Space-Like Particle Production: an Interpretation Based on the Majorana Equation

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

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,𝑀𝑀 then the maximum possible number of 𝑘𝑘 of mass, the particle is in the fundamental state. All other bradyons𝑚𝑚 𝑘𝑘 obtained𝑏𝑏 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 𝑚𝑚푎푥 푏 4) The number of 𝑏𝑏tachyons∑ 푘=1obtained𝑚𝑚푘 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 𝑚𝑚𝑚𝑚𝑚𝑚 is: 𝑚𝑚 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 𝑚𝑚푎푥 2 particles; however, there are no restrictions to apply it to Lemke concluded that푡 to ensure�𝑚𝑚푚푖푛 � 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 푞𝑘𝑘 ≥ 𝑀𝑀 concise but comprehensive way, reviews Lemke’s � 2 2 In this constraint, is−푝 the푙 ≥four 𝑀𝑀-momentum of the kth kinematics theory and the discrete mass spectrum bradyon produced in the decay. In the case of constraint (5) 𝑘𝑘 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 𝑀𝑀mass ( = 1,2, … , ) , and massless 𝑘𝑘 𝑚𝑚luxons𝑘𝑘 [13]. To 𝑏𝑏comply with covariant kinematics,푡 the 푙 2.2. Bradyons and Tachyons from the Perspective of decay must hold the following𝑚𝑚 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,𝑷𝑷 it must satisfy the following 𝑘𝑘 requires an infinite-dimensional representation of the constraints: 𝒑𝒑 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: 푝𝑘𝑘푝푙 ≤ 푙 𝑘𝑘 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 − 휷𝑚𝑚 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 𝑚𝑚0 ∑ 푝 ≤ 푙 푡 𝑚𝑚 1 tachyonic mass is imaginary. Since the momentum of the In equation (7), 𝑚𝑚 is 퐽the rest�2 +퐽mass푛� of the particle and

𝑚𝑚0 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:

𝑚𝑚 = + = 1,2, … (8) ( ) = = 1,2, … (11) 퐽 ( ) 𝑚𝑚0 In (8), is the 𝑚𝑚spin of the particle (i.e. the particle’s 퐽 푠 푛 푛 In (11), 𝑚𝑚is 푛the order𝑚𝑚+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 𝑚𝑚 𝑚𝑚 퐽 > 퐽 𝑚𝑚 ( + 1) = intrinsic angular momentum and have an 퐽 ( )( ) (12) occupation probability proportional to ( / ) [15]: 𝑚𝑚0 𝑚𝑚 𝑚𝑚+1 𝑚𝑚+2 퐽 푠 𝑚𝑚 The energy produced∆𝑚𝑚 푛 → in 푛 this transition is: = ( / ) ( / ) 푣 푐 (9) = (13) 𝑚𝑚 𝑚𝑚+1 ( )( 2 ) In (9), is the푃𝑚𝑚 particle� 푣 푐velocity− 푣 and푐 is the speed of 훾𝑚𝑚0푐 light. This equation shows that the occupation probability Equation (13) holds∆퐸 if the 𝑚𝑚+1produced𝑚𝑚+2 bradyons have the of an excited푣 state increases with the particle’s푐 velocity. same velocity of the original particle. In other words, the At a constant speed, the probability decreases with order Lorentz factor does not change after the transition. Let us of the excited state. suppose this energy leads to the production of a tachyon. The tachyonic solutions of the Majorana equation This must be a space-like solution of the Majorana 푛 instead have a continuous spectrum of mass and energy equation for the original particle: that can be both positive and negative. In addition, these states may be evident as excited states of the initial = (14) particle. In fact, when the particle’s velocity approaches 2 >2 2 2 4 with the constraint 퐸 푝 푐 − 𝑚𝑚. 0Obtaining푐 from (13) the speed of light, the bradyonic mass spectrum tends to the term and 2knowing2 2 4that = 0 become continuous to join up with the tachyonic one. (where is2 4the tachyon’s푝 푐 𝑚𝑚 velocity),푐 2 2the inequality2 2 2 2 0 0 Because of its space-like solutions, the Majorana > 𝑚𝑚 푐gives us: 푝 푐 훾 𝑚𝑚 푣 푐 equation is not local. Moreover, since the matrix must 2 2 푣2 4 0 > [( + 1)( + 2)] (15) be positive definite, the CPT theorem does not hold. For 푝 푐 𝑚𝑚 푐 2 휷 2 ∆퐸 2 instance, one of the transformations this theorem provides 4 2 2 is: Making explicit푣 the훾 Lorentz𝑚𝑚0푐 푛 factor 푛and considering that > , (15) becomes: (10) 0 0 In (10), = = . However, since must be 푣 푐 2 [( + 1)( + 2)] > 1 (16) 휸 → −휸 2 2 2 ∆퐸 �푐 −푢 � 2 positive definite,0 it−1 cannot† have negative eigenvalues. 2 8 Thus, transformation휸 휷 (10)휷 is not possible. 휷Using the Equation (16)𝑚𝑚 0yields푐 the푛 maximum푛 value of the initial Majorana equation to investigate the decay of an ordinary particle’s mass needed because the following decay process takes place: particle in bradyons, luxons and tachyons means avoiding 0 the problems arising from quantum field theory extended 𝑚𝑚 < ( + 1)( + 2) (17) 2 2 to superluminal particles. �푐 −푢 � 4 The solutions of the Majorana equation are compatible This mass 𝑚𝑚gets0 smaller∆퐸 푐 and smaller푛 푛as the velocity with the constraints that Lemke’s theory provided. For approaches the speed of light. In addition, according to (9), instance, the bounded momentum of the produced this increases the occupation probability of states with푢 bradyons and their limited number are coherent with their high , with a consequent reduction of amplitude . If discrete mass spectrum. In fact, given the velocity of the the decay leads also to the production of luxons, then the decaying particle, with (9) it is possible to evaluate which tachyon’s푛 velocity will be lower than (15). Particularly,∆퐸 occupied state is more probable and thus, which is the when = , the decay produces only bradyons and more probable mass of the produced bradyons. According luxons. to this picture, the produced bradyons are not more than Notably,푣 푐this mechanism always leads to the production excited states of the initial particle with lower mass. The of bradyons with intrinsic angular momentum higher than that of the decaying particle. It never leads to the energy emitted in this transition leads to the production of 𝑚𝑚 luxons and tachyons. production of only tachyons. This result is in agreement퐽 with the second constraint of Lemke’s theory. The rest mass of the produced bradyon is ( + 1) = 3. Decay Mechanisms of an Ordinary /( + 2). Therefore, according to (3), the maximum number of bradyons obtained from the decay is:𝑚𝑚 푛 Particle 𝑚𝑚0 푛 = = ( + 2) ( ) (18) Let us consider a fermion with rest mass and spin 𝑚𝑚0 , travelling at a subluminal velocity .We may rewrite The higher the 𝑏𝑏order𝑚𝑚푎푥 of 𝑚𝑚the𝑚𝑚+1 excited푛 state, the greater is 𝑚𝑚0 푠 푢 22 Space-like Particle Production: An Interpretation Based on the Majorana Equation

the number of bradyons produced. The eventual production of luxons does not change the When = 0 , the decaying particle is in the mechanism of this possible decay. fundamental state and the produced bradyon is the first Let us generalize the decay mechanism that leads to the excited state.푛 With this value, (13) yields: production of bradyons and tachyons. In particular, let us suppose that the produced bradyon occupies an excited = (19) 2 state with an intrinsic angular momentum with 𝑚𝑚0푐 = 2,3, …, and that its velocity is different from that of 0 𝑚𝑚+𝑘𝑘 while for other excited∆퐸 states 훾 2 . Since this is the the decaying particle. The energy emitted in the퐽 transition energy for the production of tachyons (and eventually 0 𝑘𝑘from the initial particle to the produced bradyon is: luxons), it follows that it is low∆퐸 bounded. ≤ ∆퐸 This result is in agreement with the first constraint of Lemke’s theory. = (25) ( 2 ) ( 2) From (13), it is possible to obtain the mass of the 𝑚𝑚0푐 𝑚𝑚0푐 produced tachyon. For this purpose, the energy of a The obtained energy∆퐸 훾goes2 𝑚𝑚+𝑘𝑘+1 toward− 훾the2 𝑚𝑚+1production of the tachyon is [18]: tachyon. Supposing this energy is positive, we can write: = > 0 = 1 (20) ( ) ( ) (26) 2 1 𝑚𝑚 2 푣∙푢 2 훾 훾 2 2 2 0 푡 0 푡 푣 푐 𝑚𝑚+𝑘𝑘+1 𝑚𝑚+1 퐸 � 푐 푠푔푛 � − � In turn,𝑚𝑚 (26)푐 �gives us: − � 훾 𝑚𝑚 푐 The function is푐2 −1the anomalous factor that Park’s ( ) > 0 > theory introduced [18] about the relativistic dynamic of ( ) ( ) ( ) (27) tachyons. Therefore,푠푔푛 we can write: 훾2 훾1 𝑚𝑚+𝑘𝑘+1 Equation𝑚𝑚+𝑘𝑘+1 (27)− holds𝑚𝑚+1 if velocity⇒ 훾 2 훾of1 the𝑚𝑚+1 produced 1 = + bradyon is lower than velocity of the decaying 2 ( + 1)( + 2) 2 0 particle. However, according to (9),푢 the occupation 푡 𝑚𝑚 푐 1 1 ∆퐸 퐸 ℎ휈 ⇒ 2 probability of the excited state is lower푢 than that of the 푣 푛 푛 � 2 − = 푐 1 + (21) decaying particle. Therefore, this mechanism of decay is 𝑚𝑚푡 2 푣∙푢 less favourable than the one leading to the production of a 2 2 푣 푐 bradyon with a velocity higher than that of the initial � 2−1 푐 푠푔푛 � − � ℎ휈 From this, we get푐 the tachyonic mass: particle. Things change if the decay leads to the production of a tachyon with negative energy. In this case,

= 1 = 푣2 (27) becomes: 2 ( )( ) � 풗∙풖 𝑚𝑚0푐 1 푐2−1 ( ) 푡 2 2 2 < (28) 휇 𝑚𝑚 푠푔푛 � − 푐 � � 푣 𝑚𝑚+1 𝑚𝑚+2 − ℎ휈� 푐 ( ) �푐2−1 (22) 𝑚𝑚+𝑘𝑘+1 2 1 Equation (22) proves that not only is the tachyon’s that holds if > 훾. In this훾 way,𝑚𝑚+1 the produced bradyon has an occupation probability greater than that of the energy bounded, but so is that of the possible 2 1 decaying particle.푢 푢Therefore, this mechanism is more luxons—there are no singularities concerning the energy. favourable than the ones investigated above. The result This strengthens the first constraint of Lemke’s theory. obtained fulfils the fourth constraint of Lemke’s theory. Let us consider the case in which the initial particle The investigation performed in this section is also valid decays in a bradyon without the production of tachyons in the case of a bosonic particle and must be performed and luxons. We are therefore considering a transition of using (7) with integer number . the decaying particle from an excited state to an ( ) 𝑚𝑚 excited state + = 1,2, … . In the hypothesis that 퐽 the energy is conserved, we can write: 푛 푛 𝑘𝑘 𝑘𝑘 4. Conclusions = (23) ( 2) ( 2 ) 𝑚𝑚0푐 𝑚𝑚0푐 This analysis reviewed the decay of an ordinary particle If and are훾1 𝑚𝑚+1respectively훾2 𝑚𝑚+𝑘𝑘+1 the velocity of the in bradyons, luxons and tachyons from the perspective of decaying particle and that of the produced bradyon, quantum mechanics. It used the Majorana equation for a 1 2 equation푢 (23) gives푢 us: particle with an arbitrary spin. The mechanism for the ( ) decay proposes that: = ( ) (24) ( 2) a) The initial particle must be in a high Majorana 2 2 𝑚𝑚+1 2 2 2 excited state, which corresponds to high velocity Therefore, the푢2 energy푐 − produced𝑚𝑚+1+𝑘𝑘 in푐 the− 푢decay1 gives only and high intrinsic angular momentum. a kinetic contribution to the obtained bradyon. Its velocity is much closer to the speed of light, as its intrinsic angular b) The particle decays in a bradyonic excited state momentum is higher. Notably, if , then also with lower mass and higher intrinsic angular tends to . In addition, according to (9), the particle momentum. The energy emitted in this transition 𝑚𝑚+𝑘𝑘 1 tends not to퐽 decay. The occupation probability푢 ≅ 푐 of the leads the production of tachyons and/or luxons. 2 pro푢 duced bradyon푐 compares with that of the initial particle. The investigation performed leads to the following Universal Journal of Physics and Application 12(2): 19-23, 2018 23

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