Quantum Theory of Half-integer Spin Free Particles from the Perspective of the Majorana Equation

Transcription

1 Quantum Theory of Half-integer Spin Free Particles from the Perspective of the Majorana Equation *Luca Nanni *corresponding author Pacs: 3.65Pm Abstract: In this study, the Majorana equation for particles with arbitrary spin is solved for a half-integer spin free particle. The solution for the fundamental state, corresponding to the reference frame in which the particle is at rest, is compared with that obtained using the Dirac equation, especially as regards the approximation in the relativistic limit, in which the speed of the particle is close to that of light. Furthermore, the solutions that Majorana defines unphysical, proving that their occupation probability increases with the particle velocity, are taken into consideration. The anomalous behavior exhibited by these states also shows that for high-energy particles with small mass, transitions from a bradyonic state to a tachyonic state become possible. Keywords: particles with arbitrary spin, infinite component wavevector tachyon.

2 Introduction In 932, Majorana formulated an equation for particles with arbitrary spin. This equation is relativistically invariant and valid for both bosons and fermions []. However, if the spin and mass of a particle are set, this equation leads to an infinite set of solutions with a spectrum of decreasing masses. In his article, Majorana considers these solutions unphysical or accessible only under extreme conditions, focusing the attention on the fundamental state, where the reference system is that of the particle at rest. Furthermore, the equation allows solutions with imaginary mass, compatible with the tachyonic behavior of matter. These innovative results were premature for the considered historical period and were the reason for lack of interest in the Majorana equation of quantum physics of the 93s [2 3]. Moreover, the Dirac equation was proved effective in explaining the spectrum of the hydrogen atom correctly. Additionally, the existence of the positron, whose discovery was announced in 932, was theorized using the Dirac equation. However, the Majorana theory was recently considered in the context of high-spin particle physics [4 7]. In this paper, the relativistic quantum theory of Majorana for a free massive particle with halfinteger spin is presented. The results, where possible, are compared with those obtained by solving the relativistic Dirac equation [8]. If the spin and mass of the particle are set, the Majorana equation provides the solution for both the fundamental state and all infinite states considered unphysical by Majorana and referred to as excited states with increasing angular quantum number J by us. Each solution corresponds to a given projection of J along the z- axis, and is represented by a vector with infinite components. The nonzero components of this vector correspond to the nonzero elements of infinite matrices α k and β. The method of calculation of these vectors is explained in the following sections. The Majorana equation is relativistically invariant, used for free particles, and valid for any value of the spin and velocity of the particles. This equation does not allow solutions with negative energy (Majorana antiparticle). Furthermore, unlike the Dirac equation, the Majorana equation was formulated without the need of compliance with the formula of the relativistic energy: E 2 = p 2 c 2 m 2 c 4. For each solution corresponding to the same m J (magnetic quantum number), the wavevector given by the linear combination of all infinite solutions, in compliance with the principle of superposition of states, is constructed. Then, the probability of occupation of each possible state as a function of the Lorentz factor (v/c) and the energies of transitions between any possible states are calculated. Moreover, the physical behavior of the particle in the limit v c is analyzed in detail using the Heisenberg uncertainty principle. 2 Solution of the Majorana Equation for the Fundamental State The explicit form of the Majorana equation with infinite components is []: (iħ t cα iħ x cα 2iħ y cα 3iħ z βm c 2 ) Ψ = () where α k and β are the infinite matrices while is the infinite unity matrix. In particular, we want to study the fundamental state of a free particle with half-integer spin that is consistent with that reported in the original Majorana article and corresponds to the case, in which the reference system is that of the particle at rest. Moreover, the nonzero components of matrices α k must coincide with those of Dirac. Matrix β must be definite and positive (all positive eigenvalues), and must satisfy the algebraic relationship β = β. Since these matrices are infinite, the Majorana equation, whose form is the same as the form of the Dirac equation, is reduced to an infinite system of linear differential equations. However, because the nonzero 2

3 components of the matrices are related to the first four rows and four columns, the system of infinite equations is reduced to a system similar to that of Dirac, with the difference in the structure of matrix β. The easiest and most direct way to assure positive values of the energy, as required by the Majorana theory, is to force infinite matrix β to be unitary: β = This form is consistent with the form predicted by the Majorana theory []: β = s+ 2 where s is the particle spin. The explicit form of matrices α k is given by: σ k α k = ( σ k ) where σ k are the Pauli matrices [9]. The 4-vector signature is (+,,, ). By inserting these infinite matrices into Eq. (), we obtain a system of four linear differential equations of the first order. Considering only the nonzero elements of the system, the associated Hamiltonian matrix is given by: H = E = cp z ( c(p x ip y) m c 2 cp z c(p x + ip y) m c 2 c(p x ip y) cp z c(p x + ip y) cp z m c 2 m c 2 ) (2) where operator E is represented by the matrix: E E = E E E The solutions of Eq. () are plane waves []: E Ψ j = u j (p)exp [ i c t x p ħ ] (3) 3

4 where x and p are the space-like components of the 4-position and of the 4-momentum, respectively. Using the following expressions: (E + m c 2 ) Ψ = (E + m c 2 ) Ψ p k Ψ = p k Ψ the four solutions of the Majorana equation, whose spinors (limited to the first four components) are obtained as: u M up = u M up = ( ( E+m c 2 c(p x ip y ) E+m c 2 ) m c 2 +E c(p x ip y ) m c 2 +E ), u M down = (, u M down = c(p x +ip y ) E+m c 2 E+m c 2 ) ( c(p x+ip y ) m c 2 +E m c 2 +E ) (4) (5) Since the energy is always positive, to distinguish the particle and antiparticle states, the superscript symbol on the antiparticle spinors with the opposite spin is introduced. Subscripts up and down indicate the two possible spin states. Moreover, to distinguish the Majorana spinors from the Dirac spinors unambiguously, M is used to indicate the Majorana wavevector. Similar to the solutions of the Dirac equation [8], those of the Majorana equation are reduced to the Schrödinger eigenfunctions when the velocity of the particle is much lower than that of light: cp k m c 2 ±E = c m v k = m v k m c 2 ±E m c± E c v k c Let us consider the case when v k c. Using the approximation: cp k mc 2 ±E = γm v k ± v k m c 2 ±γm c 2 c v k c we obtain the following spinors: u M up = ( ), u M down = ( ) v + i k c (6) i ( + i) M ( i) = ( ), u M down = ( ) v k c (7) u up 4

5 The phase of each spinor is the same for the solutions of the Dirac equation, i.e., e ip x/ħ. Since the matrices are infinite, the Majorana spinors have infinite components whose values, beyond the first four components, can differ from zero being multiplied with the zero values of the infinite matrices. However, as shown further, we can assume (without compromising the physical meaning of the eigenfunction) that all infinite components of the Majorana spinor beyond the fourth position are zero. Let us compare the Majorana spinors with the Dirac spinors. The latter are given by: u up(+) = u up( ) = ( ( E+m c 2 c(p x ip y ) E+m c 2 ) m c 2 E c(p x ip y ) m c 2 E ), u down(+) = (, u down( ) = c(p x +ip y ) E+m c 2 E+m c 2 ) ( c(p x+ip y ) m c 2 E m c 2 E ) (8) (9) Subscripts (+) and ( ) indicate the positive and negative values of the energy, respectively. The first two expressions exhibit the states of the particle with spin up and spin down, while the latter exhibit the states of the antiparticle with spin up and spin down. In the limit v k c, the Dirac spinors become: u up(+) = ( ), u down(+) = ( i + i ) v k c + i u up( ) = ( i ), u down( ) = ( ) v k c Although the eigenfunctions of a free particle are not normalizable, it is easy to verify that the scalar u u concerning states of the antiparticle, which is proportional to the probability that state u is occupied, changes depending on the considered spinor. For instance, for state up of the Majorana antiparticle we have: u u M up = c2 p 2 +(m c 2 +E) 2 (m c 2 +E) 2 while in the Dirac theory, such a scalar becomes: u u up( ) = c2 p 2 +(m c 2 E) 2 (m c 2 E) 2 The difference between the two scalars is: 5

6 u u D M up = 4c2 p 2 m c 2 E (m 2 c 4 E 2 ) 2 This difference is always positive, which means that the probability that an antiparticle state is occupied is always smaller in the case of the Majorana theory. For particle states, the predicted probabilities are the same. In the limit v k c, the probabilities of occupation of antiparticle states are equal for both theories, though the energies have opposite signs. 3 Majorana Excited States The Majorana theory for particles with arbitrary spin predicts the existence of excited states with total angular momentum = s + n (n =,2, ), to which the values of the mass correspond []: m Jn = m J n + 2 () Equation () shows that for the fundamental state, mass m s equals the mass of the particle at rest, and it decreases progressively for the excited states: m Jn = m () n+ Equation () does not replace the relativistic formula m = γm, where γ is the Lorentz factor, which describes the increase of the inertial mass of the particle with the speed. Let us analyze in detail the first excited state with J = 3/2. Infinite matrices σ k corresponding to this state are easily obtained from the Majorana relationships [], setting J = 3/2 and m J = 3,,, 3 : σ (3/2) = ; σ 2 (3/2) = i σ 3 (3/2) = 2 3 For convenience, we omitted all other infinite zero elements of the matrices. Using these matrices, α k are constructed: α k = ( σ k σ k ) The explicit form of matrix β is []: β(3/2) = 3 =

7 whose nonzero elements are the first eight elements on the main diagonal. By inserting these matrices into the Majorana equation, we obtain an infinite system of linear differential equations, whose first eight terms are nonzero and provide nontrivial solutions. Considering only these terms, the eight spinors can be written as: { { u(3/2) M up =(,,,,B,B 2,B 3,B 4 ) T u(/2) M up =(,,,,B 2,B,B 4,B 3 ) T u( /2) M down =(,,,,B 3, B 4,B, B 2 ) T u( 3/2) M down =(,,,,B 4, B 3,B 2, B ) T (2) u(3/2) M up =(B, B 2,B 3, B 4,,,) T u(/2) M up =(B 2, B,B 4, B 3,,,,) T u( /2) M down =( B 3, B 4, B, B 2,,,,) T u( 3/2) M down =( B 4, B 3, B 2, B,,,,,) T (3) Complex values B, B 2, B 3, and B 4 are the solutions of the system of linear equations and, similar to the solutions obtained in the previous section, they are quadratic functions of the individual components of the energy and of the linear momentum: B, B 2 ( E+m c 2)2 B 3, B 4 ( c(p 2 x+ip y ) ) E+m c 2 v c v c ( v z c )2 ( + i) 2 2i Equations (2) represent particle states, while Eqs. (3) represent antiparticle states. Thus, assuming that the possible quantum states of the particle motion are the fundamental state and the first excited state, the spinor with infinite components in the limit v c is: u(/2,3/2) M up = (,,, + i,,,,,,, ( + i) 2, ( + i) 2,,, ) T (4) Similarly, all other possible spinors are obtained by changing the quantum number m J. Nonzero components B i of the first excited state are proportional to (v/c) 2, and those of the fundamental state are proportional to (v/c). This means that for velocities significantly lower than the speed of light, the components of the excited states become smaller with the increase in their order. Therefore, the excited states of the particle and of the antiparticle have physical meaning only in the limit v c, i.e., under the conditions of high energy. Thus, the components of the excited states are zero only when the particle is in the fundamental state. When the velocity differs from zero, all excited states become alive with their components proportional to (v/c) n (n is the order of the excited state) and the particle is described by the wavevector given by superposition of all infinite states. Therefore, the solutions of the Majorana equation meet the principle of superposition of states in a natural way. According to the theory of relativity, when the velocity of the particle approaches that of light, the inertial mass tends to infinity: lim m = lim γm = v c v c 7

8 However, in the Majorana theory, the mass of the excited states gradually decreases with the total angular quantum number: m Jn = m J n + 2 m, m 2, m 3, Hence, the relativistic form of the inertial mass becomes: m Jn = γ (n+) m n =,,2,3, (5) Equation (5) shows that the mass depends on the two seemingly unrelated variables: the particle velocity (by the Lorentz factor) and the excited state order. Considering these two variables, we can write the following limits: lim m J n n = { lim m J v c n = The order of the excited state depends on (v/c) n, which can be considered as the statistical weight that the particle is described by the nth excited state. Hence, variables γ and n are mutually dependent. In principle, an upper limit for n does not exist, and if speed v is sufficiently close to the speed of light, states with high n can be populated. Therefore, in the Majorana theory, the probability that the particle is in an excited state depends on its velocity. Referring to Eq. (5), in the limit v c, the progressive increase of the Lorentz factor is mitigated by the simultaneous increase of order n of the excited state. This result is topical since it is completely new for current relativistic quantum theories. Let us give some examples to compare the classical relativistic theory with the Majorana theory. The probability that a particle is described by state Ψ i is given by c i 2, i.e., by the square of its linear combination coefficient: φ = c Ψ + + c i Ψ i + + c n Ψ n where φ is the eigenvector obtained by the linear combination of all possible states. The Majorana spinor can also be expressed as the linear combination of the fundamental state and all infinite excited states: φ M = c s + c s c n s + n + (6) This is the reason why in the previous section we considered all components of the spinors as zero that are trivial solutions of the system of infinite linear differential equations. Therefore, the spinor in Eq. (6) is a vector with all nontrivial components, similar to the spinor in Eq. (4). It is easy to verify that the coefficients of the linear combination in Eq. (6) are given by: c n = ( v c )n ( v c )n+ (7) Therefore, the probability that the nth state is occupied is [( v c )n ( v c )n+ ] (this formula is proved in the next section). 8

9 Let us consider a particle with rest mass m = 9, 3 Kg and spin s = /2 (electron) and compare its physical behavior at two different velocities corresponding to ratios (v/c) =,9 and (v/c) =,99. According to the theory of relativity, the inertial masses are [2]: m = γm { m = 2, 3 Kg m = 6,4 3 Kg The work to increase the velocity of the particle from the first value to the second value is given by: W = E 2 E = (γ 2 )m c 2 (γ )m c 2 =,275 MeV (8) Let us apply the Majorana theory to the same physical system. As discussed above, the inertial mass of the particle is calculated using formula (5). However, first, we should determine orders n of the excited states occupied at the two velocities. For ease of calculation let us assume that the particle is in one of the possible states with n =, n = 2, n = 9, and n = 4. The corresponding inertial masses are given by: m M ( v c =,9) = γ (n+) m { m M (n=)=2 3 Kg m M (n=2)=,65 3 Kg m M (n=9)=2 3 Kg m M (n=4)=,3 3 Kg p=% p=8,% p=3,8% p=2,28% where p is the probability that at a given velocity, the particle is in a particular excited state. As expected, if the velocity is set, the inertial mass gradually decreases with the excited state order, while probability p shows the opposite trend. Let us repeat the same calculation for the second velocity: m M ( v =,99) = m c (n+) { γ m M 2 (n=)=6,4 3 Kg m M 2 (n=2)=2, 3 Kg m M 2 (n=9)=,64 3 Kg m M 2 (n=4)=,425 3 Kg p=% p=,98% p=,9% p=,86% In this case, the trend mass as a function of order n shows a maximum and the probabilities of state occupation are considerably lower than those corresponding to the first velocity. The Majorana theory provides the following result: in certain combinations of the particle velocity and the excited state order, the inertial mass decreases up to zero. This result is not consistent with the prevision of the classical theory of relativity. Let us calculate the work to increase the velocity of the particle from (v/c) =,9 to (v/c) =,99. First, we have to decide from which state we start to calculate this work. For simplicity and ease of calculation of some results which will be discussed shortly, we assume that at the lowest velocity, the occupied state is the fundamental state, and we calculate the work corresponding to four states with n =, n = 2, n = 9, and n = 4: W = E 2 E = { (n= n=)=,275 MeV (n= n=2)=,56 MeV (n= n=9)=,75 MeV (n= n=4)=,88 MeV (9) As expected, if the particle remains in the fundamental state, then the value of the work is the same as the one we obtain by applying the classical theory of relativity. However, if the final 9

10 state is one of the possible excited states, then the work decreases. In particular, from the given excited state, the work becomes negative, i.e., the velocity increases without the need to provide energy to the system. This result is even more surprising than the two results mentioned above. If the increase of the velocity occurs in association with a transition to an excited state, to which a lower inertial mass corresponds, then the process takes place with an emission of energy. The yield of this process is determined by the probability that excited states with high n are occupied. As discussed before, this probability is large enough only when the velocity of the particle is close to that of light. The smaller the mass is the easier the process. Particles with very small mass are the main candidates to pass from a bradyonic state to a tachyonic state. Equation (9) shows that the Majorana theory is the gateway to the superluminal behavior of matter. However, how is it possible to achieve the tachyonic regime without crossing the barrier of infinite energy that differentiates the real particles from the superluminal particles [3]? In quantum mechanics, these problems often arise when the Heisenberg uncertainty principle is used []: p q ħ In highly confined regions of space ( q is very small), the uncertainty of the linear impulse becomes very large: p = γm v v = ħ γm q Therefore, when we say that v =,99c, in reality, we have: v =,99c ± ħ γm q (2) If we use the formula of the Majorana mass, then Eq. (2) becomes: v =,99c ± ħ γ(m /(n+)) q (2) γ and (n + ) factors play the opposite role. More precisely, the increase of the particle velocity leads to highly excited states, and the Majorana factor increases the uncertainty of the velocity, while the Lorentz factor tends to decrease it. According to Eq. (2), the particle velocities can exceed the speed of light. Since the mass is in the denominator, it is clear that very light particles enhance this uncertainty and increase the transition probability to the tachyonic regime avoiding the aforementioned problem of the infinite energy barrier. When v = c, all the excited states will be occupied with a unitary probability, and the particle state oscillates among all the infinite possible states. In fact, the general Majorana spinor (see Eq. (6)) has infinitely many components that coincide with the values obtained in Sect. 2 for v = c. In principle, when v = c, all the excited states become possible, and the particle acceleration to a superluminal speed becomes spontaneous with emission of energy. Once the speed of light is overcome, the spinor components, which are proportional to (v/c) n, become larger with an increase in the rate proportional to order n of the excited state. Evidently, in the tachyonic regime, terms (v/c) n cannot be further considered as the probability since the state of the particle is distributed over all possible infinite excited states. Because those with large n become predominant, it can be concluded that with the progressive increase of the speed, the particle state is the most excited state. When n, projections m Jn of quantum number J n on the z-axis are so numerous that it can be considered almost continuous. Let us study the

11 behavior of the inertial mass when v = c: the question arises as to whether the Lorentz factor or the Majorana factor is the predominating term? This question does not require a direct answer because of the Heisenberg uncertainty principle leading the particles beyond the limit of the speed of light. When v = c, the value of the mass is still unknown, and it is not useful to understand the quantum phenomenon that does not follow the principle of causality of the classical physics even in the Majorana theory. To understand the role played by Eq. (2) in the transition from the bradyonic to the tachyonic regime, we consider the following example. An electron is confined in a spherical region with a radius of 2.5 m. It moves with a velocity v =,99c, to which corresponds Lorentz factor γ = 7,. Applying Eq. (2), we get: v = (n + )6,6 6 m/s An electron with velocity v =,99c must have an uncertainty of ±3 Km/s to enter the tachyonic regime. Therefore, the uncertainty of the velocity is negligible, even if the particle is in a highly excited Majorana state. Let us repeat the same gedankenexperiment using a neutrino. The neutrino mass [4] is estimated as,32 ev, corresponding to 5,69 37 Kg kg. In this case the uncertainty of the particle velocity is: v = (n + ) m/s This value is greater than that obtained in the case of the electron, but still quite far from a required limit of ±3 Km/s. However, if we reduce the radius of the spherical confined region, in which the particle is forced to move, then the uncertainty can approach the required value that allows the transition to the tachyonic regime. Therefore, we calculate the limit of the confinement, to which corresponds an uncertainty of ±3 Km/s; the obtained value is: q = (n + )8,7 6 m Since the occupied states are those with large n at velocity v =,99c, q can be higher than the order of microns. Distances of this magnitude are easily accessible in matter. The real problem is to confine a particle with a very small mass in such a large space. However, even if this situation occurs for a very short time, the particle achieves the tachyonic regime. Conditions favorable to the realization of such a process are those that may occur in extremely compact cosmic bodies (such as neutron stars) hit by swarms of neutrinos [6]. Once entered in tachyonic regime, the particle begins to accelerate in an irreversible way. In the theory we are developing, the probability that an excited state of order n is occupied is given by [( v c )n ( v c )n+ ]. Therefore, it is reasonably to suggest that the uncertainty affecting the particle velocity is proportional to this probability. Then we can write the uncertainty as: v = ħ γ(m /(n+)) q [(v c )n ( v c )n+ ] (22) According to Eq. (22), when the velocity of the particle is quite lower than that of light, the uncertainty is not too high even if the particle is confined in very small spaces or is in excited states with high n. In fact, the experience shows that the transition to the tachyonic regime is not observed even in extreme cases. Let us calculate the minimum distance q, in which a relativistic electron with v =,99c should be confined to have an uncertainty of ±3 Km/s:

12 q e = (n + )5,5 2 m Such confinement is not possible under normal conditions for an electron. For instance, in a heavy atom the velocity of the inner electron is: v = KZe2 rγm where K is the Coulomb constant and Z is the atomic number [5] (we use the Bohr model). Assuming Z = 92 (uranium) and considering the first electronic orbit: n =, we get: r( orbit) = 5,7 3 m ; v( orbit) =,73 8 m/s Hence, when using the Bohr model for a heavy atom, the velocity of the innermost electron is 57% of the speed of light with orbital confinement of an order of a tenth of a picometer. Then the uncertainty affecting the velocity is: v = (n + ),5 8 m/s Multiplying the uncertainty by the probability that the occupied state is the fundamental state, we get v = 6,35 7 m/s. Therefore, for the inner electron of a uranium atom, the velocity is: v( orbit) = 73. ± 63.5 Km/s With this uncertainty, the maximum velocity of the electron is 236,5 Km/s, which is insufficient to enter the tachyonic regime. On the other hand, velocities of v,6c are considered as the upper limit for atomic electrons. This means that the acceleration of an electron to a superluminal velocity can occur only in a system with a high mass density to assure spatial confinement of an order of 3 m (in neutron stars, electrons combine with protons to form neutrons, and for a short time, the uncertainty principle can assure the possibility for free electrons to be highly confined). 4 Majorana Spinors From the solution of the Majorana equation, we get an infinite wavevector with infinite components, each concerning a given excited state. These solutions are represented by: J n, m Jn where n is the order of the excited state, J n = s + n, and m Jn is one of (2J n + ) z- components of the total angular momentum. Then we can write the general state of the particle setting component m Jn as follows: j, /2 = c /2,/2 + c 3/2,/2 + + c n (/2 + n), /2 + (23) where c n = (,, n, ) are the coefficients of the linear combination. The explicit form of the ket in the linear combination in Eq. (23) is: 2

13 /2,/2 = (a, b, c, d,,, ) 3/2,/2 = (,,,, a, b, c, d, e, f, g, h,,, ) 5/2,/2 = (,,,,,,,,,,,, a, b, c,, n,,, ) where the letters indicate the nontrivial components of the wavevectors. Obviously, the higher the quantum number J n is the greater the number of nontrivial components. Moreover, for general spinors j, /2 and j, /2, all infinite wavevectors with infinite components are the nontrivial solutions of the Majorana equation (at least all four components are not zero). In compliance with the postulates of quantum mechanics, the square of the coefficients of the linear combination are the probabilities of occupation of each state belonging to the infinite sum in Eq. (23). In the Majorana theory, this probability must be proportional to term (v/c) n, and when v c, it increases progressively with order n of the excited state. Using the definition of probability, we have: p n = (v/c)n n=(v/c) n (24) The denominator is the usual geometric series, and since the ratio v/c is lower than, it converges to [/( v/c)]. Then, Eq. (24) can be written as: (v/c)n p n = = /( v/c) [(v c )n ( v c )n+ ] (25) Equation (25) corresponds to the probabilistic coefficient that is used in Sect. 3. If n is set, function p n has a maximum at well-determined (v/c) n. Figure shows the trend of this function for values n=3 and n=6: Pn,2,,8,6,4,2,5,5 (v/c) n=3 n=6 Figure : Occupation probability vs (v/c) We can calculate velocity v, for which state n has the greatest probability to be occupied: d dv [(v c )n ( v c )n+ ] = From this equation, we get: 3

14 (v/c) max = n n+ (26) From Eq. (26), we obtain: lim (v/c) max = v c n which proves that the probability function (Eq. (25)) is well defined and physically consistent with quantum theory developed so far. The general spinor in Eq. (23) can be rewritten as: j, /2 = (v/c) /2,/2 + + [( v c )n ( v c )n+ ] (/2 + n), /2 + The trend of function (26) is shown in Fig. 2:,2,8 (v/c) max,6,4, n Figure 2: (v/c) max vs order of excited state It is clear that values (v/c) corresponding to the maximum probability tend to when the excited state order increases. This increase occurs rapidly from n= to n=9, and then becomes progressively slower. This means that when v =,9c, the maximum probability of the excited states with n > 9 remains almost constant. Let us study how the probability of state occupation changes when the particle velocity is set (i.e., changing n): 4

15 ,,8 P n,6,4,2 (v/c)=,5 (v/c)=,9 (v/c)=, n Figure 3: Occupation probability vs order of excited state From Fig. 3, we see that with an increase in (v/c), the probability that a state with low n is occupied decreases rapidly. However, this decrease is homogeneously distributed over all excited states with high n. In the limit v c, all the excited states become equally probable and the reality of the particle is delocalized over all infinite states. In such a context, the uncertainty principle can decide the fate of the particle and keep it in a bradyonic state or bring it in a tachyonic state. Since order n of the excited state is a positive integer, the trends shown in the three figures are discrete (even if they are presented using solid lines). Therefore, we can say that trend p n, as a function of n, is the trend of a line spectrum, whose density increases with n. 5 Conclusion In this study, it has been proved that the physical conditions under which the transition of a fermion with half-integer spin from the bradyonic to the tachyonic regime is possible can be determined using the Majorana equation. These conditions include the very small mass of the particle, a particle speed very close to that of light, and its confinement in a very small region of space. Such conditions can be achieved only for massive particles travelling with a quasiluminal velocity (like neutrinos) in highly compact bodies (like neutron stars). Probably, this is the reason why superluminal particles have not been detected yet. Therefore, the physics of tachyons is a branch of science that finds its application only in the regions of the universe where supercompact matter and quasi-luminal particles coexist. The verification of this theory depends on the ability of experimental physics to access similar regions of the universe and to detect superluminal particles, directly or indirectly. References. E. Majorana, Il Nuovo Cimento, 9, 335 (932). 2. D.M. Fradkin, Am. J. Phys., 34, 34 (966). 3. I.M. Gel fand, A.M. Yaglom, Zh. Eksperim. i Teor. Fiz., 8, 77 (948). 4. A.O. Barut, I.H. Duru, Proc. R. Soc. Lond. A., 333, 27 (973). 5. X. Bekaert, M.R. de Traubenberg, M. Valenzuela, arxiv : v4 [hep-th] (29). 5

16 6. X. Bekaert, S. Cnockaert, C. Iazeolla, M.A. Vasiliev, arxiv:hep-th/5328 (25). 7. L. Deleichi, M. Greselin, Univers. J. Phys. Appl., 9, 68 (25). 8. P.A.M. Dirac, Proc. R. Soc. A, 7, 6 (928). 9. W. Pauli, In Zeitschrift für Physik, 43, 6 (927).. D. McMahon, Quantum Field Theory, Mc Graw Hill, USA (28).. A. Sarkar, T.K. Bhattacharyya, arxiv: quant-ph/57239v (25). 2. P. Tipler, R. Llewellyn, Modern Physics, 4th ed., W.H. Freeman & Co (22). 3. R.L. Dawe, K.C. Hines, Aust. J. Phys., 45, 59 (992). 4. J. Schechter, J.W.F. Valle, Phys. Rev. D, 22 (9) (98). 5. N. Bohr, Philos. Mag. 26, (93). astro-ph/222v (2). 6. D.G. Yakovlev, A. D. Kaminker, O.Y. Gnedin, P. Haensel, Physics of Neutron Star Interiors, Springer-Verlag (2). 7. N. Chamel, P. Haensel, J.L. Zdunik, A.F. Fantina, Int. J. Mod. Appl. Phys.,, 28 (23). List of Figure Captions Figure : Occupation probability vs ( v c ) Figure 2: (v/c) max vs order of excited state Figure 3: Occupation probability vs order of excited state 6