In the paper "Mechanism for Vanishing Zero Point Energy" by Bob Klauber
(available in E Print archive) he points out the all the relativistic equations,
Klein Gordon, Dirac and Proca have solutions which he calls supplemental in
addition to the Traditional solutions. For example in the Dirac equation
there really has eight eigenspinor solutions instead of just four. Pauli
demonstrated that these addition solutions are unphysical, they are the negative
energy states which are a disaster in QFT.
Bruce Kellet talks about these solutions in his Paper "The Asymmetry of
Radiation," ( I will make this paper available to anyone on request) As I have
mentioned before, I highly recommend this paper for a clear description of
the arrow of time problem with regard to radiation.
Everyone agrees that these supplemental solutions are unphysical, they
will never be seen as real particles. However, Klauber has proposed that these
supplemental states can "exist" as virtual particles which can be used to
solve the ZPE problem. Similar proposals have now be made by Moffat, Randel,
Sundrum and T Hooft. These supplemental states also directly relate to the
Biverse proposal and Linde uses these negative energy states to zero out all of
the vacuum energy contributions in a paper available in the archive.
But the purpose of this post isn't to argue these points one way or the
other. Rather I will show by very simple mathematics that what really
separates the physical from the unphysical states are the sign of the invariant mass
term. I invite critical comments if this point is of interest to anyone.
Given that
Tau=t/gamma
Where t is rest frame time and Tau is proper time.
we get
tau/t= m/E c=1
So we have
TRADITIONAL STATES (physical)
Matter
tau(+)/t = m(+)/E(+)
Positive energy into the future
Antimatter
Tau(-)/t= m(+)/E(-)
Negative energy into the past
Equivalent to positive energy into the future all charges reversed.
SUPPLEMENTAL STATES (unphysical)
Negative matter
Tau( +)/t = m(-)/E(-)
Negative energy into the future
Equivalent to positive energy into the past all charges reversed.
Negative Antimatter
Tau(-)/t=m(-)/E(+)
Positive energy into the past.
So the supplemental states have negative mass terms for particles that have
mass. There is no way to avoid this, really. This may not be a profound point
but it is interesting and may relate to the Higgs question in interesting
ways.
Bob Zannelli