Expansion of the time evolution operator
In scattering theory, a part of mathematical physics, the Dyson series, formulated by Freeman Dyson, is a perturbative expansion of the time evolution operator in the interaction picture. Each term can be represented by a sum of Feynman diagrams.
This series diverges asymptotically, but in quantum electrodynamics (QED) at the second order the difference from experimental data is in the order of 10−10. This close agreement holds because the coupling constant (also known as the fine-structure constant) of QED is much less than 1.[clarification needed]
In the interaction picture, a Hamiltonian H, can be split into a free part H0 and an interacting part VS(t) as H = H0 + VS(t).
The potential in the interacting picture is
![{\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 },}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4089473fad8fc71473d0cd0e5dc19127dcbb5424)
where
is time-independent and
is the possibly time-dependent interacting part of the Schrödinger picture.
To avoid subscripts,
stands for
in what follows.
In the interaction picture, the evolution operator U is defined by the equation:
![{\displaystyle \Psi (t)=U(t,t_{0})\Psi (t_{0})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ee7bc2ef0be75b7be2503292119c98beda1eb84)
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:
[1]
- Composition:
[2]
- Time Reversal:
[clarification needed]
- Unitarity:
[3]
and from these is possible to derive the time evolution equation of the propagator:[4]
![{\displaystyle i\hbar {\frac {d}{dt}}U(t,t_{0})\Psi (t_{0})=V(t)U(t,t_{0})\Psi (t_{0}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9259fddc122cf322bc42ff0277cf791ffd8abb14)
In the interaction picture, the Hamiltonian is the same as the interaction potential
and thus the equation can also be written in the interaction picture as
![{\displaystyle i\hbar {\frac {d}{dt}}\Psi (t)=H_{\rm {int}}\Psi (t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8470eb1fc1c5f88af6a47a3b055e1740718c04ab)
Caution: this time evolution equation is not to be confused with the Tomonaga–Schwinger equation.
The formal solution is
![{\displaystyle U(t,t_{0})=1-i\hbar ^{-1}\int _{t_{0}}^{t}{dt_{1}\ V(t_{1})U(t_{1},t_{0})},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d83a01b909c9e8cccc7c333b12a1ce742a585f37)
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:
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b8cb04facd5b85cf4e63bd74133f42d0b6c0608)
Here,
, and so the fields are time-ordered. It is useful to introduce an operator
, called the time-ordering operator, and to define
![{\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}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c8d0182f2cb83691088649555e6167fe4374c96)
The limits of the integration can be simplified. In general, given some symmetric function
one may define the integrals
![{\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}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a69dd2d1c8df59b4c94e8866807e97250f8dfcac)
and
![{\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}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2cd58abe62833a9d9051f495b4e8c0d26802fc0)
The region of integration of the second integral can be broken in
sub-regions, defined by
. Due to the symmetry of
, the integral in each of these sub-regions is the same and equal to
by definition. It follows that
![{\displaystyle S_{n}={\frac {1}{n!}}I_{n}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1fc09376c5eed153912d5cd9c21f04e638594e4c)
Applied to the previous identity, this gives
![{\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}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/425c87bf877c6c4590fbd53375ad28f84db06cfe)
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]
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/26fd2c72e06cdcba755fcdedcf2a1918a2e3dee0)
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
can be expressed in terms of the state vector at time
, for
as
![{\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 .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0652699f34981d610a23a5b25d6ade8c389bebf2)
The inner product of an initial state at
with a final state at
in the Schrödinger picture, for
is:
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7436fb3ae78938d1a53e64fe8de0c4ef1bd3006)
The S-matrix may be obtained by writing this in the Heisenberg picture, taking the in and out states to be at infinity:[7]
![{\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 .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc029b8c676c67bd4331538750148e39eaa42ed8)
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