A Bivector mediation of standard field equations to form coupled system

[physicslife]

TL;DR Summary

Can the standard field equations (Yang-Mills, Einstein, Klein-Gordon, and Dirac) be mediated by a bivector and a covariant derivative to couple the system and have agreement with both the Standard Model and General Relativity?

Can a single bivector field $\mathcal{A}_\mu$, defined within a Clifford algebra framework, mediate the following four standard field equations using a covariant derivative $D_\mu = \partial_\mu + [\mathcal{A}_\mu, \cdot]$? The standard equations are:

\textbf{Yang-Mills Equation}:
$$D_\mu F^{\mu\nu, a} = J^{\nu, a} + (D^\nu \phi)^\dagger T^a \phi,$$
describing gauge field dynamics, where $F^{\mu\nu, a}$ is the gauge field strength, $J^{\nu, a}$ is the fermionic current, and $\phi$ is a scalar field.

\textbf{Einstein Field Equation}:
$$G_{\mu\nu} = \kappa T_{\mu\nu},$$
governing gravity, with $G_{\mu\nu}$ as the Einstein tensor and $T_{\mu\nu}$ as the stress-energy tensor.

\textbf{Klein-Gordon Equation}:
$$D_\mu D^\mu \phi + \frac{\partial V}{\partial \phi^\dagger} - y \sum_i \overline{\psi}^i \psi^i = 0,$$
for the scalar field $\phi$, with $V(\phi)$ as the potential and $y$ as the Yukawa coupling.

\textbf{Dirac Equation}:
$$i \gamma^\mu D_\mu \psi^i - m \psi^i - y \phi \psi^i = 0,$$
for fermionic fields $\psi^i$, generations $i=1,2,3$.

The bivector field $\mathcal{A}_\mu$ would split into gravitational $\mathcal{A}_\mu^{\text{spin}}$ and gauge $\mathcal{A}_\mu^{\text{int}}$ components, with the metric $g_{\mu\nu}$ derived from $\mathcal{A}_\mu^{\text{spin}}$ via a tetrad $e^a_\mu$.

Is this approach mathematically and physically feasible? Does it align with the Standard Model and General Relativity predictions? If so, wouldn't this effectively unify GR and SM as the above equations are now fully coupled.

 

[renormalize]

TL;DR Summary: Can the standard field equations (Yang-Mills, Einstein, Klein-Gordon, and Dirac) be mediated by a bivector and a covariant derivative to couple the system and have agreement with both the Standard Model and General Relativity?

Can a single bivector field $\mathcal{A}_\mu$, defined within a Clifford algebra framework, mediate the following four standard field equations using a covariant derivative $D_\mu = \partial_\mu + [\mathcal{A}_\mu, \cdot]$?

I'm confused by your terminology. The gauge-field $\mathcal{A}_\mu$ that appears in a general covariant derivative is a vector in 4D spacetime and therefore contains four components (each of which can in general be complex-matrix-valued). In contrast, a bivector in 4D is an antisymmetric rank-2 tensor comprised of six components. How are you relating a six-component bivector to the four-component vector $\mathcal{A}_\mu$?

 

[physicslife]

I'm confused by your terminology. The gauge-field $\mathcal{A}_\mu$ that appears in a general covariant derivative is a vector in 4D spacetime and therefore contains four components (each of which can in general be complex-matrix-valued). In contrast, a bivector in 4D is an antisymmetric rank-2 tensor comprised of six components. How are you relating a six-component bivector to the four-component vector $\mathcal{A}_\mu$?

Thank you for the comment and, yes, I now realize I left out some important structural details:


To clarify, I’m not relating a 6-component bivector to a 4-component vector. Instead, $\mathcal{A}\mu$ is a 4-component vector field in spacetime (like a standard gauge field $A\mu$), where each component is a bivector with 6 dimensions in the internal Clifford algebra $\text{Cl}(1,3)$. This gives $\mathcal{A}_\mu$ a total of $4 \times 6 = 24$ real components, designed to encode both gravitational and gauge interactions.


Each spacetime component $\mathcal{A}_\mu$ carries its own internal bivector structure. For a fixed index $\mu$, we write:

$$
\mathcal{A}\mu = \frac{1}{2},\omega\mu^{ab},\sigma_{ab},
$$

where the six parameters $\omega_\mu^{ab}$ correspond to the Lorentz generators $\sigma_{ab}$.


This construction naturally splits $\mathcal{A}_\mu$ into a gravitational (spin connection) part and an internal gauge part. The spacetime metric is then derived from the gravitational component via a tetrad field.


Let me know if this clear presentation resolves the confusion, or if you’d like more details!

 

[renormalize]

Does this resolve the confusion? Let me know if you need more details!

You still have problems with your LaTeX displaying. You can use the magnifying-glass icon in the upper right corner to preview your LaTeX and edit it as necessary prior to posting

 

[physicslife]

You still have problems with your LaTeX displaying. You can use the magnifying-glass icon in the upper right corner to preview your LaTeX and edit it as necessary prior to posting

Hi Yes, I'm having some problems formatting. Thank you for the tip on the preview but it's not working for me. In the preview it looks like this:

But when I save it the formatting doesn't work. I'll keep trying to solve it, here's the image for now.

 

[physicslife]

Ah ok, I needed to refresh my screen apparently. Thank you for the patience!

 

[renormalize]

Thanks for cleaning up your LaTeX.

$\mathcal{A}\mu = \frac{1}{2},\omega\mu^{ab},\sigma_{ab},$

However, this equation still makes no sense to me. Nevertheless, I will attempt to address your original question, where you asked:

Is this approach mathematically and physically feasible? Does it align with the Standard Model and General Relativity predictions? If so, wouldn't this effectively unify GR and SM as the above equations are now fully coupled.

Yes, your approach is quite feasible and does indeed align with both SM and GR, but it's nothing new.
Our best current model of the universe generalizes the flat-space standard-model gauge-theories of quark chromodynamics and the Weinberg-Salam model of weak and electromagnetic interactions and puts them on the classical curved background of general relativity using minimal coupling. Sometimes expectation values of the quantum particle and gauge fields are even used to construct a classical energy-momentum tensor in order to investigate the back-influence of those fields on the classical gravitational metric-tensor field. So all the fields in our current model are indeed already "fully coupled".
But by no means does this "effectively unify GR and SM" because it's only a semi-classical effective model, and will remain so until someone develops a successful theory of quantum gravity.

 

[physicslife]

I see your point, if this were just a semiclassical pairing of the SM with GR, it wouldn’t be true unification. But what I’m describing goes beyond that. Instead of inserting SM fields onto a fixed metric, I’m working with a single gauge field, $\mathcal{A}\mu$, that includes both gravitational and SM gauge components. The metric $g{\mu\nu}$ isn’t independent like in GR, it’s dynamically built from the gravitational part, $\mathcal{A}\mu^{\text{spin}}$, via a tetrad $e^a\mu$, where $g_{\mu\nu} = e^a_\mu e^b_\nu \eta_{ab}$.


This setup makes the metric vary dynamically with $\mathcal{A}_\mu$. In the action,


$$ S = \int \sqrt{-g(\mathcal{A})} \left[ R(\mathcal{A}) + \mathcal{F}_{\text{int}}^2 + \text{other terms} \right], $$


we vary with respect to $\mathcal{A}\mu$. Since $g{\mu\nu}$ depends on $\mathcal{A}\mu$, this variation involves everything at once. The volume element $\sqrt{-g(\mathcal{A})}$ shifts as the metric changes, the curvature $R(\mathcal{A})$ adjusts because it’s computed from $\mathcal{A}\mu$’s spin connection, and the gauge field strength $\mathcal{F}{\text{int}}^2$ responds to $\mathcal{A}\mu$’s gauge parts. Mathematically, this variation,


$$ \frac{\delta S}{\delta \mathcal{A}_\mu}, $$


mixes gravitational and gauge effects inseparably, unlike standard GR, where the metric stays fixed when you tweak gauge fields, or semiclassical models, where SM fields live on a static spacetime. The equations reduce to those models in limits but are fundamentally different due to this coupled variation. What do you think?

 

[mitchell porter]

$$
\mathcal{A}\mu = \frac{1}{2},\omega\mu^{ab},\sigma_{ab},
$$

I assume this is the comma notation for derivatives of tensors, but can't figure it out. Are you taking two partial derivatives of mu, one with respect to omega and one with respect to sigma?

 

[renormalize]

What do you think?

I follow your description but I'm skeptical of your claims. First, how does what you describe differ from those theories that vary both the tetrad and the spin-connection independently to arrive at generalizations of GR that include torsion and/or non-metricity? Second, to convince me you'd need to explicitly display your field equations and identify how they differ from those of SM+GR. (To keep things simple, it would suffice to write down your generalization of the vacuum Einstein equations.)
Also, I now realize that you are evidently describing a personal theory, the discussion of which is off limits according to the rules of Physics Forums:
Speculative or Personal Theories:
Physics Forums is not intended as an alternative to the usual professional venues for discussion and review of new ideas, e.g. personal contacts, conferences, and peer review before publication. If you have a new theory or idea, this is not the place to look for feedback on it or help in developing it.

 

[physicslife]

I typed up the blow before seeing you most recent comment but I wiill just say I don't claim this as a theory, it's more about whether mediating the standard equations this way is appropriate and if the resulting path integral functions normally or better as a result.

Let me address your last comment about needing a theory quantum gravity, which I agree with and how this approach differs from those you mentioned. In the semiclassical picture, gravity is treated classically via the metric $g_{\mu\nu}$, while matter fields are quantum. I totally get your point and agree that they are coupled through the expectation value of the stress-energy tensor, $\langle T_{\mu\nu} \rangle$, which sources the Einstein field equations. This works fine for low-energy approximations, but it stalls when you try to fully quantize gravity. Why? The metric $g_{\mu\nu}$ isn’t a quantum field, there’s no operator or path integral over it that plays nice with QFT. So looking at the usual GR path integral:

$$ Z = \int \mathcal{D}g_{\mu\nu} \, e^{i S[g_{\mu\nu}, \text{matter}]}, $$

GR’s action, the Einstein-Hilbert term, is non-renormalizable. Perturbative QFT techniques choke on the infinite counterterms needed to tame UV divergences.

My approach sidesteps this entirely with a single gauge field, $\mathcal{A}_\mu$, that unifies gravity and SM interactions. Unlike GR, where $g_{\mu\nu}$ is the fundamental object, here the metric is a derived quantity, constructed from $\mathcal{A}_\mu$. Think of $\mathcal{A}_\mu$ as a connection-like field, similar to those in Yang-Mills theories, but engineered to encode both spacetime geometry and particle physics. Quantizing $\mathcal{A}_\mu$ automatically quantizes gravity, because the metric emerges from it.

The path integral becomes:

$$ Z = \int \mathcal{D}\mathcal{A}_\mu \, e^{i S[\mathcal{A}_\mu]}, $$

where $S[\mathcal{A}_\mu]$ is the action built solely from $\mathcal{A}_\mu$, capturing both gravitational and SM dynamics. No separate $g_{\mu\nu}$ integration, no semiclassical split, everything’s unified under one quantum field.

The path integral over $\mathcal{A}_\mu$ doesn’t assume a background geometry, it generates it dynamically as a quantum output. The action $S[\mathcal{A}_\mu]$ isn’t split into a gravity piece and a matter piece. It’s a single, coherent functional where $\mathcal{A}_\mu$’s components dictate both curvature (gravity) and gauge interactions (SM). This unity is what makes the quantization fundamentally distinct. Does this clarify the approach somewhat?

 

[physicslife]

I assume this is the comma notation for derivatives of tensors, but can't figure it out. Are you taking two partial derivatives of mu, one with respect to omega and one with respect to sigma?

Ah sorry Mitchell it was a formatting issue, it's meant to represent a normal linear combination of the generators, you can ignore the commas.

 

[renormalize]

My approach sidesteps this entirely with a single gauge field, $\mathcal{A}_\mu$, that unifies gravity and SM interactions. Unlike GR, where $g_{\mu\nu}$ is the fundamental object, here the metric is a derived quantity, constructed from $\mathcal{A}_\mu$. Think of $\mathcal{A}_\mu$ as a connection-like field, similar to those in Yang-Mills theories, but engineered to encode both spacetime geometry and particle physics. Quantizing $\mathcal{A}_\mu$ automatically quantizes gravity, because the metric emerges from it.

Again, I can't buy any of these claims without some explicit math. You would have to demonstrate that:

• The classical theory $\delta S/\delta A_{\mu}=0$ gives rise to equations for the metric/tetrad/spin-connection that are close enough to standard general relativity to agree with all the successful empirical verifications of GR. In other words, your theory must look much like GR classically.
• The quantum theory of $A_{\mu}$ gives rise to quantum behavior for the metric/tetrad/spin-connection that is notably distinct from the unsuccessful, non-renormalizable attempts to quantize gravity . In other words, your theory must not look anything like GR quantum mechanically.

That's a tall order! (And it certainly looks to me like a recipe for a personal theory.)

 

[physicslife]

Again, I can't buy any of these claims without some explicit math. You would have to demonstrate that:

• The classical theory $\delta S/\delta A_{\mu}=0$ gives rise to equations for the metric/tetrad/spin-connection that are close enough to standard general relativity to agree with all the successful empirical verifications of GR. In other words, your theory must look much like GR classically.
• The quantum theory of $A_{\mu}$ gives rise to quantum behavior for the metric/tetrad/spin-connection that is notably distinct from the unsuccessful, non-renormalizable attempts to quantize gravity . In other words, your theory must not look anything like GR quantum mechanically.

That's a tall order! (And it certainly looks to me like a recipe for a personal theory.)

Those two I can certainly outline, this was actually the first thing I checked for when I started looking into this:

The classical action is given by

$$ S = \int d^4x \, \sqrt{-g} \left[ \frac{1}{2\kappa} R - \frac{1}{4} \langle \mathcal{F}_{\mu\nu} \mathcal{F}^{\mu\nu} \rangle \right], $$

where $\sqrt{-g} = \det(e^a_\mu)$, with $e^a_\mu$ derived from $\mathcal{A}_\mu^{\text{spin}}$, $R$ is the Ricci scalar from the tetrad’s curvature, $\mathcal{F}_{\mu\nu} = \partial_\mu \mathcal{A}_\nu - \partial_\nu \mathcal{A}_\mu + [\mathcal{A}_\mu, \mathcal{A}_\nu]$ is the gauge field strength, $\langle \cdot \rangle$ traces over the gauge algebra, and $\kappa = 8\pi G$.

The term $\frac{1}{2\kappa} R$ mirrors GR’s Einstein-Hilbert action. In vacuum or weak-field limits, where gauge effects are minimal, the $\mathcal{F}_{\mu\nu}$ term fades, leaving

$$ S_{\text{classical}} \approx \int d^4x \, \sqrt{-g} \, \frac{1}{2\kappa} R. $$

Varying this with respect to $e^a_\mu$ (determined by $\mathcal{A}_\mu^{\text{spin}}$) gives the vacuum Einstein equations

$$ G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu} = 0. $$

With matter (via $T_{\mu\nu}$), it becomes

$$ G_{\mu\nu} = \kappa T_{\mu\nu}, $$

matching GR precisely and therefor any of it's predictions.

So while $\mathcal{A}_\mu$ adds extra structure, these contributions decouple or diminish classically. The tetrad $e^a_\mu$ ensures metric compatibility, preserving GR’s empirical foundation, which is exactly what we'd want.

For quantum, this approach departs from GR. In GR quantizing the metric $g_{\mu\nu}$ via

$$ Z_{\text{GR}} = \int \mathcal{D}g_{\mu\nu} \, e^{i S_{\text{EH}}[g_{\mu\nu}]}, $$

yielding a non-renormalizable mess due to UV divergences from second-order derivatives.

Here, $\mathcal{A}_\mu$ is the quantum field, with $g_{\mu\nu}$ and $e^a_\mu$ as derived quantities. The path integral is

$$ Z = \int \mathcal{D}\mathcal{A}_\mu \, e^{i S[\mathcal{A}_\mu]}, $$

where $S[\mathcal{A}_\mu]$ includes both gravitational and gauge terms. This looks like Yang-Mills structure, which is renormalizable, a good sign.

The gauge term $-\frac{1}{4} \langle \mathcal{F}_{\mu\nu} \mathcal{F}^{\mu\nu} \rangle$ is power-counting renormalizable, with $\mathcal{F}_{\mu\nu}$ at dimension 2. The gravitational $R(\mathcal{A})$ term, though non-linear, follows from $\mathcal{A}_\mu$. Quantization uses gauge fixing (e.g., $\frac{1}{2\xi} (\partial^\mu \mathcal{A}_\mu)^2$), Faddeev-Popov ghosts, and perturbation around $\mathcal{A}_\mu = 0$ (Minkowski space).

This would, I think, yield Feynman rules akin to gauge field theories, suggesting renormalizability. The metric’s quantum behavior would emerge from $\mathcal{A}_\mu$, not as a primary field. I haven't done all of that yet, of course but it shows every sign of working and no signs of not working. Could it be this has simply not yet been tried?

 

[renormalize]

The gauge term $-\frac{1}{4} \langle \mathcal{F}_{\mu\nu} \mathcal{F}^{\mu\nu} \rangle$ is power-counting renormalizable, with $\mathcal{F}_{\mu\nu}$ at dimension 2....
This would, I think, yield Feynman rules akin to gauge field theories, suggesting renormalizability.

Those statements are naive in the context of what you propose. Because you posit that the metric is derived from the gauge potential $\mathcal{A}_{\mu}\,$, I can write that metric as $g^{\alpha\beta}\left(\mathcal{A}\right)$ to symbolize that it depends on $\mathcal{A}$ and its derivatives, and then decompose it around flat spacetime according to:$$g^{\alpha\beta}\left(\mathcal{A}\right)=\eta^{\alpha\beta}+h^{\alpha\beta}\left(\mathcal{A}\right)$$As a result, the Yang-Mills action becomes:
\begin{align}
S_{F}&\equiv -\frac{1}{4}\int d^{4}x\,\sqrt{-g\left(\mathcal{A}\right)}\,g^{\mu\sigma}\left(\mathcal{A}\right)g^{\nu\tau}\left(\mathcal{A}\right)\mathcal{F}_{\mu\nu}\left(\mathcal{A}\right)\cdot\mathcal{F}_{\sigma\tau}\left(\mathcal{A}\right) \nonumber \\
& =-\frac{1}{4}\int d^{4}x\,\eta^{\mu\sigma}\eta^{\nu\tau}\mathcal{F}_{\mu\nu}\left(\mathcal{A}\right)\cdot\mathcal{F}_{\sigma\tau}\left(\mathcal{A}\right)+\int d^{4}x\,H^{\mu\nu\sigma\tau}\left(h\left(\mathcal{A}\right)\right)\mathcal{F}_{\mu\nu}\left(\mathcal{A}\right)\cdot\mathcal{F}_{\sigma\tau}\left(\mathcal{A}\right) \nonumber
\end{align}Yes, the first term is power-counting renormalizable. But you have to actually prove that the highly-nonlinear second term is also renormalizable. (And my guess is that it's not!) Moreover, you must demonstrate that the contributions of that term to your field-equations doesn't spoil their correspondence to SM+GR.

 

[physicslife]

I agree, renormalization is the challenge of this approach. In the broader context of how $\mathcal{A}\,$ is developed which I only touched on in this thread, there are symmetries available to restrict some corrections (gauge and diffeomorphism invariance). I had originally been thinking to go the FRG route. My understanding is that in 4-d spacetime, keeping the dimension at 4 or less ensures that divergences can be controlled with a finite number of counterterms. Meaning it might get hairy but to me this indicates it should be possible? But you seem more familiar with this than me, do you see a route to renormalizing? Do you think FRG is the right approach?

 

[renormalize]

Do you think FRG is the right approach?

Sorry, the functional renormalization group is beyond my pay grade! But I do know that the FRG is used to study the asymptotic-safety scenario in "conventional" quantum gravity. So it might well apply to your theory too. But to really investigate that possibility is likely to require a team of bright PhD theoreticians and their graduate students!

 

[physicslife]

Sorry, the functional renormalization group is beyond my pay grade! But I do know that the FRG is used to study the asymptotic-safety scenario in "conventional" quantum gravity. So it might well apply to your theory too. But to really investigate that possibility is likely to require a team of bright PhD theoreticians and their graduate students!

No problem and thanks for all of the careful reviewing and insights, it's been very helpful!

And now to go recruit a team of PhD theoreticians!