Noncommutative Gravity and the Standard-Model Extension

TL;DR

This paper demonstrates that noncommutative gravity can be incorporated into the Standard-Model Extension framework, expanding the scope of noncommutative geometry in theoretical physics beyond the Standard Model.

Contribution

It shows that noncommutative gravity, like noncommutative QED, can be formulated within the SME framework, providing a unified approach to these theories.

Findings

Noncommutative QED fits within the SME framework.

Noncommutative gravity can also be incorporated into the SME.

The work broadens the applicability of the SME to include noncommutative gravity.

Abstract

Noncommutative geometry has become popular mathematics for describing speculative physics beyond the Standard Model. Noncommutative QED has long been known to fit within the framework of the Standard-Model Extension (SME). We argue in this work that noncommutative gravity also fits within the SME framework.

Equations

$$(f\cdot g)(x)\to (\hat{f}\star \hat{g})(x):={\exp \left(\frac{i}{2}{\theta }^{\mu \nu }\frac{\partial }{\partial x^{\mu }}\frac{\partial }{\partial y^{\nu }}\right)\hat{f}(x)\hat{g}(y)|}_{x=y}.$$

$$S_1\sim \mathrm{Tr}\int d^{4}x{\epsilon }^{\mu \nu \rho \sigma }{F}_{\mu \nu }{F}_{\rho \sigma }\varphi ,S_2\sim \mathrm{Tr}\int d^{4}x{\epsilon }^{\mu \nu \rho \sigma }{F}_{\mu \nu }{D}_{\rho }\varphi D_{\sigma }\varphi \varphi ,$$

$$\text{mbox}and{S}_3\sim \mathrm{Tr}\int d^{4}x{\epsilon }^{\mu \nu \rho \sigma }{D}_{\mu }\varphi D_{\nu }\varphi D_{\rho }\varphi D_{\sigma }\varphi \varphi .$$

$$S_{\mathrm{NCR}}=-\int \frac{d^{4}xe}{16\pi G_N}\left\{\left[R-\frac{6(1+c_2+2c_3)}{{\ell }^2}\right]+\frac{1}{8{\ell }^4}\sum _{u=1}^6{\theta }^{\alpha \beta }{\theta }^{\gamma \delta }{C}_{(u)}{L}_{\alpha \beta \gamma \delta }^{(u)}\right\}.$$

$$S_{\mathrm{NCR}}\supset \frac{1}{64\pi G_N}\int d^{4}x{h}_{\mu \nu }\left\{s^{(2,1)\mu \rho \nu \sigma }+s^{(4)\mu \rho \alpha \nu \sigma \beta }{\partial }_{\alpha }{\partial }_{\beta }+\cdots \right\}{h}_{\rho \sigma }.$$

$$S_{\mathrm{NCR},\mathrm{mass}}$$

$$s^{(2,1)\mu \rho \nu \sigma }$$

$$k^{(2,1)\mu \nu \rho \sigma }$$

$$S_{\mathrm{NCR},\mathrm{kinetic}}\supset \frac{1}{64\pi G_N}\int d^{4}x{h}_{\mu \nu }\left\{s^{(4)\mu \rho \alpha \nu \sigma \beta }{\partial }_{\alpha }{\partial }_{\beta }+\cdots \right\}{h}_{\rho \sigma },$$

$$s^{(4)\mu \rho \alpha \nu \sigma \beta }\sim \frac{2C_{(1)}-3C_{(2)}+C_{(4)}+C_{(5)}}{{\ell }^4}{\epsilon }^{\mu \rho \alpha \kappa }{\epsilon }^{\nu \sigma \beta \lambda }\left[{\theta }_{\kappa \gamma }{{\theta }_{\lambda }}^{\gamma }-\frac{1}{4}{\eta }_{\kappa \lambda }{\theta }^2\right].$$

$$\left|\frac{{\theta }_{\mu \nu }{\theta }^{\mu \nu }}{{\ell }^4}\right|to0.0pt\text{lower}4.0pt\sim \text{hss}\text{raise}1.0pt<10^{-15}.$$

Full text

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The original inspiration for considering noncommutative geometry in physics[1] was the desire to have a Heisenberg-like uncertainty relation for position coordinates: $\Delta x\Delta y>0$, which corresponds to noncommutativity between position coordinates, $[x,y]\neq 0$. This idea may be made compatible with observer Lorentz symmetry by assuming $\left[x^{\mu},x^{\nu}\right]=i\theta^{\mu\nu}$, where $\theta^{\mu\nu}$ is real and antisymmetric. (Note that the existence of a nonzero tensor that appears to be a property of spacetime itself violates particle Lorentz symmetry.)

A useful tool for constructing noncommutative theories is the Moyal $\star$ product.[2] Consider a commutative field theory with functions/fields $f,g,\ldots$. This may be turned into a noncommutative field theory with noncommutative functions/fields $\hat{f},\hat{g},\ldots$ by replacing all ordinary products with $\star$ products:

[TABLE]

Note: (1) This automatically gives $\left[x^{\mu},x^{\nu}\right]\rightarrow\left[\widehat{x}^{\mu},\widehat{x}^{\nu}\right]_{\star}=i\theta^{\mu\nu}$ as desired. (2) It has similar form to a multivariable Taylor series, and hence may be related to nonlocality. (3) The Moyal $\star$ product is not the only way to define a noncommutative theory; it is simply one convenient approach.

Interpretation of such noncommutative theories is nontrivial as the noncommutative fields $\widehat{\psi},\widehat{A}_{\mu},\ldots$ do not necessarily correspond to physical particles. A Seiberg-Witten map[3] $\widehat{\psi},\widehat{A}_{\mu},\ldots\rightarrow\psi,A_{\mu},\ldots$ is a method of restating noncommutative gauge theories that eases interpretation. This map guarantees that $\psi,A_{\mu}$ are ordinary fields with ordinary gauge transformations whose behavior is physically equivalent to $\widehat{\psi},\widehat{A}_{\mu}$.

This strategy has been used to show that noncommutative QED[4] fits within the flat-space SME.[5] In the rest of this work, we relate a model of noncommutative gravity to the gravitational SME.[6]

One way to model gravity is as a spontaneously broken SO(2,3) gauge theory.[7] This provides a good starting place to build a noncommutative model of gravity, as the (broken) gauge symmetry is automatically respected by the Seiberg-Witten map.

The unbroken commutative SO(2,3) action on flat (1+3)-dimensional spacetime may be written $S=c_{1}S_{1}+c_{2}S_{2}+c_{3}S_{3}$, where

[TABLE]

In this expression, $F$ is the SO(2,3) gauge field, $D$ is the associated covariant derivative, $\phi$ is a scalar field, and $c_{1},\ldots,c_{3}$ are undetermined weights.

If we then assume that $\phi$ spontaneously breaks the SO(2,3) symmetry in its ground state, $\langle{\phi}\rangle=(0,0,0,0,\ell)$, and expand the action around this ground state, then it takes a form that includes conventional gravity: $S\supset-{{1}\over{16\pi G_{N}}}\int d^{4}x\ e\left(R-{{6}\over{\ell^{2}}}(1+c_{2}+2c_{3})\right)$.

This model may then inspire a noncommutative gravitational theory[8] by following a similar prescription to that followed for NCQED: (1) Start with the unbroken SO(2,3) action. (2) Replace fields $F,\phi$ with Moyal $\star$ products of noncommutative fields $\widehat{F},\widehat{\phi}$. (3) Apply a Seiberg-Witten map to replace noncommutative fields with physically equivalent commutative fields. (4) Assume that the SO(2,3)⋆ symmetry is spontaneously broken by $\phi$ having a nonzero vacuum expectation value. The resulting action is left with a noncommutative SO(1,3)⋆ symmetry. It may be expanded in powers of $\theta^{\mu\nu}$, taking the form

[TABLE]

The initial bracketed term describes conventional General Relativity. The noncommutative modification is a sum of geometric quantities $L^{(u)}$ and their weights $C_{(u)}$, which are listed in Table Noncommutative Gravity and the Standard-Model Extension.

The action in Eq. (3) approximately works as a model for noncommutative gravity, though there are some interpretational issues. First, it assumes that $\partial_{\alpha}\theta^{\mu\nu}=0$, which is a coordinate-dependent statement. We may try to maintain coordinate independence by requiring that $\nabla_{\alpha}\theta^{\mu\nu}=0$. However, such covariant-constant tensors cannot exist in most spacetimes.[9, 10]

Second, the derivative $\widetilde{\nabla}$ that appears is covariant with respect to the SO(1,3)⋆ connection but not the Christoffel connection: $\widetilde{\nabla}_{\gamma}{e_{\alpha}}^{a}=\partial_{\gamma}{e_{\alpha}}^{a}+{\omega_{\gamma}}^{ab}e_{ab}={\Gamma^{\rho}}_{\gamma\alpha}{e_{\rho}}^{a}$. This means that the Christoffel symbols appear explicitly in the action. The troublesome terms where they appear violate observer-diffeomorphism symmetry, though they do respect local observer Lorentz transforms. For the rest of this work, we assume that these issues are negligible in experimentally relevant situations. Further, we work at quadratic order in $h_{\mu\nu}=g_{\mu\nu}-\eta_{\mu\nu}$.

To quadratic order in $h$, the gravitational SME may be written[11] $S_{\rm SME}\supset{{1}\over{64\pi G_{N}}}\int d^{4}x\ h_{\mu\nu}\sum_{d}\widehat{\cal K}^{(d)\mu\nu\rho\sigma}h_{\rho\sigma}$. The noncommutative action (3) contains many terms of this form,[6] though we only describe a few here:

[TABLE]

First, we may match the $u=3$ mass-like term in $S_{\rm NCR}$:

[TABLE]

The first term is an irrelevant constant, while the 2nd corresponds to a constant stress-energy. The bottom line contains effective values of SME coefficients:

[TABLE]

Second, we consider a sample kinetic effect with contributions from the $u=$1, 2, 4, and 5 terms:

[TABLE]

where

[TABLE]

This coefficient regulates behavior similar to the $\bar{s}^{\mu\nu}$ coefficient that appears in the minimal gravitational SME.[9] We may therefore exploit existing bounds[12] on $\bar{s}^{\mu\nu}$ to extract rough bounds on $\theta^{\mu\nu}$ (albeit bounds that depend on the gauge-breaking scale $\ell$):

[TABLE]

Acknowledgments

I would like to thank Berry College and the IU Center for Spacetime Symmetries for support during the creation of this work.