theory physics

1. Path Integration: One-Degree of Freedom Scalar Field Along Time Direction
F. J. P. Roa
In the absence of time-dependent sources, we can write the scalar field transition matrix
(1.1)
⟨𝜑′|𝑒𝑥𝑝(
−𝑖𝑇𝐻[ 𝐽 = 0 ]
ℏ
)|𝜑⟩
This matrix gives the probability (transition) amplitude that the particle in an initial state | 𝜑⟩ at
time 𝑡 = 0 will be in some other state | 𝜑′⟩ at the later time 𝑇 > 0. The system Hamiltonian
𝐻[ 𝐽 = 0 ] here does not contain time-dependent source 𝐽(𝑡) so that the time-evolution operator
simply given as
(1.2)
𝑒𝑥𝑝 (
−𝑖
ℏ
∫ 𝑑𝑡 𝐻[ 𝐽 = 0 ]
𝑇
0
) = 𝑒𝑥𝑝(
−𝑖𝑇𝐻[ 𝐽 = 0 ]
ℏ
)
The Hamiltonian contained in (1.2) is an operator although we omit the putting of a hat over it
except in confusing circumstances where we are obliged to emphasize it with a hat as an
operator. Classically, this Hamiltonian can be derived through Legendre transformation which
we shall demonstrate in the later portions of this draft.
The evaluation of (1.1) can be done via Path Integration
(1.3)
⟨𝜑′|𝑒𝑥𝑝(
−𝑖𝑇𝐻[ 𝐽 = 0 ]
ℏ
)|𝜑⟩ = ∫ 𝐷𝜑 𝑒𝑥𝑝(𝑖𝑆 𝜑/ℏ)
𝜑( 𝐵) = 𝜑′
𝜑( 𝐴) = 𝜑
and in this draft we try to get results that are close to those of doing Path Integration in
elementary particle quantum mechanics by considering a scalar field with one degree of freedom
along time direction.

2. Since the scalar field is confined to have only one degree of freedom along time direction its
action is to be given by
(1.4)
𝑆 𝜑 = ∫ 𝑑𝑡 ℓ 𝜑 = ∫ 𝑑𝑡 (
1
2
(𝜕𝑡 𝜑)2
−
1
2
𝑀2
𝜑2
)
and as cautionary, the action has the appropriate units of 𝑗𝑜𝑢𝑙𝑒𝑠 ∙ 𝑠𝑒𝑐𝑜𝑛𝑑𝑠 so we also take note
of the correct units of our one-degree of freedom scalar field in this case. This we shall do in the
later while.
Our integral limits define the interval
(1.5)
0 < 𝑡 < 𝑇
with the corresponding labelling that 𝜑( 𝐴) = 𝜑 as the initial value of the scalar field at 𝑡 = 0
(though this field value somewhat confusing with the field variable appearing in (1.4) so we must
be reminded that the former is distinct from the latter), while 𝜑( 𝐵) = 𝜑′ as the later value given
at 𝑡 = 𝑇.
In addition to confining the scalar field in one degree of freedom along time direction with the
aim to derive result that is close to that of doing path integrals involving particle quantum
mechanics, we shall also consider the scalar field as a perturbation field above the mean classical
𝜑𝑐. So
(1.6)
𝜑 = 𝜑𝑐 + 𝜑 𝑝
where 𝜑 𝑝 is the perturbation field that must vanish at the time boundaries defined in (1.5).
Given (1.6), consequently (1.4) is decomposed as
(1.7)
𝑆 𝜑 = 𝑆 𝜑𝑐 + 𝑆 𝜑𝑝 + 𝑆̅ 𝜑
(1.7.1)
𝑆 𝜑𝑐 = ∫ 𝑑𝑡 (
1
2
(𝜕𝑡 𝜑𝑐)2
−
1
2
𝑀2
𝜑𝑐
2
)
(1.7.2)

3. 𝑆 𝜑𝑝 = ∫ 𝑑𝑡 (
1
2
(𝜕𝑡 𝜑 𝑝)2
−
1
2
𝑀2
𝜑 𝑝
2
)
(1.7.3)
𝑆̅ 𝜑 = ∫ 𝑑𝑡 ((𝜕𝑡 𝜑𝑐)(𝜕𝑡 𝜑 𝑝) − 𝑀2
𝜑𝑐 𝜑 𝑝)
The last part is done through partial integration so we may write that as
(1.8)
𝑆̅ 𝜑 = [(𝜕𝑡 𝜑𝑐)𝜑 𝑝]
𝐴
𝐵
− ∫ 𝑑𝑡 ( 𝜕𝑡
2
𝜑𝑐 + 𝑀2
𝜑𝑐 ) 𝜑 𝑝
This simply vanishes given the time boundary conditions for the perturbation field
(1.9.1)
𝜑 𝑝( 𝐴) = 𝜑 𝑝( 𝐵) = 0
and the classical equation of motion
(1.9.2)
𝜕𝑡
2
𝜑𝑐 + 𝑀2
𝜑𝑐 = 0
Following this, (1.7) reduces to an effective action
(1.10)
𝑆 𝜑(𝑒𝑓𝑓) = 𝑆 𝜑𝑐 + 𝑆 𝜑𝑝
Immediately, we provide details for the derivation of the equation of motion (1.9.2).
For the classical equation of motion (1.9.2) we take the variation 𝛿𝑆 𝜑𝑐 of the classical action
(1.7.1) in terms of the variation 𝛿𝜑𝑐 of the classical field and to take note of the commutativity
(1.11)
𝛿𝜕𝑡 = 𝜕𝑡 𝛿
So after partial integration we write this variation as
(1.12)
𝛿𝑆 𝜑𝑐 = [(𝜕𝑡 𝜑𝑐)𝛿𝜑𝑐] 𝐴
𝐵
− ∫ 𝑑𝑡 ( 𝜕𝑡
2
𝜑𝑐 + 𝑀2
𝜑𝑐 ) 𝛿𝜑𝑐

4. and we shall also impose that the varied field vanishes at the time boundaries
(1.12.1)
𝛿𝜑𝑐( 𝐴) = 𝛿𝜑𝑐( 𝐵) = 0
Letting the action (1.7.1) be stationary that is,
(1.12.2)
𝛿𝑆 𝜑𝑐 = 0
we arrive at (1.9.2).
Being given with an effective action (1.10), we can now put the path integral in (1.3) in its
factored form
(1.13)
⟨𝜑′|𝑒𝑥𝑝 (
−𝑖𝑇𝐻[ 𝐽 = 0 ]
ℏ
)|𝜑⟩ = 𝑒𝑥𝑝(𝑖𝑆 𝜑𝑐/ℏ) ∫ 𝒟𝜑 𝑝 𝑒𝑥𝑝(𝑖𝑆 𝜑𝑝/ℏ)
𝜑 𝑝( 𝐴)= 𝜑 𝑝( 𝐵)=0
where the actual path integration is done on the perturbation field.
Note here that although the varied classical field must vanish at the time boundaries (see
(1.12.1)) the classical field itself assumes the non-vanishing time boundary conditions given
earlier
(1.14.1)
𝜑𝑐( 𝐴) = 𝜑( 𝐴) = 𝜑
and
(1.14.2)
𝜑𝑐( 𝐵) = 𝜑( 𝐵) = 𝜑′
Now proceeding with the actual path integration
(2)
∫ 𝒟𝜑 𝑝 𝑒𝑥𝑝(𝑖𝑆 𝜑𝑝/ℏ)
𝜑 𝑝 ( 𝐴)= 𝜑 𝑝( 𝐵)=0

5. We take note of the time-boundary conditions on the perturbation field (1.9.1) and one form of
this perturbation that immediately satisfies (1.9.1) is given by
(2.1)
𝜑 𝑝 ( 𝑡) = ∑ 𝐴𝑙 𝑠𝑖𝑛
𝜋𝑙
𝑇
𝑡
𝑙
where the summation index 𝑙 takes on both the odd and even integral values. We could have
chosen 2𝜋𝑙, which is exclusive to the even multiples of 𝜋 but since we are aiming for results that
are close to those of doing path integral in particle quantum mechanics, we simply choose 𝜋𝑙,
where 𝑙 includes both the odd and even integral values.
For convenience let us put
(2.2)
𝑘0( 𝑙) =
𝜋𝑙
𝑇
and go for the Fourier transformation of (2.1) with integral limits that define the interval
(2.3)
−𝑇 < 𝑡 < 𝑇
(2.4)
𝜑̃ 𝑝(𝑘0( 𝑙′)) =
1
𝑇
∫ 𝑑𝑡 𝜑 𝑝(𝑡) exp(−𝑖𝑘0( 𝑙′) 𝑡)
𝑇
−𝑇
The integration involves the complete set
(2.5.1)
∫ 𝑑𝑡 𝑠𝑖𝑛
𝜋𝑙
𝑇
𝑡 cos
𝜋𝑙′
𝑇
𝑡 = 0
𝑇
−𝑇
and
(2.5.2)
1
𝑇
∫ 𝑑𝑡 𝑠𝑖𝑛
𝜋𝑙
𝑇
𝑡 sin
𝜋𝑙′
𝑇
𝑡 = 𝛿𝑙𝑙′
𝑇
−𝑇

6. These for all integral values of 𝑙. Thus, we get the Fourier components
(2.6)
𝜑̃ 𝑝(𝑘0( 𝑙′)) = −𝑖𝐴𝑙′
Then for 𝜑̃ 𝑝(−𝑘0( 𝑙′)) as 𝑘0( 𝑙′) → −𝑘0( 𝑙′) we have
(2.7)
𝜑̃ 𝑝(−𝑘0( 𝑙′)) =
1
𝑇
∫ 𝑑𝑡 𝜑 𝑝( 𝑡) exp( 𝑖𝑘0( 𝑙′) 𝑡) = 𝑖𝐴𝑙′
𝑇
−𝑇
We can take the complex conjugation of (2.4)
(2.8)
[To be continued…]
Ref’s
[1] Townsend, P. K., Blackholes – Lecture Notes, http://xxx.lanl.gov/abs/gr-qc/9707012
[2]van Baal, P., A Course In Field Theory
[3]Siegel, W., Fields, http://insti.physics.sunysb.edu/~/siegel/plan.html