chiara_marletto's picture
Junior Research Fellow at Wolfson College and Postdoctoral Research Associate at the Materials Department, University of Oxford; Author, The Science of Can and Can't

The concept of "impossible" underlies all fundamental theories of physics; yet its exact meaning is little known. The impossibility of cloning, or copying, certain sets of states is at the heart of quantum theory and of quantum information. The impossibility of exceeding the speed of light is a fundamental feature of the geometry of spacetime in relativity. The impossibility of constructing perpetual motion machines is the core idea of thermodynamics: No machine can exist that produces energy without consuming any; and the second law demands the impossibility of machines converting "heat" (such as energy stored in the sea at room temperature) completely into "work" (energy that is useful, in that it can be used, for instance, to power a mill).

But what do we mean, exactly, by impossible?

The concept of impossible in physics is deep and has beautiful implications; it sharpens the everyday meaning of the word "impossible," giving to that airy nothing a solid, firm, deep connotation, rooted in the laws of physics. That something is impossible means that the laws of physics set a fundamental, draconian limit to how accurately it can be brought about.

For instance, one can construct real machines approximating a perpetual motion machine to some degree, but there is a fundamental limit to how well that can be done. This is because, since energy is conserved overall, the energy supplied by any such machine must come from somewhere else in the universe; and since there are no infinite sources of energy, perpetual motion is impossible: All finite sources eventually run out.

That something is impossible is deeply different from its not happening at all under the particular laws of motion and initial conditions of our universe. For example, it may be that, under the actual laws and initial conditions, an ice statue of the pirate Barbarossa will never arise in the whole history of our universe, and yet that statue need not be impossible. Unlike for perpetual motion machines, it might still be possible to create arbitrarily accurate approximate copies of such a statue under different initial conditions. The impossibilities I mentioned above, instead, are categorical: A perpetual motion machine cannot be brought about to arbitrarily high accuracy, under any laws of motion and initial conditions.

The exact physical meaning of the word "impossible" is illuminating also because it provides a deeper understanding of what is possible.

That something is possible means that the laws of physics set no limit to how well it can be approximated. For example, thermodynamics says that a heat engine is possible: We can come up with better and better ways of approaching the ideal behavior of the ideal engine, with higher and higher efficiencies, with no limitation to how well that can be achieved. Each realization will have a different design; they will employ different, ever improving technologies; and there is no limit to how much any given real heat engine can be improved upon.

So, once we know what is impossible under the laws of physics, we are left with plenty of room for our ideas to try and create approximations to things that are possible. This opening up of possibilities is the wonderful, unexpected implication of contemplating the fundamental physical meaning of "impossible." May it be as widely known as it is physically possible.