The Dyson operator
edit
Suppose that we have a Hamiltonian H , which we split into a free part H 0 and an interacting part VS (t) , i.e. H = H 0 + VS (t) .
We will work in the interaction picture here, that is,
V
I
(
t
)
=
e
i
H
0
⋅
(
t
−
t
0
)
V
S
(
t
)
e
−
i
H
0
⋅
(
t
−
t
0
)
,
{\displaystyle V_{I}(t)=\mathrm {e} ^{\mathrm {i} H_{0}\cdot (t-t_{0})}V_{S}(t)\mathrm {e} ^{-\mathrm {i} H_{0}\cdot (t-t_{0})},}
where
H
0
{\displaystyle H_{0}}
is time-independent and
V
S
(
t
)
{\displaystyle V_{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_{\text{I}}(t)}
in what follows.
We choose units such that the reduced Planck constant ħ is 1.
In the interaction picture, the evolution operator U defined by the equation:
Ψ
(
t
)
=
U
(
t
,
t
0
)
Ψ
(
t
0
)
{\displaystyle \Psi (t)=U(t,t_{0})\Psi (t_{0})}
is called the Dyson operator .
We have a few properties:
Identity and normalization:
U
(
t
,
t
)
=
I
,
{\displaystyle U(t,t)=I,}
[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
d
d
t
U
(
t
,
t
0
)
Ψ
(
t
0
)
=
V
(
t
)
U
(
t
,
t
0
)
Ψ
(
t
0
)
.
{\displaystyle i{\frac {d}{dt}}U(t,t_{0})\Psi (t_{0})=V(t)U(t,t_{0})\Psi (t_{0}).}
We notice again that in the interaction picture the Hamiltonian is the same as the interaction potential
V
(
t
)
{\displaystyle V(t)}
.
This equation is not to be confused with the Tomonaga–Schwinger equation
Consequently:
U
(
t
,
t
0
)
=
1
−
i
∫
t
0
t
d
t
1
V
(
t
1
)
U
(
t
1
,
t
0
)
,
{\displaystyle U(t,t_{0})=1-i\int _{t_{0}}^{t}{dt_{1}\ V(t_{1})U(t_{1},t_{0})},}
which is ultimately a type of Volterra equation .
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
∫
t
0
t
d
t
1
V
(
t
1
)
+
(
−
i
)
2
∫
t
0
t
d
t
1
∫
t
0
t
1
d
t
2
V
(
t
1
)
V
(
t
2
)
+
⋯
+
(
−
i
)
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\int _{t_{0}}^{t}dt_{1}V(t_{1})+(-i)^{2}\int _{t_{0}}^{t}dt_{1}\int _{t_{0}}^{t_{1}}\,dt_{2}V(t_{1})V(t_{2})+\cdots \\&{}+(-i)^{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 we have
t
1
>
t
2
>
⋯
>
t
n
{\displaystyle t_{1}>t_{2}>\cdots >t_{n}}
, so we can say that the fields are time-ordered , and it is useful to introduce an operator
T
{\displaystyle {\mathcal {T}}}
called time-ordering operator , defining
U
n
(
t
,
t
0
)
=
(
−
i
)
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)^{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}).}
We can now try to make this integration simpler. In fact, by the following example:
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}).}
Assume that K is symmetric in its arguments and define (look at integration limits):
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 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}}
,
t
>
t
1
>
⋯
>
t
n
{\displaystyle t>t_{1}>\cdots >t_{n}}
, etc. Due to the symmetry of K , the integral in each of these sub-regions is the same and equal to
S
n
{\displaystyle S_{n}}
by definition. So it is true that
S
n
=
1
n
!
I
n
.
{\displaystyle S_{n}={\frac {1}{n!}}I_{n}.}
Returning to our previous integral, the following identity holds
U
n
=
(
−
i
)
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)^{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, we obtain the Dyson series which is a simplified version of the Neumann series above and which includes the time ordered products:[5]
U
(
t
,
t
0
)
=
∑
n
=
0
∞
U
n
(
t
,
t
0
)
=
T
e
−
i
∫
t
0
t
d
τ
V
(
τ
)
.
{\displaystyle U(t,t_{0})=\sum _{n=0}^{\infty }U_{n}(t,t_{0})={\mathcal {T}}e^{-i\int _{t_{0}}^{t}{d\tau V(\tau )}}.}
This result is also called Dyson's formula.[6]
Application on state vectors
edit
One can then express the state vector at time t in terms of the state vector at time t 0 , for t > t 0 ,
|
Ψ
(
t
)
⟩
=
∑
n
=
0
∞
(
−
i
)
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
V
(
t
k
)
e
−
i
H
0
t
k
}
|
Ψ
(
t
0
)
⟩
.
{\displaystyle |\Psi (t)\rangle =\sum _{n=0}^{\infty }{(-i)^{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}}V(t_{k})e^{-iH_{0}t_{k}}\right\}|\Psi (t_{0})\rangle .}
Then, the inner product of an initial state (t i = t 0 ) with a final state (t f = t ) in the Schrödinger picture , for t f > t i , is as follows:
⟨
Ψ
(
t
i
)
∣
Ψ
(
t
f
)
⟩
=
∑
n
=
0
∞
(
−
i
)
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
)
V
S
(
t
1
)
e
−
i
H
0
(
t
1
−
t
2
)
⋯
V
S
(
t
n
)
e
−
i
H
0
(
t
n
−
t
i
)
∣
Ψ
(
t
i
)
⟩
.
{\displaystyle \langle \Psi (t_{i})\mid \Psi (t_{f})\rangle =\sum _{n=0}^{\infty }{(-i)^{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}}}\,\langle \Psi (t_{i})\mid e^{-iH_{0}(t_{\rm {f}}-t_{1})}V_{S}(t_{1})e^{-iH_{0}(t_{1}-t_{2})}\cdots V_{S}(t_{n})e^{-iH_{0}(t_{n}-t_{\rm {i}})}\mid \Psi (t_{i})\rangle .}
See also
edit
References
edit
^ 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