\mathbf x\cdot\mathbf y\) and form a group under multiplication.

We might be worried that there is some loss of generality in considering only linear transformations of our vectors, but in physics all our quantities should be formulated on a manifold and the vectors should be members of the tangent space (or any other vector bundle really) over each point in the manifold. Under coordinate transformations, our vectors always transform linearly.

Now, by definition the group of all matrices which leave the inner product invariant is the orthogonal group \(O(n)\). But should all of these matrices be symmetries of our physics? For example, we could consider the parity operator defined by \(P\mathbf x=-\mathbf x\). This certainly leaves the inner product invariant and because \(P^2=1\), it even forms a group by itself. Who’s to say this transformation is the only symmetry of our physics?

This is the important point I want to really drive at here. While the elements of \(O(n)\) all leave \(\mathbf x\cdot\mathbf y\) invariant, they may not all be symmetries — invariance is a necessary but not a sufficient condition. This might leave us with a somewhat sour taste in our mouths. After all, what good is thinking about the groups which leave the inner product invariant if we can’t actually rely on this to tell us what group of transformations will all be symmetries.

Not all is lost though, because if we know the maximal group which leaves the inner product invariant, we know that the true symmetry group of our physics must be a subgroup of the maximal group, and this is a rather restrictive statement.

To continue with our \(O(n)\) example, it’s common in physics to only use the special orthogonal group \(SO(n)\) as our symmetry group instead of the full \(O(n)\). If we define the operator \(P_1\) to be the matrix which just negates the first component, then for any matrix \(R_-\in O(n)\) such that \(\det(R_-)=-1\), we clearly have \(\det(P_1R_-\)=+1\), so \(P_1R_-\in S)(n)\). Since \(P_1\) is invertible, this means that any element of \(O(n)\) is either already an element of \(SO(n)\), or can be written as \(P_1R_+\) for some \(R_+\in SO(n)\). This means all we need to know to determine whether \(SO(n)\) is our symmetry group instead of \(O(n)\) is whether or not \(P_1\) is a symmetry. This is a definite question we can ask and then go test in the real world. If we find that \(P_1\) is not a symmetry, then we know true symmetry group is \(SO(n)\), or some subgroup thereof.

So while it might be a little disappointing that the specification of an invariance pronciple is not enough to tell us exactly what the symmetry group of our physics might be, there’s a silver lining. These principles give us a way to organize systematically how we investigate what the true symmetry group of nature might be. All of it, based on how restrictive a condition it is that our true symmetry group be a subgroup of some other, predefined group. Adding a further preference towards Lie groups even gives us a place to start by considering the connected and disconnected components of the Lie group, which is essentially what we did with \(O(n)\) above.

In my next post, I’ll talk about how all of this applies to special relativity. The path between the statement “the speed of light is a constant” and Lorentz invariance not straightforward, and has been in development even into the 70’s and 80’s.