CP VIOLATION AND THE ARROW OF TIME IN THE EARLY UNIVERSE
This presentation will be divided into four parts.
1) I will take a close look at CP violation and demonstrate the structural
relationship between the effective flavor oscillation of the Majorana like
Pseudoscalar mesons which exhibit CP symmetry violation and the flavor
oscillations of neutrinos also believed to have a Majorana flavor
structure.
I will also note a relationship that all the Pseudoscalar mesons share in
terms of their Fitzpatrick two space charges.
2) I will demonstrate using a simple calculation that in a Universe
consisting solely of relativistic particles the Entropy of the Universe
remains constant as the scale factor evolves.
3) I will describe a possible solution for the questions raised in section
2 about global energy conservation and propose that the energy
conservation “problem” really is just additional support for
the Biverse proposal of Linde, Sakharov, Stenger, Klauber and others.
4) Finally I will take a brief look at the whole paradox of a timeless
description of the Universe. This issue underlies the problem of a
Quantum theory of gravity. In this presentation even the supposedly
timeless evolution of the Universe is described by the time evolution
of the scale factor. How can this make sense? I hope to at least
suggest a useful way to think about this. Plus I will make a rather
bold and original suggestion.
CP VIOLATION
There is a class of Pseudoscalar mesons which directly exhibit
violation of CP symmetry in their decay process. These are
particle states where the matter-antimatter states has a
distinctive flavor charge difference but the meson states do not
contain good quantum numbers. These states serve
as our window in the physics of CP symmetry violation believed
necessary to explain the dominance of matter over anti matter
in our Universe.
This Pseudoscalar meson states are
[ K^0> = [ d sbar > [ Kbar^0 > = [ dbar s >
[ B^0> = [ d bbar > [ Bbar^0 > = [ dbar b >
[ D^0> = [ c Ubar > [ Dbar^0 > = [ cbar U >
I will use the K^0 meson as I develop the points related here.
These particles being Pseudoscalar mesons all have-1 parity
eigenvalues
P[K^0> = - [K^0> P[Kbar^0>= - [Kbar^0>
These particle states are of course not CP eigenstates since they
have distinct anti particle flavor charge but their
CP eigenstates are given by
[ K_1> = (1/sqrt[2])* ( [ K^0> - [ Kbar^0 > )
[ K_2> = (1/sqrt[2])* ( [ K^0> + [ Kbar^0 > )
These CP eigenstates which have a Majorana like flavor structure
have eigenstates with +1 and -1 eigenvalues.
CP[K_1>= [ K_1> CP[K_2> = - [ K_2 >
It should be noted that the K_1 state has a shorter half life because given
its positive CP eigenvalue it can decay into two pions while the K_2 CP
eigenstates must decay into three pions to conserve CP Symmetry.
However, because CP symmetry is not conserved the actual physical states are
superposition states of these CP eigenstates. There are the K_s state which
consists
of mostly the K_ 1 State and the K_L state which consist of mostly the K_2
state.
We can describe these physical states in terms of the CP violation
parameter where we have
{ K_s K_L } = M_ij* { K_1 K_2 }
Where
M_11= 1/sqrt[1+ epilson^2] M_12 = -epsilon / sqrt[1+ epilson^2]
M_21= epsilon / sqrt[1+ epilson^2] M_22 = 1 / sqrt[1+ epilson^2]
Where epsilon is a complex CP parameter
[epsilon] = 2.2 E-3
However, these physical flavor states can also be represented
as superposition of the mass eigenstates. The mass split between these
two states is 3.5 E-6 ev. Therefore we can write
{ K_s K_L } = exp[ -i*theta*sigma_y] * { K_1 K_2}
The probability for flavor transition as the meson travels is given by
P( s to L) = SUM { i=1, 2 } [ U_s,i]^2*[U_L_i]^2
+ SUM{ i <> j, 2} U_s,i*U^c_L_i* U^c_s,j*U_L,j*
{ Sin [ delta*pi*L/L_osc] / delta*pi*L/L_osc}*
Cos [ 2*pi*L/L_osc]
Where L is the travel distance, delta is the mass eigenstate energy
difference in the rest frame
and L_osc is the oscillation length given approximately by
L_osl ( meters) = 2.54 E(mev) / Delta_ij ( ev^2)
And
Delta_ij= M_2^2- M^2_1= (M+delta(M))^2 –M^2
Where M is the meson mass and delta(M) is the mass split. This gives us
Delta_ij= 2*Delta(M)*M + Delta(M) = approx 3484.04 ev^2
This gives us a very short decoherence length so that we can write
P( s to L) = SUM { i=1, 2 } [ U_s,i]^2*[U_L_i]^2
For all practical purposes.
Based on the experimentally determined probability for flavor oscillations
we calculate the mixing angle to be approximately 1.8 degrees for
the K meson.
The connection between the Majorana like physical Pseudoscalar mesons
and the neutrino flavor oscillations is intriguing because based on
the See Saw neutrino mass model the neutrinos are believed to be
Majorana states. But I will leave this intriguing point for later
development
if in fact such development proves possible.
In Addition it is interesting to look at the Fitzpatrick global charges
for these unique Pseudoscalar mesons.
[ K^0> = [ -1/2 +1/2 > [ Kbar^0 > = [ +1/2 -1/2 >
[ B^0 > = [ +1/2 -1/2 > [ Bbar^0> = [ -1/2 +1/2 >
[ D^0> = [ +1/2 -1/2> [ Dbar^0> = [ -1/2 +1/2 >
Interestingly this charge structure shows up in the Helon model
where the braids without any twist charges are superpositions of
[ -/1/6 +1/6 > & [ +1/6 -1/6 >
Which lead to Majorana neutrinos. This is perhaps just a coincidence
or it may point to some fundamental relationship between
matter and anti matter states. Perhaps.
ARROW OF TIME IN THE EARLY UNIVERSE
In this section I shall absorb all constants into a generic constant K to
make the argument clearer.
The cosmological arrow of time is normally considered to be related
to the scale factor of the Universe. However, as will be made clear,
the evolution of scale factor in a Universe consisting of strictly
relativistic particles has constant entropy, indicating no
effective arrow of time.
We can define the entropy of the Universe by the Boltzmann equation.
S = K*ln [W]
Where W is the number of possible states given a single energy system.
Of course the Universe is a single energy system even in quantum
mechanical terms, hence the problem of time but this relates to
other issues.
We can therefore write this equation in terms of particles as
S= K*ln [D^N]
Where D is the number of degrees of freedom and N is
the particle number.
Therefore we have
S= K* N*ln[D] = K*N
The particle number is scale dependent so that
N= E_tot/E_min = K* E_tot*lamda_max. = K*E*R
Where R is the causal radius of the Universe. Therefore
S=K*E*R
In a Universe consisting of only relativistic particles we have
Rho= K/R^4
Since
E= K*rho*R^3
We get
S= K*R^3*R/R^4 = K
S=K
Therefore a Universe consisting solely of relativistic particles has
constant entropy regardless of scale.
However once CP violation occurs together with the other
Sakharov conditions needed for matter to dominate antimatter
allowing the existence of massive particles we get
Rho=K/R^3 therefore
S= K*R^3*R/R^3 = K*R
Now one criticism of the points made here is the apparent violation of
global energy violation as demonstrated below.
Since
E_tot= K* rho*R^3
And for a Universe consisting solely of relativistic particles we get
E_tot= K/R
So that
Delta(E) = K*( R_1 –R_2 )/R_1*R_2
So as the Universe expands under a relativistic regime the total energy
of the Universe decreases as the Universe expands.
In my view this criticism is totally specious. In addition this
apparent violation of energy conservation in the early Universe
might well suggest the necessity of the Biverse model given
the inevitability of the evolution of entropy in the
early Universe that I describe here.
GLOBAL ENERGY CONSERVATION IN THE
EARLY UNIVERSE
Global energy conservation has always been an identified problem in
General Relativity. In a flat spacetime global energy conservation is
unambiguous but not so in a curved space time or a spacetime with
a future event horizon, i.e. a De Sitter spacetime. Mathematically there
are two way to define global energy conservation, via differential
equations or via
equations involving integrals. In flat space spacetime these are
equivalent
but in curved spacetime this equivalence breaks down. The integral
form becomes problematic. The exact details of this are beyond
the scope of this presentation but I believe it is helpful in pointing
out that the problem of global energy conservation are not unique to this
proposal made in this presentation. Efforts have been made using the
mathematical tool of pseudo-tensors to solve the global energy conservation
issue
but not all Physicists accept this as a solution. I will simplify this
question,
hopefully not beyond the point of usefulness, in a way that will be
helpful in relating this problem to the proposal made in this
presentation.
It’s my view that the problem of global Energy Conservation in
General Relativity is a problem of boundaries. For example
for any given volume of spacetime, energy conservation is not
globally observed. There is no principle that can insure energy
flow into or out of this spacetime volume are equal. Given a curved
spacetime we have the same boundary issues.
Therefore based on this I think the energy conservation issue identified
with this proposal is due to the conceptual boundary created by assuming
an arrow of time when none exists. The correct causal boundary up to
the point of CP violation and the emergence of the cosmological arrow
of time is the whole Biverse, not our half of the Universe. Therefore the
Biverse proposal seems thrust upon us by the global energy conservation
problem identified in an expanding Universe consisting of
only relativistic particle states.
By properly identifying the boundary of the early Universe the
solution to the Global energy conservation issue becomes trivial.
We can now write the equation of the change in total energy
as a function of the scale factor interval as
Delta(E) = K*( R_1-R_2)/R_1*R_2 + K*( R_2 –R_1)/R_1*R_2 =0
Therefore with correctly defined boundaries Global energy conservation
in the early “Biverse” is maintained.
TIMELESS REALITY WITH TIME.
Despite the fact that this proposal attempts to demonstrate the absence of
an arrow of time in the early Universe prior to CP violation in some form,
this description still involves the time evolution of the scale factor of
the
Universe. This seems like a pretty significant logical inconsistency to say
the least. To attempt to make sense of this I will make an addition and
quite speculative proposal.
I think the source of this descriptive paradox is due to the solely
classical
description employed to describe the early Universe. In other words, this
description uses a semi classical gravity description.
We have two successful ways to incorporate gravity in our physics. These are
The Semi Classical gravity theory which correlates a classical gravity
theory with
the quantum theory of matter fields can be represented as
G_mu,nu=kappa* < T_mu,nu>_psi
Where < T_mu,nu >_ psi is the expectation for the stress energy operator
given the quantum state of the matter fields and G_mu,nu is the
classical Einstein Tensor.
And the fully classical gravity theory, General Relativity
G_mu,nu=kappa* T_mu,nu
Where T_mu,nu is the classical stress energy Tensor.
However, at some scale, near or at the Planck Scale we need
a theory which gives us the equation
< G_mu,nu > _psi =kappa* < T_mu,nu>_psi
Here the Einstein Tensor is the quantum gravity operator.
We of course have no such theory yet.
Nevertheless the need for such a theory may additionally be demonstrated
by the strange parallelism of how time is observed in the both the
classical and quantum descriptions. The explanation for this is unknown
but is well demonstrated in Balbour’s treatment of time in his
landmark book “The End of Time”. In this book both classical
general relativity and the quantum theory of the whole Universe as
described by the Universe’s Schrödinger equation, are timeless
descriptions. (The Wheeler De Witt equation)
I think the very nature of the Schrödinger equation, in not being
relativistic; hence not time symmetric, provides a clearer description
of the role time plays in Quantum theory. This time symmetry is
fully present in the relativistic equations due their second order time
derivatives or in the case of the Dirac equation, its special matrix form.
The Schrödinger equation requires us to “put the time
symmetry in by hand” in describing quantum state evolution using
two arrows of time each evolving in opposite directions relative to each
other. Therefore, to describe any pure quantum state we need not one but
two
Schrödinger equations
- {hbar^2/2*m)* Del^2psi= i*hbar*(&psi/&t)
- {hbar^2/2*m)* Del^2psi= - i*hbar*(&psi/&t)
Actually we really need two addition equations to include the
supplemental solutions which are
+ {hbar^2/2*m)* Del^2psi= i*hbar*(&psi/&t)
+ {hbar^2/2*m)* Del^2psi= - i*hbar*(&psi/&t)
But we needn’t concern our selves with the meaning, if any, of these
Equations for the purposes of this presentation. They are included
for completeness only. These are unphysical states with regard to observable
particles.
Therefore it seems that we must view every pure quantum state as a
superposition of time reversed states. This includes any description
of the Universe as a pure quantum state. This is how you get a
timeless Universe and a time evolution of the scale factor. Time
as an actual emergent property of reality is the result of the
Decoherence of the wave function. Gravity based on our need
for and the success of, the Semi classical description undergoes
the process of Decoherence at a very small scale, unlike the
other quantum fields where coherence can persist to
observable scales.
Finally while the actual mechanism of CP violation is not understood,
based on the proposal being made here , perhaps the micro arrow
of time we see associated with CP violation , is the emergent arrow
of time resulting from the Decoherence of the quantum gravity field.
This seems like a very interesting possibility indeed.
Bob Zannelli
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