of paths. For higher order terms inV(q), one can get a series expan-
sion by expanding out the exponential, giving terms that are moments of
Gaussians so can be evaluated exactly.- After analytical continuation, path integrals can be rigorously defined us-
ing Wiener measure techniques, and often evaluated numerically even in
cases where no exact solution is known.
On the other hand, there are disadvantages: - Some path integrals such as phase space path integrals do not at all have
the properties one might expect for an integral, so great care is required
in any use of them. - How to get unitary results can be quite unclear. The analytic continua-
tion necessary to make path integrals well-defined can make their physical
interpretation obscure. - Symmetries with their origin in symmetries of phase space that aren’t
just symmetries of configuration space are difficult to see using the con-
figuration space path integral, with the harmonic oscillator providing a
good example. Such symmetries can be seen using the phase space path
integral, but this is not reliable.
Path integrals for anticommuting variables can also be defined by analogy
with the bosonic case, using the notion of fermionic integration discussed ear-
lier. Such fermionic path integrals will usually be analogs of the phase space
path integral, but in the fermionic case there are no points and no problem of
continuity of paths. In this case the “integral” is not really an integral, but
rather an algebraic operation with some of the same properties, and it will not
obviously suffer from the same problems as the phase space path integral.
35.5 For further reading
For much more about Lagrangian mechanics and its relation to the Hamiltonian
formalism, see [2]. More details along the lines of the discussion here can be
found in most quantum mechanics and quantum field theory textbooks. An
extensive discussion at an introductory level of the Lagrangian formalism and
the use of Noether’s theorem to find conserved quantities when it is invariant
under a group action can be found in [63]. For the formalism of constrained
Hamiltonian dynamics, see [89], and for a review article about the covariant
phase space and the Peierls bracket, see [50].
For the path integral, Feynman’s original paper [21] or his book [24] are quite
readable. A typical textbook discussion is the one in chapter 8 of Shankar [81].
The book by Schulman [77] has quite a bit more detail, both about applications
and about the problems of phase space path integrals. Yet another fairly com-
prehensive treatment, including the fermionic case, is the book by Zinn-Justin
[109].