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December 26, 2019

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 the world where 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:

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:

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

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

His field theory then has the Lagrangian

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

where λ\lambda is some constant.

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

detg \sqrt{ - det g }

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

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

is the volume form associated to the metric g μνg_{\mu \nu}. If the Ricci tensor R μν 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=Ld 4x=2λ MdetRd 4x S \; = \; \int L \; d^4 x \; = \; \frac{2}{\lambda} \int_M \sqrt{ - det R } \; d^4 x

as the volume of MM 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 μνg_{\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 μνg_{\mu \nu} by

g μν=1λR μν 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 Γ μν λ{}^\bullet \Gamma^\lambda_{\mu \nu} are related to the Christoffel symbols Γ μν λ \Gamma^\lambda_{\mu \nu} of his original connection DD as follows:

Γ μν λ=Γ μν λ+23δ μ λΓ ν {}^\bullet \Gamma^\lambda_{\mu \nu} = \Gamma^\lambda_{\mu \nu} + \frac{2}{3} \delta^\lambda_\mu \Gamma_\nu

where

Γ ν=12(Γ νλ λΓ λν λ) \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 R{}^\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

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

R ik=Γ ik σx σ+Γ iσ σx k+Γ iτ σΓ ρk τΓ ρσ σΓ ik ρ 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 Γ\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’état in 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.

Posted at December 26, 2019 7:07 PM UTC

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26 Comments & 0 Trackbacks

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

Posted by: Andrei on December 26, 2019 10:51 PM | Permalink | Reply to this

Re: Schrödinger’s Unified Field Theory

Thanks!

I know just a little about Heisenberg’s theory. For example, I know Dirac told him this in a letter sent on March 6, 1967:

My main objection to your work is that I do not think your basic (non-linear field) equation has sufficient mathematical beauty to be a fundamental equation of physics. The correct equation, when it is discovered, will probably involve some new kind of mathematics and will excite great interest among the pure mathematicians, just like Einstein’s theory of the gravitational field did (and still does). The existing mathematical formalism just seems to me inadequate.

Perhaps every really ambitious physicist needs to take a try at a ‘theory of everything’ at some point in their life.

Posted by: John Baez on December 27, 2019 1:08 AM | Permalink | Reply to this

Re: Schrödinger’s Unified Field Theory

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!

Posted by: Tom Leinster on December 26, 2019 11:06 PM | Permalink | Reply to this

Re: Schrödinger’s Unified Field Theory

Schrödinger was in regular correspondence with Einstein and other physicists, so if he fooled himself it wasn’t for lack of feedback.

After his press conference, which resulted in some newspaper articles, and Einstein’s tart public response, he sent a rather lame apology to Einstein, saying he’d done it just to get a pay raise:

I had to indulge in a little hot air in the present somewhat precarious position…. In the first place, our basic salaries have not been increased since 1940.

But there’s evidence (like the letter I quoted at the start of my post) that Schrödinger really did think he was on to something big.

Einstein wrote a curt response saying Schrödinger’s theory was essentially equivalent to one Einstein had already proposed with Straus. Schrödinger replied… but Einstein didn’t write to him again for 3 years.

I doubt there’s anything really great about this theory, but it’s sufficiently different from the sort of field theories people talk about today that I’m having trouble understanding it… yet it’s simple enough that I feel I should.

I suppose I should think about what Einstein and Straus did, but that too seems a bit mysterious.

Posted by: John Baez on December 27, 2019 1:02 AM | Permalink | Reply to this

There’s an extended discussion of the conflict between Einstein and Schrödinger over this issue in Paul Halpern’s book.

The subsequent history is described in sections 9 and 10 of the Goenner review mentioned previously. In brief, to the extent that these theories make any predictions at all, they don’t agree with experiment.

Posted by: Phil Harmsworth on December 29, 2019 3:24 AM | Permalink | Reply to this

Re: Schrödingers Unified Field Theory

I’d like to read Halpern’s book!

I read Goenner’s paper but I have trouble following the logic of the arguments concerning Schrödinger’s theory. I’m willing to accept that if it agreed with experiment, everyone would know about this theory. I’m more interested in the geometrical meaning of the theory and whether it’s mathematically well-behaved at all. I’d never thought of such a simple Lagrangian built simply from a connection on the tangent bundle! The fact that nobody talks about it anymore suggests that there’s something horrible about it. I was confused about what the Lagrangian |detR|\sqrt{| det R |} even meant (in a coordinate-free way) until Rogier explained it.

Posted by: John Baez on December 29, 2019 7:44 AM | Permalink | Reply to this

There’s an alternative treatment (in classical notation) in the second appendix (“Relativistic Theory of the Non-Symmetric Field”) of Einstein’s book, The Meaning of Relativity.

Much more recently, James Shifflett has proposed a modification to the Einstein-Schrödinger theory that apparently overcomes several of the unphysical features of the original - https://einstein-schrodinger.com/einstein-schrodinger.html.

Posted by: Phil Harmsworth on December 31, 2019 1:16 PM | Permalink | Reply to this

Re: Schrödinger’s Unified Field Theory

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 EE 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 EE, 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 EE 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).
Posted by: Todd Trimble on December 27, 2019 7:16 PM | Permalink | Reply to this

Re: Schrödinger’s Unified Field Theory

I like the first theory for why affine connections are called that. In his book Space-Time Structure, which by the way is written in a very pleasant conversational style, Schrödinger calls them “affine connexions” or “affinities”.

I remember being very relieved to hear that connections on a principal bundle form an affine space. In learning general relativity one focuses on tensor fields, and it’s upsetting to learn that the Christoffel symbols, while important, are not a tensor field. The mathematical approach to connections helps settle what’s going on, but understanding the structure of the set of all connections helped nail everything down. I think most mathematical physicists who care about gauge fields or gravity become keenly aware that connections form an affine space.

Somehow the affine space of connections feels like it came after the term ‘affine connection’. But maybe you’re suggesting that it was lurking in there from the start. After all, the lack of a distinguished origin precisely says there’s no god-given ‘best’ way to parallel transport a tangent vector along a curve.

Posted by: John Baez on December 27, 2019 8:15 PM | Permalink | Reply to this

Re: Schrödinger’s Unified Field Theory

One way it could perhaps have been lurking in there from the start is that the transformation law for the Christoffel symbols is an affine equation, in contrast to the transformation law for tensors which is a linear equation.

Posted by: Mike Shulman on December 27, 2019 9:48 PM | Permalink | Reply to this

Re: Schrödinger’s Unified Field Theory

I always imagined that it's because the structure of an affine connection makes the manifold into an affine space … if the connection is flat and torsion-free, that is. I never had any particular evidence for this, but the discussion at https://en.wikipedia.org/wiki/Affine_connection#Affine_connections_as_Cartan_connections seems relevant to both this idea and to Todd's ideas.

Posted by: Toby Bartels on January 5, 2020 1:59 AM | Permalink | Reply to this

Re: Schrödinger’s Unified Field Theory

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

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

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

If AA is symmetric and nondegenerate, we use it to define a nonzero 4-form on each tangent space T xMT_x M, and that’s what

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

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

detAd 4x? \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.

Posted by: John Baez on December 27, 2019 8:01 PM | Permalink | Reply to this

Re: Schrödinger’s Unified Field Theory

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).

Posted by: Matty Wacksen on December 27, 2019 9:29 PM | Permalink | Reply to this

Re: Schrödinger’s Unified Field Theory

Thanks!

What I really need is to see how, or whether, a nondegenerate bilinear form AA on an nn-dimensional real vector space VV gives a ‘volume form’: an element of the exterior power Λ nV *\Lambda^n V^* where n=dim(V)n = dim(V). I know how to do it when AA is either symmetric or skew-symmetric (= antisymmetric).

Posted by: John Baez on December 28, 2019 2:02 AM | Permalink | Reply to this

Re: Schrödinger’s Unified Field Theory

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!

Posted by: Jordi Gaset on December 28, 2019 1:16 PM | Permalink | Reply to this

Re: Schrödinger’s Unified Field Theory

Thanks! I’ll look at those references.

I think Schrödinger was building upon previous work of Einstein and Straus that used a Palatini-like formalism but for non-symmetric ‘metrics’. It’s explained fairly well in Chapter XII of Schrödinger’s book Space-Time Structure, so I recommend a look at that. Unlike most scientists, Schrödinger can write quite well, so this chapter is fun to read if one already knows general relativity. The worst thing about it is that each section builds on the previous one, and he just describes what’s new rather than retelling the whole story, so you’re forced to read the whole thing. But if you do that, it’s not bad.

He starts by describing the Palatini formalism for gravity, where the metric and connection are treated as independent fields, and shows how the vanishing of torsion arises from the equations of motion.

Then he describes the “Einstein–Straus theory”, saying

The singular merit of Palatini’s derivation is that it can be extended straightaway without ambiguity to a non-symmetric g ikg_{i k}.

I guess I need to study this section more carefully.

Then he introduces his own theory, “the purely affine theory”, saying

Can we not avoid introducing, with Palatini, two basic connexions of the space-time manifold, a quasi-metrical one by the g ikg_{i k} and an affinity Γ jk i\Gamma^i_{j k}? Can one not go a step beyond Palatini and base a theory on affine connexion alone […]?

(I think here he’s using ‘connexion’ in some old way that includes both the metric and what we would call the connection.)

He says Eddington tried this idea in 1921, using the square root of the determinant of the Ricci tensor of an affine connection as the Lagrangian — but assuming the affine connection was torsion-free. He wants to drop this assumption.

(Beware: he calls the Ricci tensor the ‘Einstein tensor’.)

He says Einstein also had studied such a theory, again assuming the connection was torsion-free.

It’s sort of funny that back then, these various formulations and generalizations of gravity were being studied as possible ‘theories of everything’, worthy of press conferences and announcements in newspapers. (Even Einstein was guilty of the latter.) I’m thinking that perhaps the calculations were so difficult back then that only the lure of a ‘unified field theory’ provided people with the mental energy to carry them out.

Posted by: John Baez on December 29, 2019 1:30 AM | Permalink | Reply to this

Re: Schrödinger’s Unified Field Theory

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

(1) nV nV * \wedge^n V \to \wedge^n V^*

or equivalently, a well defined element

(2)det(B):1( nV *) 2:=L \det(B): 1 \to (\wedge^n V^*)^{ \otimes 2} := L

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

(3)gdet(g) 2 g \mapsto \det(g)^{-2}

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 BB is non degenerate it makes sense to say that det(B)\det(B) is positive or negative.

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

Remark 1: The bilinear form BB gives two maps from VV to its dual but one map is the dual of the other. By functoriality this means that the two maps nV nV *\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)\det(B) the same for the two choices for the same reason that for a matrix det(A)=det(A t)\det(A) = \det(A^t) .

Remark 2: If V=T xMV = T_x M this all works out nicely bundle wise for a smooth bilinear form BB, and we should indeed get a well defined section |det(B)|\sqrt{|\det(B)|} that lives canonically in the bundle of densities | n|M|\wedge^n| M. It is smooth if BB 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 BB is symmetric, and we have a symmetric non degenerate form gg on V. We can choose a gg orthonormal basis that diagonalises B(,)=g(diag(b 1,..,b n),)B(-, -) = g(\mathrm{diag}(b_1, .., b_n) - , - ) (at least if gg is definite). Now det(B)= ib ivol g 2\det(B) = \prod_i b_i \mathrm{vol}_g^2. That does not seem to be directly related to tr g(B)=b i\mathrm{tr}_g(B) = \sum b_i , however, that would give the scalar curvature if B=RicB = Ric for a Levi-Civita connection.

Posted by: Rogier Brussee on December 28, 2019 4:42 PM | Permalink | Reply to this

Re: Schrödinger’s Unified Field Theory

Brilliant! This is just the sort of modern approach that I was wanting! Thanks ever so much.

Minor remark: it’s nice how your analysis reveals that |detB|\sqrt{|det B|} is a density, so we can integrate it without choosing an orientation; often physicists try to make their Lagrangians be nn-forms on an nn-manifold, but then you need an orientation to integrate them.

Your Remark 3 makes me want to study Schrödinger’s theory perturbatively: that is, fixing a connection D 0D_0 on the tangent bundle of MM, writing some other connection DD as

D=D 0+ϵΓ D = D_0 + \epsilon \Gamma

for some End(TM)End(T M)-valued 1-form Γ\Gamma, and Taylor-expanding

L=|detRic D| L = \textstyle{ \sqrt{ |det \, Ric_{D}| }}

in powers of ϵ\epsilon.

Your remark, and way the trace shows up when you differentiate the determinant, makes me hope something like the Ricci scalar will appear, at least in situations where we can find coordinates where Ric DRic_{D} is the matrix δ μν\delta_{\mu \nu}. In general something like Jacobi’s formula could be required.

But I guess I’m getting ahead of myself: the first question is what are the critical points of the action

S= ML? S = \int_M L ?

That is, what are the equations of motion? Schrödinger worked them out, but in a way that seems cryptic to me. As I mentioned in my blog article, he prefers to express them in terms of a connection other than the original connection DD. But I don’t understand the meaning of this other connection.

Posted by: John Baez on December 29, 2019 1:05 AM | Permalink | Reply to this

Re: Schrödinger’s Unified Field Theory

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

First, for any square matrix AA,

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

Thus

ddsdet(A+sB)| s=0 = ddsdet(A)det(1+sA 1B)| s=0 = det(A)tr(A 1B) \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

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

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

Posted by: John Baez on January 1, 2020 7:35 AM | Permalink | Reply to this

Re: Schrödinger’s Unified Field Theory

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.

Posted by: Sean Carroll on January 2, 2020 1:17 AM | Permalink | Reply to this

Re: Schrödinger’s Unified Field Theory

I agree with all your comments.

As far as I know, Schrödinger and Einstein never considered quantizing their unified field theories. (Einstein once had his assistant Valentine Bargmann try to teach him quantum field theory, but lost interest in a month.) But in March 1946, Einstein admitted a problem that’s slightly related to the effective field theory issue you mention:

the non-symmetric tensor is not the most simple structure that is covariant with respect to the group, but decomposes into the independently transforming parts g (ik)g_{(i k)} and g [ik]g_{[i k]}; the consequence of this is that one can obtain a nondescript number of systems of second-order equations.

In November 1946, Pauli wrote to Einstein:

I also believe that each tensor, e.g., the contracted curvature tensor, immediately must be split into a symmetric and a skew part (in general: tensors into their irreducible symmetry classes) and to avoid every adding sign between them. What God did separate, humans must not join.

I am completely uninterested in Schrödinger’s theory as a viable theory of physics, and I don’t want to quantize it. I just want to know what sort of classical field equations you get when you take |detR|\sqrt{|det R|} as a Lagrangian, where RR is the Ricci tensor either of an arbitrary connection on the tangent bundle (as in Schrödinger’s theory) or of a torsion-free connection (as apparently in some work of Einstein). I’m having trouble understanding what Schrödinger said about this, because he introduces another connection to formulate the field equations, and I don’t understand what this means. It’s either a bad idea or there’s some interesting geometry behind it.

Posted by: John Baez on January 2, 2020 5:37 AM | Permalink | Reply to this

Re: Schrödinger’s Unified Field Theory

Einstein and Pauli were smart cookies.

Posted by: Sean Carroll on January 2, 2020 10:46 PM | Permalink | Reply to this

Re: Schrödinger’s Unified Field Theory

This idea of adding a symmetric and antisymmetric tensor also appears in the Born-Infeld theory. There, if the symmetric part is the standard Minkowski metric and the antisymmetric part is intepreted as the EMG tensor, the determinant can be expanded as in this comment, but up to all orders of the antisymmetric part (the trace would of course vanish).

Posted by: Jan on January 6, 2020 5:54 PM | Permalink | Reply to this

Bruce

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.

Posted by: Bruce Bartlett on February 4, 2020 8:38 PM | Permalink | Reply to this

Re: Bruce

Hi, Bruce! That’s cool!

Theories of gravity with two metrics instead of one — called ‘bimetric gravity theories’ — have actually been studied. They go back to a 1940 paper by Nathan Rosen. You may know about the famous ‘Einstein–Podolsky-Rosen’ and ‘Einstein–Rosen’ papers: the first was about spooky action at a distance in quantum mechanics, while the second was about wormholes, which are kind of like spooky action at a distance in general relativity. These are all the same Nathan Rosen.

More recently, people have used bimetric gravity to try to understand dark matter. Since 2010 there’s been a renaissance of interest in bimetric gravity as a way to get massive gravitons.

The Wikipedia article I just linked to is a fairly painless way to get a taste of these theories.

Posted by: John Baez on February 5, 2020 7:57 AM | Permalink | Reply to this

Bimetric gravity

Ok, many thanks for informing me about these theories John.

Posted by: Bruce Bartlett on February 5, 2020 8:22 AM | Permalink | Reply to this

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