# How to construct general multiparticle states that respect fermionic or bosonic symmetry?

Background: The arena is fixed particle number nonrelativistic quantum mechanics. The state space is $$\mathbf{H}(1)=\mathcal H\otimes\mathcal S,$$ where $$\mathcal H$$ is an "orbital" state space ($$L^2(\mathbb R^3)$$ if one'd like), and $$\mathcal S$$ is the spin space, which for a particle of spin $$s$$ is identifiable with $$\mathbb C^{2s+1}$$.

Because in this case it is (in my opinion) more tractable notationally, I will identify $$\mathcal H$$ with $$L^2(\mathbb R^3)$$ (position-basis wavefunctions), and a basis for $$\mathcal S$$ is $$\left|\sigma\right\rangle$$, where $$\sigma=-s,-s+1/2,...,+s$$. We can identify $$\left|\sigma\right\rangle$$ with the $$\sigma$$th standard basis vector of $$\mathbb C^{2s+1}$$.

A general 1-particle state is then $$\Psi(\mathbf x)=\sum_\sigma\psi_\sigma(\mathbf x)\left|\sigma\right\rangle,$$ which is just a $$2s+1$$-component wavefunction.

The $$k$$-particle fermionic state space is $$\mathbf H_A(k)=\bigwedge^k(\mathcal H\otimes\mathcal S),$$ and the $$k$$-particle bosonic state space is $$\mathbf H_S(k)=\bigvee^k(\mathcal H\otimes\mathcal S).$$

In case $$k=2$$, the construction of a general state is easy. The "unsymmetric" state space is $$\mathcal H\otimes\mathcal S\otimes\mathcal H\otimes\mathcal S\simeq \mathcal H\otimes\mathcal H\otimes\mathcal S\otimes\mathcal S$$, and in the product spin space $$\mathcal S^2\equiv\mathcal S\otimes\mathcal S$$, and we may easily construct a basis, it is $$\Sigma_{\sigma_1\sigma_2}\equiv \left|\sigma_1\sigma_2\right\rangle\equiv\left|\sigma_1\right\rangle\otimes \left|\sigma_2\right\rangle,$$ which we may identify with $$(2s+1)\times(2s+1)$$ square matrices, particularily, $$\Sigma_{\sigma_1\sigma_2}$$ is the matrix whose $$(\sigma_1,\sigma_2)$$th element is 1, and the rest is zero.

A general "unsymmetric" state is $$\Psi(\mathbf x_1,\mathbf x_2)=\sum_{\sigma_1,\sigma_2}\psi_{\sigma_1\sigma_2}(\mathbf x_1,\mathbf x_2)\Sigma_{\sigma_1\sigma_2}.$$

To find symmetric or skew-symmetric states, one may decompose $$\Sigma_{\sigma_1\sigma_2}=S_{\sigma_1\sigma_2}+A_{\sigma_1\sigma_2},$$ where $$S_{\sigma_1\sigma_2}=\Sigma_{(\sigma_1\sigma_2)}$$ and $$A_{\sigma_1\sigma_2}=\Sigma_{[\sigma_1,\sigma_2]}$$. Here I am ignoring the normalizations of the bases, as that is conceptually not relevant. There are $$2s^2+3s+1$$ independent $$S_{\sigma_1\sigma_2}$$ states, and $$2s^2+s$$ independent $$A_{\sigma_1\sigma_2}$$ states and together they span $$\mathcal S^2$$, so we can express a general symmetric state as $$\Psi_S(\mathbf x_1,\mathbf x_2)=\sum_{\sigma_1\le\sigma_2}\psi^S_{\sigma_1\sigma_2}(\mathbf x_1,\mathbf x_2)S_{\sigma_1\sigma_2}+\sum_{\sigma_1<\sigma_2}\psi^A_{\sigma_1\sigma_2}(\mathbf x_1,\mathbf x_2)A_{\sigma_1,\sigma_2},$$ where the $$\psi^S_{\sigma_1\sigma_2}(\mathbf x_1,\mathbf x_2)$$ wavefunctions are symmetric with respect to the exchange of the positions $$\mathbf x_1,\mathbf x_2$$ and the wavefunctions $$\psi^A_{\sigma_1\sigma_2}(\mathbf x_1,\mathbf x_2)$$ are skew.

A general skew-symmetric state is expressed then by the same combination but with the $$\psi^S_{\sigma_1\sigma_2}(\mathbf x_1,\mathbf x_2)$$ being skew in the position vectors and the functions $$\psi^A_{\sigma_1\sigma_2}(\mathbf x_1,\mathbf x_2)$$ being symmetric in the position vectors.

Question:

Assume I want to do the same thing but with $$k$$ particles instead of 2. The situations seems much more complicated, mainly because the spin bases $$\Sigma_{\sigma_1...\sigma_k}$$ now cannot be split into totally symmetric and skew-symmetric parts while still spanning the set of all "unsymmetric" spin states.

So, if I want to construct all bosonic and all fermionic $$k$$-particle states, what is the analogue of what I have done with 2-particle states?

I have a feeling this might actually be quite untractable generally (we want to have the decomposition of $$\bigwedge V\otimes W$$ into direct sums of exterior (or symmetric) products, which by this answer here: https://math.stackexchange.com/questions/211561/exterior-power-of-a-tensor-product/553405 seems untractable), however I assume this is a very important situation for applied quantum mechanics, so in this case, how to construct symmetry-respecting states?