# Does Galileo's Tower of Pisa argument contradict quantum mechanics?

(My questions are at the end, but they may not mean much without explanation below.)

Galileo argued that because the mass of a falling object can always be redistributed in ways that asymptotically approximate a set of smaller falling objects, e.g. linked by light strings, then the equations of motion for an object falling in a vacuum cannot depend on its mass. That is, because dependency on mass would lead to paradoxical motion predictions if in one case you used the total mass $$m_\Sigma$$ and in another one of the asymptotically free submasses $$m_i$$ were used as the arguments to the equation of motion. Thus for example, the equation for the instantaneous velocity of any object falling in a vacuum does not include mass as a significant parameter:

$$v_i=\sqrt{2gd}$$

Galileo's argument was insightful but largely tautological. The problem is that the equations of motion for objects falling though fluids, such as terminal velocity, provide an existence proof that self-consistent equations of motion with mass as a significant parameter can be constructed. However, such self-consistency becomes possible only if the distribution of object mass in space also becomes a non-trivial parameter. This additional information allows paradoxes to be resolved by converting seemingly inconsistent state predictions into asymptotic limits of a larger continuum of predictions.

For example, in fluid mechanics a mass $$m_{\Sigma}$$ is correctly predicted to have a higher terminal velocity than the same mass configured as two smaller spheres $$m_1$$ and $$m_2$$ linked by a thin string. The latter form is an example of how the predicted behavior of a single falling sphere can be made to approach asymptotically the prediction for two smaller spheres by slowing "morphing" the original object into one that closely approximates the case of two smaller spheres.

What Galileo really proved, then, was that since experimental evidence showed that mass was not a relevant parameter for the equations of motion of objects falling in a vacuum, then the specific distribution of mass in any one object must also be irrelevant to those equations.

What Galileo could never have realized was that exists a domain of physics for which his argument is very relevant: quantum physics.

The de Broglie wavelength $$\lambda=h/p$$ of an object falling at modest speeds in a vacuum is a function of its instantaneous momentum $$p_i=mv_i$$, so:

$$\lambda_i=\frac{h}{mv_i}=\frac{h}{m \sqrt{2gd}}=\frac{h}{m}\sqrt{\frac{1}{2gd}}$$

While not an equation of motion, this equation violates Galileo's argument by predicting that the de Broglie frequency of an object falling in a vacuum is a function of its mass. As in the case of fluid dynamics, the inclusion of mass as a significant parameter results in paradoxical predictions unless and until the configuration of the mass in the object is also taken into account.

In other words, the de Broglie wavelength of an object must necessarily depend upon its shape and how mass is distributed within that object. The resulting more closely resembles solid state acoustic resonances in complexity than it does the simple singular mass parameter of pointlike particles isolated in space.

For example, if $$m_1=m_2=m$$ and $$m_\Sigma=m_1+m_2=2m$$, three state equations are relevant:

$$\lambda_{\Sigma}=\lambda_{2m}=\frac{h}{2m}\sqrt{\frac{1}{2gd}}$$

$$\lambda_1= \lambda_m=\frac{h}{m}\sqrt{\frac{1}{2gd}}$$

$$\lambda_2= \lambda_m=\frac{h}{m}\sqrt{\frac{1}{2gd}}$$

Self-consistency alone argues that if these equations are even approximately correct, then $$\lambda_{2m}$$ should dominate when the $$2m$$ mass takes the form of a compact spherical object. Conversely, reconfiguring the $$2m$$ sphere into two smaller $$m$$ spheres linked by a slender thread should asymptotically approach the case of two separate $$m$$ spheres, and so should be dominated by the $$\lambda_m$$ wavelength.

However, any single wavelength cannot be accurate in any of the cases. There should instead be a spectrum of wavelengths and intensities that is highly dependent on the particulars of how the mass is distributed and connected in the object. That is why I mentioned earlier that an accurate equation model for predicting de Broglie wavelengths for non-point object will necessarily be comparable in complexity and subtly to resonance models in solid state physics.

I do not readily see a way out of this. Galileo's argument, while largely specious in the case of the equations of motion for objects falling in a vacuum, is extraordinarily relevant to quantum mechanics, where it appears to say that the entire concept of "mass" requires revisiting if it is to be used accurately and meaningfully. A correct model should instead have more kinship with the equations of motion for objects falling though a fluid, in the sense that both must define very carefully how the distribution and "connection" of mass impacts the predictions.

So, my two main questions are:

1. Do there currently exist mathematical models of de Broglie wavelengths that pass the Galilean self-consistency test? For example, are they perhaps hidden away in the mathematical details of phenomena such as quantum models of molecular vibration? (My impression is "no", but I could certainly be wrong.)

2. What are the particle physics implications, if any, of quantum mass being dependent on form?

• very nice question, I am eagerly anticipating an answer to this! – user36538 Jan 30 '14 at 7:30
• Paticle physics has to do with elementary particles. Elementary particle are point particles, no extent. The de broglie wavelength assigned to them by their four vector is a wave in probability space and tells us what is the probability of finding the elementary particle at (x,y,z). Due to special relativity composites of elementary particles have an invariant mass given by the measure of the addition of their four vector which is bigger than m_1+m_2. The rho meson is heavier than the sum of masses of the individual pions, the pion is heavier than the sum of the masses of individual quarks+g – anna v Jan 30 '14 at 11:55
• anna v thanks, nice to hear from you again. One resolution would be to forbid composite mass frequencies, but of course that cannot be right, since very tightly bonded particles such as helium nuclei behave like point masses all the time and in all sorts of calculations. Bonding strength is in any case almost a profound component of how such the wavelength spectra of composite articles emerge. I still suspect that I'm asking something that a good quantum chemist might be able to answer easily, since they have to deal with quantum behavior with bonding all the time. – Terry Bollinger Feb 2 '14 at 5:37

## 2 Answers

There is no contradiction between quantum mechanics and the equivalence principle. There has never been any such contradiction. String theory makes the compatibility of the principles explicit but to explain the compatibility at the level of the question, we don't really need any characteristically stringy arguments. So the answers to the numbered questions are

1. Quantum mechanics with the gravitational potential has been compatible with the equivalence principle since the birth of quantum mechanics.

2. The quantum mass depends on the internal arrangement of the objects thanks to special relativity's $E=mc^2$. For example, all masses in string theory boil down to the different energies of vibrating strings in string theory. But this is just special relativity; there is no "other" dependence on the mass, there is no threat for the equivalence principle posed by the basic postulates of quantum mechanics, so as long as I correctly feel which implications the OP means, there are no implications.

The equivalence principle is how we call the general observation that all massive bodies accelerate at the same acceleration in a given gravitational field.

Even in classical physics, Galileo's argument with thin strings is just a heuristic way to think about the problem. It would only apply if all objects were composed of the same small massive bodies or "atoms", with the same mass $m$ and other parameters, that are just connected by thin strings. In such a case, the equivalence principle could be a tautology.

But in more realistic physics, objects are composed of different atoms, with different masses and other parameters, and the forces simply don't have to obey the equivalence principle a priori. In other words, the parameter $m$ in $F=ma$ doesn't have to be the same one as the parameter $m$ in $F=GMm/r^2$. In the real world, the inertial mass and the gravitational mass are the same parameter because the equivalence principle holds but Newton's theory may be formulated equally naturally while refuting this assumption i.e. it leaves the assumption unexplained. Einstein's general relativity explains the equivalence principle. Well, it was constructed so that it explains it (in this sense, "it" assumes it).

Larger bodies have a larger momentum and larger energy, and because the energy/momentum is linked to frequency/wavenumber (inverse wavelength) by quantum mechanics, larger objects also have shorter wavelengths and higher frequencies.

But these frequencies and wavenumbers are in no way "observable patterns we may directly measure" by the measurements of phases at particular points and moments. The de Broglie frequency and wave number tell us how quickly the phase of the wave function – using a particular simple convention for the phase – is changing as a function of space and time. But the phase of the wave function is not observable (and it is not an observable in the sense of a linear operator on the Hilbert space).

Just a trivial example. Nonrelativistic quantum mechanics is using the non-relativistic formula for the kinetic energy $E_k=p^2/2m$ which is completely omitting the latent energy $E=mc^2$ implied by special relativity. But that doesn't damage anything about the validity of the theory because the energy in non-relativistic physics may be additively shifted, $H\to H+\Delta E$, without any impact on the physics. In quantum mechanics (Schrodinger's picture), this energy shift has to be combined with a redefinition of the phase of the wave function, $\psi(t)\to\psi(t) \exp(\Delta E\cdot t / i\hbar )$, which doesn't change anything about the quantum physics, either (because the overall phase of the wave function is unphysical).

An energy eigenstate in quantum mechanics with energy $E$ always has the wave function depending on time as $\exp(Et/i\hbar)$, by definition of the energy, and this dependence can't be affected by any details about the internal distribution of the energy within the object. Analogously for the momentum eigenstates and the dependence of the wave function on the center-of-mass location of the object.

So if we have two identical objects of masses $m+m$ connected to a bound state, their overall de Broglie frequency will be twice the frequency of one of them (and similarly for the wavenumber) because the wave function is multiplicative over the subsystems. But the de Broglie wave of the composite system is in no way a "[classical] pattern drawn in the spacetime" that may be measured. The frequency and wave number of the wave function is being measured by the energy and momentum measurements (they're the same thing up to the $\hbar$) and you have presented no argument that such measurements inevitably violate the equivalence principle. Such an argument cannot exist because it may be easily seen that quantum mechanics with the $\sum_i m_i\Phi(x,y,z,t)$ gravitational potential added into the Hamiltonian manifestly obeys the equivalence principle.

• Luboš Motl thanks, but the equivalence principle was never in doubt. Try this: Drop three same-mass balls. Link two with a string {1+1}. Leave the third one {1} free as a wavelength reference. If the string doubles the mass {1+1}, then an observer measuring relative wavelengths of {1+1} over {1} as they pass will always get $\frac{1}{2}$. Can you see why that answer is too simple? It leads to the same geometric inconsistencies Galileo complained about, only this time for phenomena that really exist. You're a sharp fellow. How do you resolve that, preferably with no strings attached? – Terry Bollinger Feb 2 '14 at 5:31
• Sorry, @Terry, I have already tried to correct this error of yours, but you were not listening. The de Broglie wavelength measurement is nothing else than the momentum measurement (and the momentum depends on the reference frame in the obvious way, it's $mv$...). So the person just measures that the momentum of 2 balls is 2 times the momentum of 1 ball. There is no other non-momentum (in particular, "geometric") measurement of the de Broglie wavelength; the phase of the wave function is not observable. Could you please try to read the previous sentence again? – Luboš Motl Feb 2 '14 at 6:31
• I think that you are noticing that the wavelength depends on the reference frame. Of course it does. It depends in the same wave as the inverse momentum because it's the same thing. – Luboš Motl Feb 2 '14 at 6:33
• By "geometric" I was referring to Galileo's argument that using mass a non-trivial parameter in a set of equations implies dependence of that same set of equations on the geometry of the object. Hmm. OK, let's try "drop the neutrons" neutron diffraction crystallography: You would agree that if dineutrons formed in the beam, their angle of diffraction would change? Observably? Imagine if neutron bonding could be slight, medium, strong, tight: What would happen to diffraction pattern intensities? Galileo's argument, if valid, would in effect have argued "nothing would change for any cases." – Terry Bollinger Feb 2 '14 at 7:52
• Dear @Terry, dineutrons fail to be stable, by 60 keV, but they do. If you imagine a world where dineutrons form and you measure any interference or any process involving them in a freely falling frame, you will not recognize that you are in a gravitational field. That's what the equivalence principle implies. On one hand, you say that you don't question the EP; on the other hand, you clearly do. I am confused by the sequences of the words. The EP holds even in QM with Earth's uniform gravity and neutrons and all arguments that it doesn't are wrong. – Luboš Motl Feb 2 '14 at 9:16

Luboš is rigorous. Let's try simple - Einstein's inertial elevator (hard vacuum, of course). The mass is not accelerating, its observer is accelerating,

http://thinkingscifi.files.wordpress.com/2012/07/loadbinary.gif

What remains to be argued? Now, the fun part! "Drop" two masses that are different in one or more ways, a bit of goose down fluff vs. a solid lead bowling ball. Only the observer is accelerating. The Equivalence Principle is unbeatable.

If the Referees will indulge me, I am an empiricist not a theorist. If not, dump what follows.

GR superset Einstein-Cartan-Kibble-Sciama gravitation contains spacetime torsion. Spacetime torsion is chiral, like Lorentz force. Opposite shoes embed within chiral vacuum (mount a left foot) with different energies. They vacuum free fall along non-identical minimum action trajectories, exhibiting Equivalence Principle (EP) violation. It's neat a trick if it works. If not, ECKS gravitation defaults to GR with spacetime curvature only.

Chirality is an emergent property. One desires rather a lot of very small shoes. One test mass is all left shoes, the other all right shoes. Crystallography's opposite shoes are visually and chemically identical, single crystal test masses in enantiomorphic space groups: P3(1)21 vs. P3(2)21 alpha-quartz. It's chemistry.

0.113 nm^3 volume/alpha-quartz unit cell. 40 grams net as two 20 gram single crystal test masses compare 6.68×10^22 pairs of opposite shoes (pairs of 9-atom enantiomorphic unit cells). The Equivalence Principle is bulletproof inside physics, more or less.

http://www.physics.indiana.edu/~kostelec/faq.html

If we embrace chemistry, maybe only bullet-resistant. Somebody should look (geometric Eötvös experiment, or Robert Reasenberg's SR-POEM experiment).