A tachyon is a hypothetical particle whose momentum 4-vector is spacelike. Hence for any tachyon there is a frame of reference in which its velocity is infinite. This is strange behaviour for a particle.

I shall show that the basic tachyon mathematics has an interpretation in terms of impulse, with no new particles involved.

LEMMA. Any non-null 4-vector whatsoever can be put into 'particle' form.
Proof. We shall use the following notation for 4-vectors:

          Component form:           (p1, p2, p3, p4)
          Vector-scalar form:      =(p, p4)

Let (p, p4) be any timelike 4-vector and (q, q4) any spacelike 4-vector. Let the metric tensor be diag(-1, -1, -1, 1). Define Mo, M'o as the real, positive, invariant quantities given by

          Mo^2 c^2 = (p4)^2 - p^2,   M'o^2 c^2 = q^2 - (q4)^2.

Define

 
                      V = cp/|p4|, M  = |p4|/c,
                      U = cq/|q4|, M' = |q4|/c.   

Then

(1)     (p, p4) = M (V, kc), where M = Mo/sqrt(1 - V^2/c^2), k=p4/|p4|,

(2)     (q, q4) = M'(U, k'c), where M' = M'o/sqrt(U^2/c^2 - 1), k'=q4/|q4|. 

We also have |V|<c, |U|>c. But formulae (1) are formally the same as those for a particle of rest-mass Mo and velocity V, whose 4-momentum is (p,p4). Similarly, formulae (2) are formally the same as those for a tachyon of 'pseudo-mass' M'o and velocity U, whose 4-momentum is (q,q4). These interpretations will apply physically if the 4-momenta have the dimensions of mass times velocity, because in that case M, M' really are masses and V, U really are velocities.

There are three Lorentz invariant cases of (1), (2).

1. p4>0: when the principal interpretation is that (p, p4) is the momentum 4-vector of an ordinary particle with rest mass Mo, velocity V, and energy E = cp4 = Mc^2.

   Interestingly, there is another interpretation, in which p is the total momentum of a system of free particles, V is the velocity of the centre of mass of the system, and Mc^2 is the total energy of the particles. [1] Here (p, p4) has the single particle form, but does not apply to any physical particle. (Classical mechanics has a similar case.)

2. p4<0: This case appears in the study of antiparticles.

3. By choice of frame, q4 may be made positive, negative or zero. The tachyon interpretation is that (q, q4) is the 4-momentum of a particle with velocity U and energy cq4. This interpretation has not been established, but was suggested by the fact that putting M'o=iMo gives M the same form as M'. [2]

A genuine physical interpretation of Case 3 arises as follows. Let (P, P4) be the total 4-momentum of a system of free particles of typical rest mass m and velocity v. Then

         (P, P4) = Sum m(v, c)/sqrt(1 - v^2/c^2)
               
                 = Sum m[v + O(v^2/c^2), c + O(v/c)].

Let (P', P'4) be the 4-momentum of the system on completion of an interaction with some other system. Write

 
                 (q, q4) = (P',P'4) - (P, P4). 

If the rest masses of the particles are unchanged by the interaction, and each v/c remains small, then

           (q, q4) ~ Sum m[v - v' + O(v^2/c^2), O(v/c)],  

which is clearly spacelike (apart possibly from some very special cases), and may be written in 'tachyon' form, although no particle is involved.

Møller [3] dealt only with the case where this (q, q4) is timelike, which is another interpretation of Case 1.

Every tachyon has infinite velocity in some frame. Pirani [4] showed that tachyons also suffer from causal inconsistency. The particle (tachyon) interpretation of Case 3 is therefore unsatisfactory, while the 'impulse' interpretation raises no problems and actually applies physically in the relevant circumstances.

This perhaps reduces the psychological pressure to expect tachyons which the tachyon formulae produce. The logical position, however, is unchanged: if I want to invoke tachyons, I have to recognize that they may (very likely?) not exist; if I want to neglect tachyons, I have to recognize that they may (even yet?) exist. The empirical approach is paramount.

[1] C. MØLLER, The Theory of Relativity, 2nd edn., Oxford University Press, Delhi (1974), p.75. (back to [1])
[2] O. M. P. BILANUIK, V. K. DESHPANDE, and E. C. G. SUDARSHAN, '"Meta" Relativity', American Journal of Physics, 30, 718-723 (1962). (back to [2])
[3] C. MØLLER, The Theory of Relativity, 2nd edn., Oxford University Press, Delhi (1974), p.78. (back to [3])
[4] F. A. E. PIRANI, Physical Review D1, 3224-3225 (1970). (back to [4])

Last updated: 04 January 2005