Many-Body Hilbert Spaces

A many-body model must specify what its factors represent before it specifies a Hamiltonian. Tensor factors can label distinguishable subsystems, while symmetric or exterior powers describe identical particles. These choices lead to different local observables and different infinite-system limits. This page explains the operator consequences of those choices and gives a domain-correct construction of independent-particle energy.

Helpful background. Hilbert Space Tensor Products supplies the completed tensor product. Symmetric and Antisymmetric Tensor Products proves the permutation-projection construction.

Particle slots, sites, and modes are different structures

For $N$ distinguishable subsystems, the Hilbert space is $\bigotimes_{j=1}^N\mathcal h_j$. For identical particles with one-particle space $\mathcal h$, the physical fixed-number space is instead

$$\mathcal H_N^+=\operatorname{Sym}^N\mathcal h, \qquad \mathcal H_N^-=\bigwedge\nolimits^N\mathcal h.$$

A continuum spinful particle can have $\mathcal h=L^2(\mathbb R^d)\otimes\mathbb C^q$. The spin factor belongs to each one-particle state; exchange symmetry acts on the full spatial-and-spin slot. Antisymmetrizing only spatial coordinates would impose a different restriction unless a spin sector had already been fixed.

For a spin lattice, by contrast, a site is a physically specified subsystem with a local space $\mathcal h_x$. Its finite-volume Hilbert space is $\bigotimes_{x\in\Lambda}\mathcal h_x$. This tensor product does not impose exchange symmetry between lattice sites.

A mode decomposition of a particle space is a further construction. Its Fock factorization does not identify mode labels with particle slots. Fock Space Construction makes that distinction precise.

Local operators must respect the chosen space

On a distinguishable tensor product, $A_j$ acting in one factor is a well-defined local operator. Operators on disjoint factors commute.

For identical particles, an individual formal-slot operator generally does not preserve the symmetric or antisymmetric subspace. A bounded one-particle observable $A$ instead lifts to the permutation-invariant sum

$$A^{(N)}=\sum_{j=1}^N A_j, \qquad \|A^{(N)}\|\leq N\|A\|.$$

The sum commutes with every permutation, so it restricts to either statistics sector. The physical occupation-number expression and reduced-density expectations belong to One-Body Operators.

This restriction is not a ban on spatially local measurements of identical particles. A one-particle spatial projector can be summed to count particles in a region. What fails is the interpretation of an arbitrary labeled slot as a separately addressable identical particle.

Independent-particle energy uses joint spectral calculus

Let $h$ be self-adjoint on $\mathcal h$. The lifted copies $h_j$ strongly commute because their spectral projections act in different tensor factors. Their joint projection measure defines the self-adjoint operator

$$H_N^{(0)}=\int_{\mathbb R^N} (\lambda_1+\cdots+\lambda_N)\, d(E_h^{\otimes N})(\lambda).$$

Its domain consists of vectors with

$$\int_{\mathbb R^N} |\lambda_1+\cdots+\lambda_N|^2\,d\mu_\Psi(\lambda)<\infty.$$

On algebraic tensors of vectors in $D(h)$, this agrees with the familiar sum $\sum_jh_j$. The spectral construction specifies its closure and maximal domain. If $h$ is not semibounded, cancellation can make this domain larger than the intersection of the separate $D(h_j)$.

The joint multiplier is invariant under permutations. Therefore the bosonic and fermionic projections commute with every spectral projection of $H_N^{(0)}$, and its restrictions to $\mathcal H_N^\pm$ are self-adjoint. A merely invariant algebraic test set would not, by itself, establish this reducing-subspace statement.

Interactions require additional control. A symmetric potential may respect permutations while still needing operator or form bounds for self-adjointness; see Domains of Hamiltonians and Quadratic Form Methods.

Cancellation changes an operator domain

Take $\mathcal h=\ell^2(\mathbb Z)$ and $he_n=ne_n$. For two distinguishable slots, choose

$$\Psi=C\sum_{n=1}^\infty\frac1n\,e_n\otimes e_{-n}.$$

It is normalizable. Neither $h_1\Psi$ nor $h_2\Psi$ exists as a Hilbert vector because its squared norm would contain $\sum_{n\geq1}1$.

Nevertheless, the joint sum multiplier vanishes on every displayed basis vector. Hence $\Psi\in D(H_2^{(0)})$ and $H_2^{(0)}\Psi=0$. Writing $h_1\Psi+h_2\Psi$ as the sum of two individually defined vectors would miss this state.

The example explains why an exact domain belongs to the resulting self-adjoint operator, rather than being inferred from formal notation.

Increasing the system size changes the comparison problem

Fixed-$N$ particle spaces have no unique canonical embedding into the $(N+1)$-particle space. One must specify an added particle, a mode, or another comparison map. Fock space packages all finite sectors orthogonally, but that does not turn a thermodynamic limit into a Hilbert-norm limit of particle vectors.

For a lattice, local observable algebras do have canonical embeddings: tensor with identities on added sites. This motivates the quasi-local algebra and local-state convergence developed in Thermodynamic Limit.

Dimension growth, particle-number growth, spatial-volume growth, and removal of a mode cutoff are different limits. A theorem must say which one is taken and which quantities remain fixed.

Exercises

1. Why is $A_1$ not generally an observable acting within the symmetric two-particle space?

Solution

For a swap $U$, $UA_1U^*=A_2$. Unless the action happens to agree on the state under consideration, $A_1$ need not preserve the $U=1$ subspace. The sum $A_1+A_2$ commutes with $U$ and therefore preserves both statistics sectors.

2. Modify the domain-cancellation example to make a symmetric two-particle vector.

Solution

Replace each tensor by $(e_n\otimes e_{-n}+e_{-n}\otimes e_n)/\sqrt2$. These vectors remain orthonormal and have zero total energy. The same coefficients yield a vector in the symmetric total-energy domain but outside the separate slot domains.

References

• J. Dereziński, “Introduction to representations of the canonical commutation and anticommutation relations,” in Large Coulomb Systems, Lecture Notes in Physics 695, Springer, 2006, pp. 63–143, doi:10.1007/3-540-32579-4_3, author text.
• G. Teschl, Mathematical Methods in Quantum Mechanics: With Applications to Schrödinger Operators, 2nd ed., Graduate Studies in Mathematics 157, American Mathematical Society, 2014, Sections 4.2 and 4.6, author’s text.