# De Broglie theory and relativity

1. Jun 12, 2014

### PhilDSP

After suitable study I'd have to say that it is a misnomer to state that de Broglie's theory is not relativistic. But it does express relativity in such a unique way that it becomes a replacement for Newtonian mechanics within any chosen single inertial frame.

The most basic expression is covered on page 40 of "An Introduction To The Study Of Wave Mechanics". For the Lorentz group:

$x = \frac{x_0 + vt_0}{\sqrt{1 - \beta^2}} \ \ \ \ \ t = \frac{t_0 + \frac{\beta}{c}x_0}{\sqrt{1 - \beta^2}} \ \ \ \ \$ and $\ \ \ \ \ x_0 = \frac{x - vt}{\sqrt{1 - \beta^2}} \ \ \ \ \ t_0 = \frac{t - \frac{\beta}{c}x}{\sqrt{1 - \beta^2}}$

An EM or energy wave for system with reference to $x_0, y_0, z_0, t_0$ depends on time only through a factor $cos 2\pi f_0(t_0 - \tau_0)$

Switching to system $x, y, z, t$ with $\tau_0 = 0$ the factor becomes $cos 2\pi f_0(\frac{t - \frac{\beta}{c}x}{\sqrt{1 - \beta^2}})$

Since $f = \frac{f_0}{\sqrt{1 - \beta^2}}$ and $v_{phase} = \frac{c^2}{v} = \frac{c}{\beta}$

$\beta = \frac{c}{v_{phase}}$ so that the phase factor is $cos 2\pi f(t - \frac{x}{v_{phase}})$

What de Broglie has done is to account for all relativistic effects on waves, and subsequently particles, within that single chosen inertial reference. There becomes no further need to consider something in a separate inertial frame. The bulk of his thesis is developed in phase space using mathematical ideas of Hamilton and Jacobi, but he carefully relates the variables in the phase space to variables in Euclidean physical space and time. The concept of a secondary inertial frame is not even especially valid then as his theory applies even to accelerating particles.

Last edited: Jun 12, 2014
2. Jun 12, 2014

### Staff: Mentor

Jump to a frame where the particle is at rest. What's the wavelength there? What's the group velocity?

Thanks
Bill

3. Jun 12, 2014

### vanhees71

In addition the naive interpretation of "wave functions" in the sense of a single-particle description is flawed for relativistic wave equations. The reason is simply that for relativistic energies it is always possible that in reactions particles get created and destroyed. That's why relativistic quantum theory is most appropriately formulated as a quantum field theory, where such creation and annihilation processes are very natural.

4. Jun 12, 2014

### PhilDSP

If we give credence to zitterbewegung the particle will be at rest in that frame for an exceedingly short period of time. But then the wavelength would be infinite, group velocity zero while the phase velocity infinite.

I know you've expressed consternation at an infinite wavelength, but since there is no energy moving anyway in that case, one can interpret an infinite wavelength as an instance of non-oscillation.

Last edited: Jun 12, 2014
5. Jun 12, 2014

### PhilDSP

I suppose an alternative to QFT where those processes are modeled more in concert with DB theory would incorporate particle creation and destruction in a Hamiltonian/Jacobi framework...

6. Jun 12, 2014

### Staff: Mentor

The problem here is the intermixing of classical and quantum ideas. That's the cause of the issue and why its just an intermediate step to fully developed QM that was realized in the transformation theory of Dirac. Once that came out there was simply no need for it.

Thanks
Bill

7. Jun 12, 2014

### PhilDSP

Fair enough. There are cases where QM concepts have found uses and new physics in the classical realm (Berry phase for example). But going the other direction and providing QM with new insights would require a truly exceptional further refinement and development of de Broglie's ideas.

Berry phase effects on electronic properties
http://journals.aps.org/rmp/abstract/10.1103/RevModPhys.82.1959

Last edited: Jun 12, 2014
8. Jun 13, 2014

### vanhees71

If by DB you mean de Broglie-Bohm (pilote wave) mechanics, one must say that there seems to be no satisfying formulation of this concept for relativistic quantum theory.

I also never understood what might be attractive with the Bohm interpretation since it introduces trajectories that are not observable anyway and does not make any other predictions about Nature than (minimally interpreted) quantum theory. So what's the point to introduce complicated concepts that don't buy anything in our understanding of Nature?

9. Jun 15, 2014

### PhilDSP

I can't offer any insight into Bohm's interpretation. I've read his "Quantum Theory" text book (which I'd rate quite favorably). But he didn't delve into Pilot Wave theory there. Ballentine has a nice, short summary on p. 399.

Starting on p. 394 Ballentine derives the Quantum Potential starting from the SchrÃ¶dinger equation. He arrives at a different value than de Broglie.

Ballentine: $\ \ \ W_Q = -\frac{h^2}{2M} \frac{\nabla^2 A}{A} \ \ \ \ \ \$ de Broglie: $\ \ \ F_1(x, y, z, t) = -\frac{h^2}{8 \pi^2 m} \frac{\nabla^2 a}{a}$

Since de Broglie's is a covering theory which derives the SchrÃ¶dinger equation as the non-relativistic limit (along with some other stipulations), I'll assume de Broglie's derivation of the Quantum Potential (ending on p. 112) is the correct one.

However, Ballentine makes a very important observation on p. 398 in further analysis where he states

"The failure of this method to yield the expected classical limit in this case is clearly due to the formation of a standing wave, which is a manifestation of the quantum-mechanical phenomenon of interference between the leftward- and rightward-reflected waves that make up $\Psi$".

Last edited: Jun 15, 2014
10. Jun 15, 2014

### UltrafastPED

I suspect the factor of 4 * pi is due to the use of different units; thus it is not fundamental.