Algebraic form of Klein Gordon ##\phi^4## vacuum and ladder operators

[QFT1995]

In theory, does an algebraic expression exist for the ground state of the Klein Gordon equation with $\phi^4$ interactions in the same way an algebraic expression exists for the simple harmonic oscillator ground state wavefunction in Q.M.? Is it just that it hasn't been found yet or is it impossible to construct? Also, will the creation and annihilation operators have an explicit differential representation that you can explicitly construct (like for that of the simple harmonic oscillator) or is it not possible?

 

[fresh_42]

I'm no physicist so forgive me if it doesn't answer your question, but what is wrong about the formulas on Wikipedia:
https://en.wikipedia.org/wiki/Klein–Gordon_equation
https://de.wikipedia.org/wiki/Klein-Gordon-Gleichung#Lösung (can be translated in Chrome)

 

[QFT1995]

There's no explicit form for the full vacuum $|{\Omega}\rangle$ on there. I was wondering if in principle it can exist as an algebraic expression. Also the creation and annihilation operators are just defined by how they act on the free vacuum $|{0}\rangle$ as an abstract definition. I'm not sure if we can write them down because in QFT, particle number is not conserved.

 

[Keith_McClary]

ground state of the Klein Gordon equation with ϕ4ϕ4\phi^4 interactions in the same way an algebraic expression exists for the simple harmonic oscillator

The zero dimensional $\phi^4$ interaction is the simple harmonic oscillator with an $x^4$ perturbation. This has been studied. I'm pretty sure there is no explicit solution in any sense.

 

[QFT1995]

Okay however I was asking if in principle it exists and that we just haven't found it or is it impossible to construct?

 

[Keith_McClary]

It does not exist (Summers, p.5) but in 1+1 dimensional space time a new Hilbert space with a vacuum state can be defined by the GNS construction (p.7).