In the interaction picture , a Hamiltonian H , can be split into a free part H 0 and an interacting part V S (t ) as H = H 0 + V S (t ) .
The potential in the interacting picture is
V
I
(
t
)
=
e
i
H
0
(
t
−
t
0
)
/
ℏ
V
S
(
t
)
e
−
i
H
0
(
t
−
t
0
)
/
ℏ
,
{\displaystyle V_{\mathrm {I} }(t)=\mathrm {e} ^{\mathrm {i} H_{0}(t-t_{0})/\hbar }V_{\mathrm {S} }(t)\mathrm {e} ^{-\mathrm {i} H_{0}(t-t_{0})/\hbar },}
where
H
0
{\displaystyle H_{0}}
is time-independent and
V
S
(
t
)
{\displaystyle V_{\mathrm {S} }(t)}
is the possibly time-dependent interacting part of the Schrödinger picture .
To avoid subscripts,
V
(
t
)
{\displaystyle V(t)}
stands for
V
I
(
t
)
{\displaystyle V_{\mathrm {I} }(t)}
in what follows.
In the interaction picture, the evolution operator U is defined by the equation:
Ψ
(
t
)
=
U
(
t
,
t
0
)
Ψ
(
t
0
)
{\displaystyle \Psi (t)=U(t,t_{0})\Psi (t_{0})}
This is sometimes called the Dyson operator .
The evolution operator forms a unitary group with respect to the time parameter. It has the group properties:
Identity and normalization:
U
(
t
,
t
)
=
1
,
{\displaystyle U(t,t)=1,}
[ 1]
Composition:
U
(
t
,
t
0
)
=
U
(
t
,
t
1
)
U
(
t
1
,
t
0
)
,
{\displaystyle U(t,t_{0})=U(t,t_{1})U(t_{1},t_{0}),}
[ 2]
Time Reversal:
U
−
1
(
t
,
t
0
)
=
U
(
t
0
,
t
)
,
{\displaystyle U^{-1}(t,t_{0})=U(t_{0},t),}
[clarification needed ]
Unitarity:
U
†
(
t
,
t
0
)
U
(
t
,
t
0
)
=
1
{\displaystyle U^{\dagger }(t,t_{0})U(t,t_{0})=\mathbb {1} }
[ 3]
and from these is possible to derive the time evolution equation of the propagator:[ 4]
i
ℏ
d
d
t
U
(
t
,
t
0
)
Ψ
(
t
0
)
=
V
(
t
)
U
(
t
,
t
0
)
Ψ
(
t
0
)
.
{\displaystyle i\hbar {\frac {d}{dt}}U(t,t_{0})\Psi (t_{0})=V(t)U(t,t_{0})\Psi (t_{0}).}
In the interaction picture , the Hamiltonian is the same as the interaction potential
H
i
n
t
=
V
(
t
)
{\displaystyle H_{\rm {int}}=V(t)}
and thus the equation can also be written in the interaction picture as
i
ℏ
d
d
t
Ψ
(
t
)
=
H
i
n
t
Ψ
(
t
)
{\displaystyle i\hbar {\frac {d}{dt}}\Psi (t)=H_{\rm {int}}\Psi (t)}
Caution : this time evolution equation is not to be confused with the Tomonaga–Schwinger equation .
The formal solution is
U
(
t
,
t
0
)
=
1
−
i
ℏ
−
1
∫
t
0
t
d
t
1
V
(
t
1
)
U
(
t
1
,
t
0
)
,
{\displaystyle U(t,t_{0})=1-i\hbar ^{-1}\int _{t_{0}}^{t}{dt_{1}\ V(t_{1})U(t_{1},t_{0})},}
which is ultimately a type of Volterra integral .
Derivation of the Dyson series
edit
An iterative solution of the Volterra equation above leads to the following Neumann series :
U
(
t
,
t
0
)
=
1
−
i
ℏ
−
1
∫
t
0
t
d
t
1
V
(
t
1
)
+
(
−
i
ℏ
−
1
)
2
∫
t
0
t
d
t
1
∫
t
0
t
1
d
t
2
V
(
t
1
)
V
(
t
2
)
+
⋯
+
(
−
i
ℏ
−
1
)
n
∫
t
0
t
d
t
1
∫
t
0
t
1
d
t
2
⋯
∫
t
0
t
n
−
1
d
t
n
V
(
t
1
)
V
(
t
2
)
⋯
V
(
t
n
)
+
⋯
.
{\displaystyle {\begin{aligned}U(t,t_{0})={}&1-i\hbar ^{-1}\int _{t_{0}}^{t}dt_{1}V(t_{1})+(-i\hbar ^{-1})^{2}\int _{t_{0}}^{t}dt_{1}\int _{t_{0}}^{t_{1}}\,dt_{2}V(t_{1})V(t_{2})+\cdots \\&{}+(-i\hbar ^{-1})^{n}\int _{t_{0}}^{t}dt_{1}\int _{t_{0}}^{t_{1}}dt_{2}\cdots \int _{t_{0}}^{t_{n-1}}dt_{n}V(t_{1})V(t_{2})\cdots V(t_{n})+\cdots .\end{aligned}}}
Here,
t
1
>
t
2
>
⋯
>
t
n
{\displaystyle t_{1}>t_{2}>\cdots >t_{n}}
, and so the fields are time-ordered . It is useful to introduce an operator
T
{\displaystyle {\mathcal {T}}}
, called the time-ordering operator , and to define
U
n
(
t
,
t
0
)
=
(
−
i
ℏ
−
1
)
n
∫
t
0
t
d
t
1
∫
t
0
t
1
d
t
2
⋯
∫
t
0
t
n
−
1
d
t
n
T
V
(
t
1
)
V
(
t
2
)
⋯
V
(
t
n
)
.
{\displaystyle U_{n}(t,t_{0})=(-i\hbar ^{-1})^{n}\int _{t_{0}}^{t}dt_{1}\int _{t_{0}}^{t_{1}}dt_{2}\cdots \int _{t_{0}}^{t_{n-1}}dt_{n}\,{\mathcal {T}}V(t_{1})V(t_{2})\cdots V(t_{n}).}
The limits of the integration can be simplified. In general, given some symmetric function
K
(
t
1
,
t
2
,
…
,
t
n
)
,
{\displaystyle K(t_{1},t_{2},\dots ,t_{n}),}
one may define the integrals
S
n
=
∫
t
0
t
d
t
1
∫
t
0
t
1
d
t
2
⋯
∫
t
0
t
n
−
1
d
t
n
K
(
t
1
,
t
2
,
…
,
t
n
)
.
{\displaystyle S_{n}=\int _{t_{0}}^{t}dt_{1}\int _{t_{0}}^{t_{1}}dt_{2}\cdots \int _{t_{0}}^{t_{n-1}}dt_{n}\,K(t_{1},t_{2},\dots ,t_{n}).}
and
I
n
=
∫
t
0
t
d
t
1
∫
t
0
t
d
t
2
⋯
∫
t
0
t
d
t
n
K
(
t
1
,
t
2
,
…
,
t
n
)
.
{\displaystyle I_{n}=\int _{t_{0}}^{t}dt_{1}\int _{t_{0}}^{t}dt_{2}\cdots \int _{t_{0}}^{t}dt_{n}K(t_{1},t_{2},\dots ,t_{n}).}
The region of integration of the second integral can be broken in
n
!
{\displaystyle n!}
sub-regions, defined by
t
1
>
t
2
>
⋯
>
t
n
{\displaystyle t_{1}>t_{2}>\cdots >t_{n}}
. Due to the symmetry of
K
{\displaystyle K}
, the integral in each of these sub-regions is the same and equal to
S
n
{\displaystyle S_{n}}
by definition. It follows that
S
n
=
1
n
!
I
n
.
{\displaystyle S_{n}={\frac {1}{n!}}I_{n}.}
Applied to the previous identity, this gives
U
n
=
(
−
i
ℏ
−
1
)
n
n
!
∫
t
0
t
d
t
1
∫
t
0
t
d
t
2
⋯
∫
t
0
t
d
t
n
T
V
(
t
1
)
V
(
t
2
)
⋯
V
(
t
n
)
.
{\displaystyle U_{n}={\frac {(-i\hbar ^{-1})^{n}}{n!}}\int _{t_{0}}^{t}dt_{1}\int _{t_{0}}^{t}dt_{2}\cdots \int _{t_{0}}^{t}dt_{n}\,{\mathcal {T}}V(t_{1})V(t_{2})\cdots V(t_{n}).}
Summing up all the terms, the Dyson series is obtained. It is a simplified version of the Neumann series above and which includes the time ordered products; it is the path-ordered exponential :[ 5]
U
(
t
,
t
0
)
=
∑
n
=
0
∞
U
n
(
t
,
t
0
)
=
∑
n
=
0
∞
(
−
i
ℏ
−
1
)
n
n
!
∫
t
0
t
d
t
1
∫
t
0
t
d
t
2
⋯
∫
t
0
t
d
t
n
T
V
(
t
1
)
V
(
t
2
)
⋯
V
(
t
n
)
=
T
exp
−
i
ℏ
−
1
∫
t
0
t
d
τ
V
(
τ
)
{\displaystyle {\begin{aligned}U(t,t_{0})&=\sum _{n=0}^{\infty }U_{n}(t,t_{0})\\&=\sum _{n=0}^{\infty }{\frac {(-i\hbar ^{-1})^{n}}{n!}}\int _{t_{0}}^{t}dt_{1}\int _{t_{0}}^{t}dt_{2}\cdots \int _{t_{0}}^{t}dt_{n}\,{\mathcal {T}}V(t_{1})V(t_{2})\cdots V(t_{n})\\&={\mathcal {T}}\exp {-i\hbar ^{-1}\int _{t_{0}}^{t}{d\tau V(\tau )}}\end{aligned}}}
This result is also called Dyson's formula.[ 6] The group laws can be derived from this formula.
Application on state vectors
edit
The state vector at time
t
{\displaystyle t}
can be expressed in terms of the state vector at time
t
0
{\displaystyle t_{0}}
, for
t
>
t
0
,
{\displaystyle t>t_{0},}
as
|
Ψ
(
t
)
⟩
=
∑
n
=
0
∞
(
−
i
ℏ
−
1
)
n
n
!
∫
d
t
1
⋯
d
t
n
⏟
t
f
≥
t
1
≥
⋯
≥
t
n
≥
t
i
T
{
∏
k
=
1
n
e
i
H
0
t
k
/
ℏ
V
(
t
k
)
e
−
i
H
0
t
k
/
ℏ
}
|
Ψ
(
t
0
)
⟩
.
{\displaystyle |\Psi (t)\rangle =\sum _{n=0}^{\infty }{(-i\hbar ^{-1})^{n} \over n!}\underbrace {\int dt_{1}\cdots dt_{n}} _{t_{\rm {f}}\,\geq \,t_{1}\,\geq \,\cdots \,\geq \,t_{n}\,\geq \,t_{\rm {i}}}\,{\mathcal {T}}\left\{\prod _{k=1}^{n}e^{iH_{0}t_{k}/\hbar }V(t_{k})e^{-iH_{0}t_{k}/\hbar }\right\}|\Psi (t_{0})\rangle .}
The inner product of an initial state at
t
i
=
t
0
{\displaystyle t_{i}=t_{0}}
with a final state at
t
f
=
t
{\displaystyle t_{f}=t}
in the Schrödinger picture , for
t
f
>
t
i
{\displaystyle t_{f}>t_{i}}
is:
⟨
Ψ
(
t
i
)
∣
Ψ
(
t
f
)
⟩
=
∑
n
=
0
∞
(
−
i
ℏ
−
1
)
n
n
!
×
∫
d
t
1
⋯
d
t
n
⏟
t
f
≥
t
1
≥
⋯
≥
t
n
≥
t
i
⟨
Ψ
(
t
i
)
∣
e
−
i
H
0
(
t
f
−
t
1
)
/
ℏ
V
S
(
t
1
)
e
−
i
H
0
(
t
1
−
t
2
)
/
ℏ
⋯
V
S
(
t
n
)
e
−
i
H
0
(
t
n
−
t
i
)
/
ℏ
∣
Ψ
(
t
i
)
⟩
{\displaystyle {\begin{aligned}\langle \Psi (t_{\rm {i}})&\mid \Psi (t_{\rm {f}})\rangle =\sum _{n=0}^{\infty }{(-i\hbar ^{-1})^{n} \over n!}\times \\&\underbrace {\int dt_{1}\cdots dt_{n}} _{t_{\rm {f}}\,\geq \,t_{1}\,\geq \,\cdots \,\geq \,t_{n}\,\geq \,t_{\rm {i}}}\,\langle \Psi (t_{i})\mid e^{-iH_{0}(t_{\rm {f}}-t_{1})/\hbar }V_{\rm {S}}(t_{1})e^{-iH_{0}(t_{1}-t_{2})/\hbar }\cdots V_{\rm {S}}(t_{n})e^{-iH_{0}(t_{n}-t_{\rm {i}})/\hbar }\mid \Psi (t_{i})\rangle \end{aligned}}}
The S -matrix may be obtained by writing this in the Heisenberg picture , taking the in and out states to be at infinity:[ 7]
⟨
Ψ
o
u
t
∣
S
∣
Ψ
i
n
⟩
=
⟨
Ψ
o
u
t
∣
∑
n
=
0
∞
(
−
i
ℏ
−
1
)
n
n
!
∫
d
4
x
1
⋯
d
4
x
n
⏟
t
o
u
t
≥
t
n
≥
⋯
≥
t
1
≥
t
i
n
T
{
H
i
n
t
(
x
1
)
H
i
n
t
(
x
2
)
⋯
H
i
n
t
(
x
n
)
}
∣
Ψ
i
n
⟩
.
{\displaystyle \langle \Psi _{\rm {out}}\mid S\mid \Psi _{\rm {in}}\rangle =\langle \Psi _{\rm {out}}\mid \sum _{n=0}^{\infty }{(-i\hbar ^{-1})^{n} \over n!}\underbrace {\int d^{4}x_{1}\cdots d^{4}x_{n}} _{t_{\rm {out}}\,\geq \,t_{n}\,\geq \,\cdots \,\geq \,t_{1}\,\geq \,t_{\rm {in}}}\,{\mathcal {T}}\left\{H_{\rm {int}}(x_{1})H_{\rm {int}}(x_{2})\cdots H_{\rm {int}}(x_{n})\right\}\mid \Psi _{\rm {in}}\rangle .}
Note that the time ordering was reversed in the scalar product.
^ Sakurai, Modern Quantum mechanics, 2.1.10
^ Sakurai, Modern Quantum mechanics, 2.1.12
^ Sakurai, Modern Quantum mechanics, 2.1.11
^ Sakurai, Modern Quantum mechanics, 2.1 pp. 69-71
^ Sakurai, Modern Quantum Mechanics, 2.1.33, pp. 72
^ Tong 3.20, http://www.damtp.cam.ac.uk/user/tong/qft/qft.pdf
^ Dyson (1949), "The S-matrix in quantum electrodynamics" , Physical Review , 75 (11): 1736–1755, Bibcode :1949PhRv...75.1736D , doi :10.1103/PhysRev.75.1736