Preliminarily, we recall that the fermionization procedure relies on the tomographic transform presented in [20], which displays a known nonlocality drawback [15]. Nevertheless, this problem is not relevant in our case, since, as we said, we are dealing with a low energy effective bosonic field theory.
3.1. The Tomographic Transform
In this section, we review some basic properties of the tomographic transform, which we conveniently use in the fermionization process. For more details on the tomographic representation, we refer to [20].
The basic ingredient of the tomographic transform in three spatial dimensions is the generalized function
δ
′
(
y
-
n
·
r
)
, defined as
(13)
δ
′
(
y
-
n
·
r
)
=
∫
-
∞
+
∞
d
k
k
e
i
k
y
e
-
i
r
·
k
n
,
where
n
is an angular variable and
k
is a scalar. From Definition (13), it is easy to prove that
δ
′
(
y
-
n
·
r
)
satisfies the completeness and the orthonormality relation:
(14)
1
8
π
2
∫
d
y
d
2
n
δ
′
(
y
-
n
·
r
)
δ
′
(
y
-
n
·
r
′
)
=
δ
(
r
-
r
′
)
,
(15)
1
4
π
2
∫
d
3
r
δ
′
(
y
-
n
·
r
)
δ
′
(
y
′
-
n
′
·
r
)
=
δ
(
y
-
y
′
)
δ
(
n
,
n
′
)
-
δ
(
y
+
y
′
)
δ
(
n
,
-
n
′
)
.
We now review the properties of the tomographic transform of the scalar, the vector, and the fermionic field.
Following [20], the tomographic transform of the scalar field is defined as
(16)
ϕ
~
(
y
,
n
)
=
1
2
π
∫
d
3
r
δ
′
(
y
-
n
·
r
)
ϕ
(
r
)
=
-
ϕ
~
(
-
y
,
-
n
)
,
while the inverse tomographic transformation is
(17)
ϕ
(
r
)
=
1
4
π
∫
d
y
d
2
n
δ
′
(
y
-
n
·
r
)
ϕ
~
(
y
,
n
)
.
For what concerns the vector field
A
μ
, the tomographic transforms of its four space-time components are organized as
(18)
A
~
S
(
y
,
n
)
=
1
2
π
∫
d
3
r
δ
′
(
y
-
n
·
r
)
A
0
(
r
)
,
A
~
L
(
y
,
n
)
=
1
2
π
∫
d
3
r
δ
′
(
y
-
n
·
r
)
n
·
A
(
r
)
,
A
~
T
a
(
y
,
n
)
=
1
2
π
∫
d
3
r
δ
′
(
y
-
n
·
r
)
ɛ
a
(
n
)
·
A
(
r
)
,
where
A
~
L
(
y
,
n
)
and
A
~
T
a
(
y
,
n
)
are the longitudinal and the transverse transforms, respectively, and we have introduced the polarization vectors
ɛ
a
(
n
)
orthogonal to
n
, with
a
=
1,2
. The spatial antitransform is defined as
(19)
A
(
r
)
=
1
4
π
∫
d
y
d
2
n
δ
′
(
y
-
n
·
r
)
×
[
n
A
~
L
(
y
,
n
)
+
ɛ
a
(
n
)
A
~
T
a
(
y
,
n
)
]
.
The tomographic transform of the four component spinor field
ψ
α
(
r
)
is as follows:
(20)
ψ
~
b
(
y
,
n
)
=
1
2
π
∫
d
3
r
δ
′
(
y
-
n
·
r
)
u
α
†
b
(
n
)
ψ
α
(
r
)
,
u
α
†
b
is a spinor, where
α
=
{
1
,
…
,
4
}
is a spinor index. Introducing the usual
4
×
4
Dirac matrices
α
and the spin matrices
Σ
=
-
(
i
/
2
)
α
×
α
,
u
α
b
is, by definition, an eigenspinor of
α
·
n
with eigenvalue
-
1
. Moreover, since
[
Σ
·
n
,
α
·
n
]
=
0
,
u
α
b
(
n
)
is also an eigenvector of
Σ
·
n
, with
(21)
(
Σ
·
n
)
u
α
b
(
n
)
=
b
u
α
b
(
n
)
,
b
=
{
1
,
-
1
}
.
The orthogonality condition
u
α
†
b
(
n
)
u
α
c
(
n
)
=
δ
b
c
holds, so we can write the projector as
(22)
∑
b
u
α
b
(
n
)
u
β
†
b
(
n
)
=
1
2
(
1
-
α
·
n
)
α
β
.
The antitransform of a spinor field is defined as
(23)
ψ
α
(
r
)
=
1
2
π
∫
d
y
d
2
n
δ
′
(
y
-
n
·
r
)
u
α
b
(
n
)
ψ
~
b
(
y
,
n
)
.
Finally, by using the following identity:
(24)
∂
i
δ
′
(
y
-
n
·
r
)
=
-
n
i
∂
y
δ
′
(
y
-
n
·
r
)
,
and the completeness relation (14), it is possible to prove that the massless Dirac equation,
γ
μ
∂
μ
ψ
=
0
, in the tomographic representation reads:
(25)
(
∂
0
-
∂
y
)
ψ
~
b
(
y
,
n
)
=
0
.
The last equation shows that, for a fixed value of
n
,
ψ
b
(
y
,
n
)
is a “right moving” field propagating along the positive direction of
y
. It is also possible to define the tomographic transform of a spinor field as follows:
(26)
χ
~
b
(
y
,
n
)
=
1
2
π
∫
d
3
r
δ
′
(
y
-
n
·
r
)
v
α
†
b
(
n
)
ψ
α
(
r
)
,
where
v
α
†
b
is an eigenspinor of
α
·
n
with eigenvalue
+
1
. For
v
α
b
, the following relations hold:
(27)
(
Σ
·
n
)
v
α
b
(
n
)
=
b
v
α
b
(
n
)
,
b
=
{
1
,
-
1
}
,
∑
b
v
α
b
(
n
)
v
β
†
b
(
n
)
=
1
2
(
1
+
α
·
n
)
α
β
.
The antitransform of the spinor field, in this case, is defined as
(28)
ψ
α
(
r
)
=
1
2
π
∫
d
y
d
2
n
δ
′
(
y
-
n
·
r
)
v
α
b
(
n
)
χ
~
b
(
y
,
n
)
,
while the tomographic transformed Dirac equation for
χ
~
b
(
y
,
n
)
is
(29)
(
∂
0
+
∂
y
)
χ
~
b
(
y
,
n
)
=
0
.
Analogously, here, we have a “left moving” field. As we will see, the two previous constructions of the tomographic transform of the fermionic field are completely equivalent.
3.3. Fermionization
In the previous sections, we defined the properties of the tomographic transform and we wrote the duality relation (7) in terms of tomographic variables (30). Now, we implement the fermionization process following the steps described in [21, 22].
Identifying
Λ
~
(
y
,
n
)
and
ξ
~
(
y
,
n
)
as the tomographic transforms of the bosonic d.o.f. on which the
3
+
1
D
theory described by the Lagrangian (5) depends, we define a fermionic field as
(31)
Ψ
~
(
y
,
n
)
=
1
2
π
α
:
e
i
π
[
ξ
~
(
y
,
n
)
+
Λ
~
(
y
,
n
)
]
:
,
where
α
is a regularizing constant. The normal ordering prescription appearing in (31) is
{
ξ
~
(
-
)
,
Λ
~
(
-
)
,
Λ
~
(
+
)
,
ξ
~
(
+
)
}
. The ordinary time evolution of the free fields
Λ
and
ξ
allows us to write down the Heisenberg operators of the positive and negative frequency parts of
ξ
~
and
Λ
~
as follows:
(32)
ξ
~
(
±
)
(
y
,
n
)
=
∫
-
∞
∞
d
p
2
π
1
2
|
p
|
a
(
±
)
(
p
)
e
∓
(
i
p
y
-
i
|
p
|
t
)
e
-
(
α
/
2
)
|
p
|
,
Λ
~
(
±
)
(
y
,
n
)
=
∫
-
∞
∞
d
p
2
π
1
2
|
p
|
b
(
±
)
(
p
)
e
∓
(
i
p
y
-
i
|
p
|
t
)
e
-
(
α
/
2
)
|
p
|
,
and
a
(
+
)
(
p
)
(
a
(
-
)
(
p
)
) and
b
(
+
)
(
p
)
(
b
(
-
)
(
p
)
) are the creation (annihilation) operators for
ξ
~
(
y
,
n
)
and
Λ
~
(
y
,
n
)
, respectively. Because of the duality relations (30), they are related by
(33)
b
(
±
)
(
p
)
=
-
sign
(
p
)
a
(
±
)
(
p
)
with
sign
(
x
)
denoting the sign of
x
. Let us define
ϕ
(
y
,
n
)
as
(34)
ϕ
(
y
,
n
)
=
ξ
~
(
y
,
n
)
+
Λ
~
(
y
,
n
)
,
and, accordingly,
(35)
ϕ
(
±
)
(
y
,
n
)
=
∫
0
∞
d
p
π
1
2
|
p
|
a
(
±
)
(
p
)
e
∓
(
i
p
y
-
i
|
p
|
t
)
e
-
(
α
/
2
)
|
p
|
.
The presence of only positive
p
momenta reminds what happens in the
1
+
1
D
bosonization, where only the right moving components are involved [13]. The minus sign in (34) would correspond to terms with only negative momenta components, in analogy with the left movers in
1
+
1
D
.
The crucial observation is that the Lorentz scalar defined in (31) is a fermionic anticommuting variable. To see this, we compute the anticommutator
(36)
{
Ψ
~
(
y
,
n
)
,
Ψ
~
†
(
y
′
,
n
)
}
=
1
2
π
α
:
e
i
π
[
ϕ
(
+
)
(
y
,
n
)
+
ϕ
(
-
)
(
y
,
n
)
)
]
:
:
e
-
i
π
[
ϕ
(
+
)
(
y
′
,
n
)
+
ϕ
(
-
)
(
y
′
,
n
)
)
]
:
+
1
2
π
α
:
e
-
i
π
[
ϕ
(
+
)
(
y
′
,
n
)
+
ϕ
(
-
)
(
y
′
,
n
)
)
]
:
:
e
i
π
[
ϕ
(
+
)
(
y
,
n
)
+
ϕ
(
-
)
(
y
,
n
)
)
]
:
.
We observe that
(37)
:
e
i
π
[
ϕ
(
+
)
(
y
,
n
)
+
ϕ
(
-
)
(
y
,
n
)
]
:
:
e
-
i
π
[
ϕ
(
+
)
(
y
′
,
n
)
+
ϕ
(
-
)
(
y
′
,
n
)
]
:
=
e
π
〈
ϕ
(
y
,
n
)
ϕ
(
y
′
,
n
)
-
(
ϕ
2
(
y
,
n
)
+
ϕ
2
(
y
′
,
n
)
)
/
2
〉
=
α
α
-
i
(
y
-
y
′
)
.
where we have used [13]:
e
A
e
B
=
:
e
A
+
B
:
e
<
A
B
+
(
A
2
+
B
2
)
/
2
>
.
The second term in the right-hand side of (36) can be treated in the same way, with the outcome that
(38)
{
Ψ
~
(
y
,
n
)
,
Ψ
~
†
(
y
′
,
n
)
}
=
[
α
α
-
i
(
y
-
y
′
)
+
α
α
+
i
(
y
-
y
′
)
]
1
2
π
α
=
α
π
(
α
2
+
(
y
-
y
′
)
2
)
⟶
α
→
0
δ
(
y
-
y
′
)
.
Therefore, we verified that the scalar field defined in (31) satisfies the anticommutation relations for
n
=
n
′
, but the commutation relation is still bosonic for
n
≠
n
′
. The complete Fermi statistics can be achieved introducing the Klein factors [15]:
(39)
O
n
=
e
(
i
π
/
2
)
∫
(
θ
,
φ
)
d
2
n
′
[
α
(
n
′
)
+
β
(
n
′
)
]
,
where the operators
(40)
α
(
n
)
=
∫
-
∞
∞
d
y
∂
0
ξ
~
(
y
,
n
)
,
β
(
n
)
=
∫
-
∞
∞
d
y
∂
0
Λ
~
(
y
,
n
)
take the role of generalized charges [21, 22]. The angle parameterization
(
θ
′
,
ϕ
′
)
defined the versor direction
n
′
and the integration domain
∫
(
θ
,
φ
)
is
0
<
φ
′
≤
2
π
∪
0
<
θ
′
<
θ
if
θ
′
≠
θ
and
0
<
φ
′
<
φ
if
θ
′
=
θ
. Using the definitions (39)-(40), the following rule can be verified:
(41)
O
n
Ψ
~
(
y
,
m
)
=
{
-
Ψ
~
(
y
,
m
)
O
n
m
<
n
Ψ
~
(
y
,
m
)
O
n
m
≥
n
,
where
m
<
n
means
θ
m
<
θ
n
or
θ
m
=
θ
n
and
φ
m
<
φ
n
.
At this point, it is straightforward to check that the operator
(42)
ψ
~
(
y
,
n
)
≡
Ψ
~
(
y
,
n
)
O
n
satisfies the anticommutation relations
(43)
{
ψ
~
(
x
,
n
)
,
ψ
~
†
(
y
,
m
)
}
=
δ
(
n
,
m
)
δ
(
x
-
y
)
,
{
ψ
~
(
x
,
n
)
,
ψ
~
(
y
,
m
)
}
=
0
,
which allow us to conclude that
ψ
~
(
y
,
n
)
is a well-defined fermionic field in the tomographic representation.
In [20], it was shown how the tomographic transformed fields behave under space-time rotations. We have verified that definition (31) is consistent with those properties. Moreover, by using the definition (42) and the duality relations (30), it is easy to see that
ψ
~
must satisfy
(44)
(
∂
0
-
∂
y
)
ψ
~
(
y
,
n
)
=
0
,
which is the tomographic version of the massless Dirac equation (25).
An additional fermionic field can be introduced:
(45)
χ
~
(
y
,
n
)
=
1
2
π
α
:
e
i
π
[
ξ
~
(
y
,
n
)
-
Λ
~
(
y
,
n
)
]
:
P
n
,
where
(46)
P
n
=
e
(
i
π
/
2
)
∫
(
θ
,
φ
)
d
2
n
′
[
α
(
n
′
)
-
β
(
n
′
)
]
,
which satisfies the Fermi statistics:
(47)
{
χ
~
(
x
,
n
)
,
χ
~
†
(
y
,
m
)
}
=
δ
(
n
,
m
)
δ
(
x
-
y
)
,
{
χ
~
(
x
,
n
)
,
χ
~
(
y
,
m
)
}
=
0
,
together with the massless Dirac equation (29):
(48)
(
∂
0
+
∂
y
)
χ
~
(
y
,
n
)
=
0
.
Consequently,
χ
~
(
y
,
n
)
obeys the tomographic construction (26).
As we have anticipated, it is possible to reconstruct the spinor
ψ
α
(
r
)
both from
ψ
~
(
y
,
n
)
and
χ
~
(
y
,
n
)
in a completely equivalent way. In fact, choosing a fixed eigenvalue
b
of
Σ
·
n
associated with
χ
~
(
y
,
n
)
(we are not able to determine the value of
b
a priori), we have from (26)
(49)
ψ
α
(
r
)
=
∫
d
y
d
2
n
δ
′
(
y
-
n
·
r
)
v
α
b
(
n
)
χ
~
(
y
,
n
)
=
(
y
⟶
-
y
,
n
⟶
-
n
)
=
∫
d
y
d
2
n
δ
′
(
-
y
+
n
·
r
)
v
α
b
(
-
n
)
χ
~
(
-
y
,
-
n
)
.
But, keeping in mind that
δ
′
(
y
-
n
·
r
)
is an odd function under the transformation
(
y
→
-
y
,
n
→
-
n
)
and the well-known relation
v
b
(
-
n
)
=
u
-
b
(
n
)
(remember that
α
·
n
u
b
(
n
)
=
-
u
b
(
n
)
and
α
·
n
v
b
(
n
)
=
v
b
(
n
)
), we obtain that (49) is equal to
(50)
-
∫
d
y
d
2
n
δ
′
(
y
-
n
·
r
)
u
α
-
b
(
n
)
χ
~
(
-
y
,
-
n
)
.
Finally, since, by construction,
ξ
~
(
-
y
,
-
n
)
=
ξ
~
(
y
,
n
)
and
Λ
~
(
-
y
,
-
n
)
=
-
Λ
~
(
y
,
n
)
, we obtain from the definition of
χ
~
(
y
,
n
)
(45) that
(51)
χ
~
(
-
y
,
-
n
)
=
ψ
~
(
y
,
n
)
,
And, consequently, (50) is equal to
(52)
-
∫
d
y
d
2
n
δ
′
(
y
-
n
·
r
)
u
α
-
b
(
n
)
ψ
~
(
y
,
n
)
,
which proves that the construction (20) and construction (26) are completely equivalent.
Finally, we are dealing with only one independent tomographic fermionic field, from which we can only construct a Weyl spinor since, as it is well known, for massless fermion in
3
+
1
D
,
Σ
·
n
is equivalent to
γ
5
and, consequently, our tomographic transformed spinor field is an eigenvalue of
γ
5
by construction. Then, we can use it only to construct a Weyl spinor.