fields and particles

1. Very Brief Highlights On Some Key Details: Vacuum-To-Vacuum Matrix In the Presence Of
Cubic Self Interaction
Roa, F. J. P.
Remarks: This draft is based on my answer notes dealing on some basic drills and workouts in
quantum field theory so the reader is highly encouraged to figure things out independently and
compare his results to the ones presented here.
Say we start at an initial time 𝑡 = 0 for a quantum mechanical description of a scalar particle in
the vacuum state |0⟩. With the application of the time evolution operator (teo) 𝑈1(𝑇) on the
given vacuum state we evolve this vacuum state into another state | 𝜓⟩ that is given at a later time
𝑡 = 𝑇 > 0.
(1)
|0⟩ → | 𝜓⟩ = 𝑈1(𝑇)|0⟩
We take the projection of this evolved state on the vacuum state. Thus, getting the vacuum-to-
vacuum matrix
(2)
⟨0| 𝑈1(𝑇)|0⟩
for the probability amplitude that a particle initially in the vacuum state at an initial time will still
be in the vacuum state at a later time.
The time evolution operator (TEO)
(3)
𝑈1 ( 𝑇) = 𝑈1 ( 𝑇,0) = 𝑒𝑥𝑝 (−
𝑖
ℏ
∫ 𝑑𝑡 𝐻(𝑡)
𝑇
0
)
is given for a system Hamiltonian for scalar fields considered in this particular drill and this
Hamiltonian has implicit time dependence being a Hamiltonian H(J(t)) that is a functional of
time dependent source J(t). This Hamiltonian can be derived from the scalar field system
Lagrangian via Legendre transformation. However, we shall no longer give the details of such
transformation here.
In this prior set-up, we are not yet taking into account the self-interactions in our Lagrangian and
consequently, in our Hamiltonian so our initial matrix given by (2) is in the absence of the said
interactions. These self-interactions enter into the Lagrangian and Hamiltonian in the form of

2. potential functions and in my personal convenience, I don’t include in these functions the scalar
field mass terms as I put these mass terms already explicitly in the Lagrangian and Hamiltonian.
Without going into the details of how these self-interactions enter into the vacuum-to-vacuum
(VTV) matrix we will only give this VTV matrix here that comes with the presence of the said
interactions via potential operators
(4)
⟨0|𝑈1(𝑇)𝑒𝑥𝑝 (−
𝑖
ℏ
∫ 𝑑4 𝑦 𝑉(𝜑̂)
𝐵
𝐴 ) |0⟩|
𝐽=0
= 𝑒𝑥𝑝 (−
𝑖
ℏ
∫ 𝑑4 𝑦 𝑉 (𝑖ℏ
𝛿
𝛿𝐽(𝑦)
)
𝐵
𝐴
) ⟨0| 𝑈1(𝑇)|0⟩|
𝐽=0
and in this particular drill we are considering cubic self- interaction in the form given by
(5)
𝑉[ 𝜑( 𝑥, 𝐽)] =
1
3!
𝑔(3) 𝜑3( 𝑥, 𝐽)
In this current presentation, I am also not going to dwell on the lengthy details to arrive at the
end results of this exercise but this more elaborate presentation will be done in future drafts.
The VTV matrix that includes the cubic self-interaction (5) is obtained upon the setting of all
sources to zero, J = 0. This matrix is given by
(6)
⟨0|𝑈1(𝑇)𝑒𝑥𝑝 (−
𝑖
ℏ
∫ 𝑑4 𝑦 𝑉( 𝜑̂)
𝐵
𝐴 ) |0⟩|
𝐽=0
= 𝐶 [ 𝐽 = 0] ×
(1 +
1
12
𝑔(3)
2
(𝑖ℏ)3∫ 𝑑4 𝑦 ′𝑑4 𝑦
𝐺(𝑦 ′ − 𝑦)
(2𝜋)2
𝐺( 𝑦 − 𝑦 ′ )
(2𝜋)2
𝐺(𝑦 ′ − 𝑦)
(2𝜋)2
𝐵
𝐴
+
1
8
𝑔(3)
2
(𝑖ℏ)3∫ 𝑑4 𝑦 ′𝑑4 𝑦
𝐺(𝑦 ′ − 𝑦 ′)
(2𝜋)2
𝐺( 𝑦 ′ − 𝑦 )
(2𝜋)2
𝐺(𝑦 − 𝑦)
(2𝜋)2
𝐵
𝐴
)
In my personal convenience, I am used to writing the Green’s function as (for example)
(7)
𝐺( 𝑦 ′ − 𝑦) =
1
(2𝜋)2
∫ 𝑑4 𝑘
𝑒 𝑖𝑘 𝜎(𝑦′ 𝜎− 𝑦 𝜎)
−𝑘 𝜇 𝑘 𝜇 + 𝑀2 + 𝑖𝜖
where I have already inserted an imaginary part 𝑖𝜖 to shift the poles as contour integration will be
required afterwards. The Green’s functions in the matrix expression act as propagators.

3. The resulting matrix (6), upon the setting of J = 0, consists of two groups (aside from the term
with numerical 1) of relevant identical terms. These terms are as written in coordinate space.
Let us take the group of terms with the numerical factor
1
12
and write this in momentum space
(8)
∫ 𝑑4 𝑦 ′𝑑4 𝑦
𝐺(𝑦 ′ − 𝑦)
(2𝜋)2
𝐺( 𝑦 − 𝑦 ′ )
(2𝜋)2
𝐺(𝑦 ′ − 𝑦)
(2𝜋)2
𝐵
𝐴
=
1
(2𝜋)4
∫ 𝑑4 𝑘 ∫ 𝑑4 𝑘 ′∫ 𝑑4 𝑘 ′′
1
−𝑘2 + 𝑀2 + 𝑖𝜖
1
−𝑘 ′2 + 𝑀2 + 𝑖𝜖
1
−𝑘 ′′2 + 𝑀2 + 𝑖𝜖
𝛿4( 𝑘 − 𝑘 ′
+ 𝑘 ′′ ) 𝑦 ′ 𝛿4( 𝑘 − 𝑘 ′ + 𝑘 ′′ ) 𝑦
The Dirac-delta functions in this expression represent vertices at the respective space-time points
𝑦 ′ and 𝑦. As indicated above there are three four-momentum integration variables with two
initial four-momentum vertices that we have just mentioned.
As written in coordinate space (8) can be depicted with the following Feynman graph
(Fig.1)
where at the spacetime point 𝑦 ′ is the Dirac-delta function 𝛿4( 𝑘 − 𝑘 ′ + 𝑘 ′′ ) 𝑦 ′, while at the
spacetime point y the Dirac-delta function 𝛿4( 𝑘 − 𝑘 ′ + 𝑘 ′′ ) 𝑦 . (Note the symmetric property of
the Dirac-delta function, 𝛿4(−𝑎) = 𝛿4( 𝑎)).
We also have the Fourier components of the propagators
(8.1)
𝑔̃( 𝑘) =
1
−𝑘2 + 𝑀2 + 𝑖𝜖
(8.2)

4. 𝑔̃( 𝑘 ′) =
1
−𝑘 ′2 + 𝑀2 + 𝑖𝜖
and
(8.3)
𝑔̃( 𝑘 ′′) =
1
−𝑘 ′′2 + 𝑀2 + 𝑖𝜖
We can integrate over 𝑘 ′′ at the space-time point 𝑦 with a picking 𝑘 ′′ = k ’ – k so that we can
reduce (8) into (noted (k ’ – k )2
= (k – k ′ )2
)
(9)
∫ 𝑑4 𝑦 ′𝑑4 𝑦
𝐺(𝑦 ′ − 𝑦)
(2𝜋)2
𝐺( 𝑦 − 𝑦 ′ )
(2𝜋)2
𝐺(𝑦 ′ − 𝑦)
(2𝜋)2
𝐵
𝐴
=
1
(2𝜋)4
∫ 𝑑4 𝑘′
1
−𝑘 ′2 + 𝑀2 + 𝑖𝜖
∫ 𝑑4 𝑘
1
−𝑘2 + 𝑀2 + 𝑖𝜖
1
− ( k – k ′ )
2
+ 𝑀2 + 𝑖𝜖
𝛿4( 𝑘 ′ − 𝑘
+ k – k ′ ) 𝑦 ′
In momentum space as given by (9), we have the corresponding Feynman graph
(Fig.2)
where
(9.1)
𝑔̃( 𝑘) =
1
−𝑘2 + 𝑀2 + 𝑖𝜖
(9.2)
𝑔̃( 𝑘 ′) =
1
−𝑘 ′2 + 𝑀2 + 𝑖𝜖
and

5. (9.3)
𝑔̃(k – k ′ ) =
1
− (k – k ′ )
2
+ 𝑀2 + 𝑖𝜖
In (9), we have chosen the four-momentum k as internal four-momentum and k’ as the external
four-momentum and such recognition enables us to construct the Feynman graph Fig.2.
In (9) we take note of the remaining Dirac-delta function that we write as (caution: with
symmetric integral limits)
(9.4)
𝛿4( 𝑘 ′ − 𝑘 + k – k ′ ) 𝑦 ′ = 𝛿4(0 ) 𝑦 ′ =
1
(2𝜋)4
∫ 𝑑4 𝑦 ′
This might imply a singularity if we are to base this on the basic definition of a Dirac-delta
function with an argument that is vanishing, which is 𝛿4(0 ) 𝑦 ′ = ∞. However, for the moment let
us take (9.4) as a definite constant.
We continue with the other group of terms that collectively go with the numerical factor
1
8
as given by (6)
in coordinate space. This we translate into momentum space
(10)
∫ 𝑑4 𝑦 ′𝑑4 𝑦
𝐺(𝑦 ′ − 𝑦 ′ )
(2𝜋)2
𝐺( 𝑦 ′ − 𝑦 )
(2𝜋)2
𝐺(𝑦 − 𝑦)
(2𝜋)2
𝐵
𝐴
=
1
(2𝜋)4
∫ 𝑑4 𝑘 ∫ 𝑑4 𝑘 ′∫ 𝑑4 𝑘 ′′
1
−𝑘2 + 𝑀2 + 𝑖𝜖
1
−𝑘 ′2 + 𝑀2 + 𝑖𝜖
1
−𝑘 ′′2 + 𝑀2 + 𝑖𝜖
𝛿4( 𝑘 − 𝑘 "
+ 𝑘 ′′ ) 𝑦 ′ 𝛿4( 𝑘 − 𝑘 ′ + 𝑘 ′ ) 𝑦
This is depicted in the following Feynman graph
(Fig.3)

6. 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