In this post I will make bring forth some interesting facts about mass.
Often we gloss over the difference between mass and energy in physics, lumping
the two together as if they were different names for the same thing. However, I
would suggest that doing this misses, or at least de-emphasizes, some
interesting and potentially useful relationships. In particular the relationship
between mass and the arrow of time. In fact I will suggest with the examples I
will give that a deep connection exists between mass and time. However before
I start, I think we need to define mass in terms of the covariant four vector
formalism. From Special Relativity we have;
M= (1/c)*[ P^mu*P_mu}
Where P^mu and P_mu are the contravarient and covariant vector
respectably and c is the speed of light.
1) ORIGIN OF MASS , MASS WITHOUT MASS.
The property mass, which as indicated by the equation above, is an invariant
property of everything in the Universe. However, in the world of the
Quantum, there exists entities which lack this property, the massless particles of
the standard model. Physics beyond the standard model, reveal that at a
fundamental level this property we call mass, is really the result of the
interaction of quanta with a scalar field, the famous Higgs field. In fact there are
most likely several Higgs like field that create this property mass. The Higgs
associated with the electroweak symmetry breaking generates the mass for all
the standard model particles, the SUSY related Goldstino for the super
partner particles and a color Higgs which we think generates GUT scale mass for
the sterile particle in the SO (10) representation.
However, in our Universe only a few percent of the mass in the Universe is
due to this mechanism. Over 90 percent of the mass in the Universe is due to a
quite different process. We can call this process, confinement mass because
it is the mass produced by the dynamics of the strong force which creates a
composite structure we call the Hadrons. To actually calculate the effects of
this process with regard to its mass production requires complex calculation
utilizing a Lattice formalism to make the results finite. However, we can
create a toy model which makes clear how this process works. This is based on
what Frank Wilczek calls tongue in cheek, Einstein's second law of mass energy;
m=E/c^2
Hadrons, the composite particles composed of Quarks and Gluons, form
stable Quantum states. The parameters of interests for these states are the Gluon
field energy and the quark localization energy. These energies will be the
source of over 90 percent of the mass in most Hadrons based on the Einstein
Wilczek second law of mass energy.
Given any two color charged particles, there will be associated color field
energy given by;
E_c= k*R
Where R is the distance between the color charges particles and k is
an arbitrary coupling constant. From this we can see that the minimum energy
of such a system would be given for R=0. However, based on Heisenberg
uncertainty principle (HUP) the energy of quark localization is given by;
E_h= k/R
Where k is a constant based on the HUP.
So a zero gluon field energy give us an infinite quark localization
energy. We can write an equation for confinement energy as;
E_conf= SUM {i, j} ([k_i*R + k_j/r )
Were k_i and K_j are the summed over constants based on the
complex relationships in the confined system. For example in a Baryon r-antir,
b-antib and g-antig structure.
Of course the stable point of this system will be given at a value of R
where;
dM/dR=0
So that'
R= sqrt[ K_j/K_i]
We can think of R as the effective radius of the Hadron.
This mechanism depicted above by the toy model is where just about all
the mass in the Universe comes from.
2) THE "VELOCITY " OF TIME AND MASS.
Special Relativity unifies space and time. This can be easily
illustrated by the time dilation equation.
t= t_0 *sqrt [ 1=v^2/c^2]
Where t is the measure of proper time, t_0 the measure of time in the rest
frame, v the relative velocity and c the speed of light. This gives us;
c^2= (dt/dt_0)^2*c^2+ (ds/dt)^2
Based on this equation every object in the Universe in a flat metric
has an invariant space time velocity equal to the speed of light. So we can
define dt/dt_0 as the velocity an object has through the dimension of time.
If we solve the relativistic Klein-Gordon, Proca, Dirac and the
Rarita-Schwinger equations we find we get four basic solutions. These are;
Matter (Positive energy flow forward in time)
psi= psi(0)*exp[ -i*( w*t-k*x)
Anti matter (Negative energy flow backward in time)
psi= psi(0)*exp[i*( w*t-k*x)
Negative Matter (Negative energy flow forward in time)
psi= psi(0)*exp[ -i*( w*t+k*x)
Negative anti matter (Positive energy flow backward in time)
psi= psi(0)*exp[ i*( w*t+k*x)
It can easily be shown from the equations of Special Relativity that
required mass terms for matter and anti matter must be positive while the mass terms
for negative matter and negative anti matter must be negative.
As demonstrated by Pauli negative matter and negative anti matter are
unphysical states (as real particles). In the Fock space formalism they give
negative probability prediction for odd quanta number and if they existed as real
particles, energy would not be bound from below.
But we can write an interesting relationship between their mass terms and
time Velocity. We get;
dt/dt_0]_m (+) = m(+)/E(+)
dt/dt_0]_am (-) = m(+)/E(-)
dt/dt_0 ]_nm (-) = m(-)/E(-)
dt/dt_0]_nam (+) = m(-)/E(+)
Of course in the case of anti matter we re interpret per Feynman these
particles as positive energy particles with energy flow forward in time by
reversing all their charges. But for negative matter and negative anti matter there
is no way to re interpret them as positive energy particles, hence their
unphysical nature.
So based on the Feynman re interpretation principle we can write
dt/dt_0]_m (+) = m(+)
dt/dt_0]_am (+) = m(+)
dt/dt_0 ]_nm (-) = m(-)
dt/dt_0]_nam (-) = m(-)
The point of all this is show that velocity in time is a function of mass.
And of course for m=0 the particle becomes timeless. So here we see a deep
connection between mass and time.
3) THE ARROW OF TIME IN THE UNIVERSE.
The arrow of time in the Universe is generally accepted to be a statistical
of large numbers. There is
an intrinsic micro arrow of time based on CPT symmetry and CP violation.
However, we don't understand exactly how this works. But this arrow is very weak
so it can be ignored at the macro scale, though it might be reasonable to
think that this micro arrow of time is the "seed" that gives us the macro arrow
of time in the Universe.
Nevertheless in general the arrow of time is based on the increase of
entropy. For example pure Quantum states evolve such that dS/dt =0. It is only in
the case of state reduction that we have an essentially irreversible process.
This is illustrated by the Von Neuman formalism for the density matrix;
S= -k*TR{rho*ln[rho]}
From this fact comes all the weirdness of the Quantum world.
We can write the general equations
EXP[S} = DIM { Hilbert} = D^n
Where D are the degrees of freedom and n is the number of quanta. This
gives us;
S= ln [ DIM {Hilbert} ]= n*ln[D]
To simplify we I will set all constants to unity. Therefore
S= n
We can see that
n= E_unv/E_min
Where
E_min= 1/lambda_max
So that;
S= E_unv*Lamda_max = E_unv*R_unv
Since;
E_unv= rho*V
Where rho is the energy density and V is the volume. This gives us;
E_unv= rho* R^3
In the case where all quanta are highly relativistic we have ;
m=0
rho_0 = 1/R^4
In the case where quanta are have mass
m<>0
we have
rho_m = 1/R^3
Therefore
S_0= rho_0*R^3*R = (1/R^4)*R^4 =1
S_m - rho_m*R^3*R= (1/R^3)*R^4= R
We see that in a massless Universe entropy is constant and there is no
macro arrow of time , while in a Universe with mass, entropy is a function of
scale factor. Once again we see a fundamental relationship between mass and the
arrow of time. We would expect a Universe without CP violation to be highly
relativistic so absent the Sakharov conditions, there is no macro arrow of
time based on the fact that entropy is constant in such a Universe.
In this post I won't engage in speculation as to why mass and the arrow of
time seem so deeply related. These interesting points are just offered as food
for thought.