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So I have been doing some research on different types of continous symmetries (mainly interested in continuous symmetry since I have been learning abit about Noethers theorem.) and understand the basic principles behind rotational transitional symmetry and space translational symmetry. I understand that time translational symmetry basically means that the laws of physics are unchanged but the problem I have is that the proof of it is that the physics laws are the same today as they were yesterday as they will be tommorow, and that is a poor proof as we do not have any conclusive way of stating that it will be the same in very far ahead in the future (say millions of years). Is there any experiment that has been conducted (or any good analogies people can think of) that shows that time translational symmetry is a thing.

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Is there any experiment that has been conducted (or any good analogies people can think of) that shows that time translational symmetry is a thing.

It is possible to examine the spectra of atoms and determine the fine structure constant, which is the quantity $e^2/\hbar c$ (in a system of units where the Coulomb constant equals 1). The fine structure constant is unitless, and equals approximately 1/137. Because we can observe the spectra of atoms in distant galaxies, whose light took billions of years to travel to us, we can determine whether the fine structure constant has changed over time. Measurements have put a very small upper limit on any changes in the fine structure constant.

The proof of it is that the physics laws are the same today as they were yesterday as they will be tommorow, and that is a poor proof as we do not have any conclusive way of stating that it will be the same in very far ahead in the future (say millions of years)

This seems trivially obvious to me. We can't observe the future, only the past. No physical principle can be directly verified as continuing to hold true into the future.

References

H. Chand et al., 2004, Astron. Astrophys. 417: 853, http://arxiv.org/abs/astro-ph/0401094 ; See also http://arxiv.org/abs/0711.1742 , http://arxiv.org/abs/0905.1516

Rosenband et al., 2008, 319 (5871): 1808-1812, http://www.sciencemag.org/content/319/5871/1808.abstract

Duff, 2002, "Comment on time-variation of fundamental constants," http://arxiv.org/abs/hep-th/0208093

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I'll try with a very simple example:

Imagine you are timing some event in an experimental setup, using a stopwatch. During one run of the experiment you discover to your dismay that you failed to "zero" the stopwatch before the run, and it had 15 seconds showing on it at the point when you began the next run. At the end of that run, it showed 30 seconds. What to do?

In the world we inhabit, what's important is the difference between what the stopwatch read at the end and the beginning of the run. The difference here is (30 - 15) or 15 seconds for the run in question. All the basic/fundamental laws of physics exhibit this characteristic, which is time translation symmetry, which states that the results we measure do not depend on how we happen to set our clocks. this is another way of saying that the laws of physics do not depend on what time we say it is.

In the classroom, you could perform an experiment once, set the wall clock ahead one hour, and immediately repeat the experiment for your students to demonstrate this.

As an interesting aside, note that if the behavior of a physical system is governed by laws which exhibit a certain symmetry, then there will be something in that system which is conserved during the process at hand. in the example of time-translation symmetry, we call that conserved quantity energy. Hence, the symmetry with respect to time of physical laws is the reason why energy is conserved in dynamical systems.

the deep connection between symmetry and conserved quantities was discovered by the mathematician Emmy Noether; the prescription by which one solves for the conserved quantity in terms of the symmetry is called Noether's Theorem.

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Not directly (humans didn't exist a million years ago), but indirect experimental evidence can be gotten from astronomical observations, such as how GR works for galaxies that are a few million light years away. We observe those galaxies as they were a few million years ago. If GR has changed over that timescale, then we would expect our current version of GR not to work for those galaxies.

This kind of observation indicates our current theories work all the way till just after the Big Bang. If the laws of physics are changing, they're changing very slowly.

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