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
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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:
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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]
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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
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Caution: this time evolution equation is not to be confused with the Tomonaga–Schwinger equation.
The formal solution is
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which is ultimately a type of Volterra integral.
Derivation of the Dyson series
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An iterative solution of the Volterra equation above leads to the following Neumann series:
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Here, , and so the fields are time-ordered. It is useful to introduce an operator , called the time-ordering operator, and to define
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The limits of the integration can be simplified. In general, given some symmetric function one may define the integrals
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and
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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
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Applied to the previous identity, this gives
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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]
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This result is also called Dyson's formula.[6] The group laws can be derived from this formula.
Application on state vectors
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The state vector at time can be expressed in terms of the state vector at time , for as
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The inner product of an initial state at with a final state at in the Schrödinger picture, for is:
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The S-matrix may be obtained by writing this in the Heisenberg picture, taking the in and out states to be at infinity:[7]
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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