Question

0

As we know, we can formulate the phase space theory of relativistic free particles using the Hamiltonian H=p²+m²−−−−−−−√
and the Poisson bracket [x,p]=1
. So there are no problems with Lorentz symmetry here.

There is a correspondence between these phase space theories and quantum theories. However, the corresponding quantum theory does not exist even on the level of pure mathematics because the position basis does not exist in the quantum theory.

So where does this correspondece break? And where does the position basis problem in the quantum theory manifest itself in this phase space theory?

Answer 1

The correspondence between classical and quantum mechanics breaks down in the position basis of the phase space theory. In quantum mechanics, the position basis does not exist due to the uncertainty principle, which limits the precision of simultaneously measuring position and momentum.

Step-by-step explanation:

In the phase space theory of relativistic free particles, the correspondence between classical mechanics and quantum mechanics breaks down when it comes to the position basis. In classical mechanics, we can precisely measure the position of a particle, but in quantum mechanics, the position basis does not exist.

This problem manifests itself in the phase space theory when we try to represent the position operator in terms of the momentum and position variables. In classical mechanics, the position operator is a well-defined function of the position variable, but quantum mechanics doesn't have an equivalent operator. This is because the position basis is not compatible with the uncertainty principle, which states that there is a fundamental limit to the precision with which certain pairs of physical properties, such as position and momentum, can be known simultaneously.

To summarize, the correspondence between classical and quantum mechanics breaks down when it comes to the position basis. In the phase space theory, the position basis problem manifests itself as the inability to represent the position operator in terms of the momentum and position variables.