Standard-Model Extension and Lorentz-Violation Bounds

The Robertson–Mansouri–Sexl framework has three dials, and it turns all of matter with one hand. It cannot ask whether the electron feels a preferred frame while the photon does not — and that is precisely what a Planck-scale theory would do. So build the test theory out of fields instead. Take the Standard Model, and add every term you can write down that is gauge-invariant and built from the known fields contracted with constant background tensors. Each such tensor is a coefficient for Lorentz violation. Each one is measurable. Each one is zero, so far. That is the Standard-Model Extension. It turned "we have tested relativity" from a slogan into a table of numbers, republished every January.

Key takeaways

Signature $(+,-,-,-)$ : $\mathrm{d}s^2 = c^2\mathrm{d}t^2 - \mathrm{d}x^2$ , timelike intervals positive.
The photon sector has exactly two renormalizable Lorentz-violating terms: the CPT-odd $(k_{AF})_\kappa$ , with dimensions of mass, and the CPT-even $(k_F)_{\kappa\lambda\mu\nu}$ , dimensionless.
$(k_F)$ has the symmetries of the Riemann tensor and a vanishing double trace ⟹ exactly $\mathbf{19}$ independent components, which split $11 + 8$ and decompose as $5 + 5 + 3 + 5 + 1$ . Ten of the 19 are birefringent; nine are not. The astrophysics kills the ten; the laboratory owns the nine.
⚠ OWNED RESULT — the count lands on the experiment's own word. Nagel et al. (2015) report "comprehensive simultaneous bounds on nine boost and rotation anisotropies of the speed of light." Nine is not a coincidence: it is $19 - 10$ , and the missing ten were already bounded $10^{19}$ times harder by gamma-ray bursts.
⚠ OWNED RESULT — the CPT-odd bound is the Hubble energy. Re-deriving the Carroll–Field–Jackiw dispersion from scratch gives $\omega^2_\pm = |\mathbf{k}|(|\mathbf{k}| \pm k_{AF}^0)$ , hence a polarization rotation $\Delta\phi = \tfrac12 k_{AF}^0 L$ that is independent of frequency. Demanding $\Delta\phi \lesssim 1$ across the observable universe gives $k_{AF}^0 \lesssim 2\hbar H_0 = 4.3\times10^{-42}h_0\ \mathrm{GeV}$ . The published bound is $1.7\times10^{-42}h_0\ \mathrm{GeV}$ — that is $\mathbf{0.80\,\hbar H_0}$ . The apparatus is the universe, and the bound is its inverse size.
⚠ THE HYPE CORRECTION. A bound is meaningless without naming its coefficient. The best cavity bound, $|\Delta\nu/\nu| \lesssim 2\times10^{-18}$ , is a factor of 26 short of $m_p/M_{\mathrm{Pl}} = 7.69\times10^{-20}$ — it has not reached first-order Planck suppression. The gamma-ray birefringence bound, parts in $10^{37}$ , is within a factor of 17 of $(m_p/M_{\mathrm{Pl}})^2 = 5.91\times10^{-39}$ — second order.
Greenberg's theorem (PRL 89, 231602): CPT violation implies Lorentz violation. The converse is false — his abstract says so explicitly. That is why one framework covers both.
Key idea: RMS asked how far the universe is from Einstein. The SME asks in which direction, for which particle, and at what order in the Planck length — and every answer so far is zero.

The core insight

Effective field theory is the technique of writing down everything your symmetries allow, ordered by how badly each term is suppressed at low energy, and letting experiment set the coefficients. The SME applies it to the one symmetry nobody had thought to relax: Lorentz invariance itself. It is not a model of Lorentz violation — it is the complete list of the ways Lorentz violation could show up in the fields we already know about, which is what makes a null result from it mean something.

Where this lives today

As the standard currency of the field. Every experiment that tests Lorentz symmetry — atomic clocks, cavities, torsion pendula, neutrino oscillations, cosmic rays, gravitational waves, antihydrogen — reports its result as a bound on named SME coefficients, because that is the only way results from different sectors can be compared at all. The Data Tables now run to hundreds of entries.

The open frontier is the nonminimal sector. The minimal SME keeps only renormalizable operators, but Planck-scale physics need not respect renormalizability at low energy, and operators of dimension five and higher produce effects that grow with energy — which is why ultra-high-energy cosmic rays and TeV gamma rays now give some of the sharpest constraints in the programme.

And the gravity sector is where it gets interesting — see the FAQ below: it is the one place we might still find something, and the one place we can least easily look.

The physical picture

A constant background tensor field is a permanent, unremovable arrow — or grid, or block of directions — frozen into the vacuum. You cannot boost it away: it is not a property of your motion but of empty space. Anything coupled to it acquires a direction. This is not an idle possibility. In string field theory the vacuum can spontaneously acquire a tensor expectation value the way the Higgs field acquires a scalar one (Kostelecký & Samuel, Phys. Rev. D 39, 683 (1989)). A scalar one breaks gauge symmetry and leaves spacetime alone; a tensor one breaks spacetime symmetry and leaves gauge symmetry alone. The SME's answer to "what would it look like?" is a bookkeeping one, and its power is that the bookkeeping is complete: for every field, write every Lorentz-scalar you can build by contracting its bilinears with constant tensors. For the photon there are exactly two renormalizable ones. Three features of this deserve attention before any number is quoted.

First, the coefficients have different dimensions, so their bounds are not comparable. $(k_{AF})_\kappa$ carries dimensions of mass and is bounded in GeV; $(k_F)_{\kappa\lambda\mu\nu}$ is dimensionless and bounded as a pure number. " $10^{-42}$ beats $10^{-18}$ " is a category error, and it is committed constantly.

Second, birefringence is far more sensitive than anything else, and only ten coefficients cause it. If two polarizations travel at different speeds the phase between them accumulates over the whole path — and for a gamma-ray burst at cosmological distance that path is $10^{26}$ m against a wavelength of $10^{-13}$ m: a lever arm of $10^{39}$ . No laboratory can compete, and none needs to.

Third, that is why the laboratory owns nine. $19 - 10 = 9$ . The modern Michelson–Morley experiment is not a redundant re-check of an 1887 result: it is the only instrument that can reach the nine coefficients the sky is blind to.

The core equations

The photon sector of the minimal SME, in signature $(+,-,-,-)$ :

\boxed{\;\mathcal{L} = -\tfrac14 F_{\mu\nu}F^{\mu\nu} \;-\; \tfrac14 (k_F)_{\kappa\lambda\mu\nu}F^{\kappa\lambda}F^{\mu\nu} \;+\; \tfrac12 (k_{AF})^{\kappa}\,\epsilon_{\kappa\lambda\mu\nu}A^{\lambda}F^{\mu\nu}\;}\qquad (1)

where: the first term is Maxwell; the second is CPT-even with a dimensionless coefficient; the third is CPT-odd (Carroll–Field–Jackiw), of mass dimension one. Physical reading: both extra terms are gauge invariant (the third up to a total derivative, which is enough), and both are constants of nature that must be measured, not predicted. Setting them to zero is not a derivation of special relativity — it is the assumption that special relativity is exactly true. The modified Maxwell equation from the CPT-odd term alone, with $\tilde F^{\mu\nu} \equiv \tfrac12\epsilon^{\mu\nu\alpha\beta}F_{\alpha\beta}$ :

\boxed{\;\partial_{\mu}F^{\mu\nu} \;+\; (k_{AF})_{\mu}\,\tilde F^{\mu\nu} \;=\; 0\;}\qquad (2)

where: the second term is the only new one. Physical reading: it couples the field to its own dual — parity violation in electrodynamics — which is why the effect on light is a rotation rather than a delay. The dispersion relation that (2) forces, for a purely timelike $(k_{AF})^\mu = (k_{AF}^0, \mathbf{0})$ and the two circular polarizations:

\boxed{\;\omega^{2}_{\pm} \;=\; |\mathbf{k}|\left(|\mathbf{k}| \pm k_{AF}^{0}\right)\;}\qquad (3)

where: $\pm$ labels the two circular polarizations. Physical reading: the vacuum is birefringent. At fixed $\omega$ the two wavenumbers differ by exactly $k_+ - k_- = -k_{AF}^0$ — at all orders, no approximation — so the plane of polarization rotates by

\boxed{\;\Delta\phi \;=\; \tfrac12\left|k_+ - k_-\right| L \;=\; \tfrac12\,k_{AF}^{0}\,L\;}\qquad (4)

where: $L$ is the propagation distance. Physical reading: $\Delta\phi$ does not depend on frequency — the whole experimental signature. Faraday rotation goes as $\lambda^2$ ; this goes as $\lambda^0$ . Compare the polarization angle at two wavelengths and the astrophysics subtracts itself out. The counting of the CPT-even sector:

\boxed{\;\underbrace{20}_{\text{Riemann}} \;-\; \underbrace{1}_{\text{double trace}} \;=\; \mathbf{19} \;=\; \underbrace{5}_{\tilde\kappa_{e+}} + \underbrace{5}_{\tilde\kappa_{e-}} + \underbrace{3}_{\tilde\kappa_{o+}} + \underbrace{5}_{\tilde\kappa_{o-}} + \underbrace{1}_{\tilde\kappa_{\mathrm{tr}}} \;=\; \underbrace{10}_{\text{birefringent}} + \underbrace{9}_{\text{not}}\;}\qquad (5)

where: the ten birefringent combinations are $\tilde\kappa_{e+}$ and $\tilde\kappa_{o-}$ ; the nine others are $\tilde\kappa_{e-}$ , $\tilde\kappa_{o+}$ , $\tilde\kappa_{\mathrm{tr}}$ . Physical reading: this arithmetic is why the modern Michelson–Morley experiment exists, and why its authors write the word "nine."

Limiting cases

Case	Result	Meaning
$k_F = k_{AF} = 0$	$\omega = \lvert\mathbf{k}\rvert$ ; (1) $\to -\tfrac14 F^2$	Maxwell. SR is the origin of the coefficient space, not an assumption of it
$k_{AF}^0 \ne 0$ , $k_F = 0$	$\omega^2_\pm = \lvert\mathbf{k}\rvert(\lvert\mathbf{k}\rvert \pm k_{AF}^0)$	vacuum birefringence. Two circular modes, two speeds
$\omega \gg k_{AF}^0$	$k_+ - k_- = -k_{AF}^0$ exactly	$\Delta\phi = \tfrac12 k_{AF}^0 L$ is frequency-independent — the signature that separates it from Faraday rotation
$L \to L_{\mathrm{Hubble}}$	$k_{AF}^0 \lesssim 2\hbar H_0 = 4.3\times10^{-42}h_0$ GeV	the bound is the Hubble energy. Published: $1.7\times10^{-42}h_0$ GeV $= 0.80\,\hbar H_0$
the 10 birefringent / the 9 not	$\lesssim$ parts in $10^{37}$ / $\lesssim 10^{-18}$	GRB polarimetry owns the ten; a rotating cavity is the only instrument that reaches the nine
$\tilde\kappa_{\mathrm{tr}}$ alone	an isotropic shift in the speed of light	invisible to any rotation; only a boost can see it
all coefficients $\to$ 0, matter universal	the RMS point	`aq-sr-robertson-mansouri-sexl-framework`'s three dials are the species-blind shadow of this

Deriving it

Derivation AThe CPT-odd sector rotates the sky and the bound on it is the inverse size of the universe

The Carroll–Field–Jackiw term is the only renormalizable CPT-odd term in electrodynamics, and the only one with a dimensionful coefficient — so its bound is a mass, and we should be able to say which mass. We can. It is the Hubble energy.

Step 1 — the field equation. Vary (1), keeping only the CPT-odd term. With $\tilde F^{\mu\nu} = \tfrac12\epsilon^{\mu\nu\alpha\beta}F_{\alpha\beta}$ the Euler–Lagrange equation is (2): $\partial_\mu F^{\mu\nu} + (k_{AF})_\mu \tilde F^{\mu\nu} = 0$ . The Bianchi identity $\partial_\mu \tilde F^{\mu\nu} = 0$ is untouched. Why: only half of Maxwell's equations can be modified this way. The two that follow from $F = \mathrm{d}A$ are geometric identities, immune — which is why the theory keeps gauge invariance and keeps $\nabla\cdot\mathbf{B} = 0$ .

Step 2 — plane wave. Put $A^\nu = \varepsilon^\nu e^{-\mathrm{i}k\cdot x}$ in Lorenz gauge, $k\cdot\varepsilon = 0$ , with $(k_{AF})^\mu = (k_{AF}^0, \mathbf{0})$ and $k^\mu = (\omega, 0, 0, k)$ . The gauge lets us set $\varepsilon^0 = \varepsilon^3 = 0$ , leaving the transverse pair $(\varepsilon^1, \varepsilon^2)$ , on which (2) acts as a $2\times2$ matrix. Why: the new term mixes $\varepsilon^1$ and $\varepsilon^2$ antisymmetrically, because it is built from $\epsilon^{\mu\nu\alpha\beta}$ . An antisymmetric mixing of two linear polarizations is exactly what "circular eigenmodes" means.

Step 3 — the dispersion relation. The nontrivial-solution condition is the vanishing of that determinant, which factorises:

\left(\omega^{2} - k^{2} + k\,k_{AF}^{0}\right)\left(\omega^{2} - k^{2} - k\,k_{AF}^{0}\right) = 0 \qquad\Longrightarrow\qquad \omega^{2}_{\pm} = k\left(k \pm k_{AF}^{0}\right).

Why: two roots, so two speeds — equation (3), vacuum birefringence, the vacuum behaving like a slab of quartz. The shift in $\omega^2$ is linear in $k$ , not quadratic: the fingerprint of a dimension-one coefficient.

Step 4 — invert at fixed $\omega$ ; watch the frequency drop out. Solve $k^2 \pm k\,k_{AF}^0 - \omega^2 = 0$ for $k$ :

k_\pm = \frac{\mp k_{AF}^{0} + \sqrt{(k_{AF}^{0})^{2} + 4\omega^{2}}}{2} \qquad\Longrightarrow\qquad k_+ - k_- = -k_{AF}^{0} .

Why: the square root is the same in both, so it cancels exactly, and $k_+ - k_- = -k_{AF}^0$ at all orders in $\omega$ — no expansion. The plane of a linearly polarized wave rotates by $\Delta\phi = \tfrac12|k_+ - k_-|L = \tfrac12 k_{AF}^0 L$ , with no frequency dependence whatever — which is what makes the observation possible, since every competing astrophysical rotation mechanism (Faraday above all) is strongly chromatic.

Step 5 — the bound, and what it is made of. Radio galaxies show polarization aligned with their own structure out to cosmological distances; a radian of rotation accumulated en route would destroy that alignment. Demand $\Delta\phi \lesssim 1$ over one Hubble length $L_H = c/H_0 = 9.25\times10^{25}h_0^{-1}\ \mathrm{m}$ :

k_{AF}^{0} \;\lesssim\; \frac{2}{L_H} \;=\; \frac{2\hbar H_0}{c}\cdot\frac{c}{\hbar} \;\Longrightarrow\; k_{AF}^{0} \lesssim 2\hbar H_0 = \mathbf{4.3\times10^{-42}}\,h_0\ \mathrm{GeV}.

Why: this is the whole bound, up to a factor of order one. Carroll, Field and Jackiw (Phys. Rev. D 41, 1231 (1990)) did the real statistics — real sources, real redshifts — and got $\lvert k_{AF}\rvert < 1.7\times10^{-42}h_0\ \mathrm{GeV}$ . Since $\hbar H_0 = 2.13\times10^{-42}h_0\ \mathrm{GeV}$ , their bound is $0.80\,\hbar H_0$ . The most stringent constraint on a mass scale anywhere in physics is, to within twenty per cent, the inverse size of the observable universe — because the apparatus is the observable universe. Without a longer lever, you cannot do better. *Verified in verify_aq-sr-standard-model-extension-lorentz-violation.py: the $2\times2$ determinant, its factorisation, $\omega^2_\pm = k(k \pm k_{AF}^0)$ and $k_+ - k_- = -k_{AF}^0$ are all derived symbolically with sympy from the Lagrangian (1); $\hbar H_0$ , the ratio $0.80$ and the Hubble length are recomputed at 50 digits.*

Go deeper · Derivation B · 2 minWhy there are exactly nineteen dimensionless photon coefficients and why nine of them belong to the laboratory

Derivation BWhy there are exactly nineteen dimensionless photon coefficients and why nine of them belong to the laboratory

The number of independent components of $(k_F)_{\kappa\lambda\mu\nu}$ is not a matter of taste. Getting it right tells you — with no further physics, and before a single measurement — exactly how the experimental programme must divide itself between the sky and the laboratory.

Step 1 — the symmetries. $(k_F)$ multiplies $F^{\kappa\lambda}F^{\mu\nu}$ , so only the part with the symmetries of the Riemann tensor survives: antisymmetric in $[\kappa\lambda]$ and in $[\mu\nu]$ , symmetric under $(\kappa\lambda)\leftrightarrow(\mu\nu)$ , and obeying the cyclic identity. In $D$ dimensions such a tensor has

N(D) = \frac{D^{2}(D^{2}-1)}{12}, \qquad N(4) = \frac{16 \times 15}{12} = \mathbf{20}.

Why: a pure counting fact about tensors, with nothing to do with electrodynamics. It is the same 20 that counts the Riemann tensor in four-dimensional general relativity.

Step 2 — remove the double trace. The totally traced part $(k_F)^{\kappa\lambda}{}_{\kappa\lambda}$ multiplies $F_{\mu\nu}F^{\mu\nu}$ — a rescaling of the Maxwell term, absorbable into a redefinition of the field and the charge. It is unobservable, so it is imposed to vanish:

20 - 1 = \mathbf{19}\ \text{independent components}.

Why: a coefficient that can be removed by a field redefinition is not a coefficient. Leave it in and you will "measure" a quantity no experiment can distinguish from a change of units.

Step 3 — repackage into $3\times3$ blocks. Following Kostelecký & Mewes (Phys. Rev. D 66, 056005 (2002)), define $(\kappa_{DE})^{jk} = -2(k_F)^{0j0k}$ , $(\kappa_{HB})^{jk} = \tfrac12\epsilon^{jpq}\epsilon^{krs}(k_F)^{pqrs}$ , $(\kappa_{DB})^{jk} = -(\kappa_{HE})^{kj} = (k_F)^{0jpq}\epsilon^{kpq}$ . Then $\kappa_{DE}$ , $\kappa_{HB}$ are symmetric with $\mathrm{tr}(\kappa_{DE} + \kappa_{HB}) = 0$ (the double-trace condition), and $\kappa_{DB}$ is traceless (the cyclic identity):

\underbrace{6 + 6 - 1}_{\kappa_{DE},\ \kappa_{HB}} \;+\; \underbrace{9 - 1}_{\kappa_{DB}} \;=\; 11 + 8 = \mathbf{19} \;\checkmark

Why: the count survives the repackaging — the check that the repackaging is right. These four matrices put the modified Maxwell equations into the exact form of electrodynamics in an anisotropic medium: the vacuum acquires a permittivity and a permeability tensor, which is what makes the photon sector calculable with nineteenth-century crystal optics.

Step 4 — the irreducible decomposition, and the split that matters. Form the parity-even and parity-odd combinations:

\tilde\kappa_{e+} = \tfrac12(\kappa_{DE} + \kappa_{HB}),\quad \tilde\kappa_{e-} = \tfrac12(\kappa_{DE} - \kappa_{HB}) - \tfrac13\delta\,\mathrm{tr}\,\kappa_{DE},\quad \tilde\kappa_{o\pm} = \tfrac12(\kappa_{DB} \pm \kappa_{HE}),\quad \tilde\kappa_{\mathrm{tr}} = \tfrac13\mathrm{tr}\,\kappa_{DE}.

$\tilde\kappa_{e+}$ , $\tilde\kappa_{e-}$ and $\tilde\kappa_{o-}$ are symmetric traceless (5 each); $\tilde\kappa_{o+}$ is antisymmetric (3); $\tilde\kappa_{\mathrm{tr}}$ is a scalar (1):

5 + 5 + 3 + 5 + 1 = \mathbf{19}\ \checkmark

Why: $\tilde\kappa_{o+} = \tfrac12(\kappa_{DB} - \kappa_{DB}^{\mathsf T})$ is antisymmetric, hence 3, while $\tilde\kappa_{o-} = \tfrac12(\kappa_{DB} + \kappa_{DB}^{\mathsf T})$ is symmetric traceless, hence 5. Two independent counts of 19 — the Riemann formula and this decomposition — prove nothing has been lost or double-counted.

Step 5 — birefringence selects ten, and hands the laboratory nine. Kostelecký and Mewes show the birefringence of the vacuum is controlled by $\tilde\kappa_{e+}$ and $\tilde\kappa_{o-}$ alone — $5 + 5 = \mathbf{10}$ . The remaining

19 - 10 \;=\; \tilde\kappa_{e-}\,(5) \;+\; \tilde\kappa_{o+}\,(3) \;+\; \tilde\kappa_{\mathrm{tr}}\,(1) \;=\; \mathbf{9}

produce no birefringence, so no amount of polarimetry will ever see them. Why: this is where the arithmetic becomes an experimental programme. The ten birefringent coefficients are bounded to parts in $10^{37}$ by gamma-ray-burst polarimetry (Kostelecký & Mewes, Phys. Rev. Lett. 97, 140401 (2006)) — a bound no terrestrial apparatus can approach. The nine others are invisible to the sky and must be measured in a laboratory. So when Nagel et al. (2015) set "comprehensive simultaneous bounds on nine boost and rotation anisotropies of the speed of light", they are not choosing a round number. They are reporting $19 - 10$ . *Verified in verify_aq-sr-standard-model-extension-lorentz-violation.py: $N(4) = 20$ from $D^2(D^2-1)/12$ ; the component counts $6+6-1 = 11$ , $9-1 = 8$ , and $5+5+3+5+1 = 19$ ; and the closure $19 - 10 = 9$ , asserted against the number nine as printed in the article and in the experiment's abstract.*

Go deeper · Derivation C · 2 minWhat a bound of ten-to-the-minus-eighteen is actually worth

Derivation CWhat a bound of ten-to-the-minus-eighteen is actually worth

Every review says Lorentz invariance is tested "to Planck-suppressed levels." The claim is checkable — and it is true of some coefficients and false of others. Here is the yardstick, and the honest score.

Step 1 — build the yardstick. If Lorentz violation originates at the Planck scale, a dimensionless coefficient in the low-energy effective theory is naturally of order $(m/M_{\mathrm{Pl}})^n$ , with $m$ the mass scale of the sector and $n \ge 1$ the powers of suppression. With $M_{\mathrm{Pl}} = 1.220890\times10^{19}\ \mathrm{GeV}$ :

\frac{m_p}{M_{\mathrm{Pl}}} = \frac{0.938272\ \mathrm{GeV}}{1.220890\times10^{19}\ \mathrm{GeV}} = \mathbf{7.69\times10^{-20}}, \qquad \left(\frac{m_p}{M_{\mathrm{Pl}}}\right)^{2} = \mathbf{5.91\times10^{-39}} .

\frac{m_e}{M_{\mathrm{Pl}}} = \mathbf{4.19\times10^{-23}} .

Why: these are the only numbers the phrase "Planck-suppressed" can possibly mean. First order in the proton sector is $10^{-19}$ ; second order is $10^{-39}$ . Everything else is rhetoric.

Step 2 — score the laboratory. Nagel et al. (Nat. Commun. 6, 8174 (2015)) bound orientation-dependent frequency changes at $\Delta\nu/\nu = (9.2 \pm 10.7)\times10^{-19}$ at 95% confidence — an interval of $[-1.5,\ 19.9]\times10^{-19}$ , comfortably containing zero, so a magnitude bound of about $2\times10^{-18}$ . Against the yardstick:

\frac{2\times10^{-18}}{7.69\times10^{-20}} = \mathbf{26} .

Why: the best Michelson–Morley experiment ever performed has not reached first-order Planck suppression. It is a factor of 26 away — the exact opposite of the impression "tested to $10^{-18}$ " creates, and the strongest possible argument for building the next one.

Step 3 — score the sky. The birefringent coefficients are bounded to parts in $10^{37}$ . Against second-order Planck suppression in the proton sector:

\frac{10^{-37}}{5.91\times10^{-39}} = \mathbf{17} .

Why: the sky has all but reached second order. Not first — second, a factor of 17 from $(m_p/M_{\mathrm{Pl}})^2$ . The single most stringent test of a symmetry anywhere in physics, done with a polarimeter and two gamma-ray bursts.

Step 4 — the moral, and the trap. These two bounds are not comparable, because they constrain different coefficients. Nineteen orders of magnitude separate them and neither subsumes the other: the sky cannot see the nine, the laboratory cannot reach the ten. Why: this is the whole reason the SME exists. RMS had three dials and therefore three numbers; you could ask it "how well?" and get a reply. The SME has hundreds, and the correct reply to "how well?" is "which one?" That is not a weakness of the framework — it is the discovery that the question was underdetermined all along. *Verified in verify_aq-sr-standard-model-extension-lorentz-violation.py: $m_p/M_{\mathrm{Pl}}$ , $m_e/M_{\mathrm{Pl}}$ , the square, and both ratios (26 and 17) are recomputed at 50 digits from PDG masses and grepped out of this article.*

Figure

The honest scoreboard. Every bar is a bound on a DIFFERENT coefficient, which is exactly the point — the bounds are not comparable, and no single number answers "how well have we tested relativity". The dashed lines are the two yardsticks that "Planck-suppressed" can mean: first order, m_p/M_Planck = 7.69e-20, and second order, (m_p/M_Planck)^2 = 5.91e-39. The laboratory's best Michelson-Morley result, 2e-18, is a factor of 26 SHORT of first order — it has not got there. The gamma-ray-burst birefringence bound, parts in 10^37, is within a factor of 17 of SECOND order. The sky is nineteen orders of magnitude ahead of the laboratory, and it still cannot see the nine coefficients the laboratory owns

Common trapCPT violation implies Lorentz violation. The converse is FALSE — and this is the most-misquoted theorem in the subject

Greenberg's result (Phys. Rev. Lett. 89, 231602 (2002)) is a one-way implication: "An interacting theory that violates CPT invariance necessarily violates Lorentz invariance." It is routinely restated as if CPT and Lorentz symmetry stood or fell together. They do not, and Greenberg's own abstract says so in the very next sentence: "On the other hand, CPT invariance is not sufficient for out-of-cone Lorentz invariance." A theory can be perfectly CPT-invariant and still violate Lorentz symmetry — the entire CPT-even sector, all nineteen photon coefficients, is exactly that. Get it backwards and you will conclude that the $10^{-18}$ cavity bound is a CPT test. It is not. The theorem's real payload is why one framework has to cover both: you cannot break CPT in a local interacting theory without also breaking Lorentz symmetry, so a "CPT extension" separate from a Lorentz extension is not a thing that exists.

Worked examples

1 · Reconstruct the Carroll–Field–Jackiw bound from nothing but the size of the universe

The published bound is $\lvert k_{AF}\rvert < 1.7\times10^{-42}h_0\ \mathrm{GeV}$ . Get it without a telescope.

Step 1 — identify. From (4), $\Delta\phi = \tfrac12 k_{AF}^0 L$ , with no frequency dependence. Distant radio galaxies do not show their polarization scrambled relative to their own structure, so $\Delta\phi \lesssim 1$ radian over the longest available path.

Step 2 — the longest path available. $L_H = c/H_0$ , with $H_0 = 100\,h_0\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ and $1\ \mathrm{Mpc} = 3.0857\times10^{22}\ \mathrm{m}$ :

H_0 = 3.2408\times10^{-18}\,h_0\ \mathrm{s^{-1}}, \qquad L_H = \frac{2.998\times10^{8}}{3.2408\times10^{-18}h_0} = 9.25\times10^{25}\,h_0^{-1}\ \mathrm{m} .

Step 3 — convert to an energy. In natural units $1/L_H = \hbar H_0$ . With $\hbar = 6.5821\times10^{-25}\ \mathrm{GeV\,s}$ :

\hbar H_0 = (6.5821\times10^{-25})(3.2408\times10^{-18}h_0) = \mathbf{2.13\times10^{-42}}\,h_0\ \mathrm{GeV} .

So $\Delta\phi \lesssim 1$ gives $k_{AF}^0 \lesssim 2/L_H = 2\hbar H_0 = \mathbf{4.3\times10^{-42}}h_0\ \mathrm{GeV}$ .

Step 4 — the boundary number. The published bound is $1.7\times10^{-42}h_0\ \mathrm{GeV} = \mathbf{0.80\,\hbar H_0}$ . Our estimate is high by a factor of $2.5$ — exactly what a real statistical analysis buys you over "one radian, one Hubble length." The tightest constraint on a mass parameter in all of physics is the Hubble energy, because the apparatus is the universe, and an apparatus cannot beat its own size.

Physics of the portrait

Origins · 2 minEffective field theory and how this was discovered

Effective field theory

1989–2026 · Kostelecky and Samuel find tensor vacuum values in string theory · Colladay and Kostelecky write down every term · the Data Tables make it an experimental science

Why a method and not a person. Effective field theory says you do not need the true theory to make precise statements about it: write down every operator your symmetries permit, order them by dimension, and let the data set the coefficients — including, decisively, to zero. The SME's move was to notice that Lorentz invariance is a symmetry like any other. Kostelecký and Samuel, 1989. In Phys. Rev. D 39, 683 they showed, in covariant string field theory, that a string vacuum can spontaneously acquire a tensor expectation value. This is the seed of the subject: it made Lorentz violation a prediction of something rather than a hypothesis of nothing. Colladay and Kostelecký, 1997–1998. Phys. Rev. D 55, 6760 and 58, 116002 construct the extension itself — every Lorentz- and CPT-violating term that can be added to the Standard Model while preserving gauge invariance, energy–momentum conservation and (minimally) renormalizability. Its value is exactly its completeness: a null result in a complete framework is a theorem about the world; a null result in an incomplete one is an anecdote. Kostelecký and Russell, 2008–. The Data Tables for Lorentz and CPT Violation (arXiv:0801.0287; Rev. Mod. Phys. 83, 11 (2011)) are updated every January; the edition read for this article is v19, February 2026. They are the reason this is a science and not a genre: they make "relativity has been tested" auditable, coefficient by coefficient.

History

1989

— Kostelecky and Samuel (Phys. Rev. D 39, 683) show that the vacuum in string field theory can spontaneously acquire a TENSOR expectation value, breaking Lorentz symmetry the way the Higgs breaks gauge symmetry.

1990

— Carroll, Field and Jackiw (Phys. Rev. D 41, 1231) bound the CPT-odd photon coefficient using the polarization of distant radio galaxies: |k_AF| < 1.7e-42 h0 GeV.

1997

— Colladay and Kostelecky (Phys. Rev. D 55, 6760), "CPT violation and the standard model", write the first version of the extension.

1998

— Colladay and Kostelecky (Phys. Rev. D 58, 116002), "Lorentz-violating extension of the standard model", complete it. This is the SME.

2002

— Greenberg (Phys. Rev. Lett. 89, 231602) proves that CPT violation implies Lorentz violation in any interacting theory, and states explicitly that the converse does not hold.

2002

— Kostelecky and Mewes (Phys. Rev. D 66, 056005) work out the photon sector: 19 dimensionless coefficients, 10 birefringent, bounds of 2e-32 from cosmological polarimetry and 3e-16 from dispersion.

2006

— Kostelecky and Mewes (Phys. Rev. Lett. 97, 140401) use gamma-ray-burst polarimetry to push the birefringent coefficients below parts in 10^37 — an improvement by a factor of 100,000.

2008

— Kostelecky and Russell begin the Data Tables for Lorentz and CPT Violation (arXiv:0801.0287; Rev. Mod. Phys. 83, 11 (2011)), updated every January. The edition read here is v19, February 2026.

2011

— Smiciklas, Brown, Cheuk, Smullin and Romalis (Phys. Rev. Lett. 107, 171604) bound the anisotropy of the neutron's maximum attainable velocity at the 1e-29 level with a Ne-21/Rb/K comagnetometer.

2015

— Nagel et al. (Nat. Commun. 6, 8174) bound orientation-dependent frequency changes at (9.2 +/- 10.7)e-19 and report simultaneous bounds on NINE anisotropies of the speed of light.

Going further

Primary sources. Kostelecký & Samuel, Phys. Rev. D 39, 683 (1989) · Carroll, Field & Jackiw, Phys. Rev. D 41, 1231 (1990) · Colladay & Kostelecký, Phys. Rev. D 55, 6760 (1997) and 58, 116002 (1998) · Greenberg, Phys. Rev. Lett. 89, 231602 (2002) · Kostelecký & Mewes, Phys. Rev. D 66, 056005 (2002) and Phys. Rev. Lett. 97, 140401 (2006) · Smiciklas et al., Phys. Rev. Lett. 107, 171604 (2011) · Nagel et al., Nat. Commun. 6, 8174 (2015). The tables: Kostelecký & Russell, Data Tables for Lorentz and CPT Violation, Rev. Mod. Phys. 83, 11 (2011); arXiv:0801.0287, edition read: v19, February 2026.

Provenance. Signature $(+,-,-,-)$ . Every equation here is re-derived from the Lagrangian (1) and every number recomputed in verify_aq-sr-standard-model-extension-lorentz-violation.py, which reads this file, regexes the numbers out of the prose and asserts them. Numbers whose primary source is an experiment — $1.7\times10^{-42}h_0$ GeV, parts in $10^{37}$ , $2\times10^{-32}$ , $3\times10^{-16}$ , $10^{-29}$ , $(9.2 \pm 10.7)\times10^{-19}$ — are quoted from the papers cited above and nowhere else. No bound here is taken from a secondary source.

Cross-corpus consistency. The three-dial kinematic framework this one supersedes is `aq-sr-robertson-mansouri-sexl-framework`; the isotropy experiments whose nine coefficients are counted here are the subject of `aq-sr-michelson-morley-isotropy-tests`; the photon's own mass — a different deformation of electrodynamics, and a different Lagrangian — is `aq-sr-photon-mass-light-speed-tests`.

Builds on. Lorentz Invariance · Robertson–Mansouri–Sexl · Michelson–Morley and Modern Isotropy Tests. Leads to. Tests of the Photon Mass · Tachyons and Superluminal Particles.

If special relativity is exactly right, why bother building a framework in which it might not be?

Because "special relativity is exactly right" is not a statement any experiment can make. Experiments produce bounds, and a bound needs a parameter. Without the SME, the honest summary of a century of null results is a shrug; with it, it is a table of several hundred numbers, each a coefficient consistent with zero and each with an error bar you can shrink. The framework converts "we looked and saw nothing" into a quantitative statement about how hard we looked — and into a map of where nobody has looked yet.

Why is the SME's photon sector so much better constrained than the electron's or the proton's?

Because of one accident of geometry: photons travel a very long way. Birefringence is a phase effect, and phase accumulates over the whole path — so a gamma-ray burst at cosmological distance delivers a lever arm of (path)/(wavelength) $\sim 10^{26}\,\mathrm{m} / 10^{-13}\,\mathrm{m} \sim 10^{39}$ . Electrons and protons in a laboratory are not free to travel for ten billion years. This says nothing about the relative importance of the sectors, and emphatically does not mean the photon sector is "solved": the nine non-birefringent coefficients enjoy none of that leverage and are bounded a mere $10^{-18}$ . The sky is astonishing at some things and blind to others, which is why the laboratory programme did not stop in 2006.

Isn't the SME just the RMS framework with more parameters?

No — the difference is structural. RMS is kinematic: it postulates a transformation between frames and assumes all matter and light respond to it identically. It cannot even formulate the question "does the electron feel a preferred direction that the photon does not?" The SME is dynamical — a field theory, so each field carries its own coefficients, and the answer is a set of numbers you can go and measure. The price is a large parameter space; the reward is that the SME can express what a realistic Planck-scale scenario actually predicts, species by species, and RMS cannot. RMS is the shadow the SME casts when you insist that every particle be treated the same and that space be isotropic.

What does "Planck-suppressed" actually mean, numerically?

It means a dimensionless coefficient of order $(m/M_{\mathrm{Pl}})^n$ , where $m$ is the mass scale of the sector and $n$ counts the powers of suppression. With $M_{\mathrm{Pl}} = 1.220890\times10^{19}$ GeV: first order in the proton sector is $m_p/M_{\mathrm{Pl}} = 7.69\times10^{-20}$ , and second order is $5.91\times10^{-39}$ . Now compare. The best cavity bound, $2\times10^{-18}$ , is a factor of 26 above first order — the laboratory has not yet reached the Planck scale. The gamma-ray birefringence bound, parts in $10^{37}$ , is within a factor of 17 of second order. Reviews that say "Lorentz invariance is tested at Planck-suppressed levels" are describing the second of these and being read as though they described the first.

Why can't Lorentz violation just be a small explicit term added to the Lagrangian, with no deeper origin?

In flat spacetime it can, and that is what the minimal SME writes down. The trouble arrives when gravity does. Explicit Lorentz violation — a fixed background tensor simply declared — is inconsistent in a dynamical spacetime: the Bianchi identities plus covariant conservation of the energy–momentum tensor force conditions a non-dynamical background generically cannot satisfy. The way out is spontaneous breaking, where the background tensor is the vacuum expectation value of a dynamical field, exactly as in Kostelecký and Samuel's string mechanism. This is why the SME's gravity sector is so much harder than its flat-space sector, and why its bounds are so much weaker. The place where Lorentz violation is most natural theoretically is the place where it is hardest to look for.

If every coefficient measured so far is zero, has the programme failed?

It has done the only thing a test of a symmetry can do: drawn the boundary of the possible, and drawn it where it eliminates real theories. Parts in $10^{37}$ is not "no news" — it is a fact any candidate theory of quantum gravity must now reproduce, and it has already killed models that would otherwise be viable. And the map has holes: the nine non-birefringent photon coefficients are bounded only at $10^{-18}$ , a factor of 26 short of even first-order Planck suppression; the gravity sector is weaker still; the nonminimal, energy-growing operators are barely touched. A null result that names its own blind spots is not a failure. It is a research programme.