Dyson series

The Dyson series expands the interaction-picture time-evolution operator as

S = T exp (- i \int d^{4} x H_{int} (x)) = \sum_{n \geq 0} \frac{( - i ) ^{n}}{n !} \int d^{4} x_{1} \dots d^{4} x_{n} T {\prod_{j = 1}^{n} H_{int} (x_{j})} .

Table of contents
- Differentiation of the Dyson time-ordered exponential Dyson series

Differentiation of the Dyson time-ordered exponential ( $i \partial_{t} U = H_{I} (t) U$ )

For bounded norm-continuous Hamiltonian operators on

[t_{0}, t]

, the Dyson series has an

n

-th term bounded by

M^{n} ∣ t - t_{0} ∣^{n} / n!

, so it converges in operator norm. In its nested-integral form, differentiating the outer endpoint puts

H_{I} (t)

on the left of the

(n - 1)

-fold term. Summing yields

\partial_{t} U = - i H_{I} (t) U

and

U (t_{0}, t_{0}) = I

. If

H_{I}

is Hermitian, differentiating

U^{†} U

proves unitarity. For unbounded field operators the same algebra is a formal perturbative identity unless domain and convergence assumptions or a regulator justify the steps. Evolution backward uses inverse, anti-time-ordered evolution.

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The Dyson series is a mathematical tool used in quantum mechanics to describe the time evolution of quantum states, particularly in the context of time-dependent Hamiltonians. It provides a way to express the evolution operator (or propagator) as a power series in terms of the interaction Hamiltonian.

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