From: mcirvin@husc8.harvard.edu (Mcirvin)
Newsgroups: sci.physics
Subject: Re: FTL
Date: 29 Aug 92 03:44:13 GMT
Article-I.D.: husc8.mcirvin.715059853
 <17mel5INNc1n@agate.berkeley.edu> <mcirvin.715053036@husc8> <mcirvin.715057081@husc8>

mcirvin@husc8.harvard.edu (Mcirvin) writes:

[more puzzlement about why the non-disappearance of the propagator
at spacelike separation doesn't transmit information]

I think I've got it.  Here's an explanation, complete with bad
ascii graphics (I hope the noisy phone line doesn't blitz this
completely):

In the covariant formulation of QFT (let's use the theory of
electrons for familiarity), we can "create an electron" with a
current at a point in spacetime and then "destroy it" elsewhere,
with a nonzero probability at spacelike separation:

            *----------<-----------*

The asterisks are the current.  But this isn't the whole process;
the current might also create multiple particle-antiparticle pairs,
which could do all manner of things.

(This "current" isn't the electric current; it's a fictional  e
function that has been inserted into the Lagrangian to give the
electron field a handle we can grab it with.  In practice this could
be replaced by any physical process that creates or destroys electrons.)

This Feynman diagram is an expression in covariant perturbation theory (PT).
Now, let's go back to the pre-World War Two means of doing things,
with non-covariant PT of the sort that is still usually used in
nonrelativistic QM.  This works just as well for relativistic QM,
though it's harder to use.  In covariant PT energy and momentum
are always conserved, but virtual particles can violate laws of
motion to do things like what I pictured above.  In non-covariant
PT the particles (or their wave functions) always obey the laws
of motion, but energy conservation can be violated for short periods
of time, in accordance with the time-energy uncertainty principle.

In non-covariant PT the process looks like this:

                     ^ 
                    / \   
                   /   \ 
                  /     \
                 v       ^  
                /         \  
               /           \  
              *             *  

Time is increasing upwards. 
The line is supposed to be going up from the asterisk on the right,
turning a corner, and then going back down to the asterisk on the left.
The angle is less than 90 degrees.  The process is the creation of an
electron on the right and a positron on the left, which annihilate.
Energy is temporarily nonconserved, since the asterisk on the left is
supposed to be absorbing energy rather than creating it, but everything
balances out in the end, since the annihilation of the virtual pair
creates no photons.  If the distance between the asterisks
is d, then the process takes a time t > d/(2c).

Now, you might say that since the probability for the whole process
depends on the creation of the electron on the right, you could detect
a superluminal signal on the left by measuring probabilities for repeated
trials.  Not so!  This isn't the amplitude for creating an electron
on the right and a positron on the left; it's (part of) the amplitude for 
creating that initial state and *ending up with nothing.*  Like Sidney
said, the same current can produce different numbers of particles if
it's localized to any degree, by the same Fourier transform that brought
you the Uncertainty Principle in wave mechanics (although for Fermi
particles this is kind of hard to imagine, since both the field and the
current I'm using to manipulate it are made of Grassmann variables, but
that's immaterial; I could just as easily have used bosons.)  The only way  
someone standing at the position of the asterisk on the left can even
*tell* that this is the process in question, and therefore begin to
calculate the measured probability, is to wait until it can be seen
that there aren't any particles left, which is a time t' of at least
t' = t + d/(2c), which is greater than d/c.

Really, having imagined this noncovariant picture in which the particles
behave nicely, you can see it in the covariant picture too.  What
doesn't vanish at spacelike separation is the amplitude for creating
a particle at point a, seeing it at point b, and having nothing left
over.  You can't tell that more than one particle wasn't created in the
first place until you can see the creation event at point a, so you
can't extract information from probabilities until such a time as you
could have received light-speed signals anyway.

It's quite pretty.
 
-- 
Matt McIrvin, Cambridge, Massachusetts, USA 
Newsgroups: sci.physics