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Unification of Quantum Statistics ?
It's possible with quaternions to use a mathematical trick, which could put in evidence that quantum
statistics were special case of only one quantum and quaternionic statistics, or at least special case
of two more general quaternionic statistics one fermionic and one bosonic because the bosons have
integer spin and fermions half integer spin and so different spin degeneration. Now we see this
mathematical trick:
1
e
1
E−
KT
1
e
E−
KT
−1
We see from these two formulas that the difference among the Fermi- Dirac statistics (the first
formula) and Bose Einstein Statistics (the second formula) is a minus sign in the denominator of
these fractions. How we can convert one in the other ? We can use analytic continuation, but not in
the complex field but in the quaternion field, now we see how:
1
E−
e
w pc
KT
1

Where p is the modulo of momentum and w an Hamilton's vector of
coordinates with these properties :
modulo 1 i.e. with the
w= w 1 i w 2 j w 3 k
and
w12 w 22 w 23= 1
and Hamilton's versors have the properties:
i2
j 2 k 2= i j k= − 1
Now if we make the constraint pc/KT = 2n+1), then we have in the fermionic :
1
e
E−
KT
=
w
1 −e
1
E−
KT
−1
=
1
e
E−
KT
−1
In a similar way in the bosonic case:
1
e
E−
KT
w
=
−1 −e
1
E−
KT
−1
=
−1
e
E−
KT
1
in fact we have also for quaternions the Euler's Formula:
e w= cos
w sin = − 1
the cosine of  is equal to - 1 and the sine of  is equal to zero, modulo 2. So we see the
possibility of an extension of dependence of the Statistics also on momentum could make possible
that Fermi-Dirac and Bose-Einstein statistics were special case of two quaternionic ones for p=> 0
o for pc/KT ≈ with aboutzero. If the variable pc/KT is zero the quaternionic statistics is
likely the old statistics if it's around it seems possible that old statistics can flip one to another
changing the minus sign like above. But it' s a really flip in statistics or not? In this moment I not
simple to make this statement we must make deepest considerations to solve the question, but I
think that this possibility,it is an exciting possibility that can explain many questions in physics that
now they are explained differently. Now, we make some examples. First of all I think we can
explain Superconductivity when electrons “change” statistics they can condensate like bosons also
they can't change their properties of fermions like half integer spin. After a first change they
become another time “fermions”, and then they become another time “bosons” with a second BoseEinstein condensation this possibility can explain High Temperature Superconductivity Material.
But if you see well you can explain the Helium properties phase changing in low temperature
physics because when you have T=> 0 the pc/KT can assume many times the value of  and so
Helium I, II,III,IV can be phase changing due to “statistics flip”. But the more spectacular
possibility are these Fermionization of Bosons that now is impossible and explanation of masses of
fundamental particles. The first I just explain above, you must change only the point of view: if
you begin with bosons and then you change the sign you make a fermionization of bosons,
but It is important also bosonization for example if a bar of cadmium collect more Neutrons
It's perhaps because in this element the Bosonization of neutrons it's possible but if you
change the temperature or other parameters you can flip the statistics and so bars of
Cadmium can't work because all Neutrons can be expelled from the bars, so in a nuclear
unstable kernel you must change your security interventions and security engines. So in
Japan in Fukushima nuclear central it's possible very awful errors if they don't know my
considerations and if I'm right.
An explanation of the Mass problem?
Like I explain above:
pc/ KT = 2n 1
and so:
2 2
 p c = 2n 1
2
2
⋅ K 2⋅ T 2
but from a well known formula:
E 2 = p 2 c2 m2 c 4
Where m is rest mass of the electron
It imply:
E 2 = 2n 1 2⋅
2
K 2⋅ T 2 m2 c 4
So if T is stable and not variable energy E is dependent only on n and m
Then:
E= ±
2n 1
2
2
⋅ K 2⋅ T 2 m2 c 4 (1)
if we make the inverse :
1
=
En
1
=
vn
1
=
vn
2n 1
2
1
⋅ K 2⋅ T 2 m2 c 4
2n 1
2
2
2n 1
2
2
h
⋅ K 2⋅ T 2 m2 c4
h/ mc 2
=
2
⋅ K 2⋅ T 2 / m2 c 4 1
c
2n 1
2
2
⋅ K 2⋅ T 2 /m2 c 4 1
Where:
c
= h/ mc 2
is Compton length of particle for example of the electron:
1 1
− =
vn v j
v j− v n
=
vn v j
−
c
2n 1
2
2
2n 1
2
2
2
2
2 4
2
2
2 4
⋅ K ⋅ T /m c
1
c
⋅ K ⋅ T /m c
1
2j 1
2
2
c
2
2j 1
2
2
c
2
−
⋅ K ⋅ T 2 / m2 c 4 1
⋅ K ⋅ T 2 / m2 c 4 1
but we have also because the minus sign in (1) which can be interpreted like energy of antiparticle
and so the negative sign in energy becomes minus in frequency and
v j vn= v j − − vn
so
v j vn
=
vn v j
c
2n 1
2
2
2
2
2 4
⋅ K ⋅ T /m c
1
2j 1
2
2
c
2
⋅ K ⋅ T 2 / m2 c 4 1
the very important results is that these formulas are dependent on temperature and if these formula
is correlated with Compton's Inverse Effect we can control this dependence making measurements
with an experiment where we change temperature hoping we are successful.
Now if we have:
2n 1 2⋅
then:
2
K 2⋅ T 2 /m2 c 4 ≫ 1
2
1
h
c⋅ mc
≈
=
v n 2n 1 ⋅ K⋅ T
2n 1 ⋅ K⋅ T
v j− v n ≈
obviously if we have j=n:
2j 1 ⋅ K⋅ T
2n 1 ⋅ K⋅ T
j− n ⋅ K⋅ T
−
=2
h
h
h
v j − v n ≈ 0⇒v j ≈ v n
Matteo Marcelli
Crowe Micheal,J. “ A History of Vector Analisis ”, University of Notre Dame Press,
1967.
2. Sthephen L. Adler, “ Quaternionic Quantum Mechanics and Quantum Field ”, New
York Oxford, Oxford University Press,1995 .
1.
3. SILVA, Cibelle. C. “Da força ao tensor: evolução do conceito físico e
representação matemática do campo eletromagnético.” Tese (doutorado),Campinas, SP:
Universidade Estadual de Campinas. Instituto de Física "Gleb Wataghin", 2002.
4. E.Cho , “ De Moivre’s Formula for Quaternions ”, Appl. Math. Lett.,Vol.11,
n°6, 1998.
5. Silva,Cibelle Celestino, Phd Thesis University of Campinas Sao Paulo do
Brasil
6. Matteo Marcelli Graduation Thesis Uso dei Quaternioni in Fisica Università di
Roma “La Sapienza” Roma