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The uncertainty principle holds for pairs of certain observables, such as position and momentum. All these observables have a relation to spacetime. Other particle properties, by contrast, such as mass or electric charge can be measured at arbitrary precision. Quantum theories do not even model them as observables, but as parameters to the field.

So, what is the philosophical argument for why there is a uncertainty principle for some quantities, but not for others?

Qmechanic
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3 Answers3

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An uncertainty relation for mass occurs with clock variables (i.e., proper time $\tau$ as an observable) in the form

$$c^{2}\,\Delta m\,\Delta\tau\geq\frac{\hbar}{2}.$$

The problem is, with clock variables, you end up sacrificing positive-definitedness of mass, or some other desired property.

The classic paper with no-go theorems on quantum clock variables:

  • William G. Unruh and Robert M. Wald, "Time and the interpretation of canonical quantum gravity". Physics Review D 40 (1989) 2598 doi:10.1103/PhysRevD.40.2598

For the mass uncertainty relation, see, e.g.,

  • Shoju Kudaka, Shuichi Matsumoto, "Uncertainty principle for proper time and mass". Part I and Part II
Alex Nelson
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You are simply wrong on every point you are trying to make.

  1. If you really scrutinise things in modern physics, then there is no such thing as mass. What you really have, is the invariant or rest energy. When $c=1$, there is not even a unit difference between them.

Because of that, if you think a bit about it, the uncertainty relationship $$\tag1\Delta E\Delta t\geqslant\frac\hslash2\qquad\implies\qquad c^2\Delta m\Delta\tau\geqslant\frac\hslash2$$ which has a very pleasing interpretation: The proper time lifetime of a particle limits the possible determination of the invariant rest energy of the particle. So decays and resonant states are all energy-uncertain in a well-defined manner.

N.B. However, that the energy-time uncertainty relationship is controversial even within QM.

  1. The QFT charge operator is given by $$\tag2\hat Q=\int\frac{\mathrm d^3p}{(2\pi)^3}\hat b^\dagger_p\hat b_p-\hat c^\dagger_p\hat c_p$$ and its uncertainty is related to the number-phase uncertainty relation.

  2. You cannot ask some of these questions in basic QM because basic QM is a broken approximation working in fixed number representation, where some of these relationships are hidden because, by definition of fixed number, the uncertainty in there is zero, leading to complete indetermination of its conjugate variable. As is now obvious from QFT, once you let go of the basic QM restrictions, these apparent conjugate variables reappear.

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First, we are not philosophers. We compute math calculations to model esperimental observations.

In QFT charge and mass are different entities. Charge is an observable, the charge you refer to is the eigenvalue of the observable (operator). Mass is also observable but it's complicated.

You also don't understand the uncertainty principle.

You can measure a position as precise as you want. You can't measure position AND momentum together with arbitrary precision. If you don't care about momentum you can measure position as precise as you want.

The uncertainty principle then is used when you have a very special couple of observable to measure together. Those are like energy and time, momentum and position. Those are called conjugate variables and is the simultaneous measure of such a couple which has a restricted precision.

LolloBoldo
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