Quantum vaccum one dimensional?

[physicsx0rz]

Hello.
Is a quantum vacuum 1 dimensional?
Thanks.

[smallphi]

The quantum vacuum is a state in an infinite dimensional Hilbert space. To be more specific, the vacuum is the eigenstate of the Hamiltonian operator with the lowest energy eigenvalue. A more appropriate name for 'vacuum', often used in nonrelativistic QM is ground state.

If the Hilbert space of the system is spanned by the eigenstates of a single position operator X we say the system under consideration has one spatial degree of freedom. For such a system, the wavefunction corresponding to the vaccuum/ground state |O> is a function of one spatial variable x: Psi_0 (x) = <x|O> . Here |x> denotes eigenstate of the operator X with eigenvalue x.

If the system has 3 spatial degrees of freedom, its Hilbert space is spanned by eigenstates of 3 position operators X, Y, Z that mutually commute. Correspondingly the wavefunction of the ground state, Psi_0(x,y,z) = <x,y,z|O> is a function of 3 spatial variables x, y, z. Here |x,y,z> is the commont eigenstate of operators X, Y, Z of eigenvalues x, y, z.

[physicsx0rz]

I'm lost, but I expected to be. I'm very physics illiterate and just starting to learn.
Thanks for the reply.