blackholephysics

1. Summer Physics 2016: Attachment2(part 1) for the Third Update
Roa, F. J. P.
Let us refresh ourselves with the basic approaches in Path Integral and this present document
serves as a continuation of Summer Physics 2016: Attachment1 for the Second Update.
To start, say we are given with the matrix elements
⟨ 𝑥′| 𝑈(𝑇)| 𝑥⟩
of the time evolution operator
(1)
𝑈( 𝑇, 0) = 𝑈( 𝑇) = 𝑒𝑥𝑝(−
𝑖
ℏ
∫ 𝑑𝑡 𝐻̂(𝑡)
𝑇
0
)
These matrix elements are in terms of the continuous bases | 𝑥⟩ and these matrices bear the
physical interpretation that these give the probability amplitudes that a particle initially at some
initial state x at an initial time 𝑡 = 0 will be found at some other state 𝑥′
at a later time 𝑇 > 0.
That is, from the initial state vector | 𝑥⟩ given at an initial time say, 𝑡 = 0, with the application of
time evolution operator (1) we evolve this initial state vector into some other state vector | 𝜓⟩ =
𝑈( 𝑇)| 𝑥⟩ given at the later time T. Then we project this evolved state vector onto some other
arbitrary state vector | 𝑥′⟩ to form the matrix element ⟨ 𝑥′| 𝑈(𝑇)| 𝑥⟩ that gives the cited probability
amplitude.
The time evolution operator is known to observe the causality principle and owing to this
principle we can actually decompose (with a property of a group) a given time evolution operator
as product of individual time evolution operators say for instance,
(2)
𝑈( 𝑇, 𝑡1) = 𝑈( 𝑇, 𝑡2) 𝑈( 𝑡2, 𝑡1)
also assuming that these operators are unitary.
(3)
𝑈†
𝑈 = 𝑈𝑈†
= 1

2. Given (2), we may write or decompose a particular matrix element ⟨ 𝑥′| 𝑈(𝑇)| 𝑥⟩ also as product
of matrix elements
(4)
⟨ 𝑥′| 𝑈(𝑇)| 𝑥⟩ = ∫ 𝑑𝑥1 ⟨ 𝑥′| 𝑈( 𝑇, 𝑡1)| 𝑥1⟩⟨ 𝑥1| 𝑈(𝑡1,0)| 𝑥⟩
also by having to insert the completeness relation of the bases
(5)
1 = ∫| 𝑥1⟩ 𝑑𝑥1⟨ 𝑥1|
Repeating this decomposition in an appropriate number of times inside the integral of (4) we
would end up writing this matrix element as multiple integrations with the corresponding matrix
elements
(6.1)
⟨ 𝑥′| 𝑈(𝑇)| 𝑥⟩ = ∫∏ 𝑑𝑥 𝑖
𝑁−1
𝑖=1
∏⟨𝑥𝑗+1|𝑈(𝑡𝑗+1, 𝑡𝑗)|𝑥𝑗⟩
𝑁−1
𝑗 =0
In this we are making the following identifications
(6.2)
𝑥 𝑁 = 𝑥′
, 𝑥0 = 𝑥
𝑡 𝑁 = 𝑇, 𝑡0 = 0
This resulting expression has a physical interpretation that the matrix element ⟨ 𝑥′| 𝑈(𝑇)| 𝑥⟩ that
gives final probability amplitude is the integration over those paths each of which goes with its
own probability amplitude. That is, this final probability amplitude is being decomposed into
numerous probability amplitudes for various possible paths that a particle can go along and then
summing or integrating these up to give the said final probability amplitude represented by the
said matrix element.
We make use of the fact that the projection of a given momentum state vector |𝑝𝑗⟩ onto an
arbitrary position state vector |𝑥𝑗+1⟩ can be expressed as a wave packet
(7.1)

3. ⟨𝑥𝑗+1|𝑝𝑗 ⟩ =
1
√2𝜋ℏ
𝑒 𝑖 𝑥 𝑗+1 𝑝 𝑗/ ℏ
so each matrix element ⟨𝑥𝑗+1|𝑈(𝑡𝑗+1, 𝑡𝑗)|𝑥𝑗⟩ can be written using this wavepacket in the bases of
momentum space
(7.2)
⟨𝑥𝑗+1|𝑈(𝑡𝑗+1, 𝑡𝑗 )|𝑥𝑗⟩ = ∫ 𝑑𝑝𝑗 ⟨𝑥𝑗+1|𝑝𝑗 ⟩⟨𝑝𝑗 |𝑈(𝑡𝑗+1, 𝑡𝑗)|𝑥𝑗⟩
also in this, it is assumed that the momentum bases satisfy the completeness relation
(7.3)
1 = ∫|𝑝𝑗 ⟩𝑑𝑝𝑗⟨𝑝𝑗 |
Say, considering that the Hamiltonian does not contain a time-dependent source so that we may
write the time-evolution operator as
(7.4)
𝑈(𝑡𝑗+1, 𝑡𝑗 ) = 𝑒𝑥𝑝(−
𝑖
ℏ
(𝑡𝑗+1 − 𝑡𝑗)𝐻̂)
The system Hamiltonian can be obtained given the system Lagrangian L through Legendre
transformation
(7.5.1)
𝐻 = π𝑥̇ − 𝐿 =
1
2
𝑚𝑥̇ 2 +
1
2
𝑚𝜔2 𝑥2
̇
(7.5.2)
𝑝 = 𝑚𝑥̇ = π =
𝛿𝐿
𝛿𝑥̇
(7.5.3)
𝐻 =
1
2𝑚
𝑝2
+
1
2
𝑚𝜔2
𝑥2
We can use the matrix elements ⟨𝑝𝑗|𝑈(𝑡𝑗+1, 𝑡𝑗)|𝑝 𝑘⟩ in expressing ⟨𝑝𝑗|𝑈(𝑡𝑗+1, 𝑡𝑗)|𝑥𝑗⟩ as

4. (7.5.4)
⟨𝑝𝑗|𝑈(𝑡𝑗+1, 𝑡𝑗)|𝑥𝑗⟩ = ∫⟨𝑝𝑗 |𝑈(𝑡𝑗+1 , 𝑡𝑗)|𝑝 𝑘⟩𝑑𝑝 𝑘 ⟨𝑝 𝑘|𝑥𝑗⟩
=
1
√2𝜋ℏ
∫ 𝑑𝑝 𝑘 ⟨ 𝑝𝑗|𝑈(𝑡𝑗+1, 𝑡𝑗)|𝑝 𝑘⟩ 𝑒− 𝑖 𝑥 𝑗 𝑝 𝑘/ ℏ
Note that ⟨𝑝𝑗 |𝑈(𝑡𝑗+1, 𝑡𝑗)|𝑝 𝑘⟩ are the matrix elements of the time evolution operator (7.4) using
the momentum bases and in these bases we can have the eigenvalue equation
(7.5.5)
𝑒𝑥𝑝(−
𝑖
ℏ
(𝑡𝑗+1 − 𝑡𝑗)𝐻̂) | 𝑝 𝑘⟩ = 𝑒𝑥𝑝 (−
𝑖
ℏ
(𝑡𝑗+1 − 𝑡𝑗)𝐻 𝑘)| 𝑝 𝑘⟩
where the eigenvalues 𝐻 𝑘 of the Hamiltonian operator 𝐻̂ consist of the eigenvalues of the
momentum and position operators
(7.5.6)
𝐻 𝑘 =
1
2𝑚
𝑝 𝑘
2
+
1
2
𝑚𝜔2
𝑥 𝑘
2
Let us also add the orthonormality condition on these (continuous) momentum bases
(7.5.7)
⟨𝑝𝑗 |𝑝 𝑘⟩ = 𝛿 ( 𝑝𝑗 − 𝑝 𝑘 )
Thus, writing the matrix elements ⟨𝑝𝑗 |𝑈(𝑡𝑗+1, 𝑡𝑗)|𝑝 𝑘⟩ as
(7.5.8)
⟨𝑝𝑗|𝑈(𝑡𝑗+1, 𝑡𝑗)|𝑝 𝑘⟩ = 𝑒𝑥𝑝(−
𝑖
ℏ
(𝑡𝑗+1 − 𝑡𝑗)𝐻 𝑘) 𝛿 ( 𝑝𝑗 − 𝑝 𝑘 )
These are the needed matrix elements in (7.5.4) so integrating over 𝑝 𝑘 there yields,
(7.5.9)
⟨𝑝𝑗|𝑈(𝑡𝑗+1, 𝑡𝑗)|𝑥𝑗⟩ =
1
√2𝜋ℏ
𝑒− 𝑖 𝑥 𝑗 𝑝 𝑗/ ℏ
𝑒𝑥𝑝(−
𝑖
ℏ
(𝑡𝑗+1 − 𝑡𝑗)𝐻𝑗)
In turn we use this in the matrix element (7.2) to write that as

5. (7.5.10)
⟨𝑥𝑗+1|𝑈(𝑡𝑗+1, 𝑡𝑗)|𝑥𝑗⟩
=
1
2𝜋ℏ
∫ 𝑑𝑝𝑗 𝑒 𝑖( 𝑥 𝑗+1− 𝑥 𝑗 ) 𝑝 𝑗/ ℏ
𝑒𝑥𝑝(−
𝑖
ℏ
(𝑡𝑗+1 − 𝑡𝑗 )(
1
2𝑚
𝑝𝑗
2
+
1
2
𝑚𝜔2
𝑥𝑗
2
))
which involves a Gaussian integration over the variable 𝑝𝑗 .
Let us note that in some convenient (somewhat conventional) manner time is discretized in equal
intervals such as
(7.6)
𝑡𝑗+1 − 𝑡𝑗
ℏ
=
𝑇
𝑁ℏ
∆𝑡 =
𝑇
𝑁
= 𝑡𝑗+1 − 𝑡𝑗
Of important use in this Gaussian integration is the integral result
(7.7)
∫ 𝑑𝑦 𝑒 𝑎𝑦2
+𝑏𝑦
= √
𝜋
−𝑎
𝑒−𝑏2
/4𝑎
∞
−∞
so consequently, (7.5.10) results into
(7.8)
⟨𝑥𝑗+1|𝑈(𝑡𝑗+1, 𝑡𝑗)|𝑥𝑗⟩ = √
𝑚
𝑖2𝜋ℏ∆𝑡
𝑒𝑥𝑝(
𝑖∆𝑡
ℏ
(
𝑚
2
(𝑥𝑗+1 − 𝑥𝑗 ) 2
∆𝑡2
− 𝑉(𝑥𝑗)))
𝑉(𝑥𝑗) =
1
2
𝑚𝜔2
𝑥𝑗
2
We could have stopped at (7.5.10) and utilize this result to express (6.1) in momentum space but
we have opted to continue a little further at (7.8) intently so we may use this result in the partial
product indicated in (6.1) and write this directly in coordinate space.
We can write the following partial product

6. (7.9)
∏ √
𝑚
𝑖2𝜋ℏ∆𝑡
𝑁−1
𝑗 =0
= (
𝑚
𝑖2𝜋ℏ∆𝑡
)
𝑁/2
so we may write the indicated partial product in (6.1) as
(7.10)
∏⟨𝑥𝑗+1|𝑈(𝑡𝑗+1, 𝑡𝑗)|𝑥𝑗⟩
𝑁−1
𝑗=0
= (
𝑚
𝑖2𝜋ℏ∆𝑡
)
𝑁/2
𝑒𝑥𝑝(∑
𝑖∆𝑡
ℏ
(
𝑚
2
(𝑥𝑗+1 − 𝑥𝑗 )2
∆𝑡2
− 𝑉(𝑥𝑗))
𝑁−1
𝑗 =0
)
So in coordinate space we can write for the probability amplitude represented by (6.1) in path
integral formalism and in coordinate space this can be expressed as
(7.11)
⟨ 𝑥′| 𝑈(𝑇)| 𝑥⟩ = ∫ (
𝑚
𝑖2𝜋ℏ∆𝑡
)
𝑁/2
∏ 𝑑𝑥 𝑘
𝑁−1
𝑘=1
𝑒𝑥𝑝(
𝑖
ℏ
∑ ∆𝑡 (
𝑚
2
(𝑥𝑗+1 − 𝑥𝑗 )2
∆𝑡2
− 𝑉(𝑥𝑗))
𝑁−1
𝑗 =0
)
There could be an infinite number of paths that a particle can go from its initial state to the final
state so we take this path integration in the limit as 𝑁 → ∞ and in this limit, we may take it that
(7.12)
lim
𝑁 → ∞
∆𝑡 =
𝑇
𝑁
= 𝑑𝑡
so, the discrete case may pass into the continuous path integration over a continuous time
integral although we retain the discretization ∆𝑡 =
𝑇
𝑁
in the measure
(7.13)
𝔇𝑥 = (
𝑚
𝑖2𝜋ℏ∆𝑡
)
𝑁/2
∏ 𝑑𝑥 𝑘
𝑁−1
𝑘=1
but we can always resort to its normalized form.
Regarding the eigenvalue equation (7.5.5) and its corresponding matrix element (7.5.8), we can
delve shortly into these as a needed drill to be inserted in this present draft. We go the other way
around with this drill since we must use (7.5.9) first before (7.5.8) and (7.5.5) are proved and to
prove (7.5.9) in turn, we resort to the Trotter formula to obtain this ((7.5.9)) matrix element.

7. We start by writing the momentum state vector | 𝑝 𝑘⟩ in the continuous coordinate bases
(8.1)
| 𝑝 𝑘⟩ = ∫|𝑥𝑗⟩ 𝑑𝑥𝑗 ⟨𝑥𝑗|𝑝 𝑘⟩
and let this be operated with the time evolution operator 𝑈(𝑡𝑗+1, 𝑡𝑗), then write this operation
using the matrix element (7.5.9)
(8.2)
𝑈(𝑡𝑗+1, 𝑡𝑗)| 𝑝 𝑘⟩ = ∫|𝑝𝑗 ⟩𝑑𝑝𝑗 ⟨𝑝𝑗|𝑈(𝑡𝑗+1, 𝑡𝑗)|𝑥𝑗⟩ 𝑑𝑥𝑗 ⟨𝑥𝑗|𝑝 𝑘⟩
Using the result (7.5.9) together with the wavepacket
(8.3)
⟨𝑥𝑗|𝑝 𝑘⟩ =
1
√2𝜋ℏ
𝑒 𝑖 𝑥 𝑗 𝑝 𝑘 / ℏ
we can express the matrix element of (8.2) in the following form
(8.4)
⟨𝑝𝑙|𝑈(𝑡𝑗+1, 𝑡𝑗)|𝑝 𝑘⟩ = ∫⟨𝑝𝑙|𝑝𝑗 ⟩𝑑𝑝𝑗 𝑒𝑥𝑝(−
𝑖
ℏ
(𝑡𝑗+1 − 𝑡𝑗)𝐻𝑗)
1
2𝜋
∫ 𝑑𝑦𝑗 𝑒− 𝑖 (𝑝 𝑗 − 𝑝 𝑘 )𝑦𝑗
where we have introduced a change of variable, 𝑦𝑗 = 𝑥𝑗 /ℏ and consequently we have the
integral representation of a delta function,
(8.5)
𝛿( 𝑝 𝑘 − 𝑝𝑗 ) =
1
2𝜋
∫ 𝑑𝑦𝑗 𝑒− 𝑖 (𝑝 𝑗 − 𝑝 𝑘 )𝑦𝑗
If we wish to integrate over the differential variable 𝑑𝑝𝑗 , the picking would be 𝑝𝑗 = 𝑝 𝑘 and this
leads to the matrix element that we have wished to prove
(8.6)
⟨𝑝𝑙|𝑈(𝑡𝑗+1, 𝑡𝑗)|𝑝 𝑘⟩ = 𝑒𝑥𝑝(−
𝑖
ℏ
(𝑡𝑗+1 − 𝑡𝑗)𝐻 𝑘)⟨ 𝑝𝑙| 𝑝 𝑘⟩
We can then use this resulting matrix element to prove the eigenvalue equation (7.5.5),

8. (8.7)
𝑈(𝑡𝑗+1 , 𝑡𝑗)| 𝑝 𝑘⟩ = ∫| 𝑝 𝑚⟩ 𝑑𝑝 𝑚 ⟨𝑝 𝑚 |𝑈(𝑡𝑗+1, 𝑡𝑗)|𝑝𝑙⟩ 𝑑𝑝𝑙 ⟨ 𝑝𝑙| 𝑝 𝑘⟩
= ∫| 𝑝 𝑚⟩ 𝑑𝑝 𝑚 𝑒𝑥𝑝(−
𝑖
ℏ
(𝑡𝑗+1 − 𝑡𝑗)𝐻𝑙)⟨ 𝑝 𝑚 | 𝑝𝑙⟩ 𝑑𝑝𝑙 ⟨ 𝑝𝑙| 𝑝 𝑘⟩
We would integrate over the differential variable 𝑑𝑝𝑙, while noting the orthonormality condition
(7.5.7) on the momentum bases, and the picking is 𝑝𝑙 = 𝑝 𝑘 . Note that with this picking, we also
make the corresponding index change, 𝐻𝑙 → 𝐻 𝑘. Thus, we have the required eigenvalue
equation (7.5.5).
(8.8)
𝑈(𝑡𝑗+1, 𝑡𝑗)| 𝑝 𝑘⟩ = ∫| 𝑝 𝑚 ⟩ 𝑑𝑝 𝑚 𝑒𝑥𝑝(−
𝑖
ℏ
(𝑡𝑗+1 − 𝑡𝑗)𝐻 𝑘)⟨ 𝑝 𝑚 | 𝑝 𝑘⟩
= 𝑒𝑥𝑝(−
𝑖
ℏ
(𝑡𝑗+1 − 𝑡𝑗 )𝐻 𝑘)∫| 𝑝 𝑚 ⟩ 𝑑𝑝 𝑚 ⟨ 𝑝 𝑚 | 𝑝 𝑘⟩
= 𝑒𝑥𝑝(−
𝑖
ℏ
(𝑡𝑗+1 − 𝑡𝑗 )𝐻 𝑘) | 𝑝 𝑘⟩
Next, we prove (7.5.9) using the Trotter formula.
Say, we are given with a product of exponential functions that involve matrices or operators that
are non-commuting.
(9.1)
𝑦 = 𝑒 𝜖𝐴
𝑒 𝜖𝐵
where 𝜖 is some parameter that we take here as 𝜖 ∝ 𝑁−1
.
We then obtain for the approximate expansion of the right-hand-side of (9.1), up to second order
in 𝜖. This we achieve with the use of the following Mclaurin expansions
(9.2.1)
𝑒 𝑧
= 1 + 𝑧 + ∑
𝑧 𝑛
𝑛!
∞
𝑛=2
(9.2.2)
ln(1 + 𝑥 ) = ∑
(−1) 𝑘+1
𝑥 𝑘
𝑘
∞
𝑘=1

9. Without much algebraic work we then obtain the following result
(9.3)
𝑒 𝜖𝐴
𝑒 𝜖𝐵
= exp( 𝜖𝐴 + 𝜖𝐵 + 𝜖2
[ 𝐴, 𝐵]
2
)
As earlier said, this to second order in 𝜖. To get for the Trotter formula we multiply both sides
with exp (− 𝜖2 [ 𝐴,𝐵]
2
) from the right.
(9.4)
𝑒 𝜖𝐴
𝑒 𝜖𝐵
exp(− 𝜖2
[ 𝐴, 𝐵]
2
) = exp( 𝜖𝐴 + 𝜖𝐵 +
[(𝜖𝐴 + 𝜖𝐵 + 𝜖2 [ 𝐴, 𝐵]
2
), − 𝜖2 [ 𝐴, 𝐵]
2
]
2
)
On the right-hand side of (9.4), we discard terms in the parenthesis with 𝜖 higher than second
degree, while on the left-hand side, we expand the exponential exp(− 𝜖2 [ 𝐴,𝐵]
2
) up to second
order in 𝜖. Thus arriving at Trotter formula
(9.5)
𝑒 𝜖𝐴 + 𝜖𝐵
= 𝑒 𝜖𝐴
𝑒 𝜖𝐵
(1 − 𝜖2
[ 𝐴, 𝐵]
2
)
We think of the second term in the parenthesis on the right-hand-side of (9.5) as the second order
correction with 𝜖2
∝ 𝑁−2
.
In using (9.5) in the time evolution operator (7.4) we shall make the following identifications
(9.6.1)
𝜖 = −
𝑖
ℏ
(𝑡𝑗+1 − 𝑡𝑗) = −
𝑖
ℏ
𝑇
𝑁
(9.6.2)
𝐴 =
1
2𝑚
𝑝̂2
(9.6.3)
𝐵 = 𝑉(𝑥̂)

10. However, in forming for the matrix (7.5.9), we shall drop off that said second order correction in
the Trotter formula as we apply it in the time-evolution operator (7.4). So,
(9.7)
⟨𝑝𝑗|𝑈(𝑡𝑗+1, 𝑡𝑗)|𝑥𝑗⟩ = ⟨𝑝𝑗 | 𝑒 𝜖𝐴
𝑒 𝜖𝐵
|𝑥𝑗⟩
and let the operator 𝑒 𝜖𝐴
operate on ⟨𝑝𝑗 |, while 𝑒 𝜖𝐵
on |𝑥𝑗⟩.
For 𝑒 𝜖𝐴
on ⟨𝑝𝑗|, we have the eigenvalue equation
(9.8.1)
⟨𝑝𝑗| 𝑒 𝜖𝐴
= ( 𝑒 𝜖∗
𝐴
|𝑝𝑗 ⟩)
†
= ⟨𝑝𝑗 | 𝑒 𝜖𝐴 𝑗
with the corresponding eigenvalues
𝐴𝑗 =
1
2𝑚
𝑝𝑗
2
Here, the operator A is hermitian as the momentum operator is Hermitian, 𝑝̂†
= 𝑝̂. On the other
hand, the eigenvalue equation for 𝑒 𝜖𝐵
on |𝑥𝑗⟩ is given by
(9.8.2)
𝑒 𝜖𝐵
|𝑥𝑗⟩ = 𝑒 𝜖𝐵𝑗 |𝑥𝑗⟩
𝐵𝑗 =
1
2
𝑚𝜔2
𝑥𝑗
2
as the operator A is also Hermitian as the position operator is Hermitian, 𝑥̂†
= 𝑥̂.
Then making the projection of the statevector (9.8.2) on (9.8.1) would yield the required matrix
element (7.5.9) as we also take note of the Hermitian conjugate of (8.3) at 𝑗 = 𝑘.

11. [stopped: pp. IR(23p), skip notes: hawking radiation ref: townsend jacket f]
Ref’s
[1]Merzbacher, E., Quantum Mechanics, 2nd edition, 1970, Wiley & Sons, Inc.
[2]Baal, P., A COURSE IN FIELD THEORY,
http://www.lorentz.leidenuniv.nl/~vanbaal/FTcourse.html
[3]Cardy, J., Introduction to Quantum Field Theory
[4]Gaberdiel, M., Gehrmann-De Ridder, A., Quantum Field Theory