[varsigma]

I'm sorry, I'm not sure what that means.   Time varies linearly with the number of seconds that have passed rather than having some quadratic dependence?   Time is an operator that is linear - time acts on one girl to make an old lady,  so it acts on  λ girls to make λ old ladies?   I suppose all those things are true of time.

It's true in any quantum experiment, quantum computing is experimental at the moment. It's algebraically true, or a consequence of any formal system we use to describe what we're doing, and what particles are doing. It seems to be a law of quantum information-bearing systems of particles.

It seems quite prosaic, after all, time is linear in ordinary mechanics, or mechanical clocks wouldn't work. Clocks "linearise" time. There are a lot of assumptions built into clocks, though.

[evan_au]

Clocks "linearise" time.

Most clocks also quantise time.
- "Grandfather" clocks used a pendulum which swung at 1Hz, to activate the hands on the clockface.
- Mechanical wristwatches used a spring and escapement mechanism to drive the hands on the wristwatch.
- Early wristwatches & today's cheap wall clocks use a crystal vibrating at 32,768Hz, which they divide using 15 flip-flops to 1Hz, driving the display mechanism
- Atomic clocks use a Cesium 133 resonance at 9,192,631,770Hz, which they divide down to 1Hz.

Water clocks are pretty analog, but in the details, they are also quantised at the level of water molecules...

[varsigma]

Perhaps time is just what lets observers remember a stream of clock ticks. If, that is, there's a clock ticking near them.

There, the important concept I think, is difference. Whereas photons "see" no difference between any moments of time, matter does. Entropy is based on differences in some system of particles.

Whether you can say you have a system of different particles (or with different positions) is also a question that is meaningful. Go figure.

The big, um difference, between classical systems such as ticking grandfather clocks, and "ticking" particles is the particles are almost in a different kind of universe--distances between individual atoms might not make sense, particles can tunnel through barriers, it's just weird.

[varsigma]

Ok. I do have sort of an answer, but it isn't an easy one.

It's a humorous take not a serious attempt, so please excuse any mistakes.

Righty ho, what is mathematics . . .

Start with some definitions then construct some propositions,
Make commuting diagrams, obtain the proper theorems,
Exercise an algebra, quotient some polynomials,
Use the automorphisms to generate identities,
Restrict the tensor product space to extend linearities
And find the ideal radicals in strict monoidal categories.

And we are done

oops, forgot that you might need to ensure the convolution is a closed sum of monomials.