Re: Constructive Quantum Mechanics (Relativistic)

pirillo wrote:

I've seen in many books (especially on strings) when they describe
the relativstic particle
as a warmup, that they have x_0, to x_3 say, i.e., t,x,y,z operators,
and the states are wavefunctions
of txyz (the eigenvalues of the aforemantioned operators ) (in
covariant quantization--- this is after they've
eliminated tau by the constraint) and they claim that the commutation
are [x_m, p_n] = i eta_{mn} (for n, m=0123 eta00 being -1.)

Assuming these conditions implies that p_0 is an unbounded operator,
so that it cannot describe the causal time evolution. Thus while p_0
is an operator in these spaces, it is not the Hamiltonian operator
responsible for the physics.

So I was wondering, how can we
make an explicit construction of these operators, the Hilbert space

Most of current quantum field theory (i.e., everything with exception
of 2D and 3D constructive field theory - which doesn't even cover QED)
does not have a well-defined Hilbert space at all, in which a
time operator would be defined.

Well-defined are only the asymptotic Hilbert spaces of in and out
states for scattering experiments. These are Fock spaces of
free particles, and hence defined on a mass shell.
There is a basic result called Haag's theorem which states that
these asymptotic Fock spaces cannot carry a nontrivial local dynamics,
as would be required for a field theory.

The full dynamics can be defined only indirectly, via CTP (closed
time path) integration, and subject to all interpretation problems
of the renormalization procedures.

Constructing for a relativistic field theory a physical
Hamiltonian which is bounded below is really difficult, and has
been achieved only in less than 4D theories.

The construction is usually based on a preferred time coordinate
which is needed in all cases I am familiar with;
- in the Foldy-Wouthuysen transformation (for the Dirac equation,
where p_0 also fails to have the right properties),
- in the Newton-Wigner construction (for single particles in
an arbitrary massive irreducible representation of the
Poincare group) and
- in the Osterwalder-Schrader reconstruction theorem (for
Lorentz-invariant field theories from Euclidean field theories).

While the Hilbert space and the Hamiltonian depend on the choice of
the time coordinate, the physics is independent of it since all these
Hilbert spaces are isomorphic via isomorphisms that maps the
Hamiltonians into each other.

See also the entries
S2g. Particle positions and the position operator
S2h. Localization and position operators
S6c. Functional integrals, Wightman functions, and rigorous QFT
S6d. Is there a rigorous interacting QFT in 4 dimensions?
S6e. Constructive field theory
S6h. Hilbert space and Hamiltonian in relativistic quantum field
S9c. What about relativistic QFT at finite times?
in my theoretical physics FAQ at

Arnold Neumaier