11institutetext: Faculty of Physics, University of Białystok, ul.
Ciołkowskiego 1L, 15-245 Białystok, Poland
11email<EMAIL_ADDRESS>
# Constraints and Interactions in Quantization
of Yukawa Model with Higher Order Derivatives

Jan Żochowski

###### Abstract

This work is dedicated to the quantization of the light-front Yukawa model in
$D=1+3$ dimensions with higher order derivatives of the scalar field. The
problem of the computing Dirac brackets and the (anti-) commutator algebra of
interacting fields in the presence of the constraints is discussed. The Dirac
method and the Ostrogradski formalism of the higher order derivatives are
exploited. The systematic method of obtaining the inverse of the functional
Dirac-Bergmann matrix with interactions and higher order derivatives is
introduced in two variants. The discussion of applications and details of
these two variants are conducted. The results of the quantization in the form
of the (anti-) commutator algebra are presented and analyzed with special
regard to the structure of the interactions for the light-front Yukawa model,
which includes the higher order derivatives.

Keywords: higher order derivative model, constrained system, quantization,
light-front.

##  1 Yukawa Model with Higher Order Derivatives and Its Canonical Structure

It is well known, the main problem of the canonical formalism is to introduce
the relevant quantum fields in order to obtain the acceptable Hamiltonian
density from the Lagrangian one, which should be entirely agreeable with the
constraints including the assumptive interactions. The natural structure,
which indicates the adequate quantum field operators for the Hamiltonian
density is the algebra of the (anti-) commutators. This ought to be derived
via the method, introduced by Dirac, and leading from the canonical Poisson
brackets to just so Dirac ones, consistent with the constraints and being
subjected to the procedure of the quantization at the last step. In the vast
majority of cases the constraints include complete structure of the
interactions, since they are, among others, but more often than not, the
equations of motion for the non-dynamical degrees of freedom. Finally, if we
are able, despite mathematical difficulties, to incorporate the constraints
with interactions into the Dirac procedure, the detailed solution of this
allows us to obtain the commutative relationships of the interacting fields.
And furthermore, the problem of relevant selection of the dynamical and the
non-dynamical fields for the canonical formalism is especially interesting and
worth of taking trouble in the case of the models with higher order
derivatives. One of the simplest version of the theories belonging to
mentioned class is the Yukawa model in $D=1+3$ dimensions, but extended by the
minimal, just including the higher order derivative term, introduced in the
below Lagrangian density. Let’s say for order, that except this contribution
our model embraces one massive flavour of the fermionic and one of the massive
scalar field (self-coupled) and the typical Yukawa coupling term

${\cal L}={\overline{\Psi}}\left(i\\!\\!\not\\!\partial-M\right)\Psi+{1\over
2}\left(\partial\phi\right)^{2}-{1\over 2}m^{2}\phi^{2}-{1\over
4}\lambda\phi^{4}-g\phi{\overline{\Psi}}\Psi+{1\over
2}\alpha\left(\partial^{2}\phi\right)^{2}.$ (1)

The parameter $\lambda$ denotes in this formula the dimensionless coupling
constant for the self-interaction of the scalar field. Also dimensionless $g$
marks the Yukawa coupling constant at the tri-linear vertex between the scalar
field $\phi$ of the mass $m$, the incoming fermionic field $\Psi$ and the
outgoing one ${\overline{\Psi}}$, both of the unique flavour of the mass $M$.
Remarkably, the constant $\alpha$, multiplying the higher order derivative
term, has the dimension equal to $-2$. This term makes up, in fact, the
essentially simplest version b1 ; b2 ; b3  of the three-parameter Bernard-
Duncan model for the single scalar and real field

${\cal L}_{\\!\\!\\!\mbox{ }_{B-D}}={1\over
2}\alpha\left(\partial^{2}\phi\right)^{2}-{1\over
2}\varepsilon\left(\partial\phi\right)^{2}+{1\over 2}\eta\phi^{2}.$ (2)

Herein, two of these parameters, $\varepsilon$ and $\eta$, are put, to the
needs of our analysis, as equal to zero. Only $\alpha$, the important one,
remains non-trivial. It should be honestly emphasized at this point, the
implementation of the higher order term into the Yukawa model and other
theories leads to many more or less basic problems, like the energy ladder of
the states unconstrained from below and others b2 . Nevertheless, the Yukawa
model with higher order derivative term (1) stands as a handy area for testing
the Dirac formalism of quantization with the constraints, which also embrace
the interactions.

The structure of the canonical momenta for our model (1) seems to be simpler
for the light-front formulation than for the equal time one b4 , what tempts
us to write down this Lagrangian density in that’s coordinates: (151), (152),
where $x^{+}$ is the variable of evolution. The light-front Yukawa model with
higher order derivatives should now be rewritten, for convenience, on the
grounds of the fermionic bispinor notation: (161), (162) and (163). As a
consequence, we have

${\cal L}=$ (3)
$=\\!i\sqrt{2}\;\Psi^{\dagger}_{+}\partial_{+}\Psi_{+}+i\sqrt{2}\;\Psi^{\dagger}_{-}\partial_{-}\Psi_{-}+{i\over{\sqrt{2}}}\Psi^{\dagger}_{+}\gamma^{-}\gamma^{j}\partial_{j}\Psi_{-}+{i\over{\sqrt{2}}}\Psi^{\dagger}_{-}\gamma^{+}\gamma^{j}\partial_{j}\Psi_{+}-{1\over{\sqrt{2}}}M\Psi^{\dagger}_{+}\gamma^{-}\Psi_{-}-{1\over{\sqrt{2}}}M\Psi^{\dagger}_{-}\gamma^{+}\Psi_{+}+$
$+\left(\partial_{+}\phi\right)\left(\partial_{-}\phi\right)-{1\over
2}\left(\partial_{j}\phi\right)\left(\partial_{j}\phi\right)-{1\over
2}m^{2}\phi^{2}-{1\over
4}\lambda\phi^{4}-{1\over{\sqrt{2}}}g\phi\;\Psi^{\dagger}_{+}\gamma^{-}\Psi_{-}-{1\over{\sqrt{2}}}g\phi\;\Psi^{\dagger}_{-}\gamma^{+}\Psi_{+}+$
$+{1\over
2}\alpha\left(\partial_{+}\partial_{-}\phi+\partial_{-}\partial_{+}\phi-\partial_{j}\partial_{j}\phi\right)^{2}.$

From point of view of the classification of the Lagrangians, analyzed model
includes the fermionic sector composed of the fundamental fields $\Psi_{\pm}$,
their conjugation $\Psi^{\dagger}_{\pm}$ and their first order derivatives.
Thus, the principle of minimal action requires in this case only the standard
approach and leads to the ordinary Lagrange equations:

${{\partial{\cal
L}}\over{\partial{\Psi_{\pm}}}}-\partial_{\mu}\left({{\partial{\cal
L}}\over{\partial\left({\partial_{\mu}\Psi_{\pm}}\right)}}\right)=0,\;\;\;\;\;\;\;\;\;\;\;\;{{\partial{\cal
L}}\over{\partial{\Psi^{\dagger}_{\pm}}}}-\partial_{\mu}\left({{\partial{\cal
L}}\over{\partial\left({\partial_{\mu}\Psi^{\dagger}_{\pm}}\right)}}\right)=0,$
(4)

wherein the summation index runs $\mu=+,-,j$ and $j=1,2$. We also have the
second order derivatives for the scalar field within the Lagrangian density,
what obliges us to apply the Ostrogradski formalism in this sector. So, in
other words, the scalar sector of our model embraces the $\phi$ and its first
order derivative by the variable of evolution $\partial_{+}\phi$, as the
independent fields. According to the Ostrogradski approach b5 ; b6 ; b7  for
this case of the Lagrangian density, which includes the derivatives up to the
$n$th order of the single scalar field ${\cal
L}(\phi,\phi_{,\mu_{1}},\dots,\phi_{,\mu_{1}\dots\mu_{n}})$, where
$\phi_{,\mu_{1}\dots\mu_{r}}\equiv\partial_{\mu_{1}}\dots\partial_{\mu_{r}}\phi$
and $r=1,\dots,n$, the relevant equation of motion is

${{\partial{\cal
L}}\over{\partial{\phi}}}-\partial_{\mu_{1}}\left({{\partial{\cal
L}}\over{\partial\phi_{,\mu_{1}}}}\right)+\dots+(-1)^{n}\partial_{\mu_{1}}\dots\partial_{\mu_{n}}\left({{\partial{\cal
L}}\over{\partial\phi_{,\mu_{1}\dots\mu_{n}}}}\right)=0.$ (5)

We have $n=2$ in our case. Thus, the equations of motion for the Yukawa model
with higher order derivatives, inferred from the formulae (4) and (5), yield
respectively:

$2i\partial_{+}\Psi_{+}+i\gamma^{-}\\!\\!\not\\!{\stackrel{{\scriptstyle\\!\\!\rightarrow}}{{\partial_{\perp}}}}\Psi_{-}-\left(M+g\phi\right)\gamma^{-}\Psi_{-}=0,\;\;\;\;\;\;2i\partial_{+}\Psi^{\dagger}_{+}-i\Psi^{\dagger}_{-}\\!\\!\not\\!{\stackrel{{\scriptstyle\\!\\!\leftarrow}}{{\partial_{\perp}}}}\gamma^{+}+\left(M+g\phi\right)\Psi^{\dagger}_{-}\gamma^{+}=0,$
(6)
$2i\partial_{-}\Psi_{-}+i\gamma^{+}\\!\\!\not\\!{\stackrel{{\scriptstyle\\!\\!\rightarrow}}{{\partial_{\perp}}}}\Psi_{+}-\left(M+g\phi\right)\gamma^{+}\Psi_{+}=0,\;\;\;\;\;\;2i\partial_{-}\Psi^{\dagger}_{-}-i\Psi^{\dagger}_{+}\\!\\!\not\\!{\stackrel{{\scriptstyle\\!\\!\leftarrow}}{{\partial_{\perp}}}}\gamma^{-}+\left(M+g\phi\right)\Psi^{\dagger}_{+}\gamma^{-}=0,$
(7)
$\left(\partial_{+}\partial_{-}\phi+\partial_{-}\partial_{+}\phi-\partial_{j}\partial_{j}\phi\right)+m^{2}\phi+\lambda\phi^{3}+{1\over{\sqrt{2}}}g\left(\Psi^{\dagger}_{+}\gamma^{-}\Psi_{-}+\Psi^{\dagger}_{-}\gamma^{+}\Psi_{+}\right)-$
(8)
$-\alpha\left(\partial_{+}\partial_{-}+\partial_{-}\partial_{+}-\partial_{k}\partial_{k}\right)\left(\partial_{+}\partial_{-}\phi+\partial_{-}\partial_{+}\phi-\partial_{l}\partial_{l}\phi\right)=0,$

where the arrows are related to the direction of the derivative action.

The canonical structure of the Poisson brackets is determined in details by
the precise choice of the momenta. This is especially important for the
theories with higher order derivatives, like just discussed Yukawa model. In a
very general approach, more mathematical, all partial derivatives with respect
to all partial derivatives of the fields, regardless they are dynamical or
not, i.e. they are physical degrees of freedom or not, may be treated the same
manner, as the contributions to the Legendre transformation. However, this
formalism sometimes obliterates the physical distinction between dynamical and
non-dynamical degrees of freedom, especially for the theories with the flat
space-time background - in absence of the gravity b8 ; b9 . Hence, we decide
to maintain the division into dynamical and non-dynamical momenta. Only these
first belong to the canonical structure of our model (see b2  versus b6 ).
Thus, we have in our model four standard canonical momenta, conjugated with
respect to the relevant fermionic fields:

$\pi_{\pm}={{\partial{\cal
L}}\over{\partial(\partial_{+}\Psi^{\dagger}_{\pm})}},\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\pi^{\dagger}_{\pm}={{\partial{\cal
L}}\over{\partial(\partial_{+}\Psi_{\pm})}}.$ (9)

They can be obtained from the Lagrange formalism without the Ostrogradski
approach to the higher order derivatives, which are not present in this
sector. When taking into account, that our Lagrangian density has the form
(3), we may write:

$\pi_{+}=0,\;\;\;\;\;\;\;\;\;\pi^{\dagger}_{+}=i\sqrt{2}\;\Psi^{\dagger}_{+},\;\;\;\;\;\;\;\;\;\pi_{-}=0,\;\;\;\;\;\;\;\;\;\pi^{\dagger}_{-}=0.$
(10)

The scalar sector of our model is just that, which demands aforementioned
Ostrogradski method for the treatment of the higher order derivatives. This
formalism requires to introduce the $n$ canonical momenta, if the derivatives
of the field within the Lagrangian density are from the first up to the $n$th
order. They are canonically conjugated with relevant derivatives of the scalar
field:

$\pi^{\mu_{1}\dots\mu_{n}}_{\phi}={{{\partial{\cal
L}}}\over{\partial\phi_{,\mu_{1}\dots\mu_{n}}}},\;\;\;\;\;\;\pi^{\mu_{1}\dots\mu_{r}}_{\phi}={{{\partial{\cal
L}}}\over{\partial\phi_{,\mu_{1}\dots\mu_{r}}}}-\partial_{\mu_{r+1}}\pi^{\mu_{1}\dots\mu_{r}\mu_{r+1}}_{\phi},\;\;\;\;\;\;r=1,\dots,n-1.$
(11)

As indicated previously, analyzed Yukawa model includes, besides the $\phi$,
also the first order derivative $\partial_{+}\phi$, as the independent field.
The variable of evolution for this model in the light-front formulation is
$x^{+}$, what makes, according to committed remarks above, that with help of
the definition (11) we should introduce two dynamical momenta for the scalar
sector, conjugated with the fields $\partial_{+}\phi$ and $\phi$,
respectively:

$\pi^{++}_{\phi}=0,\;\;\;\;\;\;\;\;\;\;\;\;\pi^{+}_{\phi}=\partial_{-}\phi-\alpha\left(\partial_{-}\partial_{+}\partial_{-}\phi+\partial_{-}\partial_{-}\partial_{+}\phi-\partial_{-}\partial_{j}\partial_{j}\phi\right).$
(12)

In contrast, rest of the momenta: $\pi^{+-}_{\phi}$, $\pi^{+j}_{\phi}$,
$\pi^{-+}_{\phi}$, $\pi^{--}_{\phi}$, $\pi^{-j}_{\phi}$, $\pi^{j+}_{\phi}$,
$\pi^{j-}_{\phi}$, $\pi^{jk}_{\phi}$, $\pi^{-}_{\phi}$ and $\pi^{j}_{\phi}$,
where $j,k=1,2$, are the non-dynamical ones. They are not the part of the
canonical structure of discussed model and hence, the derivatives:
$\partial_{-}\phi$, $\partial_{j}\phi$ cannot be treated as the independent
fields in the approach presented here.

We have at our disposal already introduced set of the dynamical momenta,
canonically conjugated with the fundamental fields and then, we can postulate
the algebra of the Poisson brackets onto the light-front hyper-surface:

$\left\\{\Psi_{\pm}\left(x^{+},{\bar{x}}\right)\;,\;\pi^{\dagger}_{\pm}\left(x^{+},{\bar{y}}\right)\right\\}_{\\!\\!P}\\!\\!\\!=\delta^{(3)}\\!\left({\bar{x}}-{\bar{y}}\right)\Lambda_{\pm},\;\;\;\;\;\;\left\\{\Psi^{\dagger}_{\pm}\left(x^{+},{\bar{x}}\right)\;,\;\pi_{\pm}\left(x^{+},{\bar{y}}\right)\right\\}_{\\!\\!P}\\!\\!\\!=\delta^{(3)}\\!\left({\bar{x}}-{\bar{y}}\right)\Lambda_{\pm},$
(13)
$\left\\{\partial_{+}\phi\left(x^{+},{\bar{x}}\right)\;,\;\pi^{++}_{\phi}\left(x^{+},{\bar{y}}\right)\right\\}_{\\!\\!P}=\delta^{(3)}\left({\bar{x}}-{\bar{y}}\right),\;\;\;\;\;\;\left\\{\phi\left(x^{+},{\bar{x}}\right)\;,\;\pi^{+}_{\phi}\left(x^{+},{\bar{y}}\right)\right\\}_{\\!\\!P}=\delta^{(3)}\left({\bar{x}}-{\bar{y}}\right).$
(14)

Let’s observe, there is the tensor product of the spinor fields within the
fermionic Poisson brackets and due to this fact, the projectors
$\Lambda_{\pm}$, defined by the expressions (164) and (165), appear in
relevant patterns above. These canonical Poisson brackets are solely non-
vanishing elements of their algebra in the case of the light-front Yukawa
model with higher order derivatives, discussed in this work. For the reasons
mentioned in all points of the discussion, this algebra does not embrace the
brackets for the fields $\partial_{-}\phi$, $\partial_{j}\phi$ and for the
momenta, conjugated with them.

##  2 Constraints of Yukawa Model with Higher Order Derivatives

Analyzed system, the Yukawa model, is described by the Lagrangian density (3).
This embraces the dynamical equations of motion (6) and (8), which are
determining the $x^{+}$ evolution of the independent fields $\Psi_{+}$ and
$\phi$, respectively. Thus, the equation
$\pi^{\dagger}_{+}=i\sqrt{2}\Psi^{\dagger}_{+}$ defines relevant momentum to
the needs of the first formula within the algebra (13). Of course, these
equations: (6), (8) and the above pattern for $\pi^{\dagger}_{+}$ are not the
constraints.

However, comparison of the formulae (13) and (14) on the one hand to the
first, to the third, to the fourth of the (10) and to the relationship (12) on
the other, discloses that they are mutually inconsistent. We should use the
Dirac procedure for the equations of the momenta: $\pi^{\dagger}_{-}=0$,
$\pi_{-}=0$, $\pi^{++}_{\phi}=0$,
$\pi^{+}_{\phi}=\partial_{-}\phi-\alpha\partial_{-}\partial^{2}\phi$ and
$\pi_{+}=0$ to identify the constraints, which emerging in our model. We
cannot express the derivative of the fundamental field with respect to the
variable of evolution by this field and by the relevant canonically conjugated
momentum b10 ; b11 ; b12  in the case of the primary constraints. The
necessary and the sufficient condition for appearance of the primary
constraint is to have the singular matrix of the second derivative of the
Lagrangian density with respect to the fundamental derivatives of the field by
the variable of evolution b13 ; b14 . As can be seen, the light-front Yukawa
model with higher order derivatives exhibits the above property. The
Lagrangian density (3) is, so called, irregular one in the fermionic sector,
wherein we have $\partial{\cal L}/\partial(\partial_{+}\Psi_{-})=0$, what
leads to the conclusion $\det[\partial^{2}{\cal
L}/\partial(\partial_{+}\Psi_{a})\partial(\partial_{+}\Psi_{b})]=0$ with
$a,b=+,-$. The same case we have for the fundamental field
$\Psi^{\dagger}_{-}$. We may write down, in the context of these remarks, two
equations of the primary constraints of analyzed model $\Phi_{1,2}$, denoted
below with help of the symbol $\approx$. They are:

$\Phi_{1}\approx\pi^{\dagger}_{-}=0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\Phi_{2}\approx\pi_{-}=0.$
(15)

We come across, by the study on the scalar sector of the Yukawa model,
completely the same situation. The irregular Lagrangian density (3) gives
$\partial{\cal L}/\partial(\partial_{+}\partial_{+}\phi)=0$ and then
consequently infers, that $\det[\partial^{2}{\cal
L}/\partial(\partial_{+}\partial_{\mu}\phi)\partial(\partial_{+}\partial_{\nu}\phi)]=0$,
where $\mu,\nu=+,-,j$ and $j=1,2$. Therefore, we can put another primary
constraint

$\Phi_{3}\approx\pi^{++}_{\phi}=0.$ (16)

We find, by the same type of analysis, the fourth primary constraint, related
to the momentum $\pi^{+}_{\phi}$ and introduced by the second pattern (12). We
are simply convincing, that happens the condition
$\det[\partial\pi^{+}_{\phi}/\partial(\partial_{+}\phi)]=\partial\pi^{+}_{\phi}/\partial(\partial_{+}\phi)=0$,
what complies with the necessary and the sufficient condition for the primary
constraint. As a result

$\Phi_{4}\approx\pi^{+}_{\phi}-\partial_{-}\phi+\alpha\left(\partial_{-}\partial_{+}\partial_{-}\phi+\partial_{-}\partial_{-}\partial_{+}\phi-\partial_{-}\partial_{j}\partial_{j}\phi\right)=0.$
(17)

Let’s remark, the specific case provides the dynamical momentum $\pi_{+}=0$,
conjugated with the field $\Psi^{\dagger}_{+}$. This manifestly belongs to the
set of the primary constraints within analyzed model. Nevertheless, a special
treatment is allowed for this constraint. One can clearly see, that each of
the equations (6) may be obtained from another by the Hermitian conjugation.
Thus, the dynamic and the canonical structure for the field
$\Psi^{\dagger}_{+}$, conjugated with respect to the $\pi_{+}=0$, may be
derived from those for the $\Psi_{+}$. We can eliminate in consequence the
$\Psi^{\dagger}_{+}$ from the considerations and reduce by one the set of the
constraints, skipping $\pi_{+}=0$. This saturates the discussion on the
primary constraints in our case. The Yukawa model with higher order
derivatives also embraces the secondary constraints. They emerge as the
additional equations for these of the fields, which of them are conjugated
with the momenta involved in the primary constraints b13 ; b15 ; b16 . Herein,
such the conditions of the consistency for the primary constraints are the
relevant non-dynamical Lagrange equations:

$\Phi_{5}\approx
2i\partial_{-}\Psi^{\dagger}_{-}-i\Psi^{\dagger}_{+}\\!\not\\!\\!{\stackrel{{\scriptstyle\leftarrow}}{{\partial}}}_{\\!\perp}\gamma^{-}+\left(M+g\phi\right)\Psi^{\dagger}_{+}\gamma^{-}=0,$
(18) $\Phi_{6}\approx
2i\partial_{-}\Psi_{-}+i\gamma^{+}\\!\\!\not\\!\\!{\stackrel{{\scriptstyle\rightarrow}}{{\partial}}}_{\\!\perp}\Psi_{+}-\left(M+g\phi\right)\gamma^{+}\Psi_{+}=0.$
(19)

It should be made two comments at this point. Firstly, we consider both, the
primary and the secondary, constraints together, during implementation of the
Dirac procedure. Secondly, we obtain the Dirac brackets and then the (anti-)
commutators by incorporation of the complete structure of the interactions.
Let’s introduce the Dirac-Bergmann matrix, which elements should be defined as
the Poisson brackets of the constraints. Within realities of the light-front
coordinates, relevant for the formulation of our model, this satisfies

${\cal
F}_{kl}\left(x^{+},{\bar{w}},{\bar{z}}\right)=\left\\{\Phi_{k}\left(x^{+},{\bar{w}}\right)\;,\;\Phi_{l}\left(x^{+},{\bar{z}}\right)\right\\}_{\\!\\!P}.$
(20)

The constraints $\Phi_{1,\dots,6}$ are determined by the patterns: (15), (16),
(17), (18) and (19), for the case studied in this work. It is easy to
convince, that the Poisson brackets for none of the $\Phi_{k}$, where
$k=1,\dots,6$, are not zeroed with all other constraints. For instance, we
have the non-trivial Poisson brackets:
$\\{\Phi_{1},\Phi_{6}\\}_{\\!\\!\\!\\!\mbox{ }_{P}}\neq 0$,
$\\{\Phi_{2},\Phi_{5}\\}_{\\!\\!\\!\\!\mbox{ }_{P}}\neq 0$,
$\\{\Phi_{3},\Phi_{4}\\}_{\\!\\!\\!\\!\mbox{ }_{P}}\neq 0$,
$\\{\Phi_{5},\Phi_{6}\\}_{\\!\\!\\!\\!\mbox{ }_{P}}\neq 0$ and so forth.
Noticeably, the eliminated constraint $\pi_{+}=0$ also belongs to these of the
second class, due to $\\{\pi_{+},\Phi_{5}\\}_{\\!\\!\\!\\!\mbox{ }_{P}}\neq
0$. Thus, our system does not embrace the constraints of the first class.
Finally, we ensure, that the Dirac-Bergmann matrix for the Yukawa model with
higher order derivatives is non-singular. Therefore, according to the
definition, the complete set of our constraints consists of only these of the
second class. This means, that our constraints may be directly used for
calculations of the Dirac brackets and then the (anti-) commutators. The
matrix of the Poisson brackets of the constraints of the second class ${\cal
F}(x^{+},{\bar{w}},{\bar{z}})$ should be derived with help of the canonical
quantization rules (13) and (14). We take them for the needs of current
discussion and we obtain precise structure of the ${\cal
F}(x^{+},{\bar{w}},{\bar{z}})$ array

${\cal F}\left(x^{+},{\bar{w}},{\bar{z}}\right)=$ (21)
$=\left(\;\begin{array}[]{cccccc}0&\;\;0&\;\;0&\;\;0&\;\;0&\;\;{\cal
F}_{16}\left(x^{+},{\bar{w}},{\bar{z}}\right)\\\ 0&\;\;0&\;\;0&\;\;0&\;\;{\cal
F}_{25}\left(x^{+},{\bar{w}},{\bar{z}}\right)&\;\;0\\\ 0&\;\;0&\;\;0&\;\;{\cal
F}_{34}\left(x^{+},{\bar{w}},{\bar{z}}\right)&\;\;0&\;\;0\\\ 0&\;\;0&\;\;{\cal
F}_{43}\left(x^{+},{\bar{w}},{\bar{z}}\right)&\;\;{\cal
F}_{44}\left(x^{+},{\bar{w}},{\bar{z}}\right)&\;\;{\cal
F}_{45}\left(x^{+},{\bar{w}},{\bar{z}}\right)&\;\;{\cal
F}_{46}\left(x^{+},{\bar{w}},{\bar{z}}\right)\\\ 0&\;\;{\cal
F}_{52}\left(x^{+},{\bar{w}},{\bar{z}}\right)&\;\;0&\;\;{\cal
F}_{54}\left(x^{+},{\bar{w}},{\bar{z}}\right)&\;\;0&\;\;{\cal
F}_{56}\left(x^{+},{\bar{w}},{\bar{z}}\right)\\\ {\cal
F}_{61}\left(x^{+},{\bar{w}},{\bar{z}}\right)&\;\;0&\;\;0&\;\;{\cal
F}_{64}\left(x^{+},{\bar{w}},{\bar{z}}\right)&\;\;{\cal
F}_{65}\left(x^{+},{\bar{w}},{\bar{z}}\right)&\;\;0\\\ \end{array}\right).$

Herein, we have:

${\cal
F}_{16}\left(x^{+},{\bar{w}},{\bar{z}}\right)=2i\partial^{w}_{-}\delta^{(3)}\left({\bar{w}}-{\bar{z}}\right)\Lambda_{-},\;\;\;\;\;{\cal
F}_{25}\left(x^{+},{\bar{w}},{\bar{z}}\right)=2i\partial^{w}_{-}\delta^{(3)}\left({\bar{w}}-{\bar{z}}\right)\Lambda_{-},$
(22) ${\cal
F}_{34}\left(x^{+},{\bar{w}},{\bar{z}}\right)=-2\alpha\partial^{2w}_{-}\delta^{(3)}\left({\bar{w}}-{\bar{z}}\right){\rm
P},\;\;\;\;\;\;{\cal
F}_{43}\left(x^{+},{\bar{w}},{\bar{z}}\right)=2\alpha\partial^{2w}_{-}\delta^{(3)}\left({\bar{w}}-{\bar{z}}\right){\rm
P},$ (23) ${\cal
F}_{44}\left(x^{+},{\bar{w}},{\bar{z}}\right)=-2\partial^{w}_{-}\delta^{(3)}\left({\bar{w}}-{\bar{z}}\right){\rm
P}-2\alpha\partial^{w}_{-}\Delta^{w}_{\perp}\delta^{(3)}\left({\bar{w}}-{\bar{z}}\right){\rm
P},$ (24) ${\cal
F}_{45}\left(x^{+},{\bar{w}},{\bar{z}}\right)=-g\Psi^{\dagger}_{+}\left(x^{+},{\bar{z}}\right)\gamma^{-}\delta^{(3)}\left({\bar{w}}-{\bar{z}}\right),$
(25) ${\cal
F}_{46}\left(x^{+},{\bar{w}},{\bar{z}}\right)=g\gamma^{+}\Psi_{+}\left(x^{+},{\bar{z}}\right)\delta^{(3)}\left({\bar{w}}-{\bar{z}}\right),\;\;\;\;\;\;{\cal
F}_{52}\left(x^{+},{\bar{w}},{\bar{z}}\right)=-2i\partial^{w}_{-}\delta^{(3)}\left({\bar{w}}-{\bar{z}}\right)\Lambda_{-},$
(26) ${\cal
F}_{54}\left(x^{+},{\bar{w}},{\bar{z}}\right)=g\Psi^{\dagger}_{+}\left(x^{+},{\bar{w}}\right)\gamma^{-}\delta^{(3)}\left({\bar{w}}-{\bar{z}}\right),\;\;\;\;\;\;{\cal
F}_{56}\left(x^{+},{\bar{w}},{\bar{z}}\right)={\xi}\left(x^{+},{\bar{w}},{\bar{z}}\right)\delta^{(3)}\left({\bar{w}}-{\bar{z}}\right),$
(27) ${\cal
F}_{61}\left(x^{+},{\bar{w}},{\bar{z}}\right)=-2i\partial^{w}_{-}\delta^{(3)}\left({\bar{w}}-{\bar{z}}\right)\Lambda_{-},\;\;\;\;\;\;{\cal
F}_{64}\left(x^{+},{\bar{w}},{\bar{z}}\right)=-g\gamma^{+}\Psi_{+}\left(x^{+},{\bar{w}}\right)\delta^{(3)}\left({\bar{w}}-{\bar{z}}\right),$
(28) ${\cal
F}_{65}\left(x^{+},{\bar{w}},{\bar{z}}\right)=-{\xi}\left(x^{+},{\bar{w}},{\bar{z}}\right)\delta^{(3)}\left({\bar{w}}-{\bar{z}}\right).$
(29)

The differential operator $\xi(x^{+},{\bar{w}},{\bar{z}})$, acting on the
Dirac delta, is determined by the expression

${\xi}\left(x^{+},{\bar{w}},{\bar{z}}\right)=i\sqrt{2}\Delta^{w}_{\perp}\Lambda_{-}-i\sqrt{2}\left[M+g\phi\left(x^{+},{\bar{w}}\right)\right]\\!\left[M+g\phi\left(x^{+},{\bar{z}}\right)\right]\Lambda_{-}.$
(30)

The symbol ${\rm P}$ denotes the projection operator. Precise form of this
${\rm P}$ could be inferred from the equation (31). According to this pattern,
the block structure of the ${\cal F}^{-1}$ must be matched to the matrix of
the Poisson brackets. They are given by the tensor product of their arguments
$\psi_{a}$ and $\psi_{b}$. Therefore, we can have ${\rm P}={\rm
I},\Lambda_{\pm},\gamma^{\pm},\gamma^{\mu}\otimes\gamma^{\nu}$ and
$\mu,\nu=+,-$ in the case of the Yukawa model with higher order derivatives,
due to the decompositions (161) - (170) of the bispinors. The size of the unit
matrix ${\rm I}$ is $2\times 2$.

##  3 Dirac Procedure for Yukawa Model with Higher Order Derivatives

Dirac introduced the brackets being consistent with the constraints b17 ; b18
; b19  and showed, that the pass through from the Poisson brackets to those
first ought to be described by the prescription, which includes the inverse of
the matrix (20). In terms of the light-front coordinates, this prescription
yields

$\left\\{\psi_{a}\left(x^{+},{\bar{x}}\right)\;,\;\psi_{b}\left(x^{+},{\bar{y}}\right)\right\\}_{\\!\\!D}=\left\\{\psi_{a}\left(x^{+},{\bar{x}}\right)\;,\;\psi_{b}\left(x^{+},{\bar{y}}\right)\right\\}_{\\!\\!P}-$
(31)
$-\sum_{k,l=1}^{N}\int_{R^{3}}\\!{d^{3}{\bar{w}}}\int_{R^{3}}\\!{d^{3}{\bar{z}}}\;\left\\{\psi_{a}\left(x^{+},{\bar{x}}\right)\;,\;\Phi_{k}\left(x^{+},{\bar{w}}\right)\right\\}_{\\!\\!P}{\cal
F}^{-1}_{kl}\left(x^{+},{\bar{w}},{\bar{z}}\right)\left\\{\Phi_{l}\left(x^{+},{\bar{z}}\right)\;,\;\psi_{b}\left(x^{+},{\bar{y}}\right)\right\\}_{\\!\\!P},$

where $\psi_{a}(x^{+},{\bar{x}})$ and $\psi_{b}(x^{+},{\bar{y}})$ denote, with
their possible composite indexes $a$ and $b$, the fields apparent in the
model. The parameter $N$ is the number of the second class constraints herein,
showing also the size of the matrix (20) and its inverse. We have
$\psi_{a},\psi_{b}=\Psi_{+},\Psi_{-},\Psi^{\dagger}_{-},\phi,\partial_{+}\phi$
and $N=6$ for analyzed Yukawa model. The crucial point is, clearly, to perform
the inverse matrix of the Dirac-Bergmann one.

The matrix of the Poisson brackets of the constraints of the second class,
involved in the equation (31) as its inverse, is built of different
structures. Firstly, the ${\cal F}(x^{+},{\bar{w}},{\bar{z}})$ has the
standard algebraic form as the array. Secondly, each element of this matrix
contains the distributions, their derivatives, the field operators or the
functions of the continuous light-front coordinates. Thus, we encounter the
case of the functional matrix, which requires the adjustment of the definition
for its inverse. From this point forward, we take the following integral
prescription

$\int_{R^{3}}\\!{d^{3}{\bar{z}}}\;{\cal
F}^{-1}\\!\left(x^{+},{\bar{x}},{\bar{z}}\right){\cal
F}\left(x^{+},{\bar{z}},{\bar{y}}\right)=\delta^{(3)}\left({\bar{x}}-{\bar{y}}\right){\rm
I}=\int_{R^{3}}\\!{d^{3}{\bar{z}}}\;{\cal
F}\left(x^{+},{\bar{x}},{\bar{z}}\right){\cal
F}^{-1}\\!\left(x^{+},{\bar{z}},{\bar{y}}\right).$ (32)

The symbol ${\rm I}$ denotes the unit matrix of the relevant size, purely
algebraic. Also the Dirac delta plays, as usual, role of the unit operator. It
can be seen, the structure of the functional matrix ${\cal
F}(x^{+},{\bar{w}},{\bar{z}})$ for the Yukawa model with higher order
derivatives is rather complicated and therefore brings on some difficulties on
the way of computing its inverse. The application of the Laplace method is not
possible in this case at all. Nonetheless, the functional matrix ${\cal
F}^{-1}(x^{+},{\bar{x}},{\bar{y}})$ may be easily and directly calculated for
the case of free fermionic and scalar fields, when their interactions are
switched off, namely $g=0=\lambda$, but the higher order derivative
contribution still is present: $\alpha\neq 0$. Therefore, we decompose
discussed matrix ${\cal F}(x^{+},{\bar{w}},{\bar{z}})$ in order to overcome
the mathematical complications, as the sum of two other ones

${\cal F}\left(x^{+},{\bar{w}},{\bar{z}}\right)={\cal
F}_{0}\left(x^{+},{\bar{w}},{\bar{z}}\right)+{\cal
G}\left(x^{+},{\bar{w}},{\bar{z}}\right),$ (33)

where the first, ${\cal F}_{0}(x^{+},{\bar{w}},{\bar{z}})$, includes only the
free fields, but embraces the higher order derivative contribution. The whole
interactions of the model, including hard to master fermionic operators in the
relevant vertex, are moved to the matrix ${\cal
G}(x^{+},{\bar{w}},{\bar{z}})$. We are in search of the algebraic formula,
which allows to express the inverse sum of the matrices by the series of their
inverses and themselves

$\left(A+S\right)^{-1}=A^{-1}+\sum_{n=1}^{+\infty}(-1)^{n}\left(A^{-1}S\right)^{n}\\!A^{-1},$
(34)

whereas the $A^{-1}$ is not difficult to obtain, but the array $S^{-1}$ does
not have to be known. To put it another way, the mathematical demands for
existence of this equation contain invertibility of the whole matrix $A+S$ and
also of the $A$. What’s more, the second matrix, $S$, does not have to posses
its inverse $S^{-1}$; it may be singular. This is our case, as we are about to
see in the below considerations. Obviously, the functional extension of this
pattern requires the summation with respect to the composite indexes, which
are attached to each of our matrices and also entail relevant integrals with
respect to the continuous space-time coordinates.

Thus, if we are able to calculate the exact result for the inverse free matrix
${\cal F}^{-1}_{0}(x^{+},{\bar{x}},{\bar{y}})$, the general form of the ${\cal
F}^{-1}(x^{+},{\bar{x}},{\bar{y}})$ is computed from the prescription (34), as
the below chain of the matrices, where each product of them is assigned to the
integration

${\cal F}^{-1}\\!\left(x^{+},{\bar{x}},{\bar{y}}\right)=\left[{\cal
F}_{0}\left(x^{+},{\bar{x}},{\bar{y}}\right)+{\cal
G}\left(x^{+},{\bar{x}},{\bar{y}}\right)\right]^{-1}={\cal
F}^{-1}_{0}\\!\left(x^{+},{\bar{x}},{\bar{y}}\right)+$ (35)
$+\\!\\!\sum_{n=1}^{+\infty}\\!(-1)^{n}\\!\\!\\!\int_{R^{3}}\\!\\!\\!\\!\\!\\!{d^{3}{{\bar{w}}_{1}}}\\!\\!\dots\\!\\!\\!\int_{R^{3}}\\!\\!\\!\\!\\!\\!{d^{3}{{\bar{w}}_{n}}}\\!\\!\int_{R^{3}}\\!\\!\\!\\!\\!\\!{d^{3}{{\bar{z}}_{1}}}\\!\\!\dots\\!\\!\\!\int_{R^{3}}\\!\\!\\!\\!\\!\\!{d^{3}{{\bar{z}}_{n}}}\\!\\!\left\\{\\!{\cal
F}^{-1}_{0}\\!\\!\left(x^{+},{\bar{x}},{\bar{w}}_{1}\right)\\!{\cal
G}\\!\left(x^{+},{\bar{w}}_{1},{\bar{z}}_{1}\right)\\!{\cal
F}^{-1}_{0}\\!\\!\left(x^{+},{\bar{z}}_{1},{\bar{w}}_{2}\right)\\!{\cal
G}\\!\left(x^{+},{\bar{w}}_{2},{\bar{z}}_{2}\right)\\!\dots\right.$
$\left.\dots\cal
F^{-1}_{0}\\!\\!\left(x^{+},{\bar{z}}_{j-1},{\bar{w}}_{j}\right)\\!{\cal
G}\\!\left(x^{+},{\bar{w}}_{j},{\bar{z}}_{j}\right)\dots\cal
F^{-1}_{0}\\!\\!\left(x^{+},{\bar{z}}_{n-1},{\bar{w}}_{n}\right)\\!{\cal
G}\\!\left(x^{+},{\bar{w}}_{n},{\bar{z}}_{n}\right)\\!\right\\}\\!{\cal
F}^{-1}_{0}\\!\left(x^{+},{\bar{z}}_{n},{\bar{y}}\right).$

Accordingly, the decomposition (33) leads in our case to the explicit
expression for the free Dirac-Bergmann matrix

${\cal F}_{0}\left(x^{+},{\bar{w}},{\bar{z}}\right)=$ (36)
$\\!=\\!\\!\left(\\!\begin{array}[]{cccccc}0&\;\;0&\;\;0&\;\;0&\;\;0&\;\;{\cal
F}_{0}\left(x^{+},{\bar{w}},{\bar{z}}\right)_{16}\\\
0&\;\;0&\;\;0&\;\;0&\;\;{\cal
F}_{0}\left(x^{+},{\bar{w}},{\bar{z}}\right)_{25}&\;\;0\\\
0&\;\;0&\;\;0&\;\;{\cal
F}_{0}\left(x^{+},{\bar{w}},{\bar{z}}\right)_{34}&\;\;0&\;\;0\\\
0&\;\;0&\;\;{\cal F}_{0}\left(x^{+},{\bar{w}},{\bar{z}}\right)_{43}&\;\;{\cal
F}_{0}\left(x^{+},{\bar{w}},{\bar{z}}\right)_{44}&\;\;0&\;\;0\\\ 0&\;\;{\cal
F}_{0}\left(x^{+},{\bar{w}},{\bar{z}}\right)_{52}&\;\;0&\;\;0&\;\;0&\;\;{\cal
F}_{0}\left(x^{+},{\bar{w}},{\bar{z}}\right)_{56}\\\ {\cal
F}_{0}\left(x^{+},{\bar{w}},{\bar{z}}\right)_{61}&\;\;0&\;\;0&\;\;0&\;\;{\cal
F}_{0}\left(x^{+},{\bar{w}},{\bar{z}}\right)_{65}&\;\;0\\\
\end{array}\\!\\!\right)\\!\\!.$

The elements of this matrix are specified below:

${\cal
F}_{0}\left(x^{+},{\bar{w}},{\bar{z}}\right)_{16}=2i\partial^{w}_{-}\delta^{(3)}\left({\bar{w}}-{\bar{z}}\right)\Lambda_{-},\;\;\;\;\;\;\;\;\;{\cal
F}_{0}\left(x^{+},{\bar{w}},{\bar{z}}\right)_{25}=2i\partial^{w}_{-}\delta^{(3)}\left({\bar{w}}-{\bar{z}}\right)\Lambda_{-},$
(37) ${\cal
F}_{0}\left(x^{+},{\bar{w}},{\bar{z}}\right)_{34}=-2{\alpha}\partial^{2w}_{-}\delta^{(3)}\left({\bar{w}}-{\bar{z}}\right){\rm
P},\;\;\;\;\;\;\;\;\;{\cal
F}_{0}\left(x^{+},{\bar{w}},{\bar{z}}\right)_{43}=2{\alpha}\partial^{2w}_{-}\delta^{(3)}\left({\bar{w}}-{\bar{z}}\right){\rm
P},$ (38) ${\cal
F}_{0}\left(x^{+},{\bar{w}},{\bar{z}}\right)_{44}=-2\partial^{w}_{-}\delta^{(3)}\left({\bar{w}}-{\bar{z}}\right){\rm
P}-2{\alpha}\partial^{w}_{-}\Delta^{w}_{\perp}\delta^{(3)}\left({\bar{w}}-{\bar{z}}\right){\rm
P},$ (39) ${\cal
F}_{0}\left(x^{+},{\bar{w}},{\bar{z}}\right)_{52}=-2i\partial^{w}_{-}\delta^{(3)}\left({\bar{w}}-{\bar{z}}\right)\Lambda_{-},\;\;\;\;\;\;\;\;\;{\cal
F}_{0}\left(x^{+},{\bar{w}},{\bar{z}}\right)_{56}={\xi}_{0}\left(x^{+},{\bar{w}},{\bar{z}}\right)\delta^{(3)}\left({\bar{w}}-{\bar{z}}\right),$
(40) ${\cal
F}_{0}\left(x^{+},{\bar{w}},{\bar{z}}\right)_{61}=-2i\partial^{w}_{-}\delta^{(3)}\left({\bar{w}}-{\bar{z}}\right)\Lambda_{-},\;\;\;\;\;\;\;\;\;{\cal
F}_{0}\left(x^{+},{\bar{w}},{\bar{z}}\right)_{65}=-{\xi}_{0}\left(x^{+},{\bar{w}},{\bar{z}}\right)\delta^{(3)}\left({\bar{w}}-{\bar{z}}\right).$
(41)

The differential operator $\xi_{0}(x^{+},{\bar{w}},{\bar{z}})$ states the free
part of the $\xi(x^{+},{\bar{w}},{\bar{z}})$, defined by the pattern (30), in
which the Yukawa coupling constant $g$ is taken as zero

${\xi}_{0}\left(x^{+},{\bar{w}},{\bar{z}}\right)=i\sqrt{2}\Delta^{w}_{\perp}\Lambda_{-}-i\sqrt{2}M^{2}\Lambda_{-}.$
(42)

The comprehensive information about the structure of the whole interactions in
this model is involved in the matrix ${\cal G}(x^{+},{\bar{w}},{\bar{z}})$.
This contains the fermionic operators and yields

${\cal
G}\left(x^{+},{\bar{w}},{\bar{z}}\right)=\left(\;\begin{array}[]{cccccc}0&\;\;0&\;\;0&\;\;0&\;\;0&\;\;0\\\
0&\;\;0&\;\;0&\;\;0&\;\;0&\;\;0\\\ 0&\;\;0&\;\;0&\;\;0&\;\;0&\;\;0\\\
0&\;\;0&\;\;0&\;\;0&\;\;{\cal
G}_{45}\left(x^{+},{\bar{w}},{\bar{z}}\right)&\;\;{\cal
G}_{46}\left(x^{+},{\bar{w}},{\bar{z}}\right)\\\ 0&\;\;0&\;\;0&\;\;{\cal
G}_{54}\left(x^{+},{\bar{w}},{\bar{z}}\right)&\;\;0&\;\;{\cal
G}_{56}\left(x^{+},{\bar{w}},{\bar{z}}\right)\\\ 0&\;\;0&\;\;0&\;\;{\cal
G}_{64}\left(x^{+},{\bar{w}},{\bar{z}}\right)&\;\;{\cal
G}_{65}\left(x^{+},{\bar{w}},{\bar{z}}\right)&\;\;0\\\ \end{array}\right).$
(43)

The elements of the above ${\cal G}(x^{+},{\bar{w}},{\bar{z}})$ satisfy:

${\cal
G}_{45}\left(x^{+},{\bar{w}},{\bar{z}}\right)=-g\Psi^{\dagger}_{+}\left(x^{+},{\bar{z}}\right)\gamma^{-}\delta^{(3)}\left({\bar{w}}-{\bar{z}}\right),\;\;\;\;\;\;{\cal
G}_{46}\left(x^{+},{\bar{w}},{\bar{z}}\right)=g\gamma^{+}\Psi_{+}\left(x^{+},{\bar{z}}\right)\delta^{(3)}\left({\bar{w}}-{\bar{z}}\right),$
(44) ${\cal
G}_{54}\left(x^{+},{\bar{w}},{\bar{z}}\right)=g\Psi^{\dagger}_{+}\left(x^{+},{\bar{w}}\right)\gamma^{-}\delta^{(3)}\left({\bar{w}}-{\bar{z}}\right),\;\;\;\;\;\;{\cal
G}_{56}\left(x^{+},{\bar{w}},{\bar{z}}\right)={\xi}_{g}\left(x^{+},{\bar{w}},{\bar{z}}\right)\delta^{(3)}\left({\bar{w}}-{\bar{z}}\right),$
(45) ${\cal
G}_{64}\left(x^{+},{\bar{w}},{\bar{z}}\right)=-g\gamma^{+}\Psi_{+}\left(x^{+},{\bar{w}}\right)\delta^{(3)}\left({\bar{w}}-{\bar{z}}\right),\;\;\;\;\;\;{\cal
G}_{65}\left(x^{+},{\bar{w}},{\bar{z}}\right)=-{\xi}_{g}\left(x^{+},{\bar{w}},{\bar{z}}\right)\delta^{(3)}\left({\bar{w}}-{\bar{z}}\right).$
(46)

The new object
$\xi_{g}(x^{+},{\bar{w}},{\bar{z}})=\xi(x^{+},{\bar{w}},{\bar{z}})-\xi_{0}(x^{+},{\bar{w}},{\bar{z}})$
is introduced as the operator, which includes the pure term of the interacting
scalar field

${\xi}_{g}\left(x^{+},{\bar{w}},{\bar{z}}\right)=-i\sqrt{2}gM\left[{\phi}\left(x^{+},{\bar{w}}\right)+{\phi}\left(x^{+},{\bar{z}}\right)\right]\Lambda_{-}-i\sqrt{2}g^{2}{\phi}\left(x^{+},{\bar{w}}\right){\phi}\left(x^{+},{\bar{z}}\right)\Lambda_{-}.$
(47)

Now, if we take the expansion (35) as the point of departure, we have two
slightly different approaches to the further computations. The equation (35)
represents finite case for when, the matrix series on the right side is
truncated with regard to the zeroth value of the product of the relevant
matrices. If we are able to perform the infinite summation on the right side
of the above formula, we also obtain the finite results. This is the first
approach, for which is more suitable to use the pure matrix expansion, despite
the structure of the right-sided terms, proportional to the different powers
of the coupling constant $g$. This represents the exact solution for our
problem. For the infinite expansion, where we cannot dispose the general
solution of the equation (35), we may regard, if it is physically reasonable,
the matrix $S$ as comprising a small parameter - the dimensionless coupling
constant $g\ll{1}$, for instance. It sometimes allows to run effective
calculations, physically treated as the perturbations. This is the second
approach. Obviously, both these methods are fully equivalent.

We can see immediately, the first approach is just described by the
prescription (35). This may only be rewritten for the convenience in more
compact form

${\cal F}^{-1}\left(x^{+},{\bar{x}},{\bar{y}}\right)={\cal
F}^{-1}_{0}\left(x^{+},{\bar{x}},{\bar{y}}\right)+\sum_{n=1}^{+\infty}{\cal
A}_{n}\left(x^{+},{\bar{x}},{\bar{y}}\right).$ (48)

Each contribution ${\cal A}_{n}(x^{+},{\bar{x}},{\bar{y}})$ of the $n$th order
to the above series represents the relevant product of the free matrix ${\cal
F}^{-1}_{0}$ and already introduced array with interactions ${\cal G}$. This
is

${\cal A}_{n}\left(x^{+},{\bar{x}},{\bar{y}}\right)=$ (49)
$=(-1)^{n}\left\\{{\prod_{j=1}^{n}}\left[\int_{R^{3}}\\!\\!{d^{3}{\bar{w}}_{j}}\\!\\!\int_{R^{3}}\\!\\!{d^{3}{\bar{z}}_{j}}\;\\!{\cal
F}^{-1}_{0}\\!\left(x^{+},{\bar{z}}_{j-1},{\bar{w}}_{j}\right){\cal
G}\left(x^{+},{\bar{w}}_{j},{\bar{z}}_{j}\right)\right]_{{{\bar{z}}_{0}={\bar{x}}}}\right\\}{\cal
F}^{-1}_{0}\\!\left(x^{+},{\bar{z}}_{n},{\bar{y}}\right).$

Noticeably, both powers, the first and the second, of the Yukawa coupling
constant $g$ are included in the matrix ${\cal G}$. Therefore, the constant
$g$ is not a parameter of the expansion.

The second approach, devoted to obtaining the inverse matrix ${\cal
F}^{-1}(x^{+},{\bar{x}},{\bar{y}})$, relies on the decomposition of the matrix
${\cal G}(x^{+},{\bar{w}},{\bar{z}})$. This array embraces the elements, which
are determined by the terms proportional to the first and to the second powers
of the coupling constant $g$. On account of that, we can decompose the ${\cal
G}(x^{+},{\bar{w}},{\bar{z}})$ into the sum of two matrix contributions,
proportional to aforementioned $g$ and $g^{2}$. This finally allows us to
write

${\cal G}\left(x^{+},{\bar{w}},{\bar{z}}\right)=g\;\\!{\tilde{\cal
G}}_{1}\left(x^{+},{\bar{w}},{\bar{z}}\right)+g^{2}\;\\!{\tilde{\cal
G}}_{2}\left(x^{+},{\bar{w}},{\bar{z}}\right).$ (50)

Thus, we have established two matrices, the ${\tilde{\cal
G}}_{1}(x^{+},{\bar{w}},{\bar{z}})$ and the ${\tilde{\cal
G}}_{2}(x^{+},{\bar{w}},{\bar{z}})$. The first one satisfies

${\tilde{\cal
G}}_{1}\left(x^{+},{\bar{w}},{\bar{z}}\right)=\left(\;\begin{array}[]{cccccc}0&\;\;0&\;\;0&\;\;0&\;\;0&\;\;0\\\
0&\;\;0&\;\;0&\;\;0&\;\;0&\;\;0\\\ 0&\;\;0&\;\;0&\;\;0&\;\;0&\;\;0\\\
0&\;\;0&\;\;0&\;\;0&\;\;{\tilde{\cal
G}}_{1}\left(x^{+},{\bar{w}},{\bar{z}}\right)_{45}&\;\;{\tilde{\cal
G}}_{1}\left(x^{+},{\bar{w}},{\bar{z}}\right)_{46}\\\
0&\;\;0&\;\;0&\;\;{\tilde{\cal
G}}_{1}\left(x^{+},{\bar{w}},{\bar{z}}\right)_{54}&\;\;0&\;\;{\tilde{\cal
G}}_{1}\left(x^{+},{\bar{w}},{\bar{z}}\right)_{56}\\\
0&\;\;0&\;\;0&\;\;{\tilde{\cal
G}}_{1}\left(x^{+},{\bar{w}},{\bar{z}}\right)_{64}&\;\;{\tilde{\cal
G}}_{1}\left(x^{+},{\bar{w}},{\bar{z}}\right)_{65}&\;\;0\\\
\end{array}\right).$ (51)

The elements of the above array are as follows:

${\tilde{\cal
G}}_{1}\left(x^{+},{\bar{w}},{\bar{z}}\right)_{45}=-\Psi^{\dagger}_{+}\left(x^{+},{\bar{z}}\right)\gamma^{-}\delta^{(3)}\left({\bar{w}}-{\bar{z}}\right),\;\;\;\;\;\;{\tilde{\cal
G}}_{1}\left(x^{+},{\bar{w}},{\bar{z}}\right)_{46}=\gamma^{+}\Psi_{+}\left(x^{+},{\bar{z}}\right)\delta^{(3)}\left({\bar{w}}-{\bar{z}}\right),$
(52) ${\tilde{\cal
G}}_{1}\left(x^{+},{\bar{w}},{\bar{z}}\right)_{54}=\Psi^{\dagger}_{+}\left(x^{+},{\bar{w}}\right)\gamma^{-}\delta^{(3)}\left({\bar{w}}-{\bar{z}}\right),\;\;\;\;\;\;{\tilde{\cal
G}}_{1}\left(x^{+},{\bar{w}},{\bar{z}}\right)_{56}={\tilde{\xi}}_{1}\left(x^{+},{\bar{w}},{\bar{z}}\right)\delta^{(3)}\left({\bar{w}}-{\bar{z}}\right),$
(53) ${\tilde{\cal
G}}_{1}\left(x^{+},{\bar{w}},{\bar{z}}\right)_{64}=-\gamma^{+}\Psi_{+}\left(x^{+},{\bar{w}}\right)\delta^{(3)}\left({\bar{w}}-{\bar{z}}\right),\;\;\;\;\;\;{\tilde{\cal
G}}_{1}\left(x^{+},{\bar{w}},{\bar{z}}\right)_{65}=-{\tilde{\xi}}_{1}\left(x^{+},{\bar{w}},{\bar{z}}\right)\delta^{(3)}\left({\bar{w}}-{\bar{z}}\right).$
(54)

The second matrix ${\tilde{\cal G}}_{2}(x^{+},{\bar{w}},{\bar{z}})$ is already
having the simple form

${\tilde{\cal
G}}_{2}\left(x^{+},{\bar{w}},{\bar{z}}\right)=\left(\;\begin{array}[]{cccccc}0&\;\;0&\;\;0&\;\;0&\;\;0&\;\;0\\\
0&\;\;0&\;\;0&\;\;0&\;\;0&\;\;0\\\ 0&\;\;0&\;\;0&\;\;0&\;\;0&\;\;0\\\
0&\;\;0&\;\;0&\;\;0&\;\;0&\;\;0\\\ 0&\;\;0&\;\;0&\;\;0&\;\;0&\;\;{\tilde{\cal
G}}_{2}\left(x^{+},{\bar{w}},{\bar{z}}\right)_{56}\\\
0&\;\;0&\;\;0&\;\;0&\;\;{\tilde{\cal
G}}_{2}\left(x^{+},{\bar{w}},{\bar{z}}\right)_{65}&\;\;0\\\
\end{array}\right),$ (55)

wherein the elements obey:

${\tilde{\cal
G}}_{2}\left(x^{+},{\bar{w}},{\bar{z}}\right)_{56}={\tilde{\xi}}_{2}\left(x^{+},{\bar{w}},{\bar{z}}\right)\delta^{(3)}\left({\bar{w}}-{\bar{z}}\right),\;\;\;\;\;\;{\tilde{\cal
G}}_{2}\left(x^{+},{\bar{w}},{\bar{z}}\right)_{65}=-{\tilde{\xi}}_{2}\left(x^{+},{\bar{w}},{\bar{z}}\right)\delta^{(3)}\left({\bar{w}}-{\bar{z}}\right).$
(56)

Both the operators ${\tilde{\xi}}_{1}(x^{+},{\bar{w}},{\bar{z}})$ and
${\tilde{\xi}}_{2}(x^{+},{\bar{w}},{\bar{z}})$ constitute already defined
$\xi_{g}(x^{+},{\bar{w}},{\bar{z}})$, due to the power decomposition (47) of
this part of the Dirac-Bergmann matrix, which includes the interactions:

${\xi}_{g}\\!\left(x^{+},{\bar{w}},{\bar{z}}\right)=g{\tilde{\xi}}_{1}\\!\left(x^{+},{\bar{w}},{\bar{z}}\right)+g^{2}{\tilde{\xi}}_{2}\\!\left(x^{+},{\bar{w}},{\bar{z}}\right),$
(57)
${\tilde{\xi}}_{1}\\!\left(x^{+},{\bar{w}},{\bar{z}}\right)\\!=\\!-i\sqrt{2}M\\!\left[{\phi}\left(x^{+},{\bar{w}}\right)\\!+\\!{\phi}\left(x^{+},{\bar{z}}\right)\right]\\!\Lambda_{-},\;\;\;\;\;\;{\tilde{\xi}}_{2}\\!\left(x^{+},{\bar{w}},{\bar{z}}\right)\\!=\\!-i\sqrt{2}{\phi}\left(x^{+},{\bar{w}}\right){\phi}\left(x^{+},{\bar{z}}\right)\\!\Lambda_{-}.$
(58)

The expansion (35) of the inverse matrix ${\cal
F}^{-1}(x^{+},{\bar{x}},{\bar{y}})$ should now be re-expressed by way of the
effective pattern b20 . This is possible by the insertion of the decomposition
of the ${\cal G}(x^{+},{\bar{w}},{\bar{z}})$ array into two terms (50) with
relevant powers of the coupling constant $g$. As a consequence, there the
series occurs with sequent powers of the said $g$. Finally

${\cal F}^{-1}\left(x^{+},{\bar{x}},{\bar{y}}\right)={\cal
F}^{-1}_{0}\left(x^{+},{\bar{x}},{\bar{y}}\right)+\sum_{n=1}^{+\infty}{\cal
B}_{n}\left(x^{+},{\bar{x}},{\bar{y}}\right),$ (59)

where the successive orders of obtained perturbations are:

${\cal B}_{n}\left(x^{+},{\bar{x}},{\bar{y}}\right)=$ (60)
$=g^{n}\\!\\!\\!\sum_{{{\bar{\rm{V}}}^{1,\dots,n}}_{2}}\sum_{l=1}^{n}(-1)^{l}\\!\\!\;\\!\left\\{{\prod_{j=1}^{l}}{{}^{{}^{\prime}}}\\!\left[\int_{R^{3}}\\!\\!{d^{3}{\bar{w}}_{j}}\\!\\!\int_{R^{3}}\\!\\!{d^{3}{\bar{z}}_{j}}\;\\!{\cal
F}^{-1}_{0}\\!\left(x^{+},{\bar{z}}_{j-1},{\bar{w}}_{j}\right){\tilde{\cal
G}}_{{\bar{\rm{v}}}_{2}^{j}}\\!\left(x^{+},{\bar{w}}_{j},{\bar{z}}_{j}\right)\right]_{{{\bar{z}}_{0}={\bar{x}}}}\\!\right\\}\\!{\cal
F}^{-1}_{0}\\!\left(x^{+},{\bar{z}}_{l},{\bar{y}}\right).$

The first sum runs all the elements of the set
${\bar{\rm{V}}}^{1,\dots,n}_{2}={\bar{\rm{V}}}^{1}_{2}\cup\dots\cup{\bar{\rm{V}}}^{n}_{2}$,
which is the union of the sets ${\bar{\rm{V}}}^{r}_{2}$, provided that each of
them separately includes the maps being the $r$-tuples of the two-element set
${\bar{\rm{v}}}^{r}_{2}\in{\bar{\rm{V}}}^{r}_{2}$ and $r=1,\dots,n$. The prime
next to the symbol of the product shows, that the sum of the indexes in the
chain of the matrices ${\tilde{\cal G}}_{1,2}$ is
${\bar{\rm{v}}}^{1}_{2}+\dots+{\bar{\rm{v}}}^{l}_{2}=n$, wherein the
${\bar{\rm{v}}}^{j}_{2}$ denotes not only the $j$-tuple of the two-element set
$\\{1,2\\}$, but simultaneously means the value of this map:
${\bar{\rm{v}}}^{j}_{2}=1,2$, whilst $j=1,\dots,l$. This condition provides in
fact, that the sum of the indexes of the aforesaid matrices ${\tilde{\cal
G}}_{1,2}$ gives the condition for their chain length or more precisely for
their number $l$ within each contribution to the (60), determining the $n$th
fixed order of the series ${\cal B}_{n}(x^{+},{\bar{x}},{\bar{y}})$.

##  4 Free Inverse Dirac-Bergmann Matrix

We should commence searches of the free inverse matrix of the Dirac-Bergmann
one from below remarks of the mathematical nature. We observe, that the
structure of the matrix (36) is fully anti-down-triangle. Therefore, we
postulate, that its inverse has the form of the anti-up-triangle array, but
the small block of the size $2\times 2$ at the very center is modified by the
non-trivial element ${\cal F}^{-1}_{044}(x^{+},{\bar{x}}-{\bar{y}})$. This one
is necessary for the reduction of our model to the case of $\alpha=0$.
Finally, the inverse matrix is the reflection with respect to the anti-
diagonal of the initial array ${\cal F}_{0}(x^{+},{\bar{w}},{\bar{z}})$, but
not pure. Until this moment we can say, in our case, about the well defined
procedure of obtaining the inverse of the functional matrix. In general, the
computation of the functional inverse matrix may lead if its algebraic frame
is not properly introduced, to many ambiguities. Some modifications of this
algebraic frame are acceptable if there is another reason for it, like in this
instance, where we add the non-trivial element ${\cal
F}^{-1}_{044}(x^{+},{\bar{x}}-{\bar{y}})$. Therefore, we postulate the inverse
of the free Dirac-Bergmann matrix (36) as

${\cal F}^{-1}_{0}\left(x^{+},{\bar{x}},{\bar{y}}\right)={\cal
F}^{-1}_{0}\left(x^{+},{\bar{x}}-{\bar{y}}\right)=$ (61)
$=\\!\left(\begin{array}[]{cccccc}0&\;{\cal F}^{\\!\\!\mbox{
}_{-1}}_{\\!\\!012}\\!\\!\left(x^{+}\\!,{\bar{x}}-{\bar{y}}\right)&\;0&\;0&\;0&\;{\cal
F}^{\\!\\!\mbox{
}_{-1}}_{\\!\\!016}\\!\\!\left(x^{+}\\!,{\bar{x}}-{\bar{y}}\right)\\\ {\cal
F}^{\\!\\!\mbox{
}_{-1}}_{\\!\\!021}\\!\\!\left(x^{+}\\!,{\bar{x}}-{\bar{y}}\right)&\;0&\;0&\;0&\;{\cal
F}^{\\!\\!\mbox{
}_{-1}}_{\\!\\!025}\\!\\!\left(x^{+}\\!,{\bar{x}}-{\bar{y}}\right)&\;0\\\
0&\;0&\;{\cal F}^{\\!\\!\mbox{
}_{-1}}_{\\!\\!033}\\!\\!\left(x^{+}\\!,{\bar{x}}-{\bar{y}}\right)&\;{\cal
F}^{\\!\\!\mbox{
}_{-1}}_{\\!\\!034}\\!\\!\left(x^{+}\\!,{\bar{x}}-{\bar{y}}\right)&\;0&\;0\\\
0&\;0&\;{\cal F}^{\\!\\!\mbox{
}_{-1}}_{\\!\\!043}\\!\\!\left(x^{+}\\!,{\bar{x}}-{\bar{y}}\right)&\;{\cal
F}^{\\!\\!\mbox{
}_{-1}}_{\\!\\!044}\\!\\!\left(x^{+}\\!,{\bar{x}}-{\bar{y}}\right)&\;0&\;0\\\
0&\;{\cal F}^{\\!\\!\mbox{
}_{-1}}_{\\!\\!052}\\!\\!\left(x^{+}\\!,{\bar{x}}-{\bar{y}}\right)&\;0&\;0&\;0&\;0\\\
{\cal F}^{\\!\\!\mbox{
}_{-1}}_{\\!\\!061}\\!\\!\left(x^{+}\\!,{\bar{x}}-{\bar{y}}\right)&\;0&\;0&\;0&\;0&\;0\\\
\end{array}\right)\\!.$

We introduced the elements of this inverse array as the following functions to
determine:

${\cal
F}^{-1}_{012}\left(x^{+},{\bar{x}}-{\bar{y}}\right)=c\left(x^{+},{\bar{x}}-{\bar{y}}\right)\Lambda_{-},\;\;\;\;\;\;\;\;\;{\cal
F}^{-1}_{016}\left(x^{+},{\bar{x}}-{\bar{y}}\right)=-a\left(x^{+},{\bar{x}}-{\bar{y}}\right)\Lambda_{-},$
(62) ${\cal
F}^{-1}_{021}\left(x^{+},{\bar{x}}-{\bar{y}}\right)=-c\left(x^{+},{\bar{x}}-{\bar{y}}\right)\Lambda_{-},\;\;\;\;\;\;\;\;\;{\cal
F}^{-1}_{025}\left(x^{+},{\bar{x}}-{\bar{y}}\right)=-a\left(x^{+},{\bar{x}}-{\bar{y}}\right)\Lambda_{-},$
(63) ${\cal
F}^{-1}_{033}\left(x^{+},{\bar{x}}-{\bar{y}}\right)=b_{1}\left(x^{+},{\bar{x}}-{\bar{y}}\right){\rm
P},\;\;\;\;\;\;\;\;\;{\cal
F}^{-1}_{034}\left(x^{+},{\bar{x}}-{\bar{y}}\right)=d_{1}\left(x^{+},{\bar{x}}-{\bar{y}}\right){\rm
P},$ (64) ${\cal
F}^{-1}_{043}\left(x^{+},{\bar{x}}-{\bar{y}}\right)=d_{2}\left(x^{+},{\bar{x}}-{\bar{y}}\right){\rm
P},\;\;\;\;\;\;\;\;\;{\cal
F}^{-1}_{044}\left(x^{+},{\bar{x}}-{\bar{y}}\right)=b_{2}\left(x^{+},{\bar{x}}-{\bar{y}}\right){\rm
P},$ (65) ${\cal
F}^{-1}_{052}\left(x^{+},{\bar{x}}-{\bar{y}}\right)=a\left(x^{+},{\bar{x}}-{\bar{y}}\right)\Lambda_{-},\;\;\;\;\;\;\;\;\;{\cal
F}^{-1}_{061}\left(x^{+},{\bar{x}}-{\bar{y}}\right)=a\left(x^{+},{\bar{x}}-{\bar{y}}\right)\Lambda_{-}.$
(66)

The matrix ${\cal F}^{-1}_{0}(x^{+},{\bar{x}}-{\bar{y}})$ does not embrace the
interactions and therefore is translationally symmetric, what explicitly
manifests in the second argument of its elements, being the subtractions of
the relevant light-front coordinates.

Both, the left- and the right-sided conditions (32) of the unambiguous
existence of the inverse matrix, applied to our case (61), lead to the system
of the differential equations for the above functions. The beginning and the
primary form of this system, taken directly from mentioned conditions, yields:

$2i\partial_{-}a\left(x^{+},{\bar{x}}\right)=\delta^{(3)}\left({\bar{x}}\right),\;\;\;\;\;\;\;\;\;2\partial_{-}c\left(x^{+},{\bar{x}}\right)+\sqrt{2}\left(\Delta_{\perp}-M^{2}\right)a\left(x^{+},{\bar{x}}\right)=0,$
(67)
$\alpha\partial^{2}_{-}b_{1}\left(x^{+},{\bar{x}}\right)+\partial_{-}\left(\alpha\Delta_{\perp}+1\right)d_{1}\left(x^{+},{\bar{x}}\right)=0,\;\;\;\;\;\;\alpha\partial^{2}_{-}b_{1}\left(x^{+},{\bar{x}}\right)-\partial_{-}\left(\alpha\Delta_{\perp}+1\right)d_{2}\left(x^{+},{\bar{x}}\right)=0,$
(68) $\alpha\partial^{2}_{-}b_{2}\left(x^{+},{\bar{x}}\right)=0,$ (69)
$2\alpha\partial^{2}_{-}d_{1}\left(x^{+},{\bar{x}}\right)=\delta^{(3)}\left({\bar{x}}\right),\;\;\;\;\;\;\;\;\;2\alpha\partial^{2}_{-}d_{1}\left(x^{+},{\bar{x}}\right)-2\partial_{-}\left(\alpha\Delta_{\perp}+1\right)b_{2}\left(x^{+},{\bar{x}}\right)=\delta^{(3)}\left({\bar{x}}\right),$
(70)
$-2\alpha\partial^{2}_{-}d_{2}\left(x^{+},{\bar{x}}\right)=\delta^{(3)}\left({\bar{x}}\right),\;\;\;\;\;\;\;\;\;-2\alpha\partial^{2}_{-}d_{2}\left(x^{+},{\bar{x}}\right)-2\partial_{-}\left(\alpha\Delta_{\perp}+1\right)b_{2}\left(x^{+},{\bar{x}}\right)=\delta^{(3)}\left({\bar{x}}\right).$
(71)

This can be brought, after the elementary transformations, to:

$2i\partial_{-}a\left(x^{+},{\bar{x}}\right)=\delta^{(3)}\left({\bar{x}}\right),\;\;\;\;\;\;\;\;\;2\partial_{-}c\left(x^{+},{\bar{x}}\right)+\sqrt{2}\left(\Delta_{\perp}-M^{2}\right)a\left(x^{+},{\bar{x}}\right)=0,$
(72)
$2\alpha\partial^{2}_{-}d_{1}\left(x^{+},{\bar{x}}\right)=\delta^{(3)}\left({\bar{x}}\right),\;\;\;\;\;\;\;\;\;-2\alpha\partial_{-}d_{2}\left(x^{+},{\bar{x}}\right)=\delta^{(3)}\left({\bar{x}}\right),$
(73)
$\alpha\partial^{2}_{-}b_{1}\left(x^{+},{\bar{x}}\right)+\partial_{-}\left(\alpha\Delta_{\perp}+1\right)d_{1}\left(x^{+},{\bar{x}}\right)=0,\;\;\;\;\;\;\alpha\partial^{2}_{-}b_{1}\left(x^{+},{\bar{x}}\right)-\partial_{-}\left(\alpha\Delta_{\perp}+1\right)d_{2}\left(x^{+},{\bar{x}}\right)=0,$
(74)
$\alpha\partial^{2}_{-}b_{2}\left(x^{+},{\bar{x}}\right)=0,\;\;\;\;\;\;\;\;\;\;\;\;\partial_{-}\left(\alpha\Delta_{\perp}+1\right)b_{2}\left(x^{+},{\bar{x}}\right)=0.$
(75)

We have three separate subsystems of these equations: the (72), the (74) in
relation to (73) and the (75). We easily observe, that the first (72) and the
third (75) ones are consistent and not overdetermined. We have to conclude, by
comparing the (74) with both the equations (73), that this subsystem of the
equations for the function $b_{1}(x^{+},{\bar{x}})$ is consistent and not
overdetermined, under the condition

$\partial_{-}d_{2}\left(x^{+},{\bar{x}}\right)=-\partial_{-}d_{1}\left(x^{+},{\bar{x}}\right).$
(76)

We should take this into account by solving, say, the second one of the
equations (73). We conclude, by incorporating the addressed condition, that
the system of the equations (67) - (71) and these being equivalent or deduced
from this, are also consistent. From this moment on the subsystem (74)
includes only one independent equation.

The general solution for the system of the equations (72) - (75) with the
condition (76) in the domain of the distribution should not cause serious
difficulties:

$a\left({\bar{x}}\right)={1\over{4i}}\;{\rm
sgn}\left(x^{-}\right)\delta^{(2)}\left({\bf x}_{\perp}\right)+a_{0}\left({\bf
x}_{\perp}\right),$ (77)
$c\left({\bar{x}}\right)=-{1\over{4\sqrt{2}i}}\mid\\!{x^{-}}\\!\mid\left(\Delta_{\perp}-M^{2}\right)\delta^{(2)}\left({\bf
x}_{\perp}\right)-{1\over{\sqrt{2}}}\;x^{-}\left(\Delta_{\perp}-M^{2}\right)a_{0}\left({\bf
x}_{\perp}\right)+c_{0}\left({\bf x}_{\perp}\right),$ (78)
$d_{1}\left({\bar{x}}\right)={1\over{4\alpha}}\mid\\!{x^{-}}\\!\mid\delta^{(2)}\left({\bf
x}_{\perp}\right)+x^{-}d_{10}\left({\bf x}_{\perp}\right)+d_{100}\left({\bf
x}_{\perp}\right),$ (79)
$d_{2}\left({\bar{x}}\right)=-{1\over{4\alpha}}\mid\\!{x^{-}}\\!\mid\delta^{(2)}\left({\bf
x}_{\perp}\right)-x^{-}d_{10}\left({\bf x}_{\perp}\right)+d_{200}\left({\bf
x}_{\perp}\right),$ (80) $b_{1}\left({\bar{x}}\right)=$ (81)
$=-{1\over{8\alpha}}{\rm
sgn}\left(x^{-}\right)\left(x^{-}\right)^{2}\left(\Delta_{\perp}+{1\over{\alpha}}\right)\delta^{(2)}\left({\bf
x}_{\perp}\right)-{1\over
2}\left(x^{-}\right)^{2}\left(\Delta_{\perp}+{1\over{\alpha}}\right)d_{10}\left({\bf
x}_{\perp}\right)+x^{-}b_{10}\left({\bf x}_{\perp}\right)+b_{100}\left({\bf
x}_{\perp}\right)\\!,$ $b_{2}\left({\bar{x}}\right)=x^{-}B\left({\bf
x}_{\perp}\right)+b_{20}\left({\bf x}_{\perp}\right).$ (82)

We have the following differential equation for the $B({\bf x}_{\perp})$
function

$\left(\Delta_{\perp}+{1\over{\alpha}}\right)B\left({\bf x}_{\perp}\right)=0,$
(83)

inferred from the second pattern of the subsystem (75). The most general
solution of this eigenproblem is well known in many coordinate systems. At
this point we confine ourselves to quote this obvious fact, until then without
broader discussion, concluding only, that necessary is to exclude the case
$B({\bf x}_{\perp})={constant}\neq 0$, as not satisfying the (83). The
solution $B({\bf x}_{\perp})=0$ is acceptable, but together with the
requirement $b_{20}({\bf x}_{\perp})\neq 0$, due to the fact, that we
introduced the non-trivial function $b_{2}({\bar{x}})$ for reduction of our
problem to the case of $\alpha=0$. The latter describes the pure Yukawa model,
discussed from presented point of view in b20 .

Of course, the general solutions (77) - (82) are not uniquely determined. The
functions: $a_{0}({\bf x}_{\perp})$, $b_{10}({\bf x}_{\perp})$, $b_{100}({\bf
x}_{\perp})$, $b_{20}({\bf x}_{\perp})\neq 0$, $c_{0}({\bf x}_{\perp})$,
$d_{10}({\bf x}_{\perp})$, $d_{100}({\bf x}_{\perp})$ and $d_{200}({\bf
x}_{\perp})$ are for now entirely arbitrary. Just like the $B({\bf
x}_{\perp})$, which however represents the solution of the eigenproblem (83),
but requires the boundary conditions for its final obtaining. Aforementioned
boundary conditions, imposed on the studied solutions (77) - (82), should
transform them into unequivocally determined. They also allow the computation
of the unambiguous light-front Dirac brackets. From the other side, there are
many lacks of the clarity, referred to the physical correctness of these
boundary conditions and relevant to the violation of the Lorentz or the
internal symmetries of the analyzed model b21 . Now, to avoid all discussed
ambiguities, we make the precondition, that physically acceptable is the
simplest case - the special solution of the system of the equations (72) -
(75) and of the equation (76), instead of the general. Therefore, we take:

$a\left({\bar{x}}\right)={1\over{4i}}\;{\rm
sgn}\left(x^{-}\right)\delta^{(2)}\left({\bf
x}_{\perp}\right),\;\;\;\;\;\;\;\;\;\;\;\;c\left({\bar{x}}\right)=-{1\over{4\sqrt{2}i}}\mid\\!{x^{-}}\\!\mid\left(\Delta_{\perp}-M^{2}\right)\delta^{(2)}\left({\bf
x}_{\perp}\right),$ (84)
$d_{1}\left({\bar{x}}\right)={1\over{4\alpha}}\mid\\!{x^{-}}\\!\mid\delta^{(2)}\left({\bf
x}_{\perp}\right),\;\;\;\;\;\;\;\;\;\;\;\;d_{2}\left({\bar{x}}\right)=-{1\over{4\alpha}}\mid\\!{x^{-}}\\!\mid\delta^{(2)}\left({\bf
x}_{\perp}\right),$ (85) $b_{1}\left({\bar{x}}\right)=-{1\over{8\alpha}}\;{\rm
sgn}\left(x^{-}\right)\left(x^{-}\right)^{2}\left(\Delta_{\perp}+{1\over{\alpha}}\right)\delta^{(2)}\left({\bf
x}_{\perp}\right),\;\;\;\;\;\;\;\;\;\;\;\;b_{2}\left({\bar{x}}\right)=x^{-}B\left({\bf
x}_{\perp}\right),$ (86)

wherein the function $B({\bf x}_{\perp})$ obviously satisfies the same
condition (83).

Here it is the free inverse Dirac-Bergmann matrix for the Yukawa model with
higher order derivatives at our disposal, so that we can compute this matrix
in the case of the interactions being switched on.

##  5 Inverse Dirac-Bergmann Matrix with Interactions

As mentioned earlier, we employ the first approach to obtain the inverse
Dirac-Bergmann matrix with interactions. This is based on the series expansion
(48), wherein the sequent terms within this pattern are described by the
formulae (49). In turn, the matrix containing the structure of the
interactions is determined by the expression (43). We are using the approach
especially fit for that case, in which we meet the finite expansion (48). We
commence the calculations from the three lowest order terms of the expansion
of the inverse Dirac-Bergmann matrix with interactions ${\cal F}^{-1}={\cal
F}^{-1}_{0}+{\cal A}_{1}+{\cal A}_{2}+{\cal A}_{3}$. The power of the
truncation for the studied series is not accidental, as it is clear after the
algebraic computations. According to the prescription (49), we have:

${\cal
A}_{1}\\!\left(x^{+},{\bar{x}},{\bar{y}}\right)=-\left\\{\int_{R^{3}}\\!{d^{3}{\bar{z}}_{1}}\\!\int_{R^{3}}\\!{d^{3}{\bar{z}}_{2}}\\!\;{\cal
F}^{-1}_{0}\\!\left(x^{+},{\bar{z}}_{1}\right){\cal
G}\\!\left(x^{+},{\bar{z}}_{1},{\bar{z}}_{2}\right){\cal
F}^{-1}_{0}\\!\left(x^{+},{\bar{z}}_{2},{\bar{y}}\right)\right\\},$ (87)
${\cal A}_{2}\\!\left(x^{+},{\bar{x}},{\bar{y}}\right)=$ (88)
$=\left\\{\int_{R^{3}}\\!{d^{3}{\bar{z}}_{1}}\dots\\!\int_{R^{3}}\\!{d^{3}{\bar{z}}_{4}}\\!\;{\cal
F}^{-1}_{0}\left(x^{+},{\bar{x}},{\bar{z}}_{1}\right){\cal
G}\\!\left(x^{+},{\bar{z}}_{1},{\bar{z}}_{2}\right){\cal
F}^{-1}_{0}\\!\left(x^{+},{\bar{z}}_{2},{\bar{z}}_{3}\right){\cal
G}\\!\left(x^{+},{\bar{z}}_{3},{\bar{z}}_{4}\right){\cal
F}^{-1}_{0}\\!\left(x^{+},{\bar{z}}_{4},{\bar{y}}\right)\right\\},$ ${\cal
A}_{3}\\!\left(x^{+},{\bar{x}},{\bar{y}}\right)=$ (89)
$=-\left\\{\int_{R^{3}}\\!{d^{3}{\bar{z}}_{1}}\dots\\!\int_{R^{3}}\\!{d^{3}{\bar{z}}_{6}}\\!\;{\cal
F}^{-1}_{0}\\!\left(x^{+},{\bar{x}},{\bar{z}}_{1}\right){\cal
G}\\!\left(x^{+},{\bar{z}}_{1},{\bar{z}}_{2}\right){\cal
F}^{-1}_{0}\\!\left(x^{+},{\bar{z}}_{2},{\bar{z}}_{3}\right){\cal
G}\\!\left(x^{+},{\bar{z}}_{3},{\bar{z}}_{4}\right)\times\right.$
$\left.\times{\cal
F}^{-1}_{0}\\!\left(x^{+},{\bar{z}}_{4},{\bar{z}}_{5}\right){\cal
G}\\!\left(x^{+},{\bar{z}}_{5},{\bar{z}}_{6}\right){\cal
F}^{-1}_{0}\\!\left(x^{+},{\bar{z}}_{6},{\bar{y}}\right)\right\\}.$

The first order contribution to the inverse Dirac-Bergmann matrix, described
by the effective pattern (87), gives the final result after the insertion of
the free inverse (61) - (66) and of the interaction arrays (43) - (46). It is
easy to convince, that the algebraic and the analytical computations lead to

${\cal
A}_{1}\\!\left(x^{+},{\bar{x}},{\bar{y}}\right)=-\left(\begin{array}[]{cccccc}0&\;\;{\cal
A}_{112}\left(x^{+},{\bar{x}},{\bar{y}}\right)&\;\;{\cal
A}_{113}\left(x^{+},{\bar{x}},{\bar{y}}\right)&\;\;{\cal
A}_{114}\left(x^{+},{\bar{x}},{\bar{y}}\right)&\;\;0&\;\;0\\\ {\cal
A}_{121}\left(x^{+},{\bar{x}},{\bar{y}}\right)&\;\;0&\;\;{\cal
A}_{123}\left(x^{+},{\bar{x}},{\bar{y}}\right)&\;\;{\cal
A}_{124}\left(x^{+},{\bar{x}},{\bar{y}}\right)&\;\;0&\;\;0\\\ {\cal
A}_{131}\left(x^{+},{\bar{x}},{\bar{y}}\right)&\;\;{\cal
A}_{132}\left(x^{+},{\bar{x}},{\bar{y}}\right)&\;\;0&\;\;0&\;\;0&\;\;0\\\
{\cal A}_{141}\left(x^{+},{\bar{x}},{\bar{y}}\right)&\;\;{\cal
A}_{142}\left(x^{+},{\bar{x}},{\bar{y}}\right)&\;\;0&\;\;0&\;\;0&\;\;0\\\
0&\;\;0&\;\;0&\;\;0&\;\;0&\;\;0\\\ 0&\;\;0&\;\;0&\;\;0&\;\;0&\;\;0\\\
\end{array}\right).$ (90)

The elements of this array include the fermionic fields, the scalar ones and
the functions derived from the inverse of the free Dirac-Bergmann matrix:
$a(x^{+},{\bar{x}})$, $b_{2}(x^{+},{\bar{x}})$, $d_{1,2}(x^{+},{\bar{x}})$.
They embrace mixed powers of the Yukawa coupling constant, as the method used
for this considerations permits:

${\cal
A}_{112}\left(x^{+},{\bar{x}},{\bar{y}}\right)=gF_{1}\left(x^{+},{\bar{x}},{\bar{y}}\right)+g^{2}F^{2}_{1}\left(x^{+},{\bar{x}},{\bar{y}}\right),$
(91) ${\cal
A}_{113}\left(x^{+},{\bar{x}},{\bar{y}}\right)=gF_{2}\left(x^{+},{\bar{x}},{\bar{y}}\right),\;\;\;\;\;\;\;\;\;\;\;\;{\cal
A}_{114}\left(x^{+},{\bar{x}},{\bar{y}}\right)=gF_{3}\left(x^{+},{\bar{x}},{\bar{y}}\right),$
(92) ${\cal
A}_{121}\left(x^{+},{\bar{x}},{\bar{y}}\right)=-gF_{1}\left(x^{+},{\bar{x}},{\bar{y}}\right)-g^{2}F^{2}_{1}\left(x^{+},{\bar{x}},{\bar{y}}\right),$
(93) ${\cal
A}_{123}\left(x^{+},{\bar{x}},{\bar{y}}\right)=gF_{4}\left(x^{+},{\bar{x}},{\bar{y}}\right),\;\;\;\;\;\;\;\;\;\;\;\;{\cal
A}_{124}\left(x^{+},{\bar{x}},{\bar{y}}\right)=gF_{5}\left(x^{+},{\bar{x}},{\bar{y}}\right),$
(94) ${\cal
A}_{131}\left(x^{+},{\bar{x}},{\bar{y}}\right)=-gF_{6}\left(x^{+},{\bar{x}},{\bar{y}}\right),\;\;\;\;\;\;\;\;\;\;\;\;{\cal
A}_{132}\left(x^{+},{\bar{x}},{\bar{y}}\right)=-gF_{7}\left(x^{+},{\bar{x}},{\bar{y}}\right),$
(95) ${\cal
A}_{141}\left(x^{+},{\bar{x}},{\bar{y}}\right)=-gF_{8}\left(x^{+},{\bar{x}},{\bar{y}}\right),\;\;\;\;\;\;\;\;\;\;\;\;{\cal
A}_{142}\left(x^{+},{\bar{x}},{\bar{y}}\right)=-gF_{9}\left(x^{+},{\bar{x}},{\bar{y}}\right).$
(96)

Moreover, we have the following description of these functions. Their specific
expressions contain the field operators in the linear fermionic and also in
the quadratic scalar terms:

$F_{1}\left(x^{+},{\bar{x}},{\bar{y}}\right)=2\sqrt{2}iM\\!\\!\int_{R^{3}}\\!{d^{3}{\bar{z}}}\;a\left(x^{+},{\bar{x}}-{\bar{z}}\right){\phi}\left(x^{+},{\bar{z}}\right)a\left(x^{+},{\bar{z}}-{\bar{y}}\right)\Lambda_{-},$
(97)
$F^{2}_{1}\left(x^{+},{\bar{x}},{\bar{y}}\right)=\sqrt{2}i\\!\\!\int_{R^{3}}\\!{d^{3}{\bar{z}}}\;a\left(x^{+},{\bar{x}}-{\bar{z}}\right){\phi}^{2}\left(x^{+},{\bar{z}}\right)a\left(x^{+},{\bar{z}}-{\bar{y}}\right)\Lambda_{-},$
(98)
$F_{2}\left(x^{+},{\bar{x}},{\bar{y}}\right)=-\\!\int_{R^{3}}\\!{d^{3}{\bar{z}}}\;a\left(x^{+},{\bar{x}}-{\bar{z}}\right)\gamma^{+}\Psi_{+}\left(x^{+},{\bar{z}}\right)d_{2}\left(x^{+},{\bar{z}}-{\bar{y}}\right),$
(99)
$F_{3}\left(x^{+},{\bar{x}},{\bar{y}}\right)=-\int_{R^{3}}\\!{d^{3}{\bar{z}}}\;a\left(x^{+},{\bar{x}}-{\bar{z}}\right)\gamma^{+}\Psi_{+}\left(x^{+},{\bar{z}}\right)b_{2}\left(x^{+},{\bar{z}}-{\bar{y}}\right),$
(100)
$F_{4}\left(x^{+},{\bar{x}},{\bar{y}}\right)=\int_{R^{3}}\\!{d^{3}{\bar{z}}}\;a\left(x^{+},{\bar{x}}-{\bar{z}}\right)\Psi^{\dagger}_{+}\left(x^{+},{\bar{z}}\right)\gamma^{-}d_{2}\left(x^{+},{\bar{z}}-{\bar{y}}\right),$
(101)
$F_{5}\left(x^{+},{\bar{x}},{\bar{y}}\right)=\\!\int_{R^{3}}\\!{d^{3}{\bar{z}}}\;a\left(x^{+},{\bar{x}}-{\bar{z}}\right)\Psi^{\dagger}_{+}\left(x^{+},{\bar{z}}\right)\gamma^{-}b_{2}\left(x^{+},{\bar{z}}-{\bar{y}}\right),$
(102)
$F_{6}\left(x^{+},{\bar{x}},{\bar{y}}\right)=\\!\int_{R^{3}}\\!{d^{3}{\bar{z}}}\;d_{1}\left(x^{+},{\bar{x}}-{\bar{z}}\right)\gamma^{+}\Psi_{+}\left(x^{+},{\bar{z}}\right)a\left(x^{+},{\bar{z}}-{\bar{y}}\right),$
(103)
$F_{7}\left(x^{+},{\bar{x}},{\bar{y}}\right)=-\\!\int_{R^{3}}\\!{d^{3}{\bar{z}}}\;d_{1}\left(x^{+},{\bar{x}}-{\bar{z}}\right)\Psi^{\dagger}_{+}\left(x^{+},{\bar{z}}\right)\gamma^{-}a\left(x^{+},{\bar{z}}-{\bar{y}}\right),$
(104)
$F_{8}\left(x^{+},{\bar{x}},{\bar{y}}\right)=\\!\int_{R^{3}}\\!{d^{3}{\bar{z}}}\;b_{2}\left(x^{+},{\bar{x}}-{\bar{z}}\right)\gamma^{+}\Psi_{+}\left(x^{+},{\bar{z}}\right)a\left(x^{+},{\bar{z}}-{\bar{y}}\right),$
(105)
$F_{9}\left(x^{+},{\bar{x}},{\bar{y}}\right)=-\\!\int_{R^{3}}\\!{d^{3}{\bar{z}}}\;b_{2}\left(x^{+},{\bar{x}}-{\bar{z}}\right)\Psi^{\dagger}_{+}\left(x^{+},{\bar{z}}\right)\gamma^{-}a\left(x^{+},{\bar{z}}-{\bar{y}}\right).$
(106)

The second order contribution ${\cal A}_{2}$ is described by the formula
analogous to the ${\cal A}_{1}$, but includes longer chain of the ${\cal
F}^{-1}_{0}$ and of the ${\cal G}$ matrices (88). The algebraic and the
analytical computations allow us to put final, effective expression for this
contribution as the array

${\cal
A}_{2}\left(x^{+},{\bar{x}},{\bar{y}}\right)=\left(\begin{array}[]{cccccc}{\cal
A}_{211}\left(x^{+},{\bar{x}},{\bar{y}}\right)&\;\;{\cal
A}_{212}\left(x^{+},{\bar{x}},{\bar{y}}\right)&\;\;0&\;\;0&\;\;0&\;\;0\\\
{\cal A}_{221}\left(x^{+},{\bar{x}},{\bar{y}}\right)&\;\;{\cal
A}_{222}\left(x^{+},{\bar{x}},{\bar{y}}\right)&\;\;0&\;\;0&\;\;0&\;\;0\\\
0&\;\;0&\;\;0&\;\;0&\;\;0&\;\;0\\\ 0&\;\;0&\;\;0&\;\;0&\;\;0&\;\;0\\\
0&\;\;0&\;\;0&\;\;0&\;\;0&\;\;0\\\ 0&\;\;0&\;\;0&\;\;0&\;\;0&\;\;0\\\
\end{array}\right),$ (107)

which embraces the integrals consisting of the function $a(x^{+},{\bar{x}})$
and of the $b_{2}(x^{+},{\bar{x}})$, taken from the inverse of the free Dirac-
Bergmann matrix. They are proportional to the second power of the Yukawa
coupling constant. It means, that:

${\cal
A}_{211}\left(x^{+},{\bar{x}},{\bar{y}}\right)=g^{2}H_{1}\left(x^{+},{\bar{x}},{\bar{y}}\right),\;\;\;\;\;\;{\cal
A}_{212}\left(x^{+},{\bar{x}},{\bar{y}}\right)=-g^{2}H_{2}\left(x^{+},{\bar{x}},{\bar{y}}\right),$
(108) ${\cal
A}_{221}\left(x^{+},{\bar{x}},{\bar{y}}\right)=g^{2}H_{3}\left(x^{+},{\bar{x}},{\bar{y}}\right),\;\;\;\;\;\;{\cal
A}_{222}\left(x^{+},{\bar{x}},{\bar{y}}\right)=-g^{2}H_{4}\left(x^{+},{\bar{x}},{\bar{y}}\right).$
(109)

The detailed structure of the above functions reveals the bi-linear tensor
products of the fermionic fields:

$H_{1}\left(x^{+},{\bar{x}},{\bar{y}}\right)=$ (110)
$=\\!\int_{R^{3}}\\!{d^{3}{\bar{z}}_{1}}\\!\int_{R^{3}}\\!{d^{3}{\bar{z}}_{2}}\;a\left(x^{+},{\bar{x}}-{\bar{z}}_{1}\right)\gamma^{+}{\Psi}_{+}\left(x^{+},{\bar{z}}_{1}\right)b_{2}\left(x^{+},{\bar{z}}_{1}-{\bar{z}}_{2}\right)\gamma^{+}{\Psi}_{+}\left(x^{+},{\bar{z}}_{2}\right)a\left(x^{+},{\bar{z}}_{2}-{\bar{y}}\right),$
$H_{2}\left(x^{+},{\bar{x}},{\bar{y}}\right)=$ (111)
$=\int_{R^{3}}\\!{d^{3}{\bar{z}}_{1}}\\!\int_{R^{3}}\\!{d^{3}{\bar{z}}_{2}}\;a\left(x^{+},{\bar{x}}-{\bar{z}}_{1}\right)\gamma^{+}{\Psi}_{+}\left(x^{+},{\bar{z}}_{1}\right)b_{2}\left(x^{+},{\bar{z}}_{1}-{\bar{z}}_{2}\right){\Psi}^{\dagger}_{+}\left(x^{+},{\bar{z}}_{2}\right)\gamma^{-}a\left(x^{+},{\bar{z}}_{2}-{\bar{y}}\right),$
$H_{3}\left(x^{+},{\bar{x}},{\bar{y}}\right)=$ (112)
$=\int_{R^{3}}\\!{d^{3}{\bar{z}}_{1}}\\!\int_{R^{3}}\\!{d^{3}{\bar{z}}_{2}}\;a\left(x^{+},{\bar{x}}-{\bar{z}}_{1}\right){\Psi}^{\dagger}_{+}\left(x^{+},{\bar{z}}_{1}\right)\gamma^{-}b_{2}\left(x^{+},{\bar{z}}_{1}-{\bar{z}}_{2}\right)\gamma^{+}{\Psi}_{+}\left(x^{+},{\bar{z}}_{2}\right)a\left(x^{+},{\bar{z}}_{2}-{\bar{y}}\right),$
$H_{4}\left(x^{+},{\bar{x}},{\bar{y}}\right)=$ (113)
$=-\\!\int_{R^{3}}\\!{d^{3}{\bar{z}}_{1}}\\!\int_{R^{3}}\\!{d^{3}{\bar{z}}_{2}}\;a\left(x^{+},{\bar{x}}-{\bar{z}}_{1}\right){\Psi}^{\dagger}_{+}\left(x^{+},{\bar{z}}_{1}\right)\gamma^{-}b_{2}\left(x^{+},{\bar{z}}_{1}-{\bar{z}}_{2}\right){\Psi}^{\dagger}_{+}\left(x^{+},{\bar{z}}_{2}\right)\gamma^{-}a\left(x^{+},{\bar{z}}_{2}-{\bar{y}}\right).$

The form of the third order contribution ${\cal A}_{3}$ to the inverse Dirac-
Bergmann matrix with interactions is determined by the pattern (89) and
includes the pure triple product of the arrays, denoted in the parentheses.
This is just trivial $({\cal F}^{-1}_{0}{\cal G})({\cal F}^{-1}_{0}{\cal
G})({\cal F}^{-1}_{0}{\cal G})=0$. Thus, the third order contribution to the
inverse Dirac-Bergmann matrix satisfies ${\cal A}_{3}=0$. And furthermore, the
general prescription (49) unveils its iterative feature, wherein the $n$th
order of the contribution is algebraically determined by the direct
predecessor of the $(n-1)$th order, then multiplied by the matrix chain link
$({\cal F}^{-1}_{0}{\cal G})$. Therefore, all the third and the higher order
contributions to the inverse Dirac-Bergmann matrix for the interacting Yukawa
model with Bernard-Duncan term vanish ${\cal A}_{n}(n\geq 3)=0$.

It can be seen, that the final array ${\cal F}^{-1}$ is determined by the
truncated and effective expansion ${\cal F}^{-1}={\cal F}^{-1}_{0}+{\cal
A}_{1}+{\cal A}_{2}$. This depends: on the functions (77) - (82) within the
free matrix ${\cal F}^{-1}_{0}$, on the first (97) - (106) and also on the
second (110) - (113) order contributions. In view of the above, we can
conclude, that the end result for the matrix ${\cal
F}^{-1}(x^{+},{\bar{x}},{\bar{y}})$ in discussed case is entirely finite

${\cal F}^{-1}\\!\left(x^{+},{\bar{x}},{\bar{y}}\right)=$ (114)
$=\left(\begin{array}[]{cccccc}{\cal F}^{-1}_{{11}_{{\mbox{ }}_{{\mbox{
}}_{\mbox{
}}}}}\\!\\!\\!\\!\\!\\!\left(x^{+},{\bar{x}},{\bar{y}}\right)&\;\;\;{\cal
F}^{-1}_{12}\\!\left(x^{+},{\bar{x}},{\bar{y}}\right)&\;\;\;{\cal
F}^{-1}_{13}\\!\left(x^{+},{\bar{x}},{\bar{y}}\right)&\;\;\;{\cal
F}^{-1}_{14}\\!\left(x^{+},{\bar{x}},{\bar{y}}\right)&\;\;\;0&\;\;\;{\cal
F}^{-1}_{16}\\!\left(x^{+},{\bar{x}},{\bar{y}}\right)\\\ {\cal
F}^{-1}_{{21}_{{\mbox{ }}_{{\mbox{ }}_{\mbox{
}}}}}\\!\\!\\!\\!\\!\\!\left(x^{+},{\bar{x}},{\bar{y}}\right)&\;\;\;{\cal
F}^{-1}_{22}\\!\left(x^{+},{\bar{x}},{\bar{y}}\right)&\;\;\;{\cal
F}^{-1}_{23}\\!\left(x^{+},{\bar{x}},{\bar{y}}\right)&\;\;\;{\cal
F}^{-1}_{24}\\!\left(x^{+},{\bar{x}},{\bar{y}}\right)&\;\;\;{\cal
F}^{-1}_{25}\\!\left(x^{+},{\bar{x}},{\bar{y}}\right)&\;\;\;0\\\ {\cal
F}^{-1}_{{31}_{{\mbox{ }}_{{\mbox{ }}_{\mbox{
}}}}}\\!\\!\\!\\!\\!\\!\left(x^{+},{\bar{x}},{\bar{y}}\right)&\;\;\;{\cal
F}^{-1}_{32}\\!\left(x^{+},{\bar{x}},{\bar{y}}\right)&\;\;\;{\cal
F}^{-1}_{33}\\!\left(x^{+},{\bar{x}},{\bar{y}}\right)&\;\;\;{\cal
F}^{-1}_{34}\\!\left(x^{+},{\bar{x}},{\bar{y}}\right)&\;\;\;0&\;\;\;0\\\ {\cal
F}^{-1}_{{41}_{{\mbox{ }}_{{\mbox{ }}_{\mbox{
}}}}}\\!\\!\\!\\!\\!\\!\left(x^{+},{\bar{x}},{\bar{y}}\right)&\;\;\;{\cal
F}^{-1}_{42}\\!\left(x^{+},{\bar{x}},{\bar{y}}\right)&\;\;\;{\cal
F}^{-1}_{43}\\!\left(x^{+},{\bar{x}},{\bar{y}}\right)&\;\;\;{\cal
F}^{-1}_{44}\\!\left(x^{+},{\bar{x}},{\bar{y}}\right)&\;\;\;0&\;\;\;0\\\
\;\;\;\;0_{{}_{{\mbox{ }}_{{\mbox{ }}_{\mbox{ }}}}}&\;\;\;{\cal
F}^{-1}_{52}\\!\left(x^{+},{\bar{x}},{\bar{y}}\right)&\;\;\;0&\;\;\;0&\;\;\;0&\;\;\;0\\\
{\cal
F}^{-1}_{61}\\!\left(x^{+},{\bar{x}},{\bar{y}}\right)&\;\;\;0&\;\;\;0&\;\;\;0&\;\;\;0&\;\;\;0\\\
\end{array}\right)\\!\\!.$

The elements of this array are displayed below:

${\cal
F}^{-1}_{11}\left(x^{+},{\bar{x}},{\bar{y}}\right)=g^{2}H_{1}\left(x^{+},{\bar{x}},{\bar{y}}\right),$
(115) ${\cal
F}^{-1}_{12}\left(x^{+},{\bar{x}},{\bar{y}}\right)=c\left(x^{+},{\bar{x}}-{\bar{y}}\right)\Lambda_{-}+gF_{1}\left(x^{+},{\bar{x}},{\bar{y}}\right)+g^{2}F^{2}_{1}\left(x^{+},{\bar{x}},{\bar{y}}\right)-g^{2}H_{2}\left(x^{+},{\bar{x}},{\bar{y}}\right),$
(116) ${\cal
F}^{-1}_{13}\left(x^{+},{\bar{x}},{\bar{y}}\right)=gF_{2}\left(x^{+},{\bar{x}},{\bar{y}}\right),\;\;\;\;\;\;\;\;\;\;\;\;{\cal
F}^{-1}_{14}\left(x^{+},{\bar{x}},{\bar{y}}\right)=gF_{3}\left(x^{+},{\bar{x}},{\bar{y}}\right),$
(117) ${\cal
F}^{-1}_{16}\left(x^{+},{\bar{x}},{\bar{y}}\right)=-a\left(x^{+},{\bar{x}}-{\bar{y}}\right)\Lambda_{-},$
(118) ${\cal
F}^{-1}_{21}\left(x^{+},{\bar{x}},{\bar{y}}\right)=-c\left(x^{+},{\bar{x}}-{\bar{y}}\right)\Lambda_{-}-gF_{1}\left(x^{+},{\bar{x}},{\bar{y}}\right)-g^{2}F^{2}_{1}\left(x^{+},{\bar{x}},{\bar{y}}\right)+g^{2}H_{3}\left(x^{+},{\bar{x}},{\bar{y}}\right),$
(119) ${\cal
F}^{-1}_{22}\left(x^{+},{\bar{x}},{\bar{y}}\right)=-g^{2}H_{4}\left(x^{+},{\bar{x}},{\bar{y}}\right),\;\;\;\;\;\;\;\;\;\;\;\;{\cal
F}^{-1}_{23}\left(x^{+},{\bar{x}},{\bar{y}}\right)=gF_{4}\left(x^{+},{\bar{x}},{\bar{y}}\right),$
(120) ${\cal
F}^{-1}_{24}\left(x^{+},{\bar{x}},{\bar{y}}\right)=gF_{5}\left(x^{+},{\bar{x}},{\bar{y}}\right),\;\;\;\;\;\;\;\;\;\;\;\;{\cal
F}^{-1}_{25}\left(x^{+},{\bar{x}},{\bar{y}}\right)=-a\left(x^{+},{\bar{x}}-{\bar{y}}\right)\Lambda_{-},$
(121) ${\cal
F}^{-1}_{31}\left(x^{+},{\bar{x}},{\bar{y}}\right)=-gF_{6}\left(x^{+},{\bar{x}},{\bar{y}}\right),\;\;\;\;\;\;\;\;\;\;\;\;{\cal
F}^{-1}_{32}\left(x^{+},{\bar{x}},{\bar{y}}\right)=-gF_{7}\left(x^{+},{\bar{x}},{\bar{y}}\right),$
(122) ${\cal
F}^{-1}_{33}\left(x^{+},{\bar{x}},{\bar{y}}\right)=b_{1}\left(x^{+},{\bar{x}}-{\bar{y}}\right){\rm
P},\;\;\;\;\;\;\;\;\;\;\;\;{\cal
F}^{-1}_{34}\left(x^{+},{\bar{x}},{\bar{y}}\right)=d_{1}\left(x^{+},{\bar{x}}-{\bar{y}}\right){\rm
P},$ (123) ${\cal
F}^{-1}_{41}\left(x^{+},{\bar{x}},{\bar{y}}\right)=-gF_{8}\left(x^{+},{\bar{x}},{\bar{y}}\right),\;\;\;\;\;\;\;\;\;\;\;\;{\cal
F}^{-1}_{42}\left(x^{+},{\bar{x}},{\bar{y}}\right)=-gF_{9}\left(x^{+},{\bar{x}},{\bar{y}}\right),$
(124) ${\cal
F}^{-1}_{43}\left(x^{+},{\bar{x}},{\bar{y}}\right)=d_{2}\left(x^{+},{\bar{x}}-{\bar{y}}\right){\rm
P},\;\;\;\;\;\;\;\;\;\;\;\;{\cal
F}^{-1}_{44}\left(x^{+},{\bar{x}},{\bar{y}}\right)=b_{2}\left(x^{+},{\bar{x}}-{\bar{y}}\right){\rm
P},$ (125) ${\cal
F}^{-1}_{52}\left(x^{+},{\bar{x}},{\bar{y}}\right)=a\left(x^{+},{\bar{x}}-{\bar{y}}\right)\Lambda_{-},\;\;\;\;\;\;\;\;\;\;\;\;{\cal
F}^{-1}_{61}\left(x^{+},{\bar{x}},{\bar{y}}\right)=a\left(x^{+},{\bar{x}}-{\bar{y}}\right)\Lambda_{-}.$
(126)

The inverse Dirac-Bergmann matrix ${\cal F}^{-1}(x^{+},{\bar{x}},{\bar{y}})$,
described through the above functions (115) - (126), incorporates the
information about the interactions within analyzed model, originating in the
Lagrange equations of motion. From this point of view presented method gives
the exact solution of the studied problem. But this observation refers only to
the case of the finite series (48), like in the Yukawa model. When formulating
it in some other way, there the question rises, whether already introduced
method is only perturbative and whether the exact. Of course, it also depends
on the complexity of the studied model (solvable or not), but this does not
exhaust the topics, especially for the functional methods, to which our
approach belongs b22 ; b23 ; b24 ; b25 .

##  6 (Anti-) commutators for Interacting Fields

The final results of our method (114) - (126), devoted to obtaining the
inverse matrix of the constraints for the interacting Yukawa model, may be
directly applied to compute the Dirac brackets from the Poisson ones,
according to the prescription (31). We take into considerations the canonical
Poisson brackets for our model: (13), (14), the equations of the constraints:
(15), (16), (17), (18), (19) and we insert all these expressions into the
pattern (31). Next, we apply the procedure of the quantization, carrying out
the replacement
$\\{\cdot,\cdot\\}_{\\!\mbox{}_{D,P}}\rightarrow(1/i)[\cdot,\cdot]$ or
$\\{\cdot,\cdot\\}_{\\!\mbox{}_{D,P}}\rightarrow(1/i)\\{\cdot,\cdot\\}$, due
to the gradation of the operator algebra b26 ; b27 . Thus:

$\left\\{\Psi_{+}\left(x^{+},{\bar{x}}\right)\;,\;\Psi_{+}\left(x^{+},{\bar{y}}\right)\right\\}=\left\\{\Psi_{+}\left(x^{+},{\bar{x}}\right)\;,\;\Psi_{+}\left(x^{+},{\bar{y}}\right)\right\\}_{\rm{can.}}=0,$
(127)
$\left\\{\Psi_{+}\left(x^{+},{\bar{x}}\right)\;,\;\Psi^{\dagger}_{+}\left(x^{+},{\bar{y}}\right)\right\\}=\left\\{\Psi_{+}\left(x^{+},{\bar{x}}\right)\;,\;\Psi^{\dagger}_{+}\left(x^{+},{\bar{y}}\right)\right\\}_{\rm{can.}}={1\over{\sqrt{2}}}\delta^{(3)}\left({\bar{x}}-{\bar{y}}\right)\Lambda_{+},$
(128)
$\left\\{\Psi_{+}\left(x^{+},{\bar{x}}\right)\;,\;\Psi_{-}\left(x^{+},{\bar{y}}\right)\right\\}=\left\\{\Psi_{+}\left(x^{+},{\bar{x}}\right)\;,\;\Psi_{-}\left(x^{+},{\bar{y}}\right)\right\\}_{\rm{can.}}=0,$
(129)
$\left\\{\Psi_{+}\left(x^{+},{\bar{x}}\right)\;,\;\Psi^{\dagger}_{-}\left(x^{+},{\bar{y}}\right)\right\\}={\underbrace{\left\\{\Psi_{+}\left(x^{+},{\bar{x}}\right)\;,\;\Psi^{\dagger}_{-}\left(x^{+},{\bar{y}}\right)\right\\}_{\rm{can.}}}_{0}}-$
(130)
$-{1\over{\sqrt{2}}}\left\\{i\\!\not\\!\partial^{x}_{\perp}a\left(x^{+},{\bar{x}}-{\bar{y}}\right)+a\left(x^{+},{\bar{x}}-{\bar{y}}\right)\left[M+g\phi\left(x^{+},{\bar{x}}\right)\right]\right\\}\gamma^{-},$
$\left\\{\Psi_{-}\left(x^{+},{\bar{x}}\right)\;,\;\Psi_{-}\left(x^{+},{\bar{y}}\right)\right\\}={\underbrace{\left\\{\Psi_{-}\left(x^{+},{\bar{x}}\right)\;,\;\Psi_{-}\left(x^{+},{\bar{y}}\right)\right\\}_{\rm{can.}}}_{0}}+ig^{2}H_{1}\left(x^{+},{\bar{x}},{\bar{y}}\right),$
(131)
$\left\\{\Psi_{-}\left(x^{+},{\bar{x}}\right)\;,\;\Psi^{\dagger}_{-}\left(x^{+},{\bar{y}}\right)\right\\}={\underbrace{\left\\{\Psi_{-}\left(x^{+},{\bar{x}}\right)\;,\;\Psi^{\dagger}_{-}\left(x^{+},{\bar{y}}\right)\right\\}_{\rm{can.}}}_{0}}+$
(132)
$+ic\left(x^{+},{\bar{x}}-{\bar{y}}\right)\Lambda_{-}+igF_{1}\left(x^{+},{\bar{x}},{\bar{y}}\right)+ig^{2}F^{2}_{1}\left(x^{+},{\bar{x}},{\bar{y}}\right)-ig^{2}H_{2}\left(x^{+},{\bar{x}},{\bar{y}}\right),$
$\left[\Psi_{+}\left(x^{+},{\bar{x}}\right)\;,\;\phi\left(x^{+},{\bar{y}}\right)\right]=\left[\Psi_{+}\left(x^{+},{\bar{x}}\right)\;,\,\phi\left(x^{+},{\bar{y}}\right)\right]_{\rm{can.}}=0,$
(133)
$\left[\Psi_{+}\left(x^{+},{\bar{x}}\right)\;,\;\partial_{+}\phi\left(x^{+},{\bar{y}}\right)\right]=\left[\Psi_{+}\left(x^{+},{\bar{x}}\right)\;,\;\partial_{+}\phi\left(x^{+},{\bar{y}}\right)\right]_{\rm{can.}}=0,$
(134)
$\left[\Psi_{-}\left(x^{+},{\bar{x}}\right)\;,\;\phi\left(x^{+},{\bar{y}}\right)\right]={\underbrace{\left[\Psi_{-}\left(x^{+},{\bar{x}}\right)\;,\;\phi\left(x^{+},{\bar{y}}\right)\right]_{\rm{can.}}}_{0}}+igF_{3}\left(x^{+},{\bar{x}},{\bar{y}}\right),$
(135)
$\left[\Psi_{-}\left(x^{+},{\bar{x}}\right)\;,\;\partial_{+}\phi\left(x^{+},{\bar{y}}\right)\right]={\underbrace{\left[\Psi_{-}\left(x^{+},{\bar{x}}\right)\;,\;\partial_{+}\phi\left(x^{+},{\bar{y}}\right)\right]_{\rm{can.}}}_{0}}+igF_{6}\left(x^{+},{\bar{y}},{\bar{x}}\right),$
(136)
$\left[\phi\left(x^{+},{\bar{x}}\right)\;,\;\phi\left(x^{+},{\bar{y}}\right)\right]={\underbrace{\left[\phi\left(x^{+},{\bar{x}}\right)\;,\;\phi\left(x^{+},{\bar{y}}\right)\right]_{\rm{can.}}}_{0}}+ib_{2}\left(x^{+},{\bar{x}}-{\bar{y}}\right),$
(137)
$\left[\phi\left(x^{+},{\bar{x}}\right)\;,\;\partial_{+}\phi\left(x^{+},{\bar{y}}\right)\right]={\underbrace{\left[\phi\left(x^{+},{\bar{x}}\right)\;,\;\partial_{+}\phi\left(x^{+},{\bar{y}}\right)\right]_{\rm{can.}}}_{0}}-id_{1}\left(x^{+},{\bar{y}}-{\bar{x}}\right),$
(138)
$\left[\partial_{+}\phi\left(x^{+},{\bar{x}}\right)\;,\;\partial_{+}\phi\left(x^{+},{\bar{y}}\right)\right]={\underbrace{\left[\partial_{+}\phi\left(x^{+},{\bar{x}}\right)\;,\;\partial_{+}\phi\left(x^{+},{\bar{y}}\right)\right]_{\rm{can.}}}_{0}}+ib_{1}\left(x^{+},{\bar{x}}-{\bar{y}}\right).$
(139)

The remaining part of this algebra may easily be obtained from the above
expressions by the Hermitian conjugation, what leads to another, but not
displayed here (anti-) commutators, which include one or two fermionic fields
as their arguments. Such abbreviated way of dealing is allowed due to the
fact, that presented method exploits all of the constraints and finally leads
to the consistent set of the (anti-) commutators. The similar situation recurs
for the derivatives of the discussed algebra with respect to the coordinates
$x^{-}$ and to the $x^{j}$, where $j=1,2$. The fields:
$\partial_{\bar{\mu}}\Psi_{\pm}$, $\partial_{\bar{\mu}}\Psi^{\dagger}_{\pm}$,
$\partial_{\bar{\mu}}\phi$ for $\bar{\mu}=-,j$ are not fundamental in our
approach and therefore, the (anti-) commutators with presence of them can by
straightforwardly computed by the relevant differentiation.

Let us demonstrate the structure of the above algebra in greater detail. The
expressions: (131), (132), (135), (136) describe some of the elements of this
algebra and include functions taken from the procedure of computing the
inverse Dirac-Bergmann matrix: $F_{1,3,6}(x^{+},{\bar{x}},{\bar{y}})$,
$F^{2}_{1}(x^{+},{\bar{x}},{\bar{y}})$, $H_{1,2}(x^{+},{\bar{x}},{\bar{y}})$.
These functions are defined by: (97), (100), (103), (98), (110) and (111).
Herein we have, under the integral sign of these objects, the elements of the
free inverse Dirac-Bergmann matrix: $a(x^{+},{\bar{x}})$,
$b_{2}(x^{+},{\bar{x}})$, $d_{1,2}(x^{+},{\bar{x}})$. They satisfy the
differential equations in the domain of the distribution (72) - (75), which
right sides are, among others, the Dirac deltas. The general solutions of
these equations are (77) - (82) and they allow us to bring relevant (anti-)
commutators to the form of the local relationships, which consist the linear
or the bi-linear combinations of the indefinite integrals of the fundamental
fields. We introduced these integrals via the symbol $\partial^{-1}_{-}$ or
the $\partial^{-2}_{-}$, for the single or the double indefinite integral of
the field with respect to the $x^{-}$ variable. The integration by parts of:
(97), (98), (100), (103), (110), (111) with respect to the variable $z^{-}$
reveals also, that occurring limit terms for $z^{-}\rightarrow\mp\infty$ may
be singular and thus, the whole of the final expressions of this kind could
demand regularization. Therefore, we can rewrite the anti-commutator (131) as

$\left\\{\Psi_{-}\left(x^{+},x^{-},{\bf
x}_{\perp}\right)\;,\;\Psi_{-}\left(x^{+},y^{-},{\bf
y}_{\perp}\right)\right\\}=ig^{2}h^{\rm{reg.}}_{1}\left(x^{+},x^{-},{\bf
x}_{\perp},y^{-},{\bf y}_{\perp}\right)-$ (140) $-{i\over
4}g^{2}\gamma^{+}\left(\partial^{-2}_{-}\Psi_{+}\right)\left(x^{+},x^{-},{\bf
x}_{\perp}\right)B\left({\bf x}_{\perp}-{\bf
y}_{\perp}\right)\gamma^{+}\left(\partial^{-1}_{-}\Psi_{+}\right)\left(x^{+},y^{-},{\bf
y}_{\perp}\right)+$ $+{i\over
4}g^{2}\gamma^{+}\left(\partial^{-1}_{-}\Psi_{+}\right)\left(x^{+},x^{-},{\bf
x}_{\perp}\right)B\left({\bf x}_{\perp}-{\bf
y}_{\perp}\right)\gamma^{+}\left(\partial^{-2}_{-}\Psi_{+}\right)\left(x^{+},y^{-},{\bf
y}_{\perp}\right)+$ $+{i\over
4}g^{2}\gamma^{+}\left(\partial^{-1}_{-}\Psi_{+}\right)\left(x^{+},x^{-},{\bf
x}_{\perp}\right)\Big{[}\left(x^{-}-y^{-}\right)B\left({\bf x}_{\perp}-{\bf
y}_{\perp}\right)+b_{20}\left({\bf x}_{\perp}-{\bf
y}_{\perp}\right)\Big{]}\gamma^{+}\left(\partial^{-1}_{-}\Psi_{+}\right)\left(x^{+},y^{-},{\bf
y}_{\perp}\right).$

Similarly, the equation (132) takes the form

$\left\\{\Psi_{-}\left(x^{+},x^{-},{\bf
x}_{\perp}\right)\;,\;\Psi^{\dagger}_{-}\left(x^{+},y^{-},{\bf
y}_{\perp}\right)\right\\}=$ (141)
$=igf^{\;\rm{reg.}}_{1}\left(x^{+},x^{-},{\bf x}_{\perp},y^{-},{\bf
y}_{\perp}\right)+ig^{2}f^{2\;\rm{reg.}}_{1}\left(x^{+},x^{-},{\bf
x}_{\perp},y^{-},{\bf
y}_{\perp}\right)-ig^{2}h^{\rm{reg.}}_{2}\left(x^{+},x^{-},{\bf
x}_{\perp},y^{-},{\bf y}_{\perp}\right)-$
$-{i\over\sqrt{2}}\left(x^{-}-y^{-}\right)\\!\left(\Delta^{x}_{\perp}-M^{2}\right)\\!\left[{1\over{4i}}\;{\rm{sgn}}\left(x^{-}-y^{-}\right)\delta^{(2)}\left({\bf
x}_{\perp}-{\bf y}_{\perp}\right)\\!+\\!a_{0}\left({\bf x}_{\perp}-{\bf
y}_{\perp}\right)\right]\\!\Lambda_{-}+ic_{0}\left({\bf x}_{\perp}-{\bf
y}_{\perp}\right)\\!\Lambda_{-}+$ $+\\!{i\over
4}g^{2}\gamma^{+}\left(\partial^{-2}_{-}\Psi_{+}\right)\left(x^{+},x^{-},{\bf
x}_{\perp}\right)B\left({\bf x}_{\perp}-{\bf
y}_{\perp}\right)\left(\partial^{-1}_{-}\Psi^{\dagger}_{+}\right)\left(x^{+},y^{-},{\bf
y}_{\perp}\right)\gamma^{-}-$ $-{i\over
4}g^{2}\gamma^{+}\left(\partial^{-1}_{-}\Psi_{+}\right)\left(x^{+},x^{-},{\bf
x}_{\perp}\right)B\left({\bf x}_{\perp}-{\bf
y}_{\perp}\right)\left(\partial^{-2}_{-}\Psi^{\dagger}_{+}\right)\left(x^{+},y^{-},{\bf
y}_{\perp}\right)\gamma^{-}-$ $-{i\over
4}g^{2}\gamma^{+}\left(\partial^{-1}_{-}\Psi_{+}\right)\left(x^{+},x^{-},{\bf
x}_{\perp}\right)\left[\left(x^{-}-y^{-}\right)B\left({\bf x}_{\perp}-{\bf
y}_{\perp}\right)+b_{20}\left({\bf x}_{\perp}-{\bf
y}_{\perp}\right)\right]\left(\partial^{-1}_{-}\Psi^{\dagger}_{+}\right)\left(x^{+},y^{-},{\bf
y}_{\perp}\right)\gamma^{-}+$
$\\!+\sqrt{2}igM\\!\Big{[}\\!\\!\left(\partial_{-}^{-1}\phi\right)\\!\\!\left(x^{+},x^{-},{\bf
x}_{\perp}\right)\\!-\\!\left(\partial_{-}^{-1}\phi\right)\\!\\!\left(x^{+},y^{-},{\bf
y}_{\perp}\right)\\!\\!\Big{]}\\!\\!\left[\\!{1\over{4i}}{\rm{sgn}}\\!\left(x^{-}-y^{-}\right)\\!\delta^{(2)}\\!\left({\bf
x}_{\perp}-{\bf y}_{\perp}\right)\\!+\\!a_{0}\\!\left({\bf x}_{\perp}-{\bf
y}_{\perp}\right)\\!\right]\\!\\!\Lambda_{-}+$
$+{i\over\sqrt{2}}g^{2}\\!\Big{[}\\!\\!\left(\partial_{-}^{-1}\phi^{2}\right)\\!\\!\left(x^{+},x^{-},{\bf
x}_{\perp}\right)-\left(\partial_{-}^{-1}\phi^{2}\right)\\!\\!\left(x^{+},y^{-},{\bf
y}_{\perp}\right)\\!\\!\Big{]}\\!\\!\left[\\!{1\over{4i}}{\rm{sgn}}\\!\left(x^{-}-y^{-}\right)\\!\delta^{(2)}\\!\left({\bf
x}_{\perp}-{\bf y}_{\perp}\right)\\!+\\!a_{0}\\!\left({\bf x}_{\perp}-{\bf
y}_{\perp}\right)\\!\right]\\!\\!\Lambda_{-}.$

The equation (135), on the other hand, can be expressed as

$\left[\Psi_{-}\left(x^{+},x^{-},{\bf
x}_{\perp}\right)\;,\;\phi\left(x^{+},y^{-},{\bf
y}_{\perp}\right)\right]=igf^{\;\rm{reg.}}_{3}\left(x^{+},x^{-},{\bf
x}_{\perp},y^{-},{\bf y}_{\perp}\right)+$ (142) $+{1\over
2}g\gamma^{+}\left(\partial^{-2}_{-}\Psi_{+}\right)\left(x^{+},x^{-},{\bf
x}_{\perp}\right)B\left({\bf x}_{\perp}-{\bf y}_{\perp}\right)-$ $-{1\over
2}g\gamma^{+}\left(\partial^{-1}_{-}\Psi_{+}\right)\left(x^{+},x^{-},{\bf
x}_{\perp}\right)\left[\left(x^{-}-y^{-}\right)B\left({\bf x}_{\perp}-{\bf
y}_{\perp}\right)+b_{20}\left({\bf x}_{\perp}-{\bf y}_{\perp}\right)\right].$

And then, from the pattern (136), we have

$\left[\Psi_{-}\left(x^{+},x^{-},{\bf
x}_{\perp}\right)\;,\;\partial_{+}\phi\left(x^{+},y^{-},{\bf
y}_{\perp}\right)\right]=igf_{6}^{\;\rm{reg.}}\left(x^{+},y^{-},{\bf
y}_{\perp},x^{-},{\bf x}_{\perp}\right)-$ (143)
$-{i\over{2\alpha}}g\gamma^{+}\left(\partial^{-2}_{-}\Psi_{+}\right)\left(x^{+},y^{-},{\bf
y}_{\perp}\right)\left[{1\over{4i}}\delta^{(2)}\left({\bf x}_{\perp}-{\bf
y}_{\perp}\right){\rm{sgn}}\left(x^{-}-y^{-}\right)-a_{0}\left({\bf
y}_{\perp}-{\bf x}_{\perp}\right)\right]+$ $+{1\over
2}g\gamma^{+}\left(\partial^{-2}_{-}\Psi_{+}\right)\left(x^{+},x^{-},{\bf
x}_{\perp}\right)\left[{1\over{4\alpha}}\delta^{(2)}\left({\bf x}_{\perp}-{\bf
y}_{\perp}\right){\rm{sgn}}\left(x^{-}-y^{-}\right)-d_{10}\left({\bf
y}_{\perp}-{\bf x}_{\perp}\right)\right]-$ $-{1\over
2}g\gamma^{+}\\!\left(\partial^{-1}_{-}\Psi_{+}\right)\\!\\!\left(x^{+},x^{-},{\bf
x}_{\perp}\right)\\!\\!\left[{1\over{4\alpha}}\delta^{(2)}\\!\left({\bf
x}_{\perp}-{\bf
y}_{\perp}\right)\\!\mid\\!x^{-}-y^{-}\\!\mid\\!-\\!\left(x^{-}-y^{-}\right)\\!d_{10}\\!\left({\bf
y}_{\perp}-{\bf x}_{\perp}\right)\\!+\\!d_{100}\\!\left({\bf y}_{\perp}-{\bf
x}_{\perp}\right)\right]\\!.$

The contributions: $f^{\;\rm{reg.}}_{1,3,6}(x^{+},{\bar{x}},{\bar{y}})$,
$f^{2\;\rm{reg.}}_{1}(x^{+},{\bar{x}},{\bar{y}})$ and
$h^{\rm{reg.}}_{1,2}(x^{+},{\bar{x}},{\bar{y}})$ are the regularized limit
terms of relevant functions: (97), (100), (103), (98), (110) and (111). They
require further analysis.

##  7 Limit $\alpha=0$

Whenever $\alpha=0$, the Lagrangian density of the light-front Yukawa model
without higher order derivatives, discussed from presented point of view in
b20 , is restored. This turn needs to be analyzed a little more specifically,
however. The set of the constraints is reduced by one in this case and
therefore, the size of the Dirac-Bergmann matrix yields $5\times 5$, instead
of $6\times 6$. The first equations within (12), (14) and also the (16)
disappear, what causes, that the third row and the third column of the Dirac-
Bergmann matrix (21), of its decompositions: (36), (43), (51), (55) and of its
inverse: (61), (90), (107), (114) are eliminated. Thus, the functions
$b_{1}(x^{+},{\bar{x}})$ and $d_{1,2}(x^{+},{\bar{x}})$ come out from the
considerations. This follows, that also
$F_{2,4,6,7}(x^{+},{\bar{x}},{\bar{y}})$ vanish and only:
$F_{1,3,5,8,9}(x^{+},{\bar{x}},{\bar{y}})$,
$F^{2}_{1}(x^{+},{\bar{x}},{\bar{y}})$,
$H_{1,2,3,4}(x^{+},{\bar{x}},{\bar{y}})$ still are present in the Yukawa model
with $\alpha=0$. Then, the set of the equations (67) - (71), being the
conditions of the invertibility of the Dirac-Bergmann matrix, is reduced by
erasing both the formulae (68) and the first of: (70), (71). The pattern (69)
becomes trivial and the whole set of the studied equations embraces from now
both the relationships (67) and the one
$\partial_{-}b_{2}({\bar{x}})=-(1/2)\delta^{(3)}({\bar{x}})$, taken from the
second expression (70) or (71) in the limit $\alpha=0$. Therefore, the
obtained set of equations is the same as in b20 . We complete this part of the
discussion by two comments. The first is, that the function $b_{2}({\bar{x}})$
changes its character before and after the $\alpha=0$ turn. Before, we have
this function described by the (82) with $B({\bf x}_{\perp})$ determined by
the (83) and after, there is
$\partial_{-}b_{2}({\bar{x}})=-(1/2)\delta^{(3)}({\bar{x}})$, what shows, that
the limit $\alpha=0$ is consistent with the general case $\alpha\neq 0$, but
requires delicate handling. This remains in relation to the second
observation, referred to the fact, that the primary system of the equations
(67) - (71) describes both the cases: $\alpha\neq 0$ and $\alpha=0$, whereas
its consequence, the set (72) - (75) with (76), is valid only for $\alpha\neq
0$ and cannot be applied for taking the aforementioned limit $\alpha=0$.

##  8 Conclusions

This work is devoted to the quantization of the light-front Yukawa model in
$D=1+3$ dimensions with higher order derivatives of the scalar field,
described by the Bernard-Duncan term. The discussion is focused on the problem
of obtaining the Dirac brackets and then, the (anti-) commutator algebra of
interacting fields in presence of the constraints. The Dirac method for the
constraints in the quantum theories and the Ostrogradski formalism for the
higher order derivatives are used. The second purpose of this work is to
discuss the sensitivity of the results on the modifications, caused by
introduction of the aforesaid higher order derivatives, in comparison to the
same model without the Bernard-Duncan term. From the mathematical point of
view the major concern is to compute the inverse of the functional Dirac-
Bergmann matrix with the interactions and with the higher order derivative
term. The relevant approach for solving this problem in the case of the array,
which includes the differential operators and the fermionic bispinor operators
is introduced and applied, what gives the complete algebra of the (anti-)
commutators, corresponding to the constraints. Proposed approach is based on
the certain identity of the matrix series for the inverted array. This may
have two fully equivalent variants. The first, applied in this work, handles
only the pure matrix series, whereas the second one emphasizes the array
expansion in the powers of the Yukawa coupling constant between fermions and
scalar. The latter seems to have more perturbative character. Both, the
structure of the light-front Yukawa model with higher order derivatives and
the structure of the relevant constraints lead to the finite expansion of the
functional inverse of the Dirac-Bergmann matrix. Thus, the first variant of
already introduced approach is used here for the calculations. Of course, the
second variant of deliberated approach gives the same, truncated results for
the concerned (anti-) commutator algebra of interacting fields in the case of
our Yukawa model. This contains only the contributions of the first and of the
second order of the coupling constant $g$. These results agree with the work
b20 , devoted to the model without higher order derivatives. The next problem
arises at this point and demands further analysis. This is dedicated to the
general formulation of the functions, which define the free inverse Dirac-
Bergmann matrix.

On the flip side, the structure of the results for the (anti-) commutator
algebra is strongly dependent on the higher order derivative contribution and
rapidly changes its character, although finite, in the limit $\alpha=0$, i.e.
wherein the Bernard-Duncan term is switched off. This feature may initialize
the searches, interesting from the physical point of view, for such the
models, for such the constraints and interactions, which exhibit the infinite
character of the series determining aforementioned algebra. It opens the
question, for what class of the models the functional inverse Dirac-Bergmann
matrix of the constraints with interactions can be computed exactly and for
what this can only be derived approximately, giving the infinite, perturbative
solution of the established problem. This is a very good subject of the future
considerations.

Appendix A: Heisenberg Equations for Yukawa Model with Higher Order
Derivatives

The presence of $n$ higher order derivatives of the single scalar field within
${\cal L}={\cal L}(\phi,\phi_{,\mu_{1}},\dots,\phi_{,\mu_{1}\dots\mu_{n}})$
modifies the density of the canonical energy-momentum tensor. Now, this object
yields in the general case

${\cal
T}^{\mu\nu}=\sum_{r=1}^{n}\left\\{\sum_{s=0}^{n-r}(-1)^{s}\left(\prod_{t=1}^{s}\partial_{\rho_{t}}\right)\left({{\partial{\cal
L}}\over{\partial\phi_{,\mu\sigma_{1}\dots\sigma_{r-1}\rho_{1}\dots\rho_{s}}}}\right)\right\\}\left(\prod_{u=1}^{r-1}\partial_{\sigma_{u}}\right)\partial^{\nu}\phi-g^{\mu\nu}{\cal
L}.$ (144)

This pattern replaces the standard definition for the ${\cal T}^{\mu\nu}$,
which refers to the presence of only the first order derivative of the scalar
field: ${\cal T}^{\mu\nu}=(\partial{\cal
L}/\partial\phi_{,\mu})\partial^{\nu}\phi-g^{\mu\nu}{\cal L}$. The new
formulation of the density of the energy-momentum tensor leads to more
severalfold precise structure of the operators $P^{\mu}$ for the translation
in the $x^{\mu}$ directions, however their general definition, displayed
below, does not change. These operators for the light-front formulation,
wherein the variable of evolution is $x^{+}$, obey

$P^{\mu}\left(x^{+}\right)=\\!\int_{R^{3}}\\!\\!{d^{3}}{\bar{y}}\;{\cal
T}^{+\mu}\left(x^{+},{\bar{y}}\right),\;\;\;\;\;\;\mu=+,-,j,\;\;\;\;\;\;j=1,2.$
(145)

The extension of the Lagrangian density described in this work affects, as a
consequence, the results inferred from the Heisenberg equations

$i\partial_{\mu}\psi_{a}\left(x^{+},{\bar{x}}\right)=\left[\psi_{a}\left(x^{+},{\bar{x}}\right)\;,\;P_{\mu}\left(x^{+}\right)\right],\;\;\;\;\;\;\mu=+,-,j,\;\;\;\;\;\;j=1,2.$
(146)

In our case is
$\psi_{a}=\Psi_{\pm},\Psi^{\dagger}_{\pm},\partial_{\pm}\phi,\partial_{j}\phi$,
so the Heisenberg equations relate to the fields and to the derivatives of the
field, which are present in this considerations. These equations b28  may
directly allow to deduce some elements of the (anti-) commutator algebra for
discussed model, but without appealing to the canonical quantization rules or
other methods b29 ; b30 ; b31 , being to our disposal. We obtain, using the
Lagrangian density (3) and the prescription (144), that the ${\cal T}^{++}$
component of the density of the canonical energy-momentum tensor is

${\cal
T}^{++}=i\sqrt{2}\Psi^{\dagger}_{+}\partial_{-}\Psi_{+}+\left(\partial_{-}\phi\right)^{2}-\alpha\left(\partial_{-}\partial^{2}\phi\right)\left(\partial_{-}\phi\right)+\alpha\left(\partial^{2}\phi\right)\left(\partial^{2}_{-}\phi\right).$
(147)

It is not difficult to notice, that by the same method we can compute

${\cal T}^{+-}=$ (148)
$=-i\sqrt{2}\;\Psi^{\dagger}_{-}\partial_{-}\Psi_{-}-{i\over{\sqrt{2}}}\Psi^{\dagger}_{+}\gamma^{-}\gamma^{j}\partial_{j}\Psi_{-}-{i\over{\sqrt{2}}}\Psi^{\dagger}_{-}\gamma^{+}\gamma^{j}\partial_{j}\Psi_{+}+{1\over{\sqrt{2}}}M\Psi^{\dagger}_{+}\gamma^{-}\Psi_{-}+{1\over{\sqrt{2}}}M\Psi^{\dagger}_{-}\gamma^{+}\Psi_{+}+$
$+{1\over 2}\left(\partial_{j}\phi\right)\left(\partial_{j}\phi\right)+{1\over
2}m^{2}\phi^{2}+{1\over
4}\lambda\phi^{4}+{1\over{\sqrt{2}}}g\phi\;\Psi^{\dagger}_{+}\gamma^{-}\Psi_{-}+{1\over{\sqrt{2}}}g\phi\;\Psi^{\dagger}_{-}\gamma^{+}\Psi_{+}-\alpha\left(\partial_{-}\partial^{2}\phi\right)\left(\partial_{+}\phi\right)+{1\over
2}\alpha\left(\partial^{2}\phi\right)\left(\Delta_{\perp}\phi\right).$

Some of the terms in this formula may be substituted with help of the first
equations (6) and (7), what reduces the final expression for the ${\cal
T}^{+-}$ component to

${\cal T}^{+-}=i\sqrt{2}\;\Psi^{\dagger}_{+}\partial_{+}\Psi_{+}+{1\over
2}\left(\partial_{j}\phi\right)\left(\partial_{j}\phi\right)+{1\over
2}m^{2}\phi^{2}+{1\over
4}\lambda\phi^{4}-\alpha\left(\partial_{-}\partial^{2}\phi\right)\left(\partial_{+}\phi\right)+{1\over
2}\alpha\left(\partial^{2}\phi\right)\left(\Delta_{\perp}\phi\right).$ (149)

On the other hand, the ${\cal T}^{+j}$ satisfies

${\cal
T}^{+j}=-i\sqrt{2}\;\Psi^{\dagger}_{+}\partial_{j}\Psi_{+}-\left(\partial_{-}\phi\right)\left(\partial_{j}\phi\right)+\alpha\left(\partial_{-}\partial^{2}\phi\right)\left(\partial_{j}\phi\right)-\alpha\left(\partial^{2}\phi\right)\left(\partial_{-}\partial_{j}\phi\right).$
(150)

Now, we take the Heisenberg equations (146) to obtain possible (anti-)
commutators for the Yukawa model with higher order derivatives and briefly
report the results. This approach gives the same relationships as: (127),
(128), (129), (130), (133), (134), (138) or these, which are the corollaries
of the itemized. The method based on the Heisenberg equations does not give
any predictions for the rules (131) and (132). The patterns: (135), (136) and
(139) have the analogues derived from the aforementioned approach, but not
consistent with them. They are trivial:
$[\Psi_{-}(x^{+},{\bar{x}}),\partial_{-}\phi(x^{+},{\bar{y}})]=0$,
$[\Psi_{-}(x^{+},{\bar{x}}),\partial_{+}\phi(x^{+},{\bar{y}})]=0$,
$[\partial_{+}\phi(x^{+},{\bar{x}}),\partial_{+}\phi(x^{+},{\bar{y}})]=0$. The
analogue of the result (137), computed from the Heisenberg equations, splits,
due to the fact, that in this approach all the derivatives of the scalar field
are independent, whereas for our main considerations, based on the Dirac
procedure, the independent derivative is only $\partial_{+}\phi$. And that’s
why we obtained from the Heisenberg equations the following commutator
expressions: $[\phi(x^{+},{\bar{x}}),\phi(x^{+},{\bar{y}})]=0$,
$[\phi(x^{+},{\bar{x}}),\partial_{j}\phi(x^{+},{\bar{y}})]=0$,
$[\phi(x^{+},{\bar{x}}),\partial_{-}\phi(x^{+},{\bar{y}})]=(i/2)\delta^{(3)}({\bar{x}}-{\bar{y}})$.
Remarkably, only the last one matches the (137) and is valid in the limit
$\alpha=0$. For the same reason we put here the relationship
$\\{\Psi_{+}(x^{+},{\bar{x}}),\partial_{+}\Psi_{+}(x^{+},{\bar{y}})\\}=0$, but
now non-conflicting with the (127).

To sum up, the method based on the Heisenberg equations is not sensitive to
the structure of the constraints and to the interactions for the Yukawa model
with higher order derivatives. This method is not constructional, whereas the
Dirac approach is. It is therefore only the last one gives the systematic
approach for the quantization in the presence of the constraints with
interactions.

Appendix B: Light-Front Formulation

The light-front coordinates for the flat space-time of $D=1+3$ dimensions
obey:

$x=\left(x^{+},x^{-},{\bf
x}_{\perp}\right),\;\;\;\;\;\;x^{\pm}={1\over{\sqrt{2}}}\left(x^{0}\pm{x^{3}}\right),\;\;\;\;\;\;{\bf
x}_{\perp}=\left(x^{1},x^{2}\right).$ (151)

The hyper-surface of the quantization $x^{+}=0$ for the light-front and
respectively, the $x^{-}=0$ for the anti-light-front, enable us to introduce
the reduced coordinates:

${\bar{x}}=\left(x^{-},{\bf x}_{\perp}\right)\simeq\left(x^{+}=0,x^{-},{\bf
x}_{\perp}\right),\;\;\;\;\;\;{\underline{x}}=\left(x^{+},{\bf
x}_{\perp}\right)\simeq\left(x^{+},x^{-}=0,{\bf x}_{\perp}\right).$ (152)

The components of the metric tensor, relevant to the coordinates (151), are:

$g_{+-}=1,\;\;\;\;\;\;g_{++}=0=g_{--},\;\;\;\;\;\;g_{{\pm}j}=0,\;\;\;\;\;\;g_{jk}=-\delta_{jk},\;\;\;\;\;\;j,k=1,2.$
(153)

They allow to lift up and to pull down the indexes in following way:

$a_{\pm}=a^{\mp},\;\;\;\;\;\;a_{j}=-a^{j},\;\;\;\;\;\;j=1,2.$ (154)

Therefore, the scalar product of the four-vectors satisfies:

$a\cdot{b}=a_{+}b_{-}+a_{-}b_{+}-{\bf a}_{\perp}\cdot{\bf
b}_{\perp},\;\;\;\;\;\;{\bf a}_{\perp}\cdot{\bf
b}_{\perp}=a_{j}b_{j},\;\;\;\;\;\;j=1,2.$ (155)

Herein, the Einstein notation is introduced. Accordingly, the Dirac slash in
the light-front formulation yields:

$\not\\!a=a_{\mu}\gamma^{\mu}=a_{+}\gamma^{+}+a_{-}\gamma^{-}-\not\\!{\bf
a}_{\perp},\;\;\;\;\;\;\not\\!{\bf
a}_{\perp}=a_{j}\gamma^{j},\;\;\;\;\;\;\mu=+,-,j,\;\;\;\;\;\;j=1,2.$ (156)

The coordinates (151) lead to the light-front derivatives:

$\partial_{\pm}={{\partial}\over{\partial{x^{\pm}}}},\;\;\;\;\;\;\partial_{j}={{\partial}\over{\partial{x^{j}}}},\;\;\;\;\;\;j=1,2$
(157)

and to the Laplace or to the d’Alembert operators:

$\Delta_{\perp}=\partial_{j}\partial_{j},\;\;\;\;\;\;\partial^{2}\equiv\Box=\partial_{+}\partial_{-}+\partial_{-}\partial_{+}-\Delta_{\perp},\;\;\;\;\;\;j=1,2,$
(158)

wherein the last one is in the symmetrized form, required for the formalism
presented in this work. The definition of the light-front Dirac gamma matrices
is relevant to the (151). Thus:

$\gamma=\left(\gamma^{+},\gamma^{-},\gamma_{\perp}\right),\;\;\;\;\;\;\gamma^{\pm}={1\over{\sqrt{2}}}\left(\gamma^{0}\pm{\gamma^{3}}\right),\;\;\;\;\;\;\gamma_{\perp}=\left(\gamma^{1},\gamma^{2}\right).$
(159)

There are some characteristic properties of these arrays, used in this
considerations:

$\left(\gamma^{\pm}\right)^{2}=0,\;\;\;\;\;\;\left(\gamma^{\pm}\right)^{\dagger}=\gamma^{\mp},\;\;\;\;\;\;\left(\gamma^{j}\right)^{\dagger}=-\gamma^{j},\;\;\;\;\;\;j=1,2.$
(160)

There is the natural decomposition of the fermionic bispinor

$\Psi=\Psi_{+}+\Psi_{-},$ (161)

whereas the spinors $\psi_{\pm}$ arrange the components of the above pattern:

$\Psi=\left(\begin{array}[]{c}\psi_{+}\\\ \psi_{-}\\\
\end{array}\right),\;\;\;\;\;\;\Psi_{+}=\left(\begin{array}[]{c}\psi_{+}\\\
0\\\
\end{array}\right)\simeq\psi_{+},\;\;\;\;\;\;\Psi_{-}=\left(\begin{array}[]{c}0\\\
\psi_{-}\\\ \end{array}\right)\simeq\psi_{-}.$ (162)

The two constant bispinors $u_{\pm}$ allow to perform the inverse
decomposition:

$u_{+}=\left(\begin{array}[]{c}1\\\ 0\\\
\end{array}\right),\;\;\;\;\;\;u_{-}=\left(\begin{array}[]{c}0\\\ 1\\\
\end{array}\right),\;\;\;\;\;\;\psi_{\pm}=u^{\dagger}_{\pm}\Psi.$ (163)

Their tensor product defines the matrices $\Lambda_{\pm}$, due to:

$u_{\pm}\otimes{u^{\dagger}_{\pm}}=\Lambda_{\pm}=u^{\dagger}_{\pm}\otimes{u_{\pm}},\;\;\;\;\;\;\Lambda_{+}=\left(\begin{array}[]{cc}1&\;\;0\\\
0&\;\;0\\\
\end{array}\right),\;\;\;\;\;\;\Lambda_{-}=\left(\begin{array}[]{cc}0&\;\;0\\\
0&\;\;1\\\ \end{array}\right).$ (164)

These $\Lambda_{\pm}$ arrays permit us to write the decomposition (161) as:

$\Psi_{+}=\Lambda_{+}\Psi,\;\;\;\;\;\;\Psi_{-}=\Lambda_{-}\Psi.$ (165)

The aforementioned matrices are projective and Hermitian:

$\Lambda_{\pm}^{2}=\Lambda_{\pm},\;\;\;\;\;\;\Lambda_{\pm}\Lambda_{\mp}=0=\Lambda_{\mp}\Lambda_{\pm},\;\;\;\;\;\;\Lambda_{+}+\Lambda_{-}={\rm
I},\;\;\;\;\;\;\Lambda_{\pm}^{\dagger}=\Lambda_{\pm}.$ (166)

We can easily provide the $\Lambda_{\pm}$ matrices by the Dirac gamma ones

$\Lambda_{\pm}={1\over 2}\gamma^{\mp}\gamma^{\pm}.$ (167)

The spinor decomposition, applied here, enables us to put the $\gamma^{\pm}$
arrays as the following tensor products:

$u_{\mp}\otimes{u^{\dagger}_{\pm}}=\gamma^{\pm}=u^{\dagger}_{\pm}\otimes{u_{\mp}},\;\;\;\;\;\;\gamma^{+}=\left(\begin{array}[]{cc}0&\;\;0\\\
1&\;\;0\\\
\end{array}\right),\;\;\;\;\;\;\gamma^{-}=\left(\begin{array}[]{cc}0&\;\;1\\\
0&\;\;0\\\ \end{array}\right).$ (168)

There are some handling rules of the $\gamma^{\pm}$ and the $\Lambda_{\pm}$
matrices. They satisfy

$\gamma^{\pm}\Lambda_{\pm}=\gamma^{\pm}=\Lambda_{\mp}\gamma^{\pm}.$ (169)

The $\gamma^{\pm}$ arrays allow to do certain kinds of the projections of the
$\Psi$ and the $\Psi_{\mp}$ bispinors with the flips of their internal
structure:

$\gamma^{+}\Psi=\left(\begin{array}[]{c}0\\\ \psi_{+}\\\
\end{array}\right),\;\;\;\;\;\;\gamma^{-}\Psi=\left(\begin{array}[]{c}\psi_{-}\\\
0\\\ \end{array}\right),\;\;\;\;\;\;\gamma^{\pm}\Psi_{\mp}=0.$ (170)

All the computations within this work are based on the set of presented here
formulae and definitions.

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