Re: [Phys-L] Half-Life measurement

[David Bowman <David_Bowman@georgetowncollege.edu>]

JSD mentioned something that triggered me to discuss a related tangent, although I'm certain he is fully aware of it, it is possible that others here might not be.

> There is no need to question it, because for present purposes
> it's immaterial. If all you are interested in are the half-life
> values, you could shift every time in the data set by some
> arbitrary amount and the results would be the same.
>
> The fit only cares about the number of events in a given
> interval, and the length of the interval is invariant with
> respect to a time shift. Both ends of the interval get shifted.
> This is an example of gauge invariance.
>
> In the exponentials, shifting the time changes the prefactor out
> front, but doesn't affect the shape of the curve or the
> halflife.

The tangent relates to a common cause of a translational invariance in some physical variable (e.g. time).  If a phenomenon is described by a differential equation wherein the independent variable of that equation does not itself appear explicitly in it, then the solution set of that DE exhibits a translational gauge invariance in that independent variable, and (one of) the integration constant(s) from the integration of that DE is an arbitrary translational offset the value of that independent variable, whose precise value would be determined by the initial and/or boundary conditions for the problem.  (In addition, if the DE happens to be itself an Euler-Lagrange equation resulting from the extremization of an action functional, then Noether's Theorem guarantees the existence of a function, e.g. energy, of the solution of the DE whose value remains constant over the independent parameter's domain for which the DE is applicable.)

For instance, in the case of Newton's 2nd law of classical mechanics, if the generalized forces acting on a system (& masses) do not have any explicit time dependence, i.e. they depend only on the current positions & velocities of the coordinates, then the system's solution set has a built-in time invariance so that the choice of when to start one's clock is left up to the observer and the chosen initial conditions, rather than having it being dictated by the DE.  In addition, if those generalized forces are gradients of a potential (or at least don't do any non-conservative work) then the system also has a conserved energy function.

Dave Bowman