Deep Thoughts From Marco Frasca

Italian physicist Marco Frasca's latest paper leaves his usual comfort zone of QCD and discusses instead a deep and fundamental relationship between the Schrödinger equation (that describes how the quantum state of a physical system changes in time) and the equations of the observable stochastic (i.e. probablistic) processes of Brownian motion (which describes the dispersion of particles that flows from their random movements) from first principles via some mathematical leaps of insight that have eluded some of the best minds in math and physics for almost nine decades (counting from the publication of Schrödinger equation, which is younger than the equations that describe Brownian motion). Essentially the Schrödinger equation is the square root of the equation that describes Brownian motion, when the equations are properly formulated and the square root if defined in a clever way that gives rise to a complex number valued solution.

[T]he square root of Brownian fluctuations of space are responsible for the peculiar behavior observed at quantum level. This kind of stochastic process is a square root of a Brownian motion that boils down to the product of two stochastic processes: A Wiener process [i.e. "the scaling limit of a random walk"] and a Bernoulli process proper to a tossing of a coin. This aspect could be relevant for quantum gravity studies where emergent space-time could be understood once this behavior will be identified in the current scenarios.

Note, however, that there is at least one obvious step that has to be bridged between this conclusion and a rigorous theory of quantum gravity, since the Schrödinger equation is a non-relativistic effective approximation of quantum mechanics.

[T]he solutions to the Schrödinger equation are . . . not Lorentz invariant . . . [and] not consistent with special relativity. . . . Also . . . the Schrödinger equation was constructed from classical energy conservation rather than the relativistic mass–energy relation. . . . Secondly, the equation requires the particles to be the same type, and the number of particles in the system to be constant, since their masses are constants in the equation (kinetic energy terms). This alone means the Schrödinger equation is not compatible with relativity . . . [since quantum mechanics] allows (in high-energy processes) particles of matter to completely transform into energy by particle-antiparticle annihilation, and enough energy can re-create other particle-antiparticle pairs. So the number of particles and types of particles is not necessarily fixed. For all other intrinsic properties of the particles which may enter the potential function, including mass (such as the harmonic oscillator) and charge (such as electrons in atoms), which will also be constants in the equation, the same problem follows.

In order to extend Schrödinger's formalism to include relativity, the physical picture must be transformed. The Klein–Gordon equation and the Dirac equation [which provides a description of elementary spin-½ particles, such as electrons, consistent with both the principles of quantum mechanics and the theory of special relativity] are built from the relativistic mass–energy relation; so as a result these equations are relativistically invariant, and replace the Schrödinger equation in relativistic quantum mechanics. In attempt to extend the scope of these equations further, other relativistic wave equations have developed. By no means is the Schrödinger equation obsolete: it is still in use for both teaching and research - particularly in physical and quantum chemistry to understand the properties of atoms and molecules, but understood to be an approximation to real behaviour of them, for speeds much less than light.

Presumably, however, it would be possible to square the relativistic versions of the Schrödinger equation (such as Dirac's equation) in an analogous manner to derive are diffusion equations for Brownian motion in relativistic settings that is consistent with special relativity, while still illustrating how complex number valued underlying quantum mechanics can reflect an emergent and fluctuating underlying nature of space-time.

As a footnote, it is also worth noting that his paper was made possible only with the a collaboration with elite mathematicians that would have been much more difficult to facilitate without the crowdsourcing of part of the problem that the Internet made possible.