Universal Journal of Physics and Application 12(2): 19-23, 2018 http://www.hrpub.org DOI: 10.13189/ujpa.2018.120201 Space-like Particle Production: An Interpretation Based on the Majorana Equation Luca Nanni Department of Physics and Earth Sciences, University of Ferrara, Italy Copyright©2018 by authors, all rights reserved. Authors agree that this article remains permanently open access under the terms of the Creative Commons Attribution License 4.0 International License Abstract This study reconsiders the decay of an subluminal and superluminal massive particles without ordinary particle in bradyons, tachyons and luxons in the encountering the problems mentioned. For instance, if a field of the relativistic quantum mechanics. Lemke already theory imposed compliance with the CPT theorem, then investigated this from the perspective of covariant the relationship between spin and statistics would be kinematics. Since the decay involves both space-like and reversed [9]. Instead, the introduction of a privileged time-like particles, the study uses the Majorana equation reference frame would solve the problem of vacuum for particles with an arbitrary spin. The equation describes instability. However, this would weaken the theory of the tachyonic and bradyonic realms of massive particles, relativity’s second postulate on the equivalence of inertial and approaches the problem of how space-like particles reference frames [10]. might develop. This method confirms the kinematic Although theoretical physics is making progress in this constraints that Lemke’s theory provided and proves that field [11], no experiment has ever proved the existence of some possible decays are more favorable than others are. tachyons directly or indirectly. Considering the high Keywords Tachyons, Bradyons, Infinite Components technology of current measuring instruments, the lack of Wave Functions experimental proof strengthens the position of sceptics on the nonexistence of superluminal particles. However, we cannot exclude a priori the possibility that phenomena leading to the production of tachyons have a low probability of occurrence and/or take place only in 1. Introduction extreme conditions not yet accessible to current measuring The study of faster-than-light particles is a branch of apparatuses [12]. These are good reasons to continue the theoretical physics still much debated. It leads to research on tachyon physics. Their experimental speculations and discussions ranging from a purely confirmation could radically change current cosmological scientific scope to a metaphysical-philosophical one [1-5]. theories such as the inflationary one. In the second half of the last century, several physicists The purpose of this work is to study the decay of an developed an intensive effort to extend the theory of ordinary particle in bradyons (massive particle travelling relativity. Their goal was to apply it to massive particles at a speed lower than that of light), luxons (massless travelling at velocities higher than the speed of light. particle travelling at the speed of light) and tachyons Among these physicists, the names of Recami, Surdashan (massive particle travelling at a speed higher than that of and Feinberg stand out [5-7]. They introduced the light) within the framework of quantum mechanics. Such reinterpretation principle, similar to the one a source of superluminal particles could drive Feynman-Stueckelberg proposed to explain the negative experimental research in the right direction. Lemke has energy of antiparticles in quantum field theory. This investigated the covariant kinematics of this phenomenon solved the superluminal propagation dilemma by restoring [13]. However, a theory for the possible mechanism of the principle of causality. Yet things do not go as well this occurrence still does not exist. This work attempts to when we attempt to introduce the tachyon into quantum overcome this lack by using the Majorana equation for field theory. Problems such as vacuum instability and the particles with an arbitrary spin [14]. It is possible to violation of change, parity and time reversal (CPT) propose a mechanism of production through the concept symmetry are hard to solve [6,8-10]. Physicists have of excited state [15]. This avoids the difficulties arising solved these issues separately; but so far, there is not a when applying quantum field theory to superluminal field theory able to explain the quantum behaviour of particles. 20 Space-like Particle Production: An Interpretation Based on the Majorana Equation 2. Methodological Approach decaying bradyon is real and positive, it follows from constraint (2) that a decay that produces only tachyons is The Majorana equation is a powerful tool for not possible. investigating particles with an arbitrary spin. It can help 3) In accordance with constraint (1), the total energy explore phenomena leading to the production of bradyons of the produced tachyons is lower bound. It and tachyons, since it describes the behaviour of both follows that all other particles obtained from the subluminal and superluminal massive particles [14]. For decay have a bounded momentum. Therefore, if the bradyonic realm, this equation leads to a discrete mass is the rest mass of the decaying particle and spectrum that depends on the particle’s intrinsic angular ( = 1, … , ) is the rest mass of the produced momentum. When the reference frame is that of the centre particles, then the maximum possible number of of mass, the particle is in the fundamental state. All other bradyons obtained is: states with increasing intrinsic angular momentum have a decreasing mass. Their occupation probability increases = (3) 푀 with the particle’s velocity [15]. The transition from a 푎푥 푏 4) The number of tachyons∑ 푘=1obtained푘 from the decay given quantum state to another with a higher intrinsic is limited. If is the smallest mass of the angular momentum decreases the particle’s rest mass. The produced tachyons, then their maximum number transition also emits energy for the production of luxons is: and tachyons. This is the mechanism proposed for the decay of an ordinary particle that Lemke discussed [13]. < (4) 2 Majorana formulated an equation for elementary 푀 푎푥 2 particles; however, there are no restrictions to apply it to Lemke concluded that푡 to ensure�푚푖푛 � lepton and baryon ordinary particles. The scientific literature includes uses of number conservation, the following constraint must hold: the Majorana equation to investigate composite systems like the hydrogen atom [16]. This also justifies the (5) 2 2 equation’s use in this work. The discussion below, in a 푞 ≥ concise but comprehensive way, reviews Lemke’s � 2 2 In this constraint, is−푝 the푙 ≥four -momentum of the kth kinematics theory and the discrete mass spectrum bradyon produced in the decay. In the case of constraint (5) obtained by solving the Majorana equation. above, not all obtained풒 tachyons can have positive definite energy. The latter case can only allow emission of 2.1. Relativistic Kinematics for the Production of tachyons with no positive energy. Space-Like Particles In the next section, we prove that the kinematics constraints of Lemke’s theory meet the results obtained by Lemke investigated the decay of an ordinary particle solving Majorana’s equation. This demonstrates that the with rest mass in a number b of bradyons with mass equation is a valid tool to confront the problem of tachyon ( = 1,2, … , ). He also examined a number of production. tachyons with mass ( = 1,2, … , ) , and massless luxons [13]. To comply with covariant kinematics,푡 the 푙 2.2. Bradyons and Tachyons from the Perspective of decay must hold the following 푙 constraints:푡 Majorana’s Equation 1) To denote by the four-momentum of the ordinary particle, and by that of the kth A relativistic quantum theory that includes tachyons tachyon produced, it must satisfy the following requires an infinite-dimensional representation of the constraints: Lorentz group (1,3) [7-17]. Majorana formulated his 0 = 1, … , equation using just this algebraical framework [14]. (1) 0 , = 1, … , Therefore, it is suitable푆푂 to study processes where tachyons 푃푝푙 ≥ 푙 푡 In the reference� frame of the decaying particle’s and bradyons are involved. Majorana’s equation is: 푝푝푙 ≤ 푙 푡 centre of mass, these constraints ensure that the | = 0 energies of the tachyons are positive definite. They 휕 휕 휕 휕 2 ퟏ ퟐ ퟑ 0 (6) also ensure that the kinematics of the obtained �ퟙ푖ℏ 휕푡 − 휶 푖ℏ 휕푥 − 휶 푖ℏ 휕푦 − 휶 푖ℏ 휕푧 − 휷 푐 � 훹⟩ particles is finite—that there is no singularity In this equation, and are infinite matrices. To avoid solutions with negative energy, Majorana’s equation concerning momentum and energy. 풊 2) The total momentum of the tachyons produced in requires that must휶 be positive휷 definite. This constraint the decay must be space-like: leads to subluminal solutions with a discrete mass spectrum: 휷 0 = 1, … , (2) ( ) = (7) 푡 2 The negative value푙=1 of푙 (2) results from the fact that the 0 ∑ 푝 ≤ 푙 푡 1 tachyonic mass is imaginary. Since the momentum of the In equation (7), is 퐽the rest�2 +퐽mass푛� of the particle and 0 Universal Journal of Physics and Application 12(2): 19-23, 2018 21 is the intrinsic angular momentum given by: the mass spectrum expressed in (7) as: = + = 1,2, … (8) ( ) = = 1,2, … (11) 퐽 ( ) 0 In (8), is the spin of the particle (i.e. the particle’s 퐽 푠 푛 푛 In (11), is 푛the order+1 of푛 the excited state. The intrinsic angular momentum in its fundamental state) and transition from one excited state to the next occurs with a 푠 is the order of the excited state. Equation (7) holds both decrease in mass푛 and an increase in the intrinsic angular for bosons ( is an integer number) and fermions ( is a momentum : half푛 -integer number). All the excited states have an 퐽 > 퐽 ( + 1) = intrinsic angular momentum and have an 퐽 ( )( ) (12) occupation probability proportional to ( / ) [15]: 0 +1 +2 퐽 푠 The energy produced∆ 푛 → in 푛 this transition is: = ( / ) ( / ) 푣 푐 (9) = (13) +1 ( )( 2 ) In (9), is the푃 particle� 푣 푐velocity− 푣 and푐 is the speed of 훾0푐 light.