Q.3

Draw a graph, with momentum (p) on the x-axis, and energy (E) on the y-axis.  Then draw the "light cone", two lines with the equations E = +/- p.  This divides our 1+1 dimensional space-time into two regions.  Above and below are the "timelike" quadrants, and to the left and right are the "spacelike" quadrants.

Now the fundamental fact of relativity is that E² - p² = m².  (Let's take c=1 for the rest of the discussion.)  For any non-zero value of m (mass), this is a hyperbola with branches in the timelike regions.  It passes through the point (p,E) = (0,m), where the particle is at rest.  Any particle with mass m is constrained to move on the upper branch of this hyperbola.  (Otherwise, it is "off-shell", a term you hear in association with virtual particles - but that's another topic.)  For massless particles, E² = p², and the particle moves on the light-cone.

These two cases are given the names tardyon (or bradyon in more modern usage) and luxon, for "slow particle" and "light particle".  Tachyon is the name given to the supposed "fast particle" which would move with v>c. (Tachyons were first introduced into physics by Gerald Feinberg, in his seminal paper "On the possibility of faster-than-light particles" [Phys.Rev. v.159, pp.1089--1105 (1967)]).

Now another familiar relativistic equation is E = m*[1-(v/c)²]^-1/2.  Tachyons (if they exist) have v > c.  This means that E is imaginary!  Well, what if we take the rest mass m, and take it to be imaginary?  Then E is negative real, and E² - p² = m² < 0.  Or, p² - E² = M², where M is real.  This is a hyperbola with branches in the spacelike region of spacetime.  The energy and momentum of a tachyon must satisfy this relation.

You can now deduce many interesting properties of tachyons.  For example, they accelerate (p goes up) if they lose energy (E goes down).  Furthermore, a zero-energy tachyon is "transcendent", or infinitely fast.  This has profound consequences.  For example, let's say that there were electrically charged tachyons.  Since they would move faster than the speed of light in the vacuum, they should produce Cherenkov radiation.  This would lower their energy, causing them to accelerate more!  In other words, charged tachyons would probably lead to a runaway reaction releasing an arbitrarily large amount of energy.  This suggests that coming up with a sensible theory of anything except free (noninteracting) tachyons is likely to be difficult.  Heuristically, the problem is that we can get spontaneous creation of tachyon-antitachyon pairs, then do a runaway reaction, making the vacuum unstable.  To treat this precisely requires quantum field theory, which gets complicated.  It is not easy to summarize results here.  However, one reasonably modern reference is Tachyons, Monopoles, and Related Topics, E. Recami, ed. (North-Holland, Amsterdam, 1978).