# Re: Constructive Quantum Mechanics (Relativistic)

*From*: Arnold Neumaier <Arnold.Neumaier@xxxxxxxxxxxx>*Date*: Fri, 13 Jun 2008 14:28:32 +0000 (UTC)

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

relations

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

etc?

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

theory

S9c. What about relativistic QFT at finite times?

in my theoretical physics FAQ at

http://www.mat.univie.ac.at/~neum/physics-faq.txt

Arnold Neumaier

.

**References**:**Constructive Quantum Mechanics (Relativistic)***From:*pirillo

**Re: Constructive Quantum Mechanics (Relativistic)***From:*pirillo

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