What Did Schrodinger Study in Order to Build His Theory?

Schrödinger'southward Unified Field Theory

Posted by John Baez

Erwin Schrödinger fled Austria during World War II. In 1940 he constitute a position in the newly founded Dublin Constitute for Avant-garde 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, cypher quantum.

So he had a new idea. He got very excited about information technology, and January of 1947 he wrote:

At my age I had completely abased all hope of always over again making a really large important contribution to science. It is a totally unhoped-for gift from God. One could get a laic or superstitious [gläubig oder abergläubig], e.yard., could recollect that the Old Gentleman had ordered me specifically to go to Ireland to live in 1939, the just place in the world where a person like me would be able to live comfortably and without whatsoever direct obligations, costless to follow all his fancies.

He even thought he might get a second Nobel prize.

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

Someone must have written a nice paper near Schrödinger'south theory in mod differential-geometric language, even if the conclusion is that it's a complete mess. If you know of such a newspaper, delight allow me know! I'm getting my information from some sources that employ old-fashioned alphabetize 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 Two (ca. 1930 – ca. 1965), Living Reviews in Relativity 17 (2014), commodity no. 5. Section six: Affine geometry: Schrödinger as an ardent histrion.

together with Schödinger's book:

• Erwin Schrödinger, Space-Time Construction, Cambridge U. Press, Cambridge, 1950. Chapter XII: Generalizations of Einstein'southward theory.

and his first paper on this theory:

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

The thought 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 packet of $One thousand$.

(For some reason anybody calls this an 'affine' connexion — I've never understood why. I would imagine that an affine connection is one where we take the structure group to exist the affine group, simply here it's the group of linear transformations of the tangent infinite.)

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

His field theory and then has the Lagrangian

$$50 = \frac{2}{\lambda} \sqrt{ - det R }$$

where $\lambda$ is some constant.

This makes me nervous: what's really going on here? Of form we oft run into the expression

$$\sqrt{ - det thou }$$

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

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

is the book 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 non positive definite, possibly degenerate — and care for the action in Schrödinger's theory

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

every bit the volume of $M$ computed using the book class associated to this metric. Just what's the deal when the Ricci tensor is non symmetric?

Schrödinger got his ideas from previous piece of 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 $g_{\mu \nu}$. Then, 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 exist that this whole line of idea died out precisely considering 'Riemannian geometry with non-symmetric metrics' turned out to be a thoroughly unworkable idea. And if so, I'd like to know why.

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

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

Well, he doesn't proceed exactly this fashion, but that'due south the upshot. He and so works out some field equations from his Lagrangian. To write them in a way he likes, he introduces a new connectedness 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:

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

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 printing briefing to announce it! From what he said to the reporters, you tin can tell that he was in the grip of grandiosity:

The nearer ane approaches truth, the simpler things become. I have the accolade of laying earlier 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

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

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

where $\Gamma$ is a general analogousness 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 fashion home.

The story of the smashing discovery was quickly telegraphed around the earth, and the science editor of the New York Times interview Einstein to see what he idea.

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

Schrödinger'southward latest effort […] can be judged just on the ground of mathematical-formal qualities, but non from the point of view of 'truth' (i.east., agreement with the facts of experience). Even from this point of view I tin see no special advantages over the theoretical possibilities known before, rather the reverse. Every bit an incidental remark I want to stress the post-obit. It seems undesirable to me to present such preliminary attempts to the public in any form. It is fifty-fifty worse when the impression is create that 1 is dealing with definite discoveries concerning physical reality. Such communiqués given in sensational terms requite the lay public misleading ideas about the grapheme of inquiry. The reader gets the impression that every five minutes there is a revolution in science, somewhat like the coup d'état in some of the smaller unstable republics.

Ouch. Wise words even at present!

Merely Einstein didn't merits Schrödinger'southward theory was nonsense. Information technology seems role of Einstein's irritation was due to how similar Schrödinger's work was to his ain work with Straus! Indeed, Schödinger starts the first newspaper on his theory with the following curious remark:

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

So, I feel in that location 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 downwardly hither. If y'all desire to see them, check out The terminal affine field laws I.

Posted at December 26, 2019 7:07 PM UTC

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