Dirac Equation
The Dirac equation is a relativistic wave equation derived by Paul Dirac in 1928. In its free form or including electromagnetism interactions, it describes all spin-1/2 massive particles such as electrons and quarks for which parity is a symmetry. The equation also implied the existence of a new form of matter. antimatter, previously unsuspected and unobserved and which was experimentally confirmed several years later.
Although Dirac did not at first fully appreciate the importance of his results explanation of spin as a consequence of the union of quantum mechanics and relativity - and the eventual discovery of the positron - represents one of the great triumphs of theoretical physics. The accomplishment has been described as fully on par with works of Newton, Maxwell, and Einstein.[1]
According to Schrödinger Equation the Hamiltonian operator and momentum operator may be written as (𝑉: potential energy) [2]
(3.25) 𝑖ℏ ∂𝜓/∂𝑡 = Ĥ𝜓.
(3.26) Ĥ = -(ℏ2/2𝑚) ∂2/∂𝑥2 + 𝑉
(3.29) 𝑃^ = -𝑖ℏ ∂/∂𝑥.
The operator version of classical total-energy equation 𝐸 = 𝑇 + 𝑉 = 𝑝2/2𝑚 + 𝑉: (𝑇: kinetic energy)
(3.30) Ĥ = (𝑃^)2/2𝑚 + 𝑉
(3.31) (𝑃^)2𝛹 = 𝑃^(𝑃^𝛹) = -𝑖ℏ ∂/∂𝑥(-𝑖ℏ ∂/∂𝑥) = 𝑖2ℏ2 ∂2𝛹/∂𝑥2 = -ℏ2 ∂2𝛹/∂𝑥2.
(3.32) (𝑃^)2 = -ℏ2 ∂2/∂𝑥2
Hence the equation is written formally by
Ĥ𝜓(𝑟̄) = 𝐸𝜓(𝑟̄)
where 𝐸 is called the energy equivalence of the Ĥ-operator. Without potential energy Ĥ-operator can be written
Ĥ = -(ℏ2/2𝑚) 𝛻2.
According to special relativity the relativistic energy is given by (refer to 'MC Forum 61')
𝐸2 = 𝑝2𝑐2 + 𝑚2𝑐4 or 𝐸 = ∓𝑐√(𝑝2 + 𝑚2𝑐2)
where the case of 𝐸 < 0 will be for the positron which is the antimatter counterpart of the electron!
Dirac derived a realistic wave equation [3] which has the form for kinetic energy (operator) [4]
(2.2) 𝑖ℏ ∂𝜓/∂𝑡 = [-𝑖ℏ𝑐(𝛼^1∂/∂𝑥1 + 𝛼^2∂/∂𝑥2 + 𝛼^3∂/∂𝑥3) + 𝛽^𝑚𝑐2]𝜓 ≡ Ĥ𝜓,
where 𝛼^𝑗 (𝑗 = 1,2,3) and 𝛽^ cannot be a simple number, therefore matrices can be solutions.
We consider a particle, such as an electron with mass 𝑚 in field-free region of space here now.[5]
(14.5) 𝑖ℏ ∂𝜓/∂𝑡 = √ ∣𝑃^2𝑐2 + 𝑚2𝑐4∣ 𝜓
If we are to make the standard replacement 𝑃^𝑥 = -𝑖ℏ∂/∂𝑥, etc., folllowing Dirac, we write
(14.6) 𝑖ℏ ∂𝜓/∂𝑡 = [𝑐𝛼^1𝑃^𝑥 + 𝑐𝛼^1𝑃^𝑦 + 𝑐𝛼^1𝑃^𝑧 + 𝛽^𝑚𝑐2]𝜓
(14.7) [𝑐𝛼^1𝑃^𝑥 + 𝑐𝛼^1𝑃^𝑦 + 𝑐𝛼^1𝑃^𝑧 + 𝛽^2𝑚𝑐2]2 = 𝑐2𝑃^2 + 𝑚2𝑐4
Multiplying out the lef-hand side of (14.7) and equating the corresponding terms leads to
(14.8) 𝛼^12 = 𝛼^22 = 𝛼^32 = 1
𝛼^1𝛼^2 + 𝛼^2𝛼^1 = 𝛼^2𝛼^3 + 𝛼^3𝛼^2 = 𝛼^3𝛼^1 + 𝛼^1𝛼^3 = 0
𝛼^1𝛽^ + 𝛽^𝛼^1 = 𝛼^2𝛽^ + 𝛽^𝛼^2 = 𝛼^3𝛽^ + 𝛽^𝛼^3 = 0
We call them the Dirac matrices which Dirac found.
(14.9) 𝛼^1 = 4⨯4 matrix [row-1 (0 0 0 1), row-2 (0 0 1 0), row-3 (0 1 0 0), row-4 (1 0 0 0)],
𝛼^2 = 4⨯4 matrix [row-1 (0 0 0 -𝑖), row-2 (0 0 𝑖 0), row-3 (0 -𝑖 0 0), row-4 (𝑖 0 0 0)],
𝛼^3 = 4⨯4 matrix [row-1 (0 0 1 0), row-2 (0 0 0 -1), row-3 (1 0 0 0), row-4 (0 -1 0 0)],
𝛽^2 = 4⨯4 matrix [row-1 (1 0 0 0), row-2 (0 1 0 0), row-3 (0 0 -1 0), row-4 (0 0 0 -1)],
One remarkable feature of the expression is that the matrices are all made from 2⨯2 Pauli spin matrices as
(14.10) 𝜎^𝑥 = 2⨯2 matrix [row-1 (0 1), row-2 (1 0)], 𝜎^𝑦 = 2⨯2 matrix [row-1 (0 -𝑖), row-2 (𝑖 0)], 𝜎^𝑧 = 2⨯2 matrix [row-1 (1 0), row-2 (0 -1)].
where 𝛽^ can be expressed in terms of unit matrix 𝐼. Thus
(14.11) 𝛼^𝑗 = 2⨯2 matrix [row-1 (0 𝜎^𝑗), row-2 (𝜎^𝑗 0)], (𝑗 = 1,2,3) and 𝛽^ = 2⨯2 matrix [row-1 (𝐼 0), row-2 (0 -𝐼]
where 𝜎^1 ≡ 𝜎^𝑥 etc.
The fact that the Dirac equation is a matrix equation implies that 𝜓 is a now a vector formed out of four functions
(14.13) 𝜓 = column matrix [𝜓1, 𝜓2, 𝜓3, 𝜓4] ≡ column matrix [𝜓+, 𝜓-]
where 𝜓+ and 𝜓- are two-component vector defined by (14.13).
The Dirac equation is studied such that a Lorentz covariant form is given by using 𝛾𝜇 matrices instead of 𝛼^ and 𝛽^ matrices.[4]
(3.9) 𝑖ℏ(𝛾0∂/∂𝑥0 + 𝛾1∂/∂𝑥1 + 𝛾2∂/∂𝑥2 + 𝛾3∂/∂𝑥3)𝜓 - 𝑚𝑐𝜓 = 0,
where 𝛾0 = 𝛽^, 𝛾𝑖 = 𝛽^𝛼^𝑖 (𝑖 = 1,2,3). When we use Einstein summation convention, it can be written in a short form as follows
(3.16) (𝑖ℏ𝛾𝜇𝜕𝜇 - 𝑚𝑐)𝜓 = 0,
where 𝜕𝜇 = ∂/∂𝑥𝜇. And if we use natural units such that ℏ = 𝑐 = 1, the equation takes the simpler form as
(𝑖𝛾𝜇𝜕𝜇 - 𝑚)𝜓 = 0.
which can be written the simplest form using Feynman slash notation, that is, (𝜕/) ≡ 𝛾𝜇𝜕𝜇 as follows
[𝑖(𝜕/) - 𝑚]𝜓 = 0.
According to the principle of relativity 𝜓'(𝑥') must be a solution which has the form (3.16)
(3.19) (𝑖ℏ𝛾'𝜇∂/∂𝑥'𝜇 - 𝑚𝑐)𝜓'(𝑥') = 0, ※
in the primed system, too. And the Dirac equation satisfies the form invariance for the relativity.
Finally, it is worth noting that that all the coordinates 𝑥0, 𝑥1, 𝑥2, 𝑥3 have the symmetric positions and roles!
Hole Theory [1]
The negative 𝐸 solutions to the equation were problematic, but mathematically there is no reason for us to reject the negative-energy solutions. To cope this problem, Dirac introduced hypothesis, known as hole theory, that the vacuum is the many-body quantum state where all the negative-energy electron eigenstates are occupied. It is called Dirac sea. Dirac reasoned that if the negative-energy eigenstates are incompletely filled, each unoccupied eigenstate - called a hole - would behave like a positively charged particle. the hole possesses a positive energy since energy is required to create a particle-hole pair from vacuum. The hole was eventually identified as the positron, experimentally discovered by Carl Anderson in 1932.
However in quantum field theory, a Bogoliubov transformation on the creation and annihilation operators (turning an occupied negative-energy electron state into an unoccupied positive energy positron states and an unoccupied negative-energy eclectron state into an occupied positive energy positron state) allows us to bypass the Dirac sea formalism, formally it is equivalent to it.
Antimatter of the Utmost Gravity [6]
Dirac gave his Nobel lecture on 12 December 1933 and suggested that Universe could contain both matter and antimatter without us knowing. If Dirac were right, the whole Universe should be a uniform mix of matter and antimatter. Where is this antimatter? Certainly there is no accumulated antimatter on Earth, nor even in the Solar System. If the Universe contains antimatter, examples being the positron and the kaon, then cosmic ray should also contain in cosmic rays.'Fountains' of positrons have been seen by satellite-borne detectors peering into the centre of our Galaxy, but there are no fountains of any other sort of antiparticle.
Where has all the Big Bang antimatter gone? If we cannot see any antimatter, perhaps matter and antimatter are separated into distinct domains. The Universe we know is a matter domain. Maybe somewhere else there is a corresponding antimatter domain. Wherever and whenever the boundaries of the domain and antidomain briefly touched, pieces of matter and antimatter would have mutually annihilated to give powerful bursts of radiant energy - gamma rays. As the Universe subsequently cooled down, these gamma rays would have cooled down too and produced a dim but uniform cosmic gamma-ray signal all over the sky. In 1991, the Space Shuttle Atlantis placed into orbit a new eye, the Gamma Ray Observatory (GRO). GRO clearly saw bursts from out space against a faint but uniform gamma-ray backdrop. The gamma-ray background shows no sign of matter-antimatter annihilation processes ever having taken place on a large scale. the Universe we can see looks to have been eternally free of nuclear antimatter.
The evolution of the whole Universe has been and still is controlled by the all-pervading forces of gravity and gravity will ultimately seal its fate. If a certain probability of finding a transient quantum pin-point is plugged into Einstein's equation of general relativity, the equation reveal that the quantum bubble expands faster even than the speed of light, doubling its size in just 10-34 of a second. Through called 'inflation' by cosmologist, the matter and antimatter had to resolve their differences, This tug-of-war between the initial expansion phase of gravity and the subsequent pull of aggregate matter has been continuing eve since. However Black hoes caused by the collapse of an antimatter star would look just the same as any other black hole. Moreover, the expansion of the Universe now appears to be gently speeding up rather than slow down. To allow for these effects, some bold theorists suggested that the new face face of gravity is repulsive between matter and matter or antimatter, and antimatter, but is attractive between matter and antimatter. Our understanding of cosmology and the origin of the Universe would require a major rethink, a Copernican revolution for the twenty-first century.
The Mystery of the Missing Antimatter [7]
The established wisdom is that the energetic fireball of the Big Bang fourteen billion years ago spawned matter and antimatter in perfect balance. If in the first moment matter and antimatter emerged equally from the Big Bang, an instant later they should have annihilated one another. The mystery is more a question of why has matter survived? Searches for antimatter in the rays above the atmosphere are being made by the AMS (Anti Matter Spectrometer) satellites and so on. However none has been seen, not even anything as simple as antihelium but the abundance of individual positrons and antiprotons. All of the evidence suggests that, with the exception of transient antiparticle, everything within several hundred million light years of us is made of matter. Everything that we know about the early universe, from theory, observation and the results of experiments at LEP (Large Electron-Positron Collider), suggests that in the hot aftermath of the Big Bang those numbers would have been ten billion quanta of radiation, ten billion antiprotons, and ten billion and one protons. The inference is that one of the first acts after creation was a Great Annihilation such that the matter-dominated universe today is made from the surviving one out of ten billion protons.Somewhere in the first moments of the universe, earlier than the billionth of a second that was studied by the experiments LEP, an imbalance between matter and antimatter must have emerged.
* Reference: [1] T. Reding Dirac equation (Wikipedia 10 March 2021)
[2] D. A. Fleisch A Student's Guide to the Schrödinger Equation (Cambridge University Press 2020)
[3] P. A. M. Dirac The Principles of Quantum Mechanics (4th edition, Oxford University Press 1958)
[4] W. Greiner Relativistic Quantum Mechanics - Wave Equations (3rd edition, Springer 2000)
[5] Rae Alastair I. M; Napolitano J. Quantum Mechanics (6th edition CRC Press 2016)
[6] G. Fraser Antimatter - The Ultimate Mirror (Cambridge University Press 2000)
[7] F. Close Antimatter (2nd edition, Oxford University Press 2018)
※ attention: the detailed rigorous derivations are omitted here
