The n-Category Café

It is little-known outside Germany (I read quite a number of books on the history of physics and never found out until I did a postdoc in Germany and my PI told me the story; you can see it in the German Wikipedia article on him but not on the English one) that Heisenberg also had a failed attempt at a unified field theory (called a ‘Weltformel’) which received a lot of media attention at the time it was presented, in 1958 (live on TV if I remember well). You can see more here:

https://www.spektrum.de/news/heisenbergs-weltformel/1542469

That really is a sad story. Is there any indication of how he managed to kid himself into thinking he’d made a huge discovery?

Presumably he had other physicists to talk to at the Dublin Institute for Advanced Studies. But maybe no one else understood what he was doing, or maybe they understood but didn’t dare tell him they weren’t convinced, or maybe they told him but he disagreed.

Or just maybe Schrõdinger was right, and there’s more to his theory than anyone has ever appreciated… I guess John is on a mission to find out!

For some reason everyone calls this an ‘affine’ connection — I’ve never understood why. I would imagine that an affine connection is one where we take the structure group to be the affine group, but here it’s the group of linear transformations of the tangent space.

Take this with a grain of salt: first, I don’t know anything about the naming history – I imagine it goes back a long ways; second, I disclaim any expertise in the subject area. But two things come to mind:

• Back in the very old days, when manifolds were thought of living inside an ambient Euclidean space $E$ rather than having an autonomous existence via a structure of maximal atlas, the tangent spaces of the manifold would be identified with affine subspaces of $E$, and the parallel transport induced by a connection would thus be identified with a collection of affine transformations between these affine subspaces. Perhaps the “affinity” would be most keenly felt in the case where the manifold is $E$ itself.
• In general, the space of affine connections on a manifold is actually an affine space (i.e., thinking in terms of covariant derivatives, a convex combination of connections is a connection, but general linear combinations are not since they generally fail the Leibniz rule).

So, one question is this: if we have a not-necessarily-symmetric tensor $A_{\mu \nu}$ on an oriented 4-manifold $M$, does

$$\sqrt{ det A } \; d^4 x$$

define a 4-form on this manifold in a coordinate-independent way?

If $A$ is symmetric and nondegenerate, we use it to define a nonzero 4-form on each tangent space $T_x M$, and that’s what

$$\sqrt{ det A } \; d^4 x$$

is. I think this also works if $A$ is antisymmetric. But what if $A$ is neither symmetric nor antisymmetric? Is there some way that $A$ defines a 4-form on each tangent space $T_x M$, which in coordinates looks like

$$\sqrt{ det A} \, d^4 x ?$$

The fact that nobody talks about this idea suggest there’s something wrong with it. I’ve seen nice theories of Hodge duals for vector spaces with nondegenerate symmetric or antisymmetric bilinear forms.

So, he could have been using some body of wisdom on ‘Riemannian geometry with non-symmetric metrics’ that I’m missing out on

These kind of things have received some attention in the past. As far as I remember, a lot of differential geometry carries over without modification to the non-symmetric setting (I think e.g. Lang’s book “Differential and Riemannian Geometry” covers some of this). Especially the case where the metric is asymmetric (+ some additional properties) is interesting as this is the setting of symplectic manifolds (and by extension, classical mechanics).

This is an interesting old idea! I apologize in advance because I am going to cite my own work in this comment, by this topic is actually related.

The first thing I though when I saw the equations was Lovelock Lagrangians. They have some structure in common, but Lovelock makes more sense at first glance. My colleagues Leandro Salomone and Santiago Capriotti had geometrized the Lovelock Lagrangians in a clever way, using the natural structures of the bundle of connections (https://arxiv.org/abs/1911.07278). I think these structure and ideas are a good place to start in order to geometrize Schrödinger field equations. But I will need to look it in more detail.

I find interesting that he doesn’t use a metric. In my Ph.D I showed that it is possible to formalize General Relativity (in the Palatini setting), without a metric (https://arxiv.org/abs/1804.06181). In the Hamiltonian setting, using a clever use of constraints, the momenta of the connection becomes the metric. At this point, I think of it as just a hand trick, reorganizing the variables so that the metric is hidden. But it opens the idea that you only need a connection to formalize GR. And Schrödinger proposal enters this idea of only using a connection. As before, I will need to think more about it, and probably it’s just a coincidence.

As a last note, the new connection he proposes also appears naturally in the constraints of the Palatini Lagrangian. It appears essentially from the part of the curvature that depends on the connection (and not its derivatives). It is also related to the gauge freedom of the theory (as Einstein noticed in his paper of 1921 about Palatini gravity), which is related to the linear part (on the derivatives of the connection) of the curvature. So it’s not a surprise that such an expression is considered.

Thank you for digging up and sharing such compelling ideas!

Any bilinear form $B: V\otimes V \to \mathbb{R}$ gives a (in fact two) maps $V \to V^*$. Hence taking top wedge powers a gives map

or equivalently, a well defined element

The representation of $GL(V)$ that the 1 dimensional vector space $L$ carries is

Hence while it does not have a natural trivialisation, it does have a natural orientation: the direction of tensor squares is positive. In particular if $B$ is non degenerate it makes sense to say that $\det(B)$ is positive or negative.

The representation $L$ has a positive square root, the representation $|L|^{1/2}$ with representation $g \to |\det(g)|^{-1}$, and $\sqrt{|\det(B)|} \in |L|^{1/2}$ naturally lives there.

Remark 1: The bilinear form $B$ gives two maps from $V$ to its dual but one map is the dual of the other. By functoriality this means that the two maps $\wedge^n V \to \wedge^n V^*$ are dual to each other. I didn’t check this, but once we have 1 dimensional vector spaces that should make $\det(B)$ the same for the two choices for the same reason that for a matrix $\det(A) = \det(A^t)$.

Remark 2: If $V = T_x M$ this all works out nicely bundle wise for a smooth bilinear form $B$, and we should indeed get a well defined section $\sqrt{|\det(B)|}$ that lives canonically in the bundle of densities $|\wedge^n| M$. It is smooth if $B$ is non degenerate so we avoid passing through 0 and have to deal with the singularity created by taking absolute value.

Remark 3: Suppose that $B$ is symmetric, and we have a symmetric non degenerate form $g$ on V. We can choose a $g$ orthonormal basis that diagonalises $B(-, -) = g(\mathrm{diag}(b_1, .., b_n) - , - )$ (at least if $g$ is definite). Now $\det(B) = \prod_i b_i \mathrm{vol}_g^2$. That does not seem to be directly related to $\mathrm{tr}_g(B) = \sum b_i$, however, that would give the scalar curvature if $B = Ric$ for a Levi-Civita connection.

Now that I have an understanding of the geometrical meaning of $\sqrt{| det R |}$, I’m more willing to just compute with it in coordinates.

First, for any square matrix $A$,

$$det(1 + s A) = 1 + s tr(A) + O(s^2)$$

Thus

$$\begin{array}{ccl} \displaystyle{ \frac{d}{d s} det(A + s B) \Big|_{s = 0}} &=& \displaystyle{ \frac{d}{d s} det(A) det(1 + s A^{-1} B) \Big|_{s = 0}} \\ &=& det(A) tr(A^{-1} B) \end{array}$$

so in terms of variational derivatives

$$\delta det(R) = det(R) tr(R^{-1} \delta R)$$

This is a first step. Of course it only works if $R$ is invertible.

A non-symmetric metric seems like an intrinsically weird thing. You can always decompose it into symmetric and anti-symmetric pieces. From an effective field theory point of view, there’s no reason (that I know of) for those two pieces to only and ever appear in some particular combination. You should just write down all the Lagrangian terms consistent with the symmetries. And (again, as far as I know) there’s no symmetry that keeps the antisymmetric part massless, unlike with the symmetric part. So we should expect it to be a very massive field with no phenomenological consequences.

I am not super familiar with non-symmetric metrics, but we worked out the analogous story for torsion here.

It doesn’t exactly fit, but the following is related. The mathematics of “optical cloaking via metamaterials” (i.e. building an invisibility cloak) is called transformation optics. It’s fascinating because you can “do general relativity in the lab”. It turns out that the macroscopic Maxwell’s equations for how electromagnetism works inside a material can be interpreted as doing general relativity with two metrics - the “electric metric” and the “magnetic metric”. (You can also think of it as having two different Hodge star operators on the manifold). In a metamaterial, you can change the electric and magnetic properties of the material at will, so you can tune these metrics to anything you like - you can even create a black hole in the lab!

Usually people assume the material is “impedance matched” (this means that the electric metric equals the magnetic metric). This makes them happier, because it means it corresponds to the usual general relativity setup.

But I often wondered what it means, in terms of modern differential geometry, when the electric metric is not equal to the magnetic metric. It means we have a spacetime manifold with two metrics. I haven’t seen such a thing crop up in math, but there it is, lurking in Maxwell’s equations.

Having “two metrics” is not the same as having a “non-symmetric metric”, but I couldn’t help mentioning this.

Hello all,

Hello John,

Might be of some interest:

Source: The American Mathematical Monthly, Vol. 37, No. 1 (Jan., 1930), pp. 32-34. http://www.jstor.org/stable/2299987

[…] The new matrix theory was very complicated, its physical meaning very obscure. But in 1926 a quite new development was put forward by Schrödinger, at that time professor at Zurich. Inspired by the ideas of Louis de Broglie’s Paris thesis of 1924, in which a wave was associated with every material particle, Schrödinger assumed that the dynamics of an electron can not be those of a point as in classical theory, but must be those of a wave.

This wave obeys a linear partial differential equation of the second order. (Newton mechanics leads to a partial differential equation of the first order and second degree.) Such a differential equation admits continuous uniform bounded solutions only for certain discrete values of a parameter (the “energy”), the so-called characteristic values. The critical test of the hydrogen lines was also satisfied in Schrödinger’s case.

The reason that Schrödinger’s calculus was not victorious was the difficulty in the interpretation. […]

I have not accessed the content of that book, cannot tell more than that. If you already know about this, it’s just redundant information. No spam intended.

Regarding George Birtwistle, here is one interesting read:

https://www.mprl-series.mpg.de/studies/2/10/index.html