------------------------------------
S5a. QM pictures and representations
------------------------------------
QM exists in different pictures, of which the Schroedinger picture,
the Heisenberg picture, the interaction picture, and Feynman's
path integral representation are frequently invoked. There is also
the algebraic approach using unitary representations of canonical
commutation rules (CCR).
The Schroedinger picture, the Heisenberg picture, and the interaction
pictures are equivalent because there are unitary transformations
between them. They all provide different representations of the
same canonical commutation rules
i[p_j,q_k]= hbar delta_jk
between components p_j of momentum p and q_k of position q.
The Stone-von Neumann theorem guarantees that the canonical
commutation relations (or their unitary version, the Weyl relations)
have a unique unitary representation apart from unitary
transformations, and hence suffice to specify the QM of finitely many
degrees of freedom uniquely, no matter which picture is used.
The Stone-von Neumann theorem fails for systems of infinitely many
degrees of freedom (see the FAQ entry on 'Inequivalent
representations of CCR/CAR'), which in a sense 'causes' the
difficulties in quantum field theory.
Nevertheless, QFT still has a Schroedinger picture
and a Heisenberg picture, and these are still equivalent:
The Heisenberg picture can be immediately constructed from the Wightman
fields. Then the canonical procedure - fixing the Heisenberg operators
at time t=0 and instead defining dynamical states
psi(t) := exp(-itH)psi
- produces the Schroedinger picture from it.
The Feynman path integral is related to the other pictures via the
Feynman-Kac formula, which makes the often only formally stated
equivalence precise, after analytically continuing the time to purely
imaginary times. The Osterwalder-Schrader theory
[see, e.g., math-ph/0001010 or the book by Glimm and Jaffe]
shows how to go back in case of relativistic quantum field theory.
The Feynman path integral only gives time-ordered expectation values;
this suffices to compute S-matrix elements, but is inadequate for
dynamical investigations needed for nonequilibrium quantum mechanics.
The latter can be treated with the so-called closed time path (CPT)
integral within the Schwinger-Keldysh formalism.