# B Uncertainty Principle - uncertainty in mass.

1. Sep 20, 2016

### hankaaron

Physicists often speak of the uncertainty principle in terms of position and velocity (in momentum). But momentum is Mass X Velocity. Why is it that we can say something like "a particle with a well defined position has high uncertainty in velocity". But I never hear a statement like "a particle with a well defined position has high uncertainty in mass"?

Is it perhaps because position and velocity are conjugate variables while position and mass are not?

2. Sep 20, 2016

### Staff: Mentor

Mass is a scalar so has exactly the same value every measurement.

Thanks
Bill

3. Sep 21, 2016

### vanhees71

That's a deep question :-). From a modern point of view, if there is a single concept in the development of modern theoretical physics, then it's symmetry. Here we refer to space-time symmetry. For non-relativistic physics the space-time symmetry is Galilei symmetry, i.e., the independence of physics on the place it happens, the time it happens (homogeneity or translation invariance in space and time), how your experiment is oriented in space (isotropy of space or rotation invariance) and of whether it's observed in one inertial frame of another (invariance under Galilei boosts).

Now, if quantum theory should describe processes in this specific space-time you should make sure that you somehow realize this fundamental Galilei symmetry, and it turns out that quantum theory is as if made for such ideas. There's a mathematical theory that investigates the question how such symmetries (which build an algebraic structure of a group) are realized in the mathematical formalism of quantum theory (socalled linear representations of groups on vector spaces). Now you can analyze the Galilei group, and it turns out that mass enters as a parameter of any allowed realization of this group in quantum theory. Now you can in a sense "define" an elementary particle to be described by a special class of such representations, the socalled irreducible representations. These can all be constructed from the formalism, and mass is one parameter that characterizes such a representation and in that sense the elementary particle whose behavior is described by this representation.

Another, more abstract, quantity characterizing an irreducible representation is the spin. It is a kind of intrinsic angular momentum of an elementary particle like an electron which has spin 1/2. It's related to the magnetic moment of the electron, but it's hard to describe it in any classical terms without getting it wrong. Nevertheless from the point of view of space-time symmetries the elementary particles are characterized by their mass and their spin, and these are thus fixed values for any particle.

There are more parameters characterizing the particles. These are defined by the fundamental interactions in the sense of charges. One most familiar of those is electric charge, which governs the strength of the electromagnetic interaction between particles, and it is strictly conserved in reactions. Also these charges have a foundation in symmetry principles governing the Standard Model of elementary particle physics, but that's another story.

For an excellent treatment of symmetries on the popular-science level, see

L. Ledermann, D. Teresi, The God Particle

Don't mind the stupid title. It's a really excellent book and last but not least fun to read.

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