Can Physical Laws Be Truly Independent?

[simoncks]

There is a famous example that electron couldn't absorb the whole incoming photon without emitting another one. Instead of the normal way, I try to prove it simply by argument ( which might be wrong ).
There are four constraints in the process, one from energy conservation, three from momentum. If only electron is left after so, there are only three variables (momentum in different directions, energy could be derived from the three variables). Further 'assume' the constraints are all independent, and (Corollary)
Given there are n independent constraints with m variables, if m < n, there will be no solution.

The photon-all-absorbed configuration doesn't have enough variables, thus impossible to exist.

Questions to raise :
1. Is the proof fine? Limit it to at least the case of the photon absorption first.
2. Are the physical laws, especially the energy-momentum conservation, always independent? If not, any example?
3. Is the corollary true?

Thank you.

 

[voko]

Say the electron has initial momentum (x, y, z), which corresponds to electron's energy e(x, y, z). Ditto for the photon: (a, b, c) -> f(a, b, c). Suppose the electron absorbs the photon entirely and so its momentum is now (X, Y, Z) = (x + a, y + b, z + c), and its energy is e(X, Y, Z) = e(x + a, y + b, z + c) = e(x, y, z) + f(a, b, c). So we end up with this equation:

e(x + a, y + b, z + c) = e(x, y, z) + f(a, b, c)

Can we state, without looking at the details of functions e() and f() that the equation above has no solutions?

 

[jbriggs444]

Questions to raise :
1. Is the proof fine? Limit it to at least the case of the photon absorption first.

The proof assumes without justification that the constraints are independent.

2. Are the physical laws, especially the energy-momentum conservation, always independent? If not, any example?

As a matter of economy, we would not usually want more physical laws than we need. But it's not a rule.

3. Is the corollary true?

For a system of linear equations, yes. If you have n independent equations in n unknowns, an n+1'st equation will either be a linear combination of them or will be inconsistent with them.

There is no need to go this deep to obtain a proof, however. Just adopt a frame of reference in which the electron ends at rest.

 

[ShayanJ]

1- The proof isn't fine!
Before impact, we have energy of the photon and its momentum components and energy of the electron and its momentum components. Here also you can derive each particle's energy from its momentum, but because you're going to use it in conservation of energy, you should have it anyway. So you have four equations of constraint and four quantities.
The reason you can't have the process \gamma+e \rightarrow e, is that in the centre of mass frame of reference, the net momentum is zero before impact, and because the photon-electron system is isolated, the net momentum of the system is conserved and so it should be zero after the impact too. But if there is only one particle left after the impact, it should be at rest w.r.t. the centre of mass otherwise the net momentum won't be zero after the impact. So what happened to the kinetic energy?!

2- Independence of physical laws means that they're not each others' consequences. Well, some physical laws are consequences of other laws, that's for sure. But energy conservation and momentum conservation are independent from each other.

3- That's not a corollary of your arguments. That's a known mathematical fact that over-determined linear algebraic systems, have no solutions.(I don't know about non-linear and non-algebraic systems.)