# Physical Hilbert space of dimension $N$ factorial?

In many-body physics, Hilbert spaces are usually equipped with a tensor structure (ie: $$\mathcal{H}=\mathcal{V}^{\otimes N}$$).

If the dimension of local degrees of freedom is set to be $$dim(\mathcal{V})=G$$, it is obvious that the dimension of total space $$\mathcal{H}$$ exhibit an exponential dependence of system size: $$dim(\mathcal{H})=G^N$$.

But could there be a Hilbert space whose dimension is order $$O(N!)$$ ? Surely it can never be found in a 1-D many-body system, but it seems proper to be a subspace of a 2-D system. Since $$G^{N^2}>N!>G^N$$, it looks like something lies between 1-D and 2-D, and unlike those systems live on fractals.

And if there is such a Hilbert space, what are the base vectors? Shall they be some combinatorial objects like elements of $$S_N$$ or chord diagrams?

Any example would be welcomed, and please feel free to edit my question if it is needed.

• One simple example would be a chain of eg spins where the first site has only one state, the second one two etc. The many-body eigenstates then follow from the combinatorics of combining the single-site spectra. But of course, in this system the sites are not equivalent – Wouter Sep 9 at 15:05
• @Wouter I think the problem is: interactions in your system are non-local; At the thermodynamic limit, you have to combine infinite many spins together, which I dont think is reasonable. Also, to make total dimension $N!$, you have assumed system size to be $1+2+......N=\frac{N(N+1)}{2}$, which is not typically a 1-D chain, whose size is linear in N. Thanks anyway :-) – SSSSiwei Sep 9 at 15:31

The way to construct such a system would be to take a classical system that can be in $$N!$$ states, and then consider the corresponding quantum state space.
In this case, a very natural example would be that of $$N$$ copies of a system with $$G = N$$, i.e. one that can be in $$N$$ different classical states, such that the different subsystems cannot be in the same state. This is a subspace of the full tensor product space.
It is e.g. the state space of $$N$$ distinguishable systems that can each be in $$N$$ different states, no two in the same state but otherwise non-interacting, like "quantum disks" stacked on top of each other.
• Sounds like something in 3-D. Since the original space is larger ($G^{N^3}$), it might be easier to find such a subspace. But I think there`s no reason that you cannot find such a thing in 2-D......Maybe the trick is hidden in the 'stack operation' you described. – SSSSiwei Sep 9 at 15:42