Quantum Mechanics for Mathematicians

Hamilton’s equations are

d

dt

A(pj,t) ={A(pj,t),h}=i

p^2 j

2 m

A(pj,t)

with solutions

A(pj,t) =ei

p^2 j

2 mt(pj,0)

In the continuum formalism, one can write

h=

∫+∞

−∞

p^2

2 m

A(p)A(p)dp

which should be interpreted as a limit of the finite cutoff version. We will
not try and give a rigorous continuum interpretation of a quadratic product
of distributions such as this one. As discussed at the end of section 36.3, the
quantization ofhcan be given a rigorous interpretation as a bilinear form. We
will however assume theA(pj,t) have as continuum limits distributionsA(p,t)
that satisfy
d
dt

A(p,t) =i

p^2

2 m

A(p,t)

so have time-dependence

A(p,t) =ei

2 pm^2 t

A(p,0)

Note that this time-dependence is opposite to that of the Schr ̈odinger solutions,
since, as a distribution,dtdA(p,t) evaluated on a functionfisA(p,t) evaluated

on−dtdf.

36.6 For further reading

The material of this chapter is just the conventional multi-particle formalism
described or implicit in most quantum field theory textbooks. Many do not
explicitly discuss the non-relativistic case, two that do are [35] and [45]. Three
books aimed at mathematicians that cover this subject much more compre-
hensively than done here are [28] and [17]. For example, section 4.5 of [28]
gives a detailed description of the bosonic and fermionic Fock space construc-
tions in terms of tensor products. For a rigorous version of the construction
of annihilation and creation operators as operator-valued distributions, see for
instance section 5.4 of [17], chapter X.7 of [73], or [64]. The construction of the
Fock space in the infinite dimensional case using Bargmann-Fock methods of
holomorphic functions (rather than tensor products) goes back to Berezin [6],
and Segal (for whom it is the “complex-wave representation”, see [4]) and is
explained in chapter 6 of [61].
Some good sources for learning about quantum field theory from the point
of view of non-relativistic many-body theory are Feynman’s lecture notes on
statistical mechanics [23], as well as [53] and [86].