Re: Negative Mass
From: Bill Hobba (bhobba_at_rubbish.net.au)
Date: 08/25/04
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Date: Wed, 25 Aug 2004 02:58:45 GMT
"Satya Das" <srdas@adobe.com> wrote in message
news:2p0qsgFfq1irU1@uni-berlin.de...
> If the mass equals energy divided by square of speed of light and when
> negative energy is possible then doesn't it seem that negative mass is
> inevitable?
>
Landau - Mechanics - page 7 proves mass must be positive in classical
mechanics. He also shows in the classical theory of fields by considering
the classical limit of the most reasonable Lagrangeian it is positive in
relativistic mechanics as well - see page 24 from that source. In
relativity for a free particle E = M/sqrt (1 - v2) (in units c = 1) and
follows from the Lagrangeian thus derived. (Landau - Classical Theory of
Fields - page 26). But be careful about what the equation is actually
saying, which is E = M for a stationary massive particle - it does not say a
hunk of energy (say that contained in an EM field) has an equivalent mass -
it says a stationary massive particle has energy. Thus the fact we have a
hunk of negative energy (eg gravitational energy in often considered
negative) does not imply it is equivalent to mass.
However in QM Schrodengers equation is based on the Hamiltonian which in SR
is H2 = p^2 + m^2 (again Landau - page 26). The problem is how we handle
that square when we write the equation - it leads to solutions that are
negative in energy (either in the Klein Gordon equation or the Dirac
equation). But a mathematical analysis shows such solutions are in fact
equivalent to positive solutions of its conjugate equation (remembering we
deal with complex fields in QM) - see page 9 - Sterman - Introduction to
QFT. Thus we have no reason to question that mass is always positive
exactly as classical mechanics predicts ie solutions of particles with
negative energy are exactly the same as solutions to its conjugate with
positive energy so why not make life easy on ourselves and chose the
positive one?
Now I will let you in a little secret - Landaus proof that mass is positive
is based on the extremum of the PLA being a minimum - which it often is.
But in reality it only has to be an extremum. So what really is happening
is order to keep our usual notions of lagrangians and classical mechanics we
prefer positive solutions. To me that indicates that mass being positive
is, at least in part, purely a convention - but like all conventions we
should be consistent in its use.
Thanks
Bill
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