### Schrödinger’s Unified Field Theory

#### Posted by John Baez

Erwin Schrödinger fled Austria during World War II. In 1940 he found a position in the newly founded Dublin Institute for Advanced Studies. This allowed him to think again. He started publishing papers on unified field theories in 1943, based on earlier work of Eddington and Einstein. He was trying to unify gravity, electromagnetism and a scalar ‘meson field’… all in the context of classical field theory, nothing quantum.

Then he had a new idea. He got very excited about it, and January of 1947 he wrote:

At my age I had completely abandoned all hope of ever again making a really big important contribution to science. It is a totally unhoped-for gift from God. One could become a believer or superstitious [

gläubig oder abergläubig], e.g., could think that the Old Gentleman had ordered me specifically to go to Ireland to live in 1939, the only place in theworldwhere a person like me would be able to live comfortably and without any direct obligations, free to follow all his fancies.

He even thought he might get a second Nobel prize.

He called a press conference… and the story of how it all unraveled is a bit funny and a bit sad. But what was his theory, actually?

Someone must have written a nice paper about Schrödinger’s theory in modern differential-geometric language, even if the conclusion is that it’s a complete mess. If you know of such a paper, please let me know! I’m getting my information from some sources that use old-fashioned index notation, which makes it harder for me to tell what’s really going on. Namely, this very nice paper:

- Hubert F. M. Goenner,
On the history of unified field theories II (ca. 1930 – ca. 1965),
*Living Reviews in Relativity***17**(2014), article no. 5. Section 6: Affine geometry: Schrödinger as an ardent player.

together with Schödinger’s book:

- Erwin Schrödinger,
*Space-Time Structure*, Cambridge U. Press, Cambridge, 1950. Chapter XII: Generalizations of Einstein’s theory.

and his first paper on this theory:

- Erwin Schödinger, The final affine field laws I,
*Proceedings of the Royal Irish Academy A***51**(1945 - 1948), 163–171.

The idea seems to be this. He starts with a 4-manifold $M$. The only field in his theory is a linear connection $D$ on the tangent bundle of $M$.

(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.)

He defines the Riemann curvature tensor of $D$ in the usual way and contracts two indices to get the Ricci tensor $R_{\mu\nu}$. You don’t need a metric to do this. When $D$ is the Levi-Civita connection of a Riemannian metric, $R_{\mu \nu}$ is symmetric, but in the situation at hand it needn’t be.

His field theory then has the Lagrangian

$L = \frac{2}{\lambda} \sqrt{ - det R }$

where $\lambda$ is some constant.

This makes me nervous: what’s really going on here? Of course we often see the expression

$\sqrt{ - det g }$

in general relativity, but that has a nice geometrical explanation: the 4-form

$\sqrt{ - det g } \; d^4 x$

is the volume form associated to the metric $g_{\mu \nu}$. If the Ricci tensor $R_{\mu \nu}$ were a *symmetric* tensor I could happily pretend it’s a metric — perhaps not positive definite, perhaps degenerate — and treat the action in Schrödinger’s theory

$S \; = \; \int L \; d^4 x \; = \; \frac{2}{\lambda} \int_M \sqrt{ - det R } \; d^4 x$

as the volume of $M$ computed using the volume form associated to this metric. But what’s the deal when the Ricci tensor is not symmetric?

Schrödinger got his ideas from previous work of Eddington, Einstein and Straus, all of whom had been studying variants of general relativity where the Riemannian metric is replaced by a not-necessarily-symmetric tensor $g_{\mu \nu}$. So, he could have been using some body of wisdom on ‘Riemannian geometry with non-symmetric metrics’ that I’m missing out on. Or, it could be that this whole line of thought died out precisely because ‘Riemannian geometry with non-symmetric metrics’ turned out to be a thoroughly unworkable idea. And if so, I’d like to know why.

Anyway, starting from his Ricci tensor, Schrödinger then proceeds to *define* a not-necessarily-symmetric tensor $g_{\mu \nu}$ by

$g_{\mu \nu} = \frac{1}{\lambda} R_{\mu \nu}$

Well, he doesn’t proceed exactly this way, but that’s the upshot. He then works out some field equations from his Lagrangian. To write them in a way he likes, he introduces a new connection whose Christoffel symbols ${}^\bullet \Gamma^\lambda_{\mu \nu}$ are related to the Christoffel symbols $\Gamma^\lambda_{\mu \nu}$ of his original connection $D$ as follows:

${}^\bullet \Gamma^\lambda_{\mu \nu} = \Gamma^\lambda_{\mu \nu} + \frac{2}{3} \delta^\lambda_\mu \Gamma_\nu$

where

$\Gamma_\nu = \frac{1}{2} \left( \Gamma^\lambda_{\nu \lambda} - \Gamma^\lambda_{\lambda \nu} \right)$

He says that the field equations are simpler to understand using this new connection and *its* Ricci tensor, which he calls ${}^\bullet R$.

What’s going on here?

On January 27, 1947, Schrödinger gave a lecture on his new theory. He even called a press conference to announce it! From what he said to the reporters, you can tell that he was in the grip of grandiosity:

The nearer one approaches truth, the simpler things become. I have the honour of laying before you today the keystone of the Affine Field Theory and thereby the solution of a 30 year problem: the competent generalization of Einstein’s great theory of 1915. The solution is

$\delta \int \mathcal{L} d\tau = 0 \quad with \quad \mathcal{L} = \sqrt{- \mathrm{det} R_{i k} }$

$R_{i k} = - \frac{\partial \Gamma^\sigma_{i k}}{\partial x_\sigma} + \frac{\partial \Gamma^\sigma_{i\sigma}}{\partial x_k} + \Gamma^\sigma_{i\tau}\Gamma^\tau_{\rho k} - \Gamma^\sigma_{\rho\sigma}\Gamma^\rho_{i k}$

where $\Gamma$ is a general affinity with 64 components. That is all. From these three lines my friends would reconstruct the theory, supposing the paper I am handing in got hopelessly lost, and I died on my way home.

The story of the great discovery was quickly telegraphed around the world, and the science editor of the *New York Times* interview Einstein to see what he thought.

Einstein was not impressed. In a carefully prepared statement he said:

Schrödinger’s latest effort […] can be judged only on the basis of mathematical-formal qualities, but not from the point of view of ‘truth’ (i.e., agreement with the facts of experience). Even from this point of view I can see no special advantages over the theoretical possibilities known before, rather the opposite. As an incidental remark I want to stress the following. It seems undesirable to me to present such preliminary attempts to the public in any form. It is even worse when the impression is create that one is dealing with definite discoveries concerning physical reality. Such communiqués given in sensational terms give the lay public misleading ideas about the character of research. The reader gets the impression that every five minutes there is a revolution in science, somewhat like the

coup d’étatin some of the smaller unstable republics.

Ouch. Wise words even now!

But Einstein didn’t claim Schrödinger’s theory was nonsense. It seems part of Einstein’s irritation was due to how similar Schrödinger’s work was to his own work with Straus! Indeed, Schödinger starts the first paper on his theory with the following curious remark:

The reason it has taken me so long to find out the correct Lagrangian is, that it is the most obvious one and had been tried more than once by others.

So, I feel there could be at least *something* interesting about Schrödinger’s field equations. But they’re complicated enough, and I understand them so poorly, that I don’t even want to copy them down here. If you want to see them, check out The final affine field laws I.

## Re: Schrödinger’s Unified Field Theory

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