Volume 36B, number 6 PHYSICS LETTERS 18 October 1971 ENERGY OF A TACHYON IN A MAGNETIC FIELD B. A. HUBERMAN Xerox Palo Alto Research Center, Palo Alto, California, 94304, USA Received 24 May 1971 The energy levels of a charged tachyon in a magnetic field are obtained and the restrictions imposed on the ground state level for a given magnetic field are discussed. The possibility of observing transi- tions between these magnetic energy levels is discussed. Some time ago, the possibility that faster than light particles, or tachyons, might exist was proposed [1], and it was shown that their pos- sible existence does not contradict the special theory of relativity [2]. The fact that their proper mass is imaginary, and that they travel with speed greater than that of light, makes the properties of tachyons very interesting and quite different from those of ordinary particles. It is the purpose of this letter to determine the energy eigenvalues that charged tachyons, if they exist, would have in the presence of a constant mag- netic field, and to investigate the restrictions imposed on their possible experimental observa- tion. We assume that spinless tachyons with a proper imaginary mass ~o (i.e. m o = i~ o) are described by a scalar field solution to the Klein- Gordon equation. In the presence of an external electromagnetic potential this equation reads [(iVy - eA ) 2 + ~2o]t/~x ) = 0 , (1) where Vp is the four dimensional gradient, A. the four vector potential and e the charge of t~'e tachyon. If we take the direction of the magnetic field B along the Z axis, it is convenient to choose a gauge such that Ap = (Ax, Ay, O, 0). Eq. (1) then becomes E 2 = (Px-eAx)2+(py A .2+ 2 2 -e y) Pz- Uo (2) In order to obtain an exact solution for the eigen- values of eq. (2), it is useful to define new vari- ables (3) Q = (eB)-l/2 (Px-eAx); P = (eB)-l/2(py-eAy)" It is easy to show that the transformation defined by eq. (3) is a canonical one, i.e., [Q, P] = i. In terms of the new variables, eq. (2) reads 2 eB[Q2 +p2] (4) ~-p2+.o = As can be seen, the right hand side of eq. (4) re- presents a harmonic oscillator in QP space with energy eigenvalues en = (n + 1/2)2eB. Hence, from eq. (4) we obtain for the energy of a tachyon in a magnetic field, the following expression 2 2 1/2 E n = [2eB(n + 1/2) +pz -go] (5) In order to analyze some of the features of eq. (5) [3] let us assume first that the motion of the tachyon takes place in a plane perpendilar to the applied field, i.e., Pz = O. We then have E n = [2eS(n + 1/2) -~2o] 1/2 (6) Since the energy has to be real (i.e., we are in- terested in solutions oscillatory in time), eq. (6) restricts the values of the magnetic field to the condition 2eB(n+l/2) >U 2 , (7) that is, for a given magnetic field only real solu- tions will exist for values for the quantum num- ber n > no, where n o is given by .~/2e8-1/2 < % < .~/2~B +1/2. (81 As can be seen, the presence of an imaginary mass has the effect of shifting the ground state of the system from n = 0 (as in the case of or- dinary particles) to n = no. It is interesting to note, however, that if the tachyon has a non-zero component of its momen- tum along the direction of the applied field, such that Ipzl > ~o, the previous arguments no longer 573