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The Question

(Submitted October 05, 2000)

I was taught that the Uncertainty Principle disallows the measurement of any quantity exactly. I was also taught that this property manifests itself in all of nature and not just in our lack of measuring sophistication.

On the other hand, Einstein's Theory of Relativity uses quantities such as mass, energy and time as quantities of exact definition (certainties) and not as statistical averages of the probabilities given by the Quantum Theory.

My question is, while at our human level of observation, the uncertainty in particle velocity/position is extremely negligible, wouldn't these uncertainties have fundamental effects at relativistic frames of references?

Surely the probabilistic nature of these quantities (mass, energy, time) would have fundamental impact when dealing with Relativity!

The Answer

That is a very good question.

First we should probably clarify the Uncertainty Principle (UP). This has to do with the possibility of measuring "conjugate pairs" of variables simultaneously --- these include momentum and position, time and energy --- any pair whose product has the dimensions of "action" (the dimensions of Planck's constant), as such measuring mass doesn't fall under the UP.

You are correct, quantum mechanics (as opposed to quantum field theory) is an instantaneous model, which violates causality. If you try to make it "covariant," even in flat space-time, you quickly run into the need to introduce additional particles and pair creation/annihilation to take up negative-energy states. If you don't, you find that your theory predicts every particle is an infinite source of energy and there are no ground states.

The "standard" quantum field theory (QFT) approach is an attempt to fix this problem. You can also apply QFT in curved space-time, but that isn't often necessary, except near black holes. What you can't do (yet) is quantum gravity, where space-time isn't just curved, but has spin-2 particle carriers.

In situations where both general Relativity and quantum uncertainty are important (e.g., the Big Bang before the Planck time), important things that we do not understand completely must be happening. E.g., we don't know the initial conditions for inflation, and have to assume something "generic" and make appeals to Occam's Razor.

Hope this helps,

Michael Gross, Mark Kowitt and Michael Arida
for Ask an Astrophysicist

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