Limitations of science: Dirac Equation
It is the next article in series on “Limitations of scientific language” and limitations of physics, that describes how quantum physics is limited and where it fails. The advantages and failures of the Dirac equation are the first topic in the agenda, before quantum field theory.
First quantization quantum theory
In a previous article I have written extensively why description of the electron as a wavefunction in the ‘first quantization’ paradigm (developed by Heisenberg, Bohr and others), as the first original quantum mechanics theory is inadequate. First quantization theory describes particles as if they have existed forever, and can be used to develop simple technology or understand simple non-relativistic problems such as the Hydrogen atom, although only until some precision, because electrons in the atom move with 0.1% of speed of light. In the current article I will show that, as it is incomplete, it can also be defined better, a development which is also necessary to complete the standard model of elementary particles, as the most successful theory of science in human history.
Important conclusions from the previous article were that all of science is development of mathematical models with increasing precision to describe complexity of the world. And what is extremely important is where does the theoretical model being used is failing, where are the limits of the model? Because only then, right conclusions can be made.
The wavefunction of the electron represent probability of it being in certain position in space. Although the momentum is also not precisely defined and the electron has a wavefunction for the velocity/momentum (they are related to one another by a Fourier transform). Velocity is a first derivative of the position, as the rate of change of position of the particle. First derivative of the wavefunction is also the rate of change of the wavefunction at the same location. If you have a working theory and initial conditions — the evolution of the system and the experiment can be determined. In quantum physics classical trajectory of the particle is smeared out by uncertainty — which can be represented by a wavefunction of probability: Greek letter Psi. Even though it is not the same kind of determinism as with classical physics, quantum physics is still is very efficient at both building technology and performing experiments.
Dirac equation is a very important development, and a more adequate and accurate description of microscopic reality then ‘first quantization’ equations, but still is just an imperfect limited theoretical model of reality. Quantum field theory is the best attempt at describing microscopic reality we have so far. But before describing the same, it is necessary to understand the Dirac equation.
Consequences of Dirac equation
Dirac equation was developed as a consequence of the fact that electrons can not travel with speed no more then speed of light c. In order for to have a quantum equation that describes elementary particles that can only travel with speed of light c as a maximum value, the equation has to be consistent with Einstein’s special relativity and can only use first derivatives, as rate of change of the wavefunction (the velocity of the wavefunction).
Math of Dirac equation:
One of important consequences of Dirac equation is that it can describe silicon semiconductor technology, as interaction between negative charge particles and positive charge antiparticles, at a certain energy. Which is basis of modern computing technology.
First quantization quantum mechanics equations by Heisenberg and others can be derived from the Dirac equation, including Pauli equation that included particle spin, in a certain limit. Which indicates that quantum physics is just one mathematically consistent complete theory — with increasing complexity.
Zitterbewegung — velocity of light
Solving the Dirac equation is a difficult task not possible to present completely here, but I will present some incredible consequences of even the simplest solution for the free electron — which is an electron that only has kinetic energy and is not interacting with any other particle and field.
The result is that the spin of the electron is not constant — the spin of the electron is changing without being measured. How can that be possible? First quantization by Heisenberg, Bohr and Pauli states that the electron has definite spin when being measured. The measurement collapses the wavefunction and spin is to be determined to have some value.
Nevertheless it is a mathematical consequence of Dirac equation which works perfectly with incredible precision in all experiments (when used as a quantum field).
Another result from Dirac equation is that every electron is constantly moving with speed of light c. Yes, the only velocity that the electron can have is the speed of light c. In fact, then every elementary particle moves with speed of light, so the only allowed velocity in the universe is speed of light c.
How is that even possible, considering that human beings and planet Earth are made of electrons and are not moving with speed of light c, or changing its spin? Why would the spin of a particle change in free motion, a particle that even travels with speed of light c?
The explanation is that the electron is zigzagging through space changing its direction with frequency of about 10²¹ times per second — a process called Zitterbewegung. So the electron is moving forwards and backwards with speed of light c, changing direction and spin constantly, while the observed velocity is only an accumulative average velocity of the electron, which is less then speed of light c. The same process is not unusual, and in agreement with other properties of elementary particles, including modern understanding of what is mass of a particle, which is caused by longitudinal oscillations of a massive elementary particle.
Velocity of the electron is uncertain, and the electron moves alternatively with velocity v=c and v=-c. That is completely at odds with static wavefunction description from first quantization, that has no properties until being measured. There is an interpretation that Zitterbewegung zigzagging causes the collapse of wavefunction — although a hypothesis not testable until now. Penrose interpretation is that there are different massless components for the electron wavefunction, each with opposite speed of light c, and the mixing causes the zitterbewegung.
Dirac equation was an important transition from first quantization by Heisenberg (by combining special relativity with quantum mechanics), towards quantum field theory. Dirac equation and first quantization quantum mechanics fails because the larger the energy is, there is a increasing probability of spontaneous creation and annihilation of particles in empty space.
Mathematical interpretation and solutions of the Dirac equation can be very complicated to show for laymen. It is best to use the Lagrangian or Hamiltonian as total energy of the system to show what is happening to the electron.
Hamiltonian and Lagrangian
Hamiltonian evolves the quantum state , math explanation:
Hamiltonian and Lagrangian are equivalent to one another by mathematical transformation, and Hamiltonian is basically kinetic energy plus potential energy.
In the Lagrangian potential energy terms V are negative:
In first quantization theory by Heisenberg, kinetic, gradient and potential energy are just functions or derivatives. In quantum field theory they are actions that measure the energy values of the quantum state of the electron or any other state.
Kinetic energy of the particle
The result from Dirac equation is; every particle travels with speed of light c, which is actually speed of information. Kinetic energy of a quantum elementary particle can have a mathematical interpretation.
Next article on quantum field theory will continue expanding on the idea of Dirac equation, existence of particles and antiparticles and show how particles are created and annihilated.
