# Best current bounds on nonconservation of momentum?

It's not straightforward to test conservation of momentum experimentally, and many experiments that seem like tests really aren't. For example, in a Newtonian system of identical particles that interact through collisions, conservation of momentum follows simply from Galilean invariance and the symmetry of the collisions in the center of mass frame. For reasons such as these, many freshman-physics tests of conservation of momentum may not actually test it, even approximately -- that is, they may not present even the logical possibility of falsifying the conservation law.

An added problem is that both of the fundamental theories of physics, quantum mechanics and general relativity, have local conservation of the energy-momentum four-vector pretty thoroughly embedded in their structures. In general, it's hard to test a hypothesis unless you have a test theory that is consistent with the failure of the hypothesis.

In the case of GR, we have the PPN formalism, which, although not really a scientific theory, does allow for nonconservation of momentum. The best test I know of within this framework is a lunar laser ranging experiment by Bartlett (1986), which verified the equality of active and passive gravitational mass to a precision of about $10^{-10}$. The validity of this test depended on the inhomogeneity of the moon -- otherwise, for reasons similar to those described in the first paragraph, an anomalous acceleration is forbidden by symmetry. More recent observations of pulsars constrain the momentum-nonconserving PPN parameter $|\alpha_3|$ to be less than $5\times10^{-16}$ (Bell 1995).

What about tests at the microscopic scale, in the electromagnetic sector? Of course it's hard to imagine theoretically how conservation of momentum could fail, since it seems to follow directly from translation invariance and Noether's theorem. But this isn't the same thing as verifying it experimentally.

Have there been nongravitational tests at the macroscopic scale, e.g., empirical limits on spontaneous self-accelerations of inhomogeneous kilogram-scale masses? (Seems like the kind of thing the Eot-Wash group would do.)

The interpretation of this kind of thing depends on whether or not your test theory allows Lorentz violation. E.g., the PPN formalism explicitly allows for both momentum nonconservation and Lorentz violation. If Lorentz invariance is valid, then any test of conservation of energy is also a test of conservation of momentum. So there might be one bound on nonconservation of momentum if you don't assume Lorentz invariance, and some other, tighter bound if you do.

[EDIT] It seems like the atomic-physics tests are conventionally described as tests of local position invariance (LPI), although by Noether's theorem that's equivalent to conservation of energy-momentum. The highest-precision experiments compare the rates of atomic clocks of different atomic species and watch for a variation in the ratio of their rates over time. One can also test for universality of gravitational redshifts, or compare atomic clocks (microscopic) with electromagnetic resonators (macroscopic). Some recent papers are Guéna 2012 and Agachev 2010. When I asked the question, I hadn't hit upon the right search term to turn up these experiments. I'd still be interested in seeing a synoptic answer, or one that touched on the strong-force sector, or one that provided an interesting review of the history of such tests.

Bartlett and van Buren, Phys. Rev. Lett. 57 (1986) 21, summarized in Will, http://relativity.livingreviews.org/Articles/lrr-2006-3/

Bell, "A Tighter Constraint on post-Newtonian Gravity using Millisecond Pulsars," http://arxiv.org/abs/astro-ph/9507086

Guéna 2012, http://arxiv.org/abs/1205.4235

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I am puzzled by this question, since conservation of momentum is a basic tool in particle physics in order to study the interactions. For example in four constraint fits of interactions all the momenta are accounted for. In one constrained fits a neutrino or photon may be missing and is thus identified. If it did not hold the masses etc of all the standard model would be all over the place. Are you saying that the accuracy in particle physics is not enough for what you are looking for? –  anna v May 20 '13 at 15:33
@annav: Like every law of physics, it has to be tested by experiment. As with every law of physics, it may be only an approximation. For example, conservation of mass was tested by experiment starting in the 18th century and was found to be valid to some precision. Then in the 20th century we found out that it was nonconserved at a level too low to have been detected. –  Ben Crowell May 20 '13 at 20:27
I am trying to say that conservation of momentum and energy are continually experimentally tested in particle physics, because they are the lynch pin of mathematically analyzing the data. Another example, the mass of the Psi. pdg.lbl.gov/2012/listings/rpp2012-list-J-psi-1S.pdf Within less than MeV it has a very stable value and used as a gauge for the measurement errors in higher energy experiments. These numbers can be converted to momentum conservation estimates. That it was not done explicitly does not mean it is not an experimental error for momentum conservation too. –  anna v May 21 '13 at 3:25
Momentum conservation is linked to space translation invariance, and energy conservation is linked to time translation invariance. Have you a particle physics model with no spatial or no time translation invariance ? For Lorentz violation, if you mean that the relations are not linear, it is a non-sense. Linearity is mandatory : take one system S=(S1,S2), with S1 and S2 indepedent subsystems. The quantities, like momentum/energy are additive $P = P_1 +P_2$. But this does not depend on the observers, so you will have (if f is the transformation law) $f(P) = f(P_1 + P_2) = f(P_1) + f(P_2)$ –  Trimok May 31 '13 at 10:57
Maybe I'm missing something, but it looks to me like momentum conservation has attained the status of an axiom—if the calculated momentum is shown experimentally not to be conserved, then the definition of momentum, rather than its conservation, is changed. To break that, you'd need to show that no realistic theory could possibly have anything that behaves anything like momentum, in some subjective sense. I don't think that can be done by an experiment; rather a new theory lacking such a quantity would have to be supported by overwhelming evidence. –  dfeuer Sep 9 '13 at 23:58

Stability of the photon against decay to $e^+$ $e^-$ is only insured by conservation of momentum and the fact that $m_\gamma=0$. There are very good limits on the mass of the photon ($<10^{-19}$eV). If $E_\gamma > 2m_e$ all the conservation laws are fulfilled, even energy conservation can be satisfied but not momentum conservation. This decay can indeed take place in material where the momentum kick can be accommodated by other particles. So simple observation of photons of high energy coming from cosmological distance, like TeV photon from MRK 421 can be transformed into a very strong limit against momentum non conservation. Exact limit will depend on the particular theory.
Can you provide a reference for the bounds on $m_\gamma$? –  Emilio Pisanty Oct 1 '13 at 15:45
An easy to understand limits come from existence of very low energy photons. From E²-p²=m² we get m<E and E=hc/$\lambda$, if you see very long wavelength photon (km long radio waves exists) this immediately imply very low mass for photon. –  Nicolas Produit Oct 2 '13 at 7:57