When "unphysical" solutions are not actually unphysical

Question

When solving problems in physics, one often finds, and ignores, "unphysical" solutions. For example, when solving for the velocity and time taken to fall a distance h (from rest) under earth gravity:

$\Delta t = \pm \sqrt{2h/g}$

$\Delta v = \pm \sqrt{2gh}$

One ignores the "unphysical" negative-time and positive-velocity solutions (taking x-axis as directed upwards normal to the earth's surface). However, this solution is not actually unphysical; it is a reflection of the fact that the equation being solved is invariant with respect to time-translation and time-reversal. The same equation describes dropping an object with boundary conditions ($t_i$ = 0, $x_i$ = h, $v_i$ = 0) and ($t_f$ = $|\Delta t|$, $x_f$ = 0, $v_f$ = $-\sqrt{2gh}$), or throwing an object backward in time with boundary conditions ($t_i$ = $-|\Delta t|$, $x_i$ = 0, $v_i$ = $+\sqrt{2gh}$) and ($t_f$ = 0, $x_f$ = h, $v_f$ = 0). In other words, both solutions are physical, but they are solutions to superficially different problems (though one implies the other), and this fact is an expression of the underlying physical time-translation and time-reversal invariance.

My question is: is there a more general expression of this concept? Is there a rule for knowing when or if an "unphysical" solution is or is not truly unphysical, in the sense that it may be a valid physical solution corresponding to alternate boundary conditions?

Answer 1

It is a very delicate matter to decide when solutions are unphysical or not. A classic example is Dirac's discovery of anti-particles: he found them as negative energy eigenstates for a relativistic Hamiltonian. A less insightful theorist might have discarded the negative energy solutions as unphysical, although we now know that those solutions mean a great deal, indeed.

Generally all solutions to an equation you write down have meaning within the context of the theory you used to write the equations in the first place. But you also must be wary of exceeding the scope of applicability of that theory.

Answer 2

An example of unphysical situations that lead to interesting phenomena is complexification of classical phase space, which may lead to tunneling (e.g., tunneling between integrable islands in phase space separated by chaotic sea). See for example:

http://www.sciencedirect.com/science/article/pii/S016727899700239X

http://prl.aps.org/abstract/PRL/v71/i14/p2167_1

http://www.lmpt.univ-tours.fr/~mouchet/recherche/MouchetDelande03a.pdf

Answer 3

Recently, I have allowed myself to argue in similar way, saying that if some physical theory is right then all solutions of the corresponding differential equation with physically admissible boundary conditions should be physical as well. As example, I have named the Dirac equation. However, the prof told me I am not right. He said: "solutions with negative energy have been accepted not because they must be physical as solutions, but because the mathematical postulate of quantum mechanics is that linearly independent solutions of the equation of evolution must form the basis of an adequate Hilbert space, and solutions with E > 0 do not form a basis."

Thus, I would say, the question of physicality reduces to question whether some solution is mathematically correct. For example, a particular solution of a second order linear ODE - even with physically correct boundary conditions - is, or can be, nonphysical because it does not form the basis of solution's two-dimensional vector space.

Physical can be only a linear combination of the two linearly independent functions that fulfill the equation with coefficients defined by the boundary conditions.