Dyson formula



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In quantum mechanics

In quantum mechanics what is called the Dyson formula is what in mathematics is called the iterated integral-expression for parallel transport: It an expression for the solution to a differential equation of the form

ddtψ(t)=iH(t)ψ(t) \frac{d}{d t} \psi(t) \;=\; i H(t) \psi(t)

where tψ(t)t \mapsto \psi(t) \in \mathcal{H} is a one-parameter family of elements of some Hilbert space and tH(t)t \mapsto H(t) is a one-parameter family of linear operators on this Hilbert space.

This appears prominently in the expression of the Schrödinger equation in the interaction picture, in which case H(t)H(t) is the interaction Hamiltonian in the Heisenberg picture of the free theory. In this case the Dyson series gives the S-matrix. This is the context in which the term “Dyson formula” orginates. But of course also the plain Schrödinger equation (“in the Schrödinger picture”) is already of this form if the Hamiltonian is time-dependent.

The idea is to think of the solution as given by the limit

ψ(t)=limN(id+iNH(t))(id+iNH(t(N1)/N))(id+iNH(t/N))Nfactorsψ(0) \psi(t) \;=\; \underset{N \to \infty}{\lim} \underset{N \, \text{factors}}{ \underbrace{ \left( id + \tfrac{i}{N}H(t) \right) \left( id + \tfrac{i}{N}H(t(N-1)/N) \right) \cdots \left( id + \tfrac{i}{N}H(t/N) \right) } } \psi(0)

and to think of this as the result of forming the exponential expression exp( [0,t]H(t)dt)\exp(\int_{[0,t] } H(t)\,d t) and then re-ordering in the resulting sum of products all the factors of H(t)H(t) such that they are time ordered with larger values of tt ordered to the left of smaller values.

Accordingly, in the physics literature solutions to this equation are written

ψ(t)=T(exp( [0,t]iH(t)dt))ψ(0), \psi(t) \;=\; T\left( \exp\left( \int_{[0,t]} i H(t) \,d t \right) \right) \psi(0) \,,

with T()T(-) the time ordering operator producing time-ordered products. The corresponding series of iterated integrals

ψ(t)=(n=0(i) n0t 1t ntH(t n)H(t n1)H(t 1))ψ(0) \psi(t) \;=\; \left( \underoverset{n = 0}{\infty}{\sum} (i)^n \underset{0 \leq t_1 \leq \cdots \leq t_n \leq t}{\int} H(t_n) H(t_{n-1})\cdots H(t_1) \right)\psi(0)

is called the Dyson series.

In quantum field theory

The idea generalizes to quantum field theory in Lorentzian spacetime if due care is exercised (including adiabatic switching and point extension of operator-valued distributions). Here the “time ordering” is generalized to a causal ordering, this is the content of the construction of the S-matrix and the interacting field algebra in causal perturbation theory.

See at S-matrix for details.

product in perturbative QFT\,\, induces
normal-ordered productWick algebra (free field quantum observables)
time-ordered productS-matrix (scattering amplitudes)
retarded productinteracting quantum observables

Revised on September 19, 2017 07:42:47 by Urs Schreiber (