P. A. M. Dirac conceived antimatter in 1928 as having <i>negative</i> <i>energy</i> by allowing a consistent representation of matter-antimatter annihilation into light. To achieve compatibility with special relativity, particle physics of the early 20th century made the theoretical assumption that antiparticles have <i>positive</i> <i>energy</i>, an assumption that remains in effect as of today. In this note we prove apparently for the first time a theorem stating that positive mass antiparticles violate Dirac’s particle-antiparticle annihilation into light. We then show the consequential unsettled character of the recent gravity test of the anti-Hydrogen atom due to the positive mass of its nucleus. We conclude by suggesting that a final scientific claim on matter-antimatter gravity requires tests on particles with clear antimatter character, such as the 1994 resolutory proposal for the comparative test of the gravity of very low energy electron and positron in horizontal flight on a supercooled vacuum tube.
Following epistemological studies dating back to the 19th century, the first quantitative formulation of antimatter was achieved in 1928 by P. A. M. Dirac [
( i η μ ν γ μ ∂ ν − m c ) | ψ ( x ) 〉 = 0, (1)
where the 4 × 4 matrices γ ^ μ
γ ^ k = ( 0 σ k , − σ k 0 ) , γ 4 = i ( + I 2 × 2 0 0 − I 2 × 2 ) , (2)
verify the anti-commutation rules
{ γ μ , γ ν } = 2 η μ ν , (3)
while Pauli’s matrices σ k , k = 1,2,3 represent the spin S = 1 / 2 of the particles.
According to Dirac’s conception, the electron e = e − is represented by the top 22 component of Equation (1) with unit + I 2 × 2 , thus having positive rest energy m e > 0 , while the antielectron e ¯ = e + (now called the positron), is represented by the bottom 2 × 2 component of Equation (1) with unit − I 2 × 2 , thus having negative rest energy m ¯ e = − m e < 0 .
An important aspect of Dirac’s conception of antimatter is that it allows a consistent representation of particle-antiparticles annihilation into light, such as that for the electron-positron pair [
e − + e + → γ + γ , (4)
which requires the elimination of the masses for the transition from particles to light. Such a transition is evidently possible for Equation (1) in view of the opposite values of the rest energies, while being incompatible with special and general relativities on various counts.
To achieve compatibility with special and general relativities (which was an understandable task for the time), particle physics of the early 20th century [
The above view was implemented by assuming that antiparticles are the image of particles under the PCT theorem, namely, by assuming that the antiparticle image of a particle with Hilbert state | p , s , ψ 〉 (where p is the linear momentum, s is the value of the spin along the third axis and ψ is the particle wave-function), is given by [
P C T | p , s , ψ 〉 = ( − 1 ) J − s | p , − s , n c 〉 , (5)
where parity P, charge conjugation C and time reversal operator T are characterized by
C | p , s , ψ 〉 = | p , s , − ψ † 〉 , C P | p , s , ψ 〉 = | − p , − s , − ψ † 〉 , T | p , s , ψ 〉 = | − p , − s , ψ 〉 . (6)
The 20th century interpretation of Dirac’s equation can be technically seen at the level of its symmetries. Rather than identifying the symmetry of Equation (1) as the Kronecker product of the symmetries of its two distinct 2 × 2-components (as done in the next section), physicists of the early 20th century solely considered the symmetry characterized by the 4 × 4 gamma matrices. By recalling that the electron and the positron have spin 1/2, the most important space-time symmetry is the spinorial covering of the Poincaré symmetry [
P ( 3.1 ) = S L ( 2. C ) × T ( 3.1 ) , (7)
with realization of the generators in terms of Dirac’s gamma matrices
S L ( 2. C ) : S k = 1 2 γ k × γ 4 , R k = 1 2 ε i j k γ i × γ j , T ( 3.1 ) : P μ , (8)
where S k , R k and T μ are the generators of spin, boosts and linear momentum, respectively, with the familiar commutation rules for J μ , ν = { S k , R k }
[ J μ ν , J α β ] = i ( η ν α J β μ − η μ α J β ν − η ν β J α μ + η μ β J α ν ) , [ J μ ν , P α ] = i ( η μ α P ν − η ν α P μ ) , [ P μ , P ν ] = 0 , (9)
and Casimir invariants
C 1 = I , C 2 = P 2 = P μ P μ = ( η μ ν P μ P ν ) , C 3 = W 2 = W μ W μ , W μ = ε μ α β ρ J α β P ρ . (10)
Unfortunately, the above 20th century theory of antimatter is incompatible with Dirac’s matter-antimatter annihilation into light according to the following:
THEOREM 1: The PCT definition of antiparticles, Equation (6), under the spinorial Poincaré symmetry, Equation (7), is incompatible with Dirac’s particle-antiparticle annihilation into light, Equation (4).
PROOF: Within the class of unitary equivalence of relativistic quantum mechanics, there exists no possibility of rendering null the sum of two positive masses in Equations (10).
The above Theorem suggests a moment of reflection as to whether the recent test of the gravity of the anti-hydrogen atom [
1) Theorem 1 creates doubts as to whether the antiproton, which is the nucleus of the anti-hydrogen atom [
2) In regard to the production of antiprotons via the collision of high energy protons against a matter-target (usually an iridium rod), it should be noted that Rutherford’s [
3) The isodual mathematica and related isodual theory of antimatter [
In particular, the isodual mathematics maintain the full validity of special and general relativities only formulated on isodual spaces over isodual numeric fields. Matter-antimatter gravitational repulsion them emerges rather forceful in the projection of the isoidual special and general relativities in the space-time of conventional relativities [
Additionally, the isodual theory of antimatter implies that, to be true antimatter and allow valid gravity tests, antiparticles should solely annihilate into electromagnetic radiation without massive residues, as apparently confirmed by very large explosions in our skies without solid debris in the ground (for brevity, see the detailed analysis of Ref. [
In conclusion, Theorem 1 casts shadow on the true antimatter character of 20th century antiprotons in view of the rather natural expectation that negative energy antiparticles and positive energy particles cannot annihilate into light, but merely experience ordinary scattering with the production of massive particles as established by the proton-antiproton scattering of the Bose-Einstein correlation [
In the preceding sections, we have presented various arguments essentially implying doubts as to whether the recent test of the gravity of the anti-Hydrogen aton [
By keeping in mind that the sole antiparticle with proved antimatter character is the positron in view of annihilation (4), R. M. Santilli [
In particular, the length of the tube can be selected for given eV in such a way to allow a visible deviation when the beams hit a terminal scientillator, and the radius of the tube can be selected to render stray field fluctuations smaller than the expected terminal deviation upward (downward) due to Earth’s gravity when the positron beam hits a terminal scintillator, by therefore clearly indicating antigravity (gravity)
Detailed studies by A. P. Mills [
As indicated at the 1996 international Workshop on Antimatter, Sepino, Italy, the 2011 San Marino Workshop on Antimatter, the 2016 SIPS Conference, Hainan Island, China, the 2023 SIPS Conference in Panama and other meetings, the gravity of antimatter should be tested for particles with independently proved antimatter character, such as the positrons, prior to any final claim in favor or against antigravity.
The author would like to thanks the participants of the 2020 International Teleconference on the EPR argument, the 2021 International Conference on Applied Category Theory and Graph-Operad-Logic dedicated to the memory of Prof. Zbigniew Oziewicz, the Seminars on Fundamental Problems in Physics and all participants of the 2023 International Conference on the Sustainability through Science and Technology. Additional thanks are due to Simone Beghella-Bartoli, Jonah Lissner, Valeri V. Dvoeglazov, Peter Rowlands, Piero Chiarelli, Arun Muktibodh and other colleagues for penetrating critical comments. Thanks are finally due to Mrs. Sherri Stone for linguistic control of the manuscript.
The author declares no conflicts of interest regarding the publication of this paper.
Santilli, R.M. (2024) Implications for Antigravity Tests of Dirac’s Negative Energy Antiparticles. Journal of Applied Mathematics and Physics, 12, 1661-1667. https://doi.org/10.4236/jamp.2024.125103