Posted on another list. It may be of interest here.
I intend to do a post on Sakharov's original 1967 paper on the asymmetry of
matter and anti matter. But in working through this it occurred to me that
perhaps a presentation on the symmetries of energy flow and time might be
interesting because they relate to Sakharov's paper. I will be using some
(perhaps) unorthodox use of the parity and time operators. I will use the
electron as my example particle.
These symmetries will be defined by the following SR equations.
POSITIVE ENERGY FLOW IN THE FUTURE { +E + t } The electron
This is of course the way we observe energy flow in our Universe. A paper
made available by Bruce proposed that the property of energy flow into the
future is the reason for the arrow of time which is why we never observe advanced
events in the Universe. It's my view that this may be true , but if so, it
is explained by Decoherence which is a thermodynamic process.
So given our electron we can define it as the state
[ e(-)_L >
With the following relationships
MATTER w(+) & k(+) (Positive Energy into Future )
t (+)= m(+)/E(+)
v (+)=P(+)/E(+)
gamma (+) = E(+)/m(+)
E (+) = gamma (+)*m (+)
p (+) =gamma(+)*m(+)*v(+)
S(+)=v(+)*t(+)
In describing the particle's travel in terms of a path integral we can write
Amplitude { t_0 t_1 } = Integral D(histories) exp [ (i/hbar)*Integral { t_0
to t_1} L (x(t) dt
Probability = l^2* Mod [ Amplitude { t_0 t_1 } ]^2
Where l^2 is an area since the probabilities in path integrals are
probability densities in space.
Now I will apply a special version of the Parity operator , which generates a
space reflection of the wave function.
P[e(-)_L > = [ ~ e(-)_R >
Where ~ means negative energy.
This operation gives us;
NEGATIVE ENERGY FLOW INTO THE FUTURE. { -E + t} Negative electron
Which gives us
NEGATIVE MATTER w (-) k (+)
t (+)= m (-)/E (-)
v (-)=P(+))/E(-)
gamma (+) = E(-)/m(-)
E (-) = gamma (+)*m (-)
p (+) =gamma(+)*m(-)*v(-)
S (-) = v (-)*t (+)
And
Amplitude { t_0 t_1 } = Integral D(histories) exp [ -(i/hbar)*Integral {
t_0 to t_1} L (x(t) dt
Probability = l^2* Mod [ Amplitude { t_0 t_1 } ]^2
This is obviously an unphysical state. This is in fact an example of what
Dr Bob Klauber calls a supplemental state and Moffat calls a negaton
particle.
Now we will apply the special time operator to the electron state.
T[e(-)_L > = [~e(+)_L> ( The reason for the electric charge sign change
will be explained below.
POSITIVE ENERGY FLOW INTO THE PAST. {+E -t } anti negative electron
Which give us
NEGATIVE ANTI MATTER w (+) k (-)
t (-) = m(-)/E(+)
v (-)=P(-)/E(+)
gamma (-) = E(+)/m(-)
E (+) = gamma (-)*m (-)
p (-) =gamma(-)*m(-)*v(-)
S (+) =v (-)*t (-)
and
Amplitude { t_1 t_0 } = Integral D(histories) exp [
(i/hbar)*Integral { t_1 to t_0} L (x(t) dt
Probability = l^2* Mod [ Amplitude { t_1 t_0 } ]^2
However, we must apply the Stuckelberg- Feynman reinterpretation
principle. So we get
NEGATIVE ENERGY FLOW INTO THE FUTURE. { -E + t} Negative
anti electron ( all Noether charges reversed in polarity)
Once again a unphysical supplemental state or as Moffat would call it, a
negaton particle.
Finally we will apply the combined P and T operators to our
electron state.
PT [e(-)_L>= [ e(+)_R>
NEGATIVE ENERGY INTO THE PAST { -E -t } Anti electron
Which gives us
ANTI MATTER w (-) k (-)
t(-)= m(+)/E(-)
v (+)=P(-)/E(-)
gamma (-) = E(-)/m(+)
E (-) = gamma (-)*m (+)
p (-) =gamma(-)*m(+)*v(+)
S(-) = v(+)*t(-)
And
Amplitude { t_0 t_1 }= Integral D(histories) exp [- (i/hbar)*Integral { t_0
to t_1} L (x(t) dt
Probability = l^2* Mod [ Amplitude { t_1 t_0 } ]^2
But once again we must apply the Stuckelberg- Feynman reinterpretation
principle. So we get
POSITIVE ENERGY FLOW INTO THE FUTURE. { +E + t} Negative anti electron (
all Noether charges reversed in polarity)
This is of course a physical particle.
We should note that in this formalism
PT=C which is how Feynman defines these operators in his essay " The
Reason for Anti Particles" and the way these symmetries are defined in
Bilson-Thompson's paper " A topological Model of Composite Preons" ( hep-ph/0503213)