# Supersymmetry in the geometric representation the universe wave function

B. I. Lev  Bogolyubov Institute for Theoretical Physics, NAS of Ukraine,
Metrolohichna 14-b, Kyiv 03143, Ukraine, e-mail<EMAIL_ADDRESS>
###### Abstract

An approach to the geometric description of the birth and evolution of the
universe is proposed. The wave function as the fundamental field is
represented by a Clifford number with the transfer rules that possess the
structure of the Dirac equation for any manifold. In terms of this geometric
representation of the universe wave function, the nature of super-symmetry is
explained. A probable mechanism of spontaneous symmetry breaking of the
universe excitations with Bose and Fermi degrees of freedom is proposed.

###### pacs:

73.21.Fg, 78.67.De

##  I Introduction

The problem of the geometric representation of the universe and of how to
incorporate it in the scheme of the general relativity theory is far from
being solved. Modern ideas (rather hypotheses) about the cause of the
formation of the state of the universe suggest the instability of some scalar
field associated with the quantum nature of the matter LIN . The causes and
physical mechanism of the appearance of this field, and hence the origin of
the universe, remain open for discussion for many years. Earlier, an approach
was proposed Lev6  to describe the origin and evolution of the universe in
terms of the first principles of statistical mechanics and quantum field
theory. In this approach, we can answer the question of probable emergence of
such a field, but nothing can be said about its physical and geometric nature.
The purpose of this paper is to describe it in terms of some physical entity
that has a simple geometric interpretation and to use a known mathematical
apparatus that may describe its probable changes.

We have to define a mathematical representation of the physical entity with
which we could describe the initial state of the universe. The spinor
representation of the wave function is rather unsuitable for the description
of the origin of the universe as a quantum object lev . It was shown kar  that
there is no finite-dimensional representation of the complete linear
coordinate group of transformations for spinors. Furthermore, Dirac spinors
preserve the structure of a linear vector space, but not the structure of a
ring. It was also shown fro  that there can exist only separate associative
algebras with a partition on the field of real numbers the real number, the
complex number, and the Clifford number. Just these algebras possess the ring
structure kas .

The Clifford algebra is a vector space over the field of real numbers and is
presented as the additive group where multiplication of elements is
distributive rather than commutative with respect to addition. This ring has
ideals that may be obtained by multiplying the separated element on the right
or on the left by ring elements kas . The ideals after this procedure are just
the Dirac spinors in the standard approach. Thus the representation of the
Clifford algebra contains more information on the physical properties than
spinor representations. The Clifford algebra may be extended hil , dor  to
include in the description the initiation and evolution of the Universe.

As it was shown earlier con  \- cin , the application of the Clifford algebra
covers all standard functions of quantum mechanics and provides lev  a
unifying basis for the physical knowledge including the general relativity and
electromagnetism. When we introduce the Clifford algebra in the scheme of
quantum mechanics con2 , we should not ignore the specifics of this
formulation. Actually, in this case we obtain a quantum-mechanical theory that
provides only an algebraic structure and does not contain any further specific
requirements. It is possible to show con  \- cin  that Clifford algebra
formalism is completely equivalent to the traditional approach to quantum
mechanics.

First of all in cosmology, the question arises about the geometric nature of
the fundamental field. It may be scalar as well as have other geometric
images. It is natural that its geometric characteristics should follow from
the space that would be created due to the distribution of the matter. In
terms of relevant physical characteristics, the Clifford number is the most
suitable at the moment lev ; Lev6 . The main idea of this paper is to describe
the origin of the universe in terms of the Clifford number and to find an
explanation for the dark matter and energy, as well as to explain the observed
behaviour for the visible matter. To do this, we first focus on the basic
properties of the Clifford algebra and show its advantages for the physical
subject under consideration.

In this paper we also show that the Hamiltonian of the universe in terms of
the geometric interpretation of the wave function fully corresponds to the
theory of supersymmetry. Thus it is possible to explain the asymmetry between
the Bose and Fermi degrees of freedom of the universe and to obtain non-
standard conditions of spontaneous symmetry breaking with the condensation of
fields of different tensor dimensions. This opens up the possibility of
another interpretation of the quantum phenomena in the early universe with the
emergence of the dark matter.

##  II Clifford algebra

We use the basic idea of correspondence between the matrices and elements of
the external algebra and thus determine the space for the Clifford algebras.
In the case of space-time in the special theory of relativity, the Clifford
algebra is constructed by Dirac matrices $\gamma_{\mu}$ that are associated
with unit vectors. The linear combination product of this matrices possesses
all the properties of the structure of the Clifford algebra with three
imaginary unit vectors because one time matrix $\gamma^{2}_{0}=1$ and three
space matrices $\gamma^{2}_{\mu}=-1$. Therefore, we can reproduce any element
belonging to the induced vector space in the form of a direct sum through all
probable tensor representations. The existence of a unique set of linear
independent forms defined at an arbitrary point of the space suggests that the
nature of the forms translated over the manifold is similar to the nature of
forms that characterize such manifold lev . This may be also determined by the
similar form of the geometric entities as functions of elements of the induced
space. Making use of this basis, we consider the realization of any geometric
object in the ordinary Euclidean space. In this case the full geometrical
entity may be written in terms of the direct sum of a scalar, a vector, a bi-
vector, a three-vector, and a pseudo-scalar, i.e.,
$\Psi=\Psi_{s}\oplus\Psi_{v}\oplus\Psi_{b}\oplus\Psi_{t}\oplus\Psi_{p}$, that
is given by

$\Psi=\psi_{0}\oplus\Psi_{\mu}\gamma_{\mu}\oplus\Psi_{\mu\nu}\gamma_{\mu}\gamma_{\nu}\oplus\Psi_{\mu\nu\lambda}\gamma_{\mu}\gamma_{\nu}\gamma_{\lambda}\oplus\Psi_{\mu\nu\lambda\rho}\gamma^{\mu}\gamma_{\mu}\gamma_{\nu}\gamma_{\lambda}\gamma_{\rho}$
(1)

With the reverse order of composition we may change the direction of each
basis vector, and thus obtain
$\overline{\Psi}=\Psi_{s}\ominus\Psi_{v}\oplus\Psi_{b}\ominus\Psi_{t}\oplus\Psi_{p}$.
Another element of the symmetry is the change of multiplication of the basis
vectors to the inverse in the representation of the Clifford numbers, that
turns its into
$\tilde{\Psi}=\Psi_{s}\oplus\Psi_{v}\ominus\Psi_{b}\ominus\Psi_{t}\oplus\Psi_{p}$.
As long as the symmetry element is introduced, there should be present a
mathematical operation on the field of Clifford numbers. The direct sum of the
tensor subspaces may be attributed with a ring structure with the use of a
direct tensor product in the symbolic notation given by
$\Psi\Phi=\Psi\cdot\Phi+\Psi\wedge\Phi$, where $\Psi\cdot\Phi$ is an inner
product or convolution that decreases the number of basis vectors and
$\Psi\wedge\Phi$ is an external product that increases the number of basis
vectors. If each Clifford number is multiplied by a spatial fixed matrix that
has one column with elements and other zeros, then we obtain the Dirac spinor
with four elements. Making use of this column we can reproduce the spinor
representation of each Clifford number. There occurs full correspondence
between the spinor column and the elements of the exterior algebra.

Now let us determine the rule of comparing two Clifford numbers in different
points of the manifold. For this purpose we have to consider the deformation
of the coordinate system and the rule of parallel tanslations on different
manifolds. An arbitrary deformation of the coordinate system may be set in
terms of the basis deformations $e_{\mu}=\gamma_{\mu}X$, where $X$ is the
Clifford number that describes arbitrary changes of the basis (including
arbitrary displacements and rotations), that do not violate its normalization,
i.e., provided $\tilde{X}X=1$. It is not difficult to verify that
$e^{2}_{\mu}=\gamma_{\mu}\tilde{X}\gamma_{\mu}X=\gamma^{2}_{\mu}\tilde{X}X=I$,
and this relation does not violate the normalization condition kas . Now, for
an arbitrary basis, we may set, at each point of the space, a unique complete
linearly independent form as a geometric entity that characterizes this point
of the manifold. For a four-dimensional space, such geometric entity may be
given by

$\Psi=\Psi_{0}\oplus\Psi_{\mu}e_{\mu}\oplus\Psi_{\mu\nu}e_{\mu}e_{\nu}\oplus\Psi_{\mu\nu\lambda}e_{\mu}e_{\nu}e_{\lambda}\oplus\Psi_{\mu\nu\lambda\rho}e_{\mu}e_{\nu}e_{\lambda}e_{\rho}$
(2)

If this point of the manifold is occupied by matter, then its geometric
characteristics may be described by the coefficients of this representation. A
product of arbitrary forms of this type is given by a similar form with new
coefficients, thus providing the ring structure. This approach makes it
possible to consider the mutual relationship of fields of different physical
nature lev ,tak . To determine the characteristics of the manifold as a point
function implies to associate each point of the set with a Clifford number and
to find its value. If this function is differentiable with respect to its
argument, we can introduce the differentiation operation. To define a transfer
operation on an arbitrary manifold, you have to define a derivative operator,
e.g., as given by $D=\gamma_{\ mu}\frac{\partial}{\partial x_{\mu}}$ where
$\frac{\partial}{\partial x_{\mu}}$, represents the change along the curves
passing through a given point in the space. The result of action this operator
for any Clifford number may be represented as

$D\Psi=D\cdot\Psi+D\wedge\Psi$ (3)

where $D\cdot\Psi$ and $D\wedge\Psi$ may be called the ”divergence” and the
”rotor” of the relevant Clifford number. Within the context of the definition
of a differentiated variety, it is not enough to have one non-special
coordinate system covering a variety whose topology differs from the topology
of an open set in the Euclidean space. The structure of such a geometric
construction should be supplemented by a correlation between the values of the
transferred forms at different points of the manifold. When assigning internal
values to the characteristics of the manifold, we have to introduce the
transformation of Clifford numbers by changing the coordinate system. It may
be identified by displaying the corresponding Clifford numbers under the
action of a certain group that causes the relevant transformation. Conversion
is possible if caused by any geometric characteristic changing the coordinate
system accordingly, as well as by transforming the geometric objects. This
requires the use of Clifford algebras as elements of a group of the internal
vector space (simple group $Sp(n)$) $XY=fZ$, where $X$,$Y$,$Z$ have
representation constructed from similar elements of the group, $f$ are the
structure constants. A certain transformation group transforms each form
according to the law $\Psi^{\prime}=\Psi X$, where $X$ determines the mapping
elements and satisfies the condition $\tilde{X}X=1$. For such an algebra, we
may write the first structure equation that define the co-variant derivative
as given by lev

$\Omega=d\Psi-\omega\Psi$ (4)

with the gauge transformation law for the connectivity $\omega$

$\omega^{\prime}=X\omega\tilde{X}+Xd\tilde{X}$ (5)

for the conservation co-variant transformation according to the same law
$\omega^{\prime}=\omega X$. Here the tensor representation is similar to that
of an arbitrary form of the Clifford algebra. The latter equation is referred
to as the first structure equation but now the form assumes the value in the
Clifford algebra. In this case the arbitrary Clifford number may always be
reduced to the canonical form, but local deformations of the proper basis
become, however, not observable since the Tetroude form $Xd\tilde{X}$
corresponds to the second term of the gauge transformation. Then the second
structure equation that defines the ”curvature” form may be written as

$F=d\omega-\omega\omega$ (6)

with the law of transformation under the algebra being given by
$F^{\prime}=XF\tilde{X}$. The transfer equation for the curvature tensor with
the present transformation law may be written as

$dF-F\omega+\omega F=J$ (7)

where $J$ is the source form with analogous general representation that
complies the transformation $J^{\prime}=XJ\widetilde{X}$. The equation thus
obtained may be regarded as the field equation, its form is apparently similar
to the analogous equation for the connectivity form obtained in the Lie
algebra. Those equations have a more general character as their structure
contains interrelation of geometric characteristics of different tensor
nature. In this approach we may write the fourth structure equation that
demonstrates the relationship between the covariant derivation and the form
curvature, i.e., $d\Omega-\Omega\omega+\Psi F=0$. It is the Bianca relation
but in the form of Clifford numbers.

##  III Wave function of the Universe

We may assume that each elementary formation at an arbitrary point of the
manifold may be described in terms of a Clifford number. Then the particle
wave function is represented by a complete geometric object - the sum of
probable direct forms of the induced space of the Clifford algebra 2.
Moreover, by attributing to the wave function a geometric interpretation we
can obtain the correct transfer rules for an arbitrary variety lev  and come
up with new results associated with the geometric nature of the wave function.
For the wave function as a geometric entity, we may write the first structure
equation 4 in the standard form given by

$d\Psi-\omega\Psi=m\Psi$ (8)

The latter reproduces the form of the Dirac equation but with fuller meaning
than in the spinor representation. Such Dirac equation for the wave function 4
may be obtained by minimizing the action constructed from the geometric
invariant

$S=\int d\tau\left\\{\Omega\widetilde{\Omega}+m\Psi\widetilde{\Psi}\right\\}$
(9)

The Lagrange multiplier $m$ provides the normalization condition for the wave
function $\int d\tau\left\\{\Psi\widetilde{\Psi}\right\\}=1$. The above action
is nondegenerate on the solution of the Dirac equation in contrast to the
standard approach. In our approach the dynamic equation for the wave function
is presented as a rule for the parallel translations for the characteristic on
an arbitrary manifold.

To apply this equation to the description of the birth of the universe, we
have to make another assumption. The new vacuum contains only the born
formation. For this reason, all changes associated with the wave function are
due only to its changes in vacuum presence. Therefore, its behavior may be
influenced only by one characteristic of the new vacuum, namely, the wave
function. In this case, the wave function itself acts as a field that changes
its characteristics, or as a connectivity of the space with the new vacuum.
The equation for the wave function looks natural as given by

$d\Psi-\lambda\Psi\Psi=F$ (10)

when the first structure equation is at the same time the second structure
equation 6 with the equation for the ”curvature” given by

$dF-F\lambda\Psi+\lambda\Psi F=J$ (11)

It is assumed that $\omega\sim\lambda\Psi$. Among the new results, we indicate
that the Dirac equation in the geometric representation in the general theory
of relativity is just the equation of transfer on an arbitrary variety,
therefore, its solution may be interpreted purely geometrically. Moreover, the
geometric representation of the wave function yields other results that may
simply reveal the geometric nature of the wave function kle . Later, these
equations will be derived from the least action principle in the geometric
interpretation. As follows from the previous analysis, a complete coordinates
transformation group associated with the structure equation exists only in the
Clifford-number representation of the wave function. The first structure
equation for the wave function reproduces the form of the Dirac equation and,
as it was shown in kas , its solutions are similar to those for the spinor
representation. The latter observation solves the problem of the finite-
dimensional representation of the wave function under the complete linear
group of coordinate transformations.

It was proved in kas  that each even number $\Psi=\overline{\Psi}$ for
$\Psi\tilde{\Psi}\neq 0$, and the Clifford number in the Euclidean space may
be reduced to the canonical form, i.e.,

$\Psi=\left\\{\mu(x)\exp(i\beta)\right\\}^{\frac{1}{2}}X$ (12)

where $X\tilde{X}=1$ describes all the coordinate transformations associated
with the translation and rotation of coordinates and with the Lorentz
transformation in the Euclidean space. It is not difficult to show that
$\int\widetilde{\Psi}\Psi d\tau$ is scalar and in the physical interpretation
of this geometric entity it is rather evident since $\mu(x)$ may be associated
with the probability density of finding a particle in an arbitrary spatial
point, and $\beta$ is the angle that determines the eigenvalue of a particle
with positive or negative energy. We can take $\beta=0$ for the matter and
$\beta=\pi$ for the antimatter and thus provide a possibility to describe the
intermediate states of the particle since the form of the wave function of an
arbitrary ensemble of particles is analogous dor . According to the proof in
the book kas , the odd part of the general Clifford number may be presented as
the even part multiplied by an separate element of this algebra and thus there
is no problem with the manipulation of the full Clifford number. For an
arbitrary wave function we take into account only the scalar part of this
product that may be explicitly written as

$\rho=\Psi^{2}_{0}+\Psi_{\mu}\Psi^{\mu}+\Psi_{\mu\nu}\Psi^{\mu\nu}+\Psi_{\mu\nu\lambda}\Psi^{\mu\nu\lambda}+\Psi_{\mu\nu\lambda\rho}\Psi^{\mu\nu\lambda\rho}$
(13)

Thus it becomes possible to describe the intermediate states of the particle
inasmuch as the form of the wave function of an arbitrary particle ensemble is
analogous dor ,tre -Sch . The structure equation thus obtained is written in
terms of the introduced variable and is completely equivalent to the Dirac
equation that has the well known solutions both for the calculation of the
hydrogen atom spectrum and for the interpretation of the electron states kas .

##  IV Modernization of the standard cosmological model

In the case of spontaneous generation of an additional field in vacuum, the
ground state energy of the ”new” vacuum for fields of different nature should
be lower than the ground state energy of the ”initial” vacuum Lev6 . Moreover,
the interaction of this field with the fluctuations of any other field
provides energy conservation for the new state of the system. We assume that
occurrence in vacuum of the fundamental field that is generated spontaneously
and interacts with the fluctuations of all other fields may be described in
terms of the Clifford number lev . The probable stationary states of the
fundamental field are generated by the multiplicative noise produced by the
nonlinear self-interaction with fluctuations of this field. The generator of
these fluctuations is the vacuum itself for each point of the Planck size on
the manifold.

The present model differs from the widely studied scenario of the stochastic
inflation of the Universe LIN  that takes into account fluctuations of the
fundamental field but disregards the fluctuations of the unstable vacuum. The
internal fluctuations generate the stochastic behavior of the system that may
cause changes of its stationary state. The most significant point is that now
the fundamental field is not a scalar but a Clifford number and contains all
the geometric characteristics of the space that may be born as a result of the
emergence of the matter. Only the distribution of the matter can describe the
space that arises.

We start with the assumption that the phase transition from the ”initial”
vacuum to the new state of the vacuum generates a new non-zero field. This
means that the new field generates the ”new” vacuum different from the
”primary” vacuum for any field of an arbitrary geometric characteristic that
may exist. The resulting field must reduce the energy of the ”new” vacuum with
respect to the energy of the ”primary” vacuum. Therefore, the energy density
of the ground state of the ”new” vacuum may be supplied through

$\epsilon=\epsilon_{v}-\frac{\mu_{0}^{2}}{2}\widetilde{\Psi}\Psi,$ (14)

where the second part is the field energy in terms of the wave function with
the geometric representation as the Clifford number; the coefficient
$\mu_{0}^{2}$ describes the coupling of the new field and the ”primary”
vacuum, i.e., the self-consistent interaction of the new field with the
probable fluctuations that may exist in the ”primary” vacuum. Here we should
make two remarks. The first concerns the decrease in the initial energy of the
ground state with the appearance of the new field, and the second is related
to the coupling coefficient that is now positive, so that explanations of the
appearance of such a sign used in the standard approach are not needed. The
energy of the new system may be presented in the form

$E=E_{v}-\int\frac{\mu_{0}^{2}}{2}\widetilde{\Psi}\Psi d\tau,$ (15)

If we want to describe the evolution of the system $\langle out|\exp
iHt|in\rangle$, we still need to average all probable fluctuations with which
the new field can interact. For this purpose it is sufficient to present the
nonlinear coupling in the form $\mu_{0}^{2}=\mu^{2}+\xi$, where
$\langle\xi(t)\xi(0)\rangle=\sigma^{2}$ and $\sigma^{2}$ is the dispersion of
the coupling coefficient fluctuations, and to carry out averaging

$\langle out|\exp iHt|in\rangle\sim\langle out|\int D\Psi\int D\xi\exp
i\left\\{E_{v}-\frac{1}{2}\mu^{2}\widetilde{\Psi}\Psi+\frac{1}{2}\xi\widetilde{\Psi}\Psi+\frac{\xi^{2}}{\sigma^{2}}\right\\}|in\rangle$
(16)

Integration over fluctuation fields yields

$\langle out|\exp iHt|in\rangle\sim\langle out|\int D\Psi\exp
i\left\\{E_{v}-\frac{1}{2}\mu^{2}\widetilde{\Psi}\Psi+\frac{\sigma^{2}}{4}(\widetilde{\Psi}\Psi)^{2}\right\\}|in\rangle$
(17)

This implies that we have a system with the effective energy (averaged over
the fluctuations of the other field coupled with the scalar field) given by

$E=E_{v}-\int\left[\frac{1}{2}\mu^{2}\widetilde{\Psi}\Psi-\frac{\sigma^{2}}{4}(\widetilde{\Psi}\Psi)^{2}\right]d\tau$
(18)

where
$V(\Psi)=-\frac{1}{2}\mu^{2}\widetilde{\Psi}\Psi+\frac{\sigma^{2}}{4}(\widetilde{\Psi}\Psi)^{2}$
is the well-known expression for the energy of the fundamental field with the
nonlinear coupling coefficient determined by the dispersion of fluctuations.
This implies that with no field $\Psi=0$, $E=E_{v}$ while for
$\widetilde{\Psi}\Psi=\frac{\mu^{2}}{\sigma^{2}}$ the expression for the
effective ground state energy of the ”new” vacuum reduces to
$E=E_{v}-\frac{\mu^{4}}{4\sigma^{2}}\tau$. As follows from the latter
relation, the energy of the ”new” vacuum is lower than the energy of the
primary vacuum, i.e. the phase transition results in the formation of a new
vacuum ground state. If $\sigma^{2}$ tends to infinity, then the energy of the
new state tends to the initial energy of the ground state. The energy of the
new state may vanish for $E_{v}=\frac{\mu^{4}}{4\sigma^{2}}$. This relation
may be applied to estimate the maximum dispersion of the field fluctuations if
the vacuum temperature is given by $\Theta_{v}=\frac{\mu^{4}}{2\sigma^{2}}$.

The effective potential may now be given in terms of the probability density
$\widetilde{\Psi}\Psi=\rho$ of the material field
$V(\rho)=-\frac{1}{2}\mu^{2}\rho+\frac{\sigma^{2}}{4}\rho^{2}$ that may be
useful for the interpretation of different compositions of the energy and
matter produced by the spontaneous symmetry breaking. It should be noted that
this is the total probability density of the material field, and whether it is
”dark” depends on the tensor characteristics of the field in which we feel it.
It may be invisible in the vector electromagnetic field but should definitely
be felt in the gravitation field and probably in the fields of other tensor
representations. As will be shown below, this value $\rho$ is at the same time
the supercharge and characterizes the transformation between the bosonic and
femionic degrees of freedom in the general representation of the wave
function.

##  V Supersymmetry in the geometric representation and probable predictions

We can now propose a slightly different scenario for the birth of the Universe
based on the representation of its wave function as a geometric entity. What
appears as the result of the birth of the matter should contain a geometric
image. Only the birth of the matter and its distribution may be interpreted in
terms of geometry. Such an entity might be a Clifford number with an
appropriate physical interpretation. An additional field is required for the
emergence of the matter whose spontaneous excitation leads to the emergence of
elementary particles. In our case, such a field is the wave function $\Psi$ in
terms of different tensor representations, i.e., it has all probable tensor
representations with the dimension of the space to be created. That is, the
geometry is laid down from the very beginning in the characteristics of the
point of the variety on which we describe it.

Minimizing the expression for the energy of the system 15 by independent
functions $\Psi$ and $\widetilde{\Psi}$, we obtain for the wave function in
the homogeneous case the Gross-Pitaevskii equation with the physical
consequences on its solution, i.e.,

$\frac{\delta
E}{\delta\widetilde{\Psi}}=\left[-\mu^{2}+\sigma^{2}(\widetilde{\Psi}\Psi)\right]\Psi=0$
(19)

Similar in content, but richer in nature one can obtain equation by the
dynamics of changes of the wave function in the geometric interpretation. To
do this, we consider the action written for the wave function of the universe
in the presence of the matter. As it was mentioned earlier, the action in
terms of the geometric invariant may be given by

$S=\int
d\tau\left\\{\frac{1}{2}\left(F\widetilde{F}+\widetilde{F}F\right)+m\Psi\widetilde{\Psi}\right\\}$
(20)

The Lagrange multiplier $m$ provides the normalization condition for the wave
function $\int d\tau\left\\{\Psi\widetilde{\Psi}\right\\}=1$ and the ”general”
curvature in the representation of Clifford numbers takes the form;
$F=d\Psi-\Psi\Psi$. Minimization of this functional yields an equation that is
at the same time the second structure equation, i.e.,

$dF-F\lambda\Psi+\widetilde{\lambda}\Psi F=J$ (21)

for the change of the ”curvature” at the parallel transfer under the influence
of the complete group of transformations of the coordinate system. In the
homogeneous case $d\Psi=0$ and since $dF=0$ such equation is transformed to
the previous Gross- Pitaevskii equation, i.e.,

$\left[-\mu^{2}+\sigma^{2}(\widetilde{\Psi}\Psi)\right]\Psi=0$ (22)

with $-\mu^{2}=m$ and $\sigma^{2}=\widetilde{\lambda}-\lambda$

We have suggested that in the presence of the spontaneously generated
fundamental field the energy of the vacuum state for any other field is lower
than the ground state energy of the primary vacuum and that the energy of the
fundamental field is influenced by its nonlinear interactions with the
fluctuations of the physical fields of different nature. To avoid the question
about the influence of the gravitation field on the Universe evolution at the
stage of the spontaneous generation of the fundamental field we notice that
the energy of the primary vacuum is not contained in the Einstein equation and
the dynamics of the Universe is determined only by the potential energy of the
fundamental field that produces the matter. The distribution of the matter in
its turn determines the geometry.

According to the Dirac theory we may move from the classical Poisson brackets
to the quantum ones and rewrite the Hamiltonian in terms of the secondary
quantization, where instead of classical geometric representations of the wave
function we introduce the operators of creation and destruction of quanta of
this field. In the operator form the Hamiltonian of the Universe may be
written as:

$H=E_{v}-\frac{1}{2}\mu^{2}\hat{\Psi}^{+}\hat{\Psi}+\frac{\sigma^{2}}{4}\hat{\Psi}^{+}\hat{\Psi}\hat{\Psi}^{+}\hat{\Psi}$
(23)

For the field operators of the general form thus introduced, the commutation
relations may be unusual (q-deformed). We return to this question somewhat
later, and here we just note that this field by definition has a
representation of both bosons and fermions fields that should include probable
transformations of bosons into fermions and vice versa. For this reason, we
may assume that the product of the birth and destruction operators of the
field in the general representation may be written in the form

$\hat{\Psi}^{+}\hat{\Psi}=Q=\hat{B}^{+}\hat{F}+\hat{F}^{+}\hat{B}$ (24)

where $\hat{B}^{+}$ and $\hat{B}$ denote Bose creation and annihilation
operators and $\hat{F}^{+}$ and $\hat{F}$ denote Fermi creation and
annihilation operators with the (anti)commutation relation
$[\hat{B},\hat{B}^{+}]={\hat{F},\hat{F}^{+}}=1$,
$[\hat{B},\hat{F}^{+}]=[\hat{F},\hat{B}^{+}]=\hat{F}^{2}=(\hat{F}^{+})^{2}=0$.
Now it is not difficult to verify that the required Hamiltonian may be written
in the supersymmetry form given by

$H=E_{v}-\frac{1}{2}\mu^{2}Q+\frac{\sigma^{2}}{4}Q^{2}$ (25)

where the square of the supercharge is
$Q^{2}=\hat{B}^{+}\hat{B}+\hat{F}^{+}\hat{F}^{+}=n_{B}+n_{F}$ and presents the
total number of bosons and fermions. It is obvious that all the elements of
the supersymmetry with the commutation relations
$[\hat{Q},\hat{Q}^{2}]=0,\hat{Q}=\hat{Q}^{+}$ are contained in the presented
form, where individual parts of the Hamiltonian are associated with the
integrals of motion and are preserved both separately and together. The
interpretation of the supercharge in our case is that the geometric
representation of the wave function makes it possible to consider a physical
mixture of bosons and fermions, and the charge itself describes the probable
transformation of particles into each other. That is, the initial wave
function describes a mixed state of bosons and fermions with probable mutual
transformations of individual components. If we calculate the partition
function Kap

$Z=Tr\exp\beta[\frac{1}{2}\mu^{2}Q-\frac{\sigma^{2}}{4}Q^{2}]$ (26)

then we obtain the thermodynamic quantities of interest as given by

$\langle Q\rangle=\frac{2}{\beta Z}\frac{\partial Z}{\partial\mu^{2}}$ (27)
$\langle Q^{2}\rangle=\frac{1}{Z}\frac{\partial
Z}{\partial\beta}+\mu^{2}\langle Q\rangle$ (28)

with the average value of the number of fermions in the universe being given
by

$\langle n_{F}\rangle=\frac{1}{2}(1-Z^{-1})$ (29)

and for bosons

$\langle n_{B}\rangle=\frac{\langle Q^{2}\rangle}{E}-\langle n_{F}\rangle$
(30)

As it was shown in the paper Kap  for small $\mu^{2}$, the relation between
bosons and fermions in the universe may be presented as

$\frac{\langle n_{B}\rangle}{\langle
n_{F}\rangle}=\frac{\exp\beta\sigma^{2}+1}{\exp\beta\sigma^{2}-1}$ (31)

and may take arbitrary predetermined values of this relation, depending on
when the phase transition of the condensation of the bosonic part of the
general representation of the wave function occurs. This observation indicates
the reason for the bosonic asymmetry of the universe that is closely related
to the baryonic asymmetry

##  VI Conclusion

We consider the Clifford algebra formalism as a suitable method for describing
the initial state of a vacuum with probable birth of a fundamental field. This
field should contain probable geometric characteristics and be completely
equivalent to the traditional approach in the quantum field theory. The
approach makes it possible to explain the existence of supersymmetric
properties of the initial fundamental field, as well as the spontaneous
violation of symmetry between bosons and fermions in the universe. Moreover,
it provides a possibility to explain the appearance of the ”dark matter” due
to the influence of fields of tensor dimensions different from the
electromagnetic field. Unfortunately, a rigorous mathematical proof of such an
approach does not exist at the moment, but for purely physical reasons such
representations may favor better understanding of the scenario of the birth
and evolution of the universe.

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