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Old timey unified theories

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There are many resources on the topic of modern day unified theories, such as the various extensions of the standard model, supersymmetry, string theory, and so forth. But people do not often talk much more anymore of the theories developed in the early days of general relativity

Due to their age, their scope was much narrower than modern day theories. One of the main goal was the unification of electromagnetism and gravity, and occasionally, quantum mechanics, Dirac fields, Yukawa fields and other entities thrown into the mix.

Geometric unified theories

The very common thread of the olden unified theories were their geometric character, which was calqued from general relativity.

The main idea was to broaden the setting of general relativity. This included :

• Connections with torsions
• Non-metric connections
• Non-symmetric metrics
• More dimensions

The incredibly awful general metric and connection

As with general theory, most unified theories happened on a manifold $M$. Rather than a Lorentz metric, though, we consider here a more complex object as a basic dynamical field. Rather than a symmetric rank $(0,2)$ tensor, we are taking that tensor to be arbitrary. As with all matrices, we can still split it into a symmetric and antisymmetric part. Let's define

\begin{equation} g_{\mu\nu} = \gamma_{(\mu\nu)} + \phi_{[\mu\nu]} \end{equation}

with $\gamma$ a symmetric tensor and $\phi$ antisymmetric. The basic idea behind this was that, as general relativity was the theory of a symmetric tensor field, and electromagnetism of an antisymmetric tensor field, such a combination might produce a theory for both. Due to this decomposition, everything will be much harder to work out. First, we need to define some quantities. The inverse metric $g^{\mu\nu}$ will be, as usual, the metric such that $g^{\mu\nu} g_{\nu\sigma} = \delta^\mu_\sigma$. Our inverse metric can, as any matrix, also be decomposed thusly,

\begin{equation} g^{\mu\nu} = h^{(\mu\nu)} + f^{[\mu\nu]} \end{equation}

But $h$ and $f$ do not have to bear any link to the inverse of $\gamma$ and $\phi$. We're also going to need the various inverses of every matrix we have

\begin{eqnarray} \gamma_{\mu\nu} \gamma^{\mu\sigma} &=& \delta_\nu^\sigma\\ \phi_{\mu\nu} \phi^{\mu\sigma} &=& \delta_\nu^\sigma\\ h_{\mu\nu} h^{\mu\sigma} &=& \delta_\nu^\sigma\\ f_{\mu\nu} f^{\mu\sigma} &=& \delta_\nu^\sigma\\ \end{eqnarray}

We also define the various determinants. $g$, $\gamma$, $\phi$, $f$ and $h$ are the determinants of $g_{\mu\nu}$, $\gamma_{\mu\nu}$, $\phi_{\mu\nu}$, $f_{\mu\nu}$ and $h_{\mu\nu}$. Those are defined as usual by

\begin{eqnarray} \det(g) = \frac{1}{n!} \varepsilon^{\alpha_1 \ldots \alpha_n} \varepsilon^{\beta_1 \ldots \beta_n} g_{\alpha_1 \beta_1} \ldots g_{\alpha_n \beta_n} \end{eqnarray}

For our antisymmetric tensors, $\phi$ and $f$,

First, let's find out the determinant of our metric. We'll denote the determinant of the two components by $\det(\gamma_{\mu\nu}) = \gamma$ and $\det{\phi_{\mu\nu}} = \phi$. From the definition of the determinant,

\begin{eqnarray} \det(\gamma_{\mu\nu} + \phi_{\mu\nu}) &=& \frac{1}{n!} \sum_{i, j} \varepsilon_{i_1 \ldots i_n} \varepsilon_{j_1 \ldots j_n} (\gamma_{i_1 j_1} + \phi_{i_1 j_1}) \ldots (\gamma_{i_n j_n} + \phi_{i_n j_n}) \\ &=& \end{eqnarray}

We can check that, by induction, this is true in one dimension :

\begin{eqnarray} \det(\gamma_{\mu\nu} + \phi_{\mu\nu}) &=& \det(\phi_{\mu\nu}) \end{eqnarray}

And if true in $n$ dimensions, we have, in $n+1$ dimensions,

\begin{eqnarray} \det(\gamma_{\mu\nu} + \phi_{\mu\nu}) &=& \frac{1}{(n + 1)!} \varepsilon^{i_1 \ldots i_{n + 1}} \varepsilon^{j_1 \ldots j_{n + 1}} (\gamma_{i_1 j_1} + \phi_{i_1 j_1}) \ldots (\gamma_{i_{n + 1} j_{n + 1}} + \phi_{i_{n + 1} j_{n + 1}}) \\ &=& \frac{1}{(n + 1)} \frac{1}{n!} \left[ \varepsilon^{i_1 \ldots i_{n + 1}} \varepsilon^{j_1 \ldots j_{n + 1}} (\gamma_{i_1 j_1} + \phi_{i_1 j_1}) \ldots (\gamma_{i_{n + 1} j_{n + 1}} + \phi_{i_{n + 1} j_{n + 1}}) \right]\\ &=& \frac{1}{n+1} \det^{n}(\gamma_{\mu\nu} + \phi_{\mu\nu}) + \frac{1}{(n+1)!} \sum_{i_{n+1} = 0}^{n+1} \end{eqnarray} \begin{eqnarray} \det(\gamma_{\mu\nu} + \phi_{\mu\nu}) &=& \det(\phi_{\mu\nu}) \end{eqnarray}

The affine connection

A much more common feature still found in modern theories is the generalized connection. In other words, we allow any connection rather than the Levi-Civita connection from general relativity. As is well known, a general connection can be split into three components,

\begin{equation} \nabla \end{equation}

The Einstein field equations

There are many possibilites for the Lagrangian, but if we stick to the Einstein-Hilbert action,

\begin{equation} S[g, \Gamma] = \int_\Omega g^{\mu\nu} R_{\mu\nu}[\Gamma] \sqrt{g} d^nx \end{equation} \begin{equation} \delta S = \int_\Omega \left[ (\delta g^{\mu\nu}) R_{\mu\nu}[\Gamma] \sqrt{g} + g^{\mu\nu} (\delta R_{\mu\nu}[\Gamma]) \sqrt{g} + g^{\mu\nu} R_{\mu\nu}[\Gamma] \delta \sqrt{g}\right] d^nx \end{equation}

Classification of theories

From our various geometric objects, there are many theories we can build, depending on what we allow or do not allow.

The parameters on which we can act are :

1. A theory only of the metric tensor
   1. The metric is symmetric
   2. The metric is skew-symmetric
   3. The metric is mixed
2. A theory only of the connection
   1. The connection is metric and torsion-free
3. A theory of both the metric tensor and the connection

Geometrodynamics

A slightly more recent theory of the same type, although borrowing on ideas of those original unified theories, is the idea of geometrodynamics, where deformations of spacetime are used to describe a variety of physical phenomenons, such as particles.

Bibliography

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Last updated : 2021-08-24 11:19:19