field theory

1. Sweeping Discussions On Dirac Fields
Roa, F. J. P.
We write
(1)
tni
LL
ffe LLL ),()()( 2122 ψψψψ +=
as a Lagrangian for a basic electron theory that can be outlined from an SU(2)XU(1)
construction that excludes as yet the additional fermions needed in the full electroweak
theory. Lagrangian (1) includes an effective Lagrangian ffeL )( 2ψ for the electron 2ψ
and an interaction Lagrangian tni
LL
L ),( 21 ψψ exclusive only for the lefthanded
(electron) neutrino L
1ψ , which is massless, and the lefthanded electron L
2ψ that has mass.
The effective Lagrangian (for the electron 2ψ ), aside from the free Lagrangian 02 )(ψL
present in it, also contains tniL )( 2ψ , which serves as an interaction Lagrangian in the
said effective lagrangian ( ffeL )( 2ψ ) that includes the interactions of the electron with
the Higgs boson η , the electromagnetic field me
Aµ that is massless, and one massive field
µZ .
(2.1)
tniffe LLL )()()( 2022 ψψψ +=
(2.2)
( )
( )LRR
RLL
mi
mi
miL
222
222
222202 )(
ψψγψ
ψψγψ
ψψψγψψ
ψµ
µ
ψµ
µ
ψµ
µ
−∂
+−∂=
=−∂=
Let it be noted that all equations appearing in this document are in Heaviside units in
which 1== ch and that the Dirac mass ψm is simply given in terms of the Yukawa
coupling constant y and the non-zero vacuum expectation value (vev) β of the Higgs
field,

2. (2.3)
βψ ym =
(2.4)
( ) 22222 tan)( ψαγψψψηψ µµ
µ
ZeAeyL me
tni ++−=
This Lagrangian can ofcourse be viewed in terms of the lefthanded electron and its
counter-part, R
2ψ .
(2.5.1)
( ) 2
5
2 1
2
1
ψγψ +=L
(2.5.2)
( ) 2
5
2 1
2
1
ψγψ −=R
(I must emphasize that in this document, our definitions for L
2ψ and R
2ψ are as given by
the preceding equations above.)
Given (2.5.1) and (2.5.2), the said effective Lagrangian can be decomposed under the
following expressions
(2.6.1)
222222 ψψψψψψ =+ RLLR
(2.6.2)
222222 ψγψψγψψγψ µµµ
=+ RRLL
(2.6.3)
222222 ψγψψγψψγψ µ
µ
µ
µ
µ
µ
∂=∂+∂ RRLL
For the interaction Lagrangian tni
LL
L ),( 21 ψψ , one thing to notice in this is that this
excludes the righthanded electron and represents the interactions of the massless
lefthanded neutrino L
1ψ and lefthanded electron with the weak gauge bosons.
(2.7)

3. ( ))(
12
)(
21
1
22
1
21
)sin2(
)cossin2(),(
+−−
−
+
−−=
µ
µ
µ
µ
µ
µ
ψγψψγψα
ψγψααψψ
WWe
ZeL
LLLL
LL
tni
LL
Let us take ffeL )( 2ψ from (1) and with it form the action
(3.1)
ffeffe LxdS )()( 2
4
2 ψψ ∫=
We may think of it as an effective electron theory that we can outline from a basic
SU(2)XU(1) construction without as yet the additional fermions of electroweak theory.
In the said action (3.1), we may proceed with the four integral contained in it
(3.2)
∫ ∑∫ 





∂+∂=∂
=
3
1
220
0
2
30
22
4
l
l
l
ixdxdixd ψγψγψψγψ µ
µ r
From (3.2), say we perform the one integral over time ( tx =0
) first, assuming we are
given with two end-points ( 00
BA xandx ) in time and thus, by integration by-parts we
have
(3.3)
[ ] ∫∫ ∂−=∂ 2
0
20
0
)(
)(
2
0
220
0
2
0
)(
0
0
ψγψψγψψγψ ixdiixd
Bx
Ax
We can use this to write one four integral in (3.2) as
(3.4)
[ ] ∫∫ ∫ ∂−=∂ 2
0
20
4
)(
)(
2
0
2020
0
2
4
)(
0
0
ψγψψγψσψγψ ixdidixd
Bx
Ax
where for example, 0
σd is a short for
(3.5)
32130
dxdxdxxdd ==
r
σ
In the same way we can manipulate for any given space part in (3.2)
(3.6)

4. [ ] ∫∫ ∫ ∂−=∂ 22
4
)(
)(
2222
4
)( ψγψψγψσψγψ l
l
Bx
Ax
l
ll
l
ixdidixd
l
l
Thus, by integration by-parts we are able to re-write (3.2) into a desired form
(3.7)
[ ] ∫∫ ∫ ∂−=∂ 22
4
)(
)(
2222
4
)( ψγψψγψσψγψ µ
µ
µ
µµ
µ µ
µ
ixdidixd
Bx
Ax
Consequently, using (3.7) we may put (3.1) in the following form
(3.8.1)
[ ] ( )∫∫ ++∂−= 222222
4)(
)(222 )()( ψψψψβψγψψγψσψ µ
µ
µ
µ
µ
µ q
Bx
Axffe FyixdidS
(3.8.2)
µ
µ
µ
µ
αγγη ZeeAyF me
q tan−−=
What we have done here in using integration by-parts (3.7) in (3.1) is a recasting of this
action, which is held as an effective action for the spinor field 2ψ , into an action
ffeS )( 2ψ for the adjoint spinor field 2ψ .
(3.9)
( )∫
∫
++∂−=
==
222222
4
2
4
2
)(
)()(
ψψψψβψγψ
ψψ
µ
µ q
ffeffe
Fyixd
LxdS
We may think of (3.1) and (3.9) as two separate theories that are adjoint to each other. In
these actions, we can take 2ψ and 2ψ as two independent spinor fields and we can obtain
the total simultaneous variations of these actions in terms of the variations of the cited
field variables.
From (3.1) we obtain the total variation
(3.10.1)
[ ]
( )
( ) 2222
4
2222
4
)(
)(222
)(
)(
δψψψβγψ
ψψβψγψδ
δψγψσψδ
µ
µ
µ
µ
µ
µ
µ
µ
∫
∫
∫
++∂
−−−∂
+=
q
q
Bx
Ax
Fyixd
Fyixd
idS

5. while from (3.9) we have
(3.10.2)
[ ]
( )
( ) 2222
4
2222
4
)(
)(222
)(
)(
δψψψβγψ
ψψβψγψδ
ψγψδσψδ
µ
µ
µ
µ
µ
µ
µ
µ
∫
∫
∫
++∂
−−−∂
+−=
q
q
Bx
Ax
Fyixd
Fyixd
idS
In the classical context, we can impose the boundary condition that the varied fields
( 2δψ and 2ψδ ) vanish at the two end-points A and B. Thus, rendering the effective
forms of (3.10.1) and (3.10.2) equivalent
(3.11)
ffeffe SS )()( 22 ψδψδ =
To get Dirac’s first order equation of motion for the independent field 2ψ , we vary
action (3.1) with respect to the adjoint spinor 2ψ alone and this yields
(4.1)
)(
)(
2
2
2
2
2
2
2
2
4
22
)(
)()(
)/(
Bx
Ax
ffe
ffeffe
L
d
LL
xdS
µ
µ
ψδ
ψδ
ψδσ
ψδ
ψδ
ψδ
ψδ
ψδψδψδ
µ
µ
µ
µ








∂
+








∂
∂−=
∫
∫
This, obtained by invoking the commutativity in
(4.2)
δδ µµ ∂=∂
and after integration by-parts. Again, in the classical context we impose the spacetime
boundary condition on the varied field 2ψδ to obtain the effective form of (4.1)

6. (4.3.1)
∫ 







∂
∂−=
2
2
2
2
2
4
22
)()(
)/(
ψδ
ψδ
ψδ
ψδ
ψδψδψδ
µ
µ
ffeffe
ffe
LL
xdS
(4.3.2)
0)()( 22 == BA ψδψδ
Also, in the classical context, we render this effective variation stationary
(4.4.1)
0)/( 22 =ffeS ψδψδ
we obtain the Euler-Lagrange equation
(4.4.2)
0
)()(
2
2
2
2
=
∂
∂−
ψδ
ψδ
ψδ
ψδ
µ
µ
ffeffe LL
and upon noting that
(4.4.3)
0
)(
2
2
=
∂ ψδ
ψδ
µ
ffeL
and
(4.4.4)
222
2
2 )(
ψψβψγ
ψδ
ψδ
µ
µ
q
ffe
Fyi
L
−−∂=
we arrive at the said first order equation
(4.4.5)
0222 =−−∂ ψψβψγ µ
µ
qFyi

7. Ref’s:
[1]W. Hollik, Quantum field theory and the Standard Model, arXiv:1012.3883v1 [hep-
ph]
[2]Baal, P., A COURSE IN FIELD THEORY,
http://www.lorentz.leidenuniv.nl/~vanbaal/FTcourse.html
[3]’t Hooft, G., THE CONCEPTUAL BASIS OF QUANTUM FIELD THEORY,
http://www.phys.uu.nl/~thooft/
[4]Siegel, W., FIELDS, arXiv:hep-th/9912205 v2
[5]Wells, J. D., Lectures on Higgs Boson Physics in the Standard Model and Beyond,
arXiv:0909.4541v1
[6]Cardy, J., Introduction to Quantum Field Theory
[7]Gaberdiel, M., Gehrmann-De Ridder, A., Quantum Field Theory