# Why does the classic wave equation for a non-relativistic string look like the Klein-Gordon equation?

There is a very old equation known as the "wave equation". It's an ordinary classical non-relativistic differential equation which applies to just about every kind of ordinary wave you can imagine (fluid dynamics, acoustics, mechanical waves, etc). It can be derived from Hooke's law simply by imagining a sequence of beads on a string, where each bead has the same mass. In the continuum limit of a string with constant tension and mass per unit length, such as that on a violin, you get this: $$\displaystyle {\frac {\partial ^{2}u}{\partial t^{2}}}=c^{2}{\frac {\partial ^{2}u}{\partial x^{2}}}$$ where $$c = \sqrt{T/\rho}$$ is the velocity of traveling waves on the string, T is the tension of the string, and $$\rho$$ is the mass per unit length of the string.

This equation was discovered by d’Alembert in 1746 (see, eg. https://en.wikipedia.org/wiki/Wave_equation) The 3D version of this, for ordinary fluids, mechanical waves, etc. is: $$\displaystyle {\frac {\partial ^{2}}{\partial t^{2}}}u -c^2\mathbf {\nabla } ^{2}u =0.$$

Fast forward 150+ years, and we have Einstein's theory of special relativity, which says that the energy of any body in motion is: $$\displaystyle E = \sqrt{(pc)^2 + (mc^2)^2}$$

Moving forward a few more decades, we can combine this with the assumption from quantum mechanics that momentum should not be viewed as an ordinary real number, but instead a non-commuting functional operator in a complex Hilbert space $$\displaystyle p = -i\hbar \frac{d}{dx}$$. This gives us the Klein-Gordon equation, the relativistic version of the Schrodinger Equation:

$$\displaystyle {\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}\psi -\mathbf {\nabla } ^{2}\psi +{\frac {m^{2}c^{2}}{\hbar ^{2}}}\psi =0$$ For the ultra-relativistic case when $$E >> m$$, this reduces to exactly the classic wave equation we derived from Hooke's law: $$\displaystyle {\frac {\partial ^{2}\psi}{\partial t^{2}}}=c^{2}{\frac {\nabla ^{2}\psi}{\partial x^{2}}}$$

My question is: why does the non-relativistic non-quantum wave equation for a string on a violin look more like the wave equation for a relativistic quantum particle than a non-relativistic quantum particle? Is that purely coincidence, or could it be seen as an early hint of the equations of relativity showing up in the 1700's? Without working out the math, a priori I would have guessed that the wave equation for a non-relativistic string would look more like the Schrodinger Equation than Klein-Gordon, since relativity is not involved, ie there is no reason to expect it to have a Lorentz invariant form: $$\displaystyle i\hbar {\frac {\partial \psi }{\partial t}}=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi.$$

Also, it seems interesting that this equation was first derived when thinking about the motion of a 1-dimensional string. This is a long shot, but... does this by any chance relate to the fact that quantum gravity turned out to have a consistent embedding in a quantum mechanics of strings, but not in a quantum mechanics of particles?

• Everything before the last paragraph is a good description of how one system without gravity looks a lot like another system without gravity. So the last paragraph is a long shot indeed! Commented Nov 16, 2021 at 18:54
• Related, possibly addresses the question: Why does nature favor the Laplacian? In particular, the second part of knzhou's answer there is closely related to why the same operator pops up in both places. Commented Nov 16, 2021 at 18:54
• As a very simple note: I guess asking for the time reversibility of your wave equation is reasonable. To satisfy this, you need an even order derivative in time (if you don't go in the realm of imaginary number and complex conjugates anyway, which is plausible for classical mechanics), so something which would look like a schrodinger equation would be a diffusion equation and won't be invariant by a time reversal transformation. Commented Nov 16, 2021 at 18:55
• Klein Gordon is NOT the relativistic Sch EQ , it's just the equation of motion for a (classical) massive scalar field. Remove the mass term (that is more like a quadratic self interaction of the field) and you have the same classical framework that gives rise to the wave equation... nothing exotic. Commented Nov 16, 2021 at 19:08
• @Quillo It most certainly is. "The Klein–Gordon equation was first considered as a quantum wave equation by Schrödinger in his search for an equation describing de Broglie waves. The equation is found in his notebooks from late 1925, and he appears to have prepared a manuscript applying it to the hydrogen atom. " en.wikipedia.org/wiki/Klein–Gordon_equation#History Commented Nov 16, 2021 at 19:47

With one time dimension and one space dimension, the wave equation's most general solution is $$f(t-x/c)+g(t+x/c)$$ with $$f,\,g$$ twice-differentiable, i.e. each solution is a sum of two speed-$$c$$ waves, one moving right, the other left. Equivalently, the former's value at a time $$t$$ depends on the behaviour of the wave's source a time $$x/c$$ earlier, so is a "retarded" solution, while the latter is an "advanced" solution.

I've moved from the maths to the physics, but we can just as easily go the other way. If we only want the retarded them, a solution is of the form $$f(t-x/c)$$, and the PDE this fits is $$u_t=-u_x/c$$. But as soon as we work in multiple space dimensions, e.g. because we're thumping a drum skin, the requirement that both sides are scalars effects a generalization such as $$u_t=-\hat{e}\cdot\nabla u/c$$, with $$\hat{e}$$ an arbitrary unit vector. Too arbitrary. It picks out a privileged direction in space. (We can fix that with a spacetime-unifying idea like $$\gamma^\mu u_\mu=0$$, but let's put that aside for a moment.) If we use second-order derivatives too we avoid this problem, viz. $$u_{tt}=u_{ii}$$. The only downside is there are now advanced solutions too. (The general selection in multiple space dimensions looks a fair bit more complicated, but the principle is similar; we just need a $$k$$-space integral instead of a $$2$$-term sum.)

But one of the most important ideas in science is there's more than one place you can start your theory. We're not choosing axioms, then seeing which theorems they give us, like a mathematician with a preferred set theory. The universe just "is", all at once. So let's do something very ahistorical: suppose you tried to write down a relativistic quantum field theory without first knowing what classical relativity looks like. I know that sounds crazy, but bear with me.

Uniting time and space means actions become spacetime integrals of Lagrangian densities expressible in terms of fields and spacetime derivatives, rather than just time integrals of Lagrangians expressible in terms of coordinates and their time derivatives. Even before you work out ideas like Lorentz-invariance, you already realize a field ought to vibrate in a similar retarded-plus-maybe-advanced way. This quickly gives you the Klein-Gordon equation before you know Einstein's energy-momentum relation! Or if you work with a spinor with enough components, the privileged-direction idea comes off as more workable, because of a $$\gamma^\mu\partial_\mu$$ operator. (Hilariously, though, it actually doesn't get rid of the advanced solutions; or on a related point, it doesn't get rid of $$E<0$$ solutions to $$E=m^2c^4+p^2c^2$$.) But Lagrangians tend to give us even-order time derivatives in our EOMs, although there's a way round that.

So the real question is why does plucking a string look like Klein-Gordon rather than Dirac? (Schrödinger is out of the running because of the $$f(t-x/c)$$ Ansatz; we need $$\nabla$$ to have the same degree as $$\partial_t$$.) Well, the unhelpful answer is "because this derivation says so". The intuitive summary of that unhelpful answer is the string's behaviour is built on top of Newton's second law, so you need a second-order time derivative. If you think we should make our inference in the opposite direction, that won't satisfy you, in which case I'll need to motivate why the string amplitude shouldn't be a spinor with at least $$4$$ components. However, I suspect you don't find that negative fact surprising enough to warrant an "explanation".

• Thanks... I think that mostly answers it, and covers several interesting aspects I hadn't considered. And yes, no need to explain why the Dirac equation was not needed to describe a violin string, lol! That would have been more surprising, not less ;-) I think the only two aspects of this I'm still a bit fuzzy on is 1. Why is the Schrodinger equation allowed to be time-asymmetric, but not the Klein Gordan equation or the classical wave equation? And is it just a coincidence that Maxwell's equations with no sources satisfy the same equations as those describing mechanical waves, with v -> c Commented Nov 17, 2021 at 1:51
• @reductionista 1. The TDSE requires you to also conjugate $\psi$ for time symmetry to work, and that's the C and T in CPT, which has a (complicated!) relativistic QFT justification. 2. I shouldn't have claimed the relativistic field theory one invents prematurely by thinking about Lagrangian densities need be quantum; the same logic applies to classical field theories, and is how you could predict the electromagnetic wave equation (although it has a constant speed due to relativity, and not a medium as per strings' sound waves).
– J.G.
Commented Nov 17, 2021 at 8:02
• I think I see. The Klein Gordon equation is symmetric under both parity and time-reversal (and trivially, complex conjugation) individually. The Schrödinger Equation is symmetric under parity $P$. It's not symmetric under "naive time reversal" $T_{n} : t \rightarrow -t$ or under $C$ (complex conjugation). But since the phase of $\psi$ is not physically measurable anyway, it doesn't matter that it's technically not time-symmetric since it's the dynamics of $|psi|^2$ which matters. So we're free to just define the time reversal operator to be $T = CT_{n}$, restoring the symmetry. Commented Nov 19, 2021 at 0:51
• But wait... if $T = CT_n$, and by the CPT theorem $P = CT$, then wouldn't that imply $P = T_n$? That doesn't seem right... I think I'm still missing something here. Maybe it's true in the case of a simple single-particle Schrödinger Equation only because both P and CT are essentially the identity operation (when applied to both sides). But not true in general for QFT, because in addition to complex conjugation, you have to replace each field with one that transforms oppositely under the gauge symmetry? Commented Nov 19, 2021 at 1:14
• @reductionista If $S$ is the solution set, the correct condition is $x\in S\implies PCT_nx\in S$.
– J.G.
Commented Nov 19, 2021 at 5:52

Wave equation is the simplest equation that describes wave-like solutions in an isotropic homogeneous medium. In case of Klein-Gordon equation these correspond to the symmetries of the space-time, whereas in case of elasticity or hydrodynamics they result from making some simplifying assumptions - in fact in both latter fields one frequently comes up with equations that are different from the standard form $$\frac{1}{c^2}\frac{\partial^2 u(\mathbf{x},t)}{\partial t^2} - \nabla^2 u(\mathbf{x},t)=0$$ (See this answer for overview of some of the equatiions.)

E.g., in hydrodynamics one needs to do quite some work to get from the basic Navier-Stokes equation to a wave equation, and one gets somewhat different answers depending on whether one looks at the gravity waves in a shallow water, or deep water, or the pressure waves. Moreover, the Navier-Stokes equations are themselves an approximation to a more precise description in terms of Boltzmann equation (assuming local equilibrium).

In elasticity one often has to deal with anisotropy, and in both fields the standard wave equation can be obtained only under the assumption of non-linear response (aka Hooke's law in elasticity).

Finally, Schrödinger equation can be obtained from the Klein-Gordon equation as an equation for the envelope function, i.e., assuming that $$\psi(\mathbf{x},t)=\phi(\mathbf{x},t)e^{-imc^2 t/\hbar},$$ where $$\phi(\mathbf{x},t)$$ is slowly varying with time. Plugging the above into the Klein-Gordon equation and neglecting the second time derivative of the envelope function $$\phi(\mathbf{x},t)$$ one ends up with the Schrödinger equation for a free particle. Note that this "diffusion" approximation is not unqiue to quantum mechanics - e.g., it is used for propagation of EM waves via an inhomogeneous media.