Quantum Mechanics for Mathematicians

36.5 Dynamics

To describe the time evolution of a quantum field theory system, it is generally
easier to work with the Heisenberg picture (in which the time dependence is in
the operators) than the Schr ̈odinger picture (in which the time dependence is in
the states). This is especially true in relativistic systems where one wants to as
much as possible treat space and time on the same footing. It is however also
true in the case of non-relativistic multi-particle systems due to the complexity
of the description of the states (inherent since one is trying to describe arbitrary
numbers of particles) versus the description of the operators, which are built
simply out of the annihilation and creation operators.
In the Heisenberg picture the time evolution of an operatorf̂is given by

f̂(t) =eiHtf̂(0)e−iHt

and such operators satisfy the differential equation

d

dt

f̂= [f,̂−iH]

For the operators that create and annihilate states with momentumpjin the

finite cutoff formalism of section 36.2,H=Ĥ(given by equation 36.10) and we
have
d
dt

a†(pj,t) = [a†(pj,t),−iĤ] =i

p^2 j

2 m

a†(pj,t)

with solutions

a†(pj,t) =ei

p^2 j

2 mta†(pj,0) (36.18)

Recall that in classical Hamiltonian mechanics, the Hamiltonian functionh
determines how an observablefevolves in time by the differential equation

d

dt

f={f,h}

Quantization takesfto an operatorf̂, andhto a self-adjoint operatorH.
In our case the “classical” dynamical equation is meant to be the Schr ̈odinger
equation. In the finite cutoff formalism, one can take as Hamiltonian

h=

∑

j

p^2 j

2 m

A(pj)A(pj)

Here theA(pj) should be interpreted as linear functions that on a solution given
byαtake the valueαj, andha quadratic function on solutions that takes the
value

h(α) =

∑

j

p^2 j

2 m

α(pj)α(pj)