Hi everyone. Many texts when describing QFT start immediately discussing about free field theories, Fock spaces etc.. I want to understand general properties of the Hilbert space, and how to find a basis of it, and how to find a particle interpretation. I know there are very mathematical formulations with test functions etc.. but i'm not interested in that.(adsbygoogle = window.adsbygoogle || []).push({});

I'll tell you my guess. In a QFT we must have Poincare' invariance. So, from it we can have its noether charges, and use them to label the Hilbert space with their eigensates. How many commuting objects can we create? [itex]P_{\mu}[/itex], [itex]P^2[/itex],[itex]W^2[/itex], [itex]W_3[/itex], and plus some commuting [itex]Q_a[/itex] of an internal symmetry. [itex]W_{\mu}[/itex] is [itex]W_{\mu}=\frac{1}{2}\epsilon_{\mu\nu\rho\sigma}M^{\nu\rho}P^{\sigma}[/itex] So, I could create

[itex]P^{2}|m^{2},p_{\mu},q_{a},s,\xi_{s}>=m^{2}|m^{2},p_{\mu},q_{a},s,\xi_{s}>[/itex]

[itex]P_{\mu}|m^{2},p_{\mu},q_{a},s,\xi_{s}>=p_{\mu}|m^{2},p_{\mu},q_{a},s,\xi_{s}>[/itex]

[itex]W^{2}|m^{2},p_{\mu},q_{a},s,\xi_{s}>=-m^{2}s\left(s+1\right)|m^{2},p_{\mu},q_{a},s,\xi_{s}>[/itex]

[itex]W_{3}|m^{2},p_{\mu},q_{a},s,\xi_{s}>=\xi_{s}|m^{2},p_{\mu},q_{a},s,\xi_{s}>[/itex]

[itex]Q_{a}|m^{2},p_{\mu},q_{a},s,\xi_{s}>=q_{a}|m^{2},p_{\mu},q_{a},s,\xi_{s}>[/itex]

where [itex]m^2[/itex] is real non negative (if it is null i should consider helicity), [itex]p_{\mu}[/itex] is any, [itex]s=0,1,2..[/itex] and [itex]\xi_{s}=-s,..s[/itex].

So I can express a generic element of the Hilbert space as linear combination of [itex]|m^{2},p_{\mu},q_{a},s,\xi_{s}>[/itex] right?

[itex]|m^{2},p_{\mu},q_{a},s,\xi_{s}>[/itex] could be interpreted as a particle state if it is onshell.

Is there a way to build a commuting number operator whose eigenvalues identify which particle we are considering? The vacuum would be when the number operator is zero.

Then, when we impose a certain Lagrangian, its equation of motion will give some contraints on the states, right?

Any suggestions or text references explicitly dealing with what said are welcome.

Thank you very much.

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# Hilbert space of QFT

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