Identical Particles and Exchange Symmetry

Identical particles require a restriction on the allowed many-particle states. For ordinary bosons the complete state is symmetric under exchange; for fermions it is antisymmetric. This rule acts on all one-particle labels together, including position and spin. It is not an uncertainty about which classical particle has which name.

Helpful background. Tensor products explain the auxiliary particle-coordinate slots. Quantum symmetries explain unitary transformations and invariant observables.

Enter this chapter

Task	Page	Main output
understand why labels are not identities	Indistinguishability	physical meaning of slot labels and exchange-invariant observables
state the boson/fermion sector rule	Symmetrization Postulate	symmetric and antisymmetric physical state spaces
develop symmetric many-particle states	Bosons	repeated occupation and symmetric exchange behavior
develop antisymmetric many-particle states	Fermions	antisymmetry and at-most-one occupation per one-particle state
derive the exclusion rule	Pauli Exclusion Principle	vanishing of repeated fermionic one-particle states
calculate with permutations	Exchange Operators	transpositions, projectors, permutation groups, and observables
construct coordinate-space states	Symmetric and Antisymmetric Wavefunctions	explicit two-particle forms and normalization
represent fermionic products	Slater Determinants	compact antisymmetric $N$-fermion states
represent bosonic products	Permanents	symmetric sums and occupation-dependent normalization
combine exchange with spin	Spin and Spatial Wavefunctions	singlet/triplet pairing and total exchange symmetry
avoid false entanglement claims	Identical-Particle Entanglement Cautions	operational modes, regions, and accessible algebras

What is being exchanged?

For two particles, exchanging the complete one-particle states sends $|a\rangle_1|b\rangle_2$ to $|b\rangle_1|a\rangle_2$. The subscripts label tensor slots used in the calculation. They do not identify separately addressable particles. Physical operations must respect the exchange symmetry.

For a one-particle Hilbert space $\mathcal h$, the allowed fixed-number spaces are $\operatorname{Sym}^N\mathcal h$ for bosons and $\bigwedge^N\mathcal h$ for fermions. For $N\geq3$, these two sectors do not exhaust a general auxiliary tensor product. The ordinary boson/fermion postulate selects the physical sector. The relativistic spin–statistics theorem and two-dimensional braid statistics require additional physics beyond this construction.

Two electrons can share a spatial orbital

Pauli exclusion forbids repeated occupation of the same complete spin-orbital. Two electrons can share a spatial orbital $\phi$ if their combined state is

$$\Psi=\phi(x_1)\phi(x_2) \frac{|\uparrow\downarrow\rangle-|\downarrow\uparrow\rangle}{\sqrt2}.$$

The spatial factor is symmetric, while the spin singlet is antisymmetric; their product has the required total antisymmetry. Conversely, a factorized triplet state needs an antisymmetric spatial factor. When spin and position do not factorize, impose exchange on the combined wavefunction directly.

The spin and spatial wavefunctions page develops this reasoning. For explicit $N$-particle constructions, compare Slater determinants and permanents. A single determinant or symmetrized product is a useful state, not the most general correlated state.

Choose physical subsystems before discussing entanglement

The symmetrized expression for one boson in each of two orthogonal modes $L$ and $R$ is nonfactorized in the artificial particle slots. In occupation notation, however, the same state is simply $|1_L,1_R\rangle$, a product across the two modes. Exchange symmetry alone therefore does not imply entanglement between two laboratories.

Specify the controlled modes, spatial regions, or observable algebras first. Internal spin degrees of freedom in separately controlled modes can carry entanglement, but it must be present in the state rather than inferred from particle exchange. See identical-particle entanglement cautions for the role of allowed local operations and number restrictions.

Continue with occupation numbers

Once exchange symmetry is clear, Fock Space and Occupation Number replaces long permutation sums with occupation tuples. Quantum Statistics and Ideal Gases later combines these allowed occupations with thermal probabilities. Exchange symmetry and a thermal distribution are separate ingredients.

Exercises

An overlapping pair of orbitals. Two normalized orbitals have overlap $s=\langle a|b\rangle$. Why does the familiar factor $1/\sqrt2$ fail to normalize $|a\rangle|b\rangle\pm|b\rangle|a\rangle$ when $s\ne0$?

Solution

The two diagonal inner products contribute 2; the cross terms contribute $\pm2|s|^2$. The squared norm is $2(1\pm|s|^2)$, so divide by its square root. At $|s|=1$ the antisymmetric vector vanishes and cannot be normalized, expressing exclusion of a repeated complete state.

Two descriptions of one state. Does one boson in each of the modes $L$ and $R$ necessarily entangle the two mode laboratories?

Solution

No. In occupation notation the state is $|1_L\rangle\otimes|1_R\rangle$. Its nonfactorized first-quantized slot expression is required by exchange symmetry. Entanglement depends on the physical partition and state; it does not follow from that expression alone.

References

• Fetter, A. L., and Walecka, J. D., Quantum Theory of Many-Particle Systems, Dover (2003).
• Messiah, A., Quantum Mechanics, vol. II, North-Holland (1962).
• Wiseman, H. M., and Vaccaro, J. A., “Entanglement of Indistinguishable Particles Shared between Two Parties,” Physical Review Letters 91, 097902 (2003), doi:10.1103/PhysRevLett.91.097902.