Particles and Fields – a neverending story

[meopemuk]

Do you also believe that there is something wrong in defining momentum eigenstates |p\rangle, which can be acted on by Lorentz transformation operators? I believe I just showed how this approach can be made equivalent with the approach where the Lorentz transformation is postulated with \psi'(t',x')=\psi(t,x). Logically, you should either believe that the (t,x) transforming approach is right, or that the |p> transforming approach is wrong, or then that there is something wrong with my calculation.

Hi jostpuur,

I don't see anything wrong in your calculation. However, I would be more convinced if you performed some consistency checks. For example, your transformation (1) must conserve the norm of the wavefunction. In other words, boost transformation must be unitary. Can you prove that?

Eugene.

 

[strangerep]

[...]
In any particular frame, with initial configuration \psi(0,\boldsymbol{x}), \hat{\psi}(0,\boldsymbol{p}), the time evolution is fixed with (by definition
equivalent) equations

<br /> \psi(t,\boldsymbol{x}) = e^{-it\sqrt{-\nabla^2 + m^2}}\psi(0,\boldsymbol{x}),\quad\quad \hat{\psi}(t,\boldsymbol{p})=e^{-it\sqrt{|\boldsymbol{p}|^2 + m^2}}\hat{\psi}(0,\boldsymbol{p}).<br />

These are my assumptions. [...]

Those equations above, and your later one:

i\partial_t\psi = \sqrt{-\nabla^2 + m^2}\psi

are just a representation of the Poincare Casimir operator
M^2 := H^2 - P^2.

By assuming m^2 to be a scalar constant in this way,
one is already assuming special relativity.

I guess it's a matter of personal taste which formalism
one follows -- provided one ends up at the correct destination.
(I've found that my own tastes tend to change over time.)

BTW, Eugene's suggested exercise of showing that the boosts
preserve the norm is a good one (and an important one).

 

[Hans de Vries]

This debate keeps popping up on these boards, and I'll side with Strangereps point of view on the subject.

I thought i'd add a classic thought experiment about localization often taught in early QFT courses (eg Coleman's lectures). This very much relates to the nonvanishing of the propagator outside lightcone and things like that, as well as the early tensions between field theory and relativistic QM.

If you take a one particle state in a box (defined at the moment, purely in terms of relativistic qm, not field theory), and you squeeze it past the compton wavelength of the particle. That is exactly the regime where pair creation processes start becoming kinematically important on purely dimensional analysis grounds. So the more you squeeze, the more the occupancy number becomes a bad 'state', and you are forced to no longer deal with 1 particle, but rather many. You've essentially lost localization and your would be position operator becomes mathematically illdefined.

A closely related version of this is the vacuum polarization which occurs below
the Compton radius. The divergence of the electric field is not zero anymore so
there remains an extended charge density (from virtual pairs) with a constant size
even if the particle itself becomes confined to a smaller and smaller volume.


Regards, Hans