Symmetrization Postulate

The symmetrization postulate states how identical particles are represented in ordinary nonrelativistic quantum mechanics:

$$\text{identical bosons:} \quad U(\pi)\lvert\Psi\rangle=\lvert\Psi\rangle,$$ $$\text{identical fermions:} \quad U(\pi)\lvert\Psi\rangle = \operatorname{sgn}(\pi)\lvert\Psi\rangle.$$

Here $\pi$ is a permutation of the particle slots, $U(\pi)$ is its unitary action on the slot-labeled tensor product, and $\operatorname{sgn}(\pi)$ is $+1$ for even permutations and $-1$ for odd permutations.

In words: identical boson states are symmetric under exchange, while identical fermion states are antisymmetric under exchange. The postulate is not a convention about notation. It restricts which vectors in the formal tensor-product space represent physical states of a given species.

Helpful background. Indistinguishability explains why particle-slot labels do not identify observable individuals.

Exchange Operator

For two particles built from the same one-particle Hilbert space $\mathcal h$, the formal two-slot space is

$$\mathcal h\otimes\mathcal h.$$

The exchange operator $P_{12}$ swaps the two slots:

$$P_{12} \bigl( \lvert\alpha\rangle_1\otimes\lvert\beta\rangle_2 \bigr) = \lvert\beta\rangle_1\otimes\lvert\alpha\rangle_2.$$

In a coordinate-spin representation with $q=(\mathbf x,s)$,

$$(P_{12}\Psi)(q_1,q_2) = \Psi(q_2,q_1).$$

The exchange operator is unitary and Hermitian:

$$P_{12}^{-1}=P_{12}, \qquad P_{12}^\dagger=P_{12}, \qquad P_{12}^2=I.$$

Thus its eigenvalues are $\pm1$. The symmetrization postulate says which eigenspace is used by which kind of identical particle.

Symmetric States

A two-particle state is symmetric if

$$P_{12}\lvert\Psi\rangle = \lvert\Psi\rangle.$$

Equivalently, in the coordinate-spin representation,

$$\Psi(q_2,q_1) = \Psi(q_1,q_2).$$

For two orthonormal one-particle states $\lvert\alpha\rangle$ and $\lvert\beta\rangle$, the corresponding symmetric two-slot state is

$$\lvert\alpha,\beta\rangle_S = \frac{1}{\sqrt2} \bigl( \lvert\alpha\rangle_1\lvert\beta\rangle_2 + \lvert\beta\rangle_1\lvert\alpha\rangle_2 \bigr).$$

If both bosons occupy the same one-particle state, the symmetric state is simply

$$\lvert\alpha\rangle_1\lvert\alpha\rangle_2,$$

up to normalization and notation. This is the elementary reason many bosons can occupy the same mode.

The symmetric projector for two slots is

$$\Pi_S = \frac12(I+P_{12}).$$

It extracts the exchange-symmetric part of a two-slot vector.

The coordinate-space construction and normalization caveats for nonorthogonal one-particle states are worked out in Symmetric and Antisymmetric Wavefunctions.

Antisymmetric States

A two-particle state is antisymmetric if

$$P_{12}\lvert\Psi\rangle = -\lvert\Psi\rangle.$$

Equivalently,

$$\Psi(q_2,q_1) = -\Psi(q_1,q_2).$$

For two orthonormal one-particle states $\lvert\alpha\rangle$ and $\lvert\beta\rangle$, the corresponding antisymmetric state is

$$\lvert\alpha,\beta\rangle_A = \frac{1}{\sqrt2} \bigl( \lvert\alpha\rangle_1\lvert\beta\rangle_2 - \lvert\beta\rangle_1\lvert\alpha\rangle_2 \bigr).$$

The antisymmetric projector is

$$\Pi_A = \frac12(I-P_{12}).$$

If $\lvert\alpha\rangle=\lvert\beta\rangle$, the antisymmetric combination vanishes:

$$\lvert\alpha\rangle_1\lvert\alpha\rangle_2 - \lvert\alpha\rangle_1\lvert\alpha\rangle_2 = 0.$$

This is the shortest algebraic preview of the Pauli exclusion principle. The full exclusion page spells out the physical consequences for orbitals, spin, and many-electron states.

Physical Meaning

The postulate should be read together with indistinguishability. Since identical-particle slot labels are not observable names, physical states must respond consistently when those slots are exchanged.

For identical bosons, exchange leaves the state vector unchanged. For identical fermions, exchange changes the state vector by a minus sign. Although a single overall sign is not observable by itself, fermionic minus signs are physically real in interference, antisymmetric wavefunctions, Slater determinants, and operator reordering rules.

Physical observables for identical particles commute with permutations. For two particles,

$$[O,P_{12}]=0.$$

If the Hamiltonian is exchange-invariant, then symmetric and antisymmetric sectors are preserved by time evolution. Indeed, if $[H,P_{12}]=0$ and $P_{12}\lvert\Psi(0)\rangle=\eta\lvert\Psi(0)\rangle$, then

$$P_{12}\lvert\Psi(t)\rangle = \eta\lvert\Psi(t)\rangle.$$

Exchange symmetry is therefore not an optional afterthought added to a solution. It is a restriction on the physical state space for a species of identical particles.

Spin and Space Are Exchanged Together

For particles with spin, exchange acts on all degrees of freedom belonging to the particles. If $q=(\mathbf x,s)$ includes position and spin, the exchange rule is imposed on $q_1$ and $q_2$ together:

$$\Psi(q_2,q_1) = \eta\Psi(q_1,q_2).$$

It is often useful to factor a state schematically into spatial and spin parts,

$$\Psi(q_1,q_2) = \psi(\mathbf x_1,\mathbf x_2)\, \chi(s_1,s_2),$$

but the required symmetry applies to the product. For two identical fermions, a symmetric spatial wavefunction must be paired with an antisymmetric spin state, or an antisymmetric spatial wavefunction with a symmetric spin state. The combined state must be antisymmetric.

This is why the singlet and triplet structure of two spin-$1/2$ particles becomes physically important for identical fermions. The spin state alone does not determine the exchange symmetry of the full state.

More than two identical particles

The physical fixed-$N$ spaces are $\operatorname{Sym}^N\mathcal h$ for bosons and $\wedge^N\mathcal h$ for fermions. All transpositions act with the same species-dependent sign, so an arbitrary permutation acts by $1$ or $\operatorname{sgn}(\pi)$.

For example, a three-cycle is a product of two exchanges and therefore has sign $+1$ for fermions. Antisymmetry means a minus sign for every odd permutation, not for every nonidentity permutation. Exchange Operators defines the composition convention and constructs the $N$-particle projectors.

Permutation-invariant observables alone do not select only these two sectors: the unrestricted $N$-slot tensor product can have other permutation symmetries when $N\ge3$. Restricting ordinary particles to the completely symmetric or antisymmetric sector is the postulate. This logical distinction is developed by Messiah and Greenberg.

Occupation-number notation then records the occupied modes within a sector; Fock space collects sectors with different particle numbers.

Relation to the Spin-Statistics Theorem

In nonrelativistic quantum mechanics, the symmetrization rule is a postulate. It is supported by experiment and by its consistency with many-particle spectroscopy, quantum statistics, and matter stability.

Relativistic quantum field theory explains a deeper connection: under assumptions including Lorentz covariance, spacelike locality, positive energy, and a positive physical Hilbert-space inner product, integer-spin particles are bosons and half-integer-spin particles are fermions. That result is the spin-statistics theorem.

This page does not prove that theorem. It uses the ordinary nonrelativistic rule needed for atoms, molecules, condensed matter, AMO physics, and the bridge to second quantization. In two spatial dimensions, braid statistics allow additional possibilities; Anyons and Braiding gives their configuration-space and fusion-space treatment.

Exercises

1. Verify that the state $\lvert\alpha,\beta\rangle_S$ defined above is symmetric under $P_{12}$.

Solution

Apply $P_{12}$ to each product vector:

$$P_{12} \bigl( \lvert\alpha\rangle_1\lvert\beta\rangle_2 \bigr) = \lvert\beta\rangle_1\lvert\alpha\rangle_2,$$

and

$$P_{12} \bigl( \lvert\beta\rangle_1\lvert\alpha\rangle_2 \bigr) = \lvert\alpha\rangle_1\lvert\beta\rangle_2.$$

The two terms are interchanged, but their sum is unchanged. Therefore $P_{12}\lvert\alpha,\beta\rangle_S=\lvert\alpha,\beta\rangle_S$.

2. Two identical spin-$1/2$ fermions are in an antisymmetric spin singlet. What exchange symmetry must the spatial wavefunction have?

Solution

The total two-fermion state must be antisymmetric. The spin singlet is antisymmetric under exchange. Therefore the spatial part must be symmetric, so that

$$(\text{symmetric spatial}) \times (\text{antisymmetric spin}) = \text{antisymmetric total}.$$

3. A permutation of three slots is a single transposition. What sign does it produce on a fermionic state?

Solution

A single transposition is an odd permutation, so $\operatorname{sgn}(\pi)=-1$. A fermionic state therefore transforms as

$$U(\pi)\lvert\Psi\rangle = -\lvert\Psi\rangle.$$

References

• L. E. Ballentine, Quantum Mechanics: A Modern Development, 2nd ed., World Scientific, 2014.
• P. A. M. Dirac, The Principles of Quantum Mechanics, 4th ed., Oxford University Press, 1958.
• A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, McGraw-Hill, 1971.
• A. M. L. Messiah and O. W. Greenberg, “Symmetrization Postulate and Its Experimental Foundation”, Physical Review 136, B248–B267 (1964).
• W. Pauli, “The Connection Between Spin and Statistics”, Physical Review 58, 716–722 (1940).
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.