The Spin-Statistics Theorem

Derivation

The Spin-Statistics Theorem

D-381 Home PU-402 Threads symmetry · matter · fields Depends on Mode Expansion and Fock Space, Dirac Equation from the Spinor Lagrangian, The Feynman Propagator

Statement

For a free relativistic field, the requirements of microcausality (fields, or their bilinears, commuting at spacelike separation), Lorentz invariance, and a Hamiltonian bounded below fix the quantization uniquely: fields carrying integer spin must be quantized with commutators (Bose statistics) and fields carrying half-integer spin with anticommutators (Fermi statistics). The opposite choice makes the Pauli–Jordan structure fail to vanish outside the light cone, or drives the energy spectrum unbounded below. We derive this for the spin-0 scalar and the spin-\(\tfrac12\) Dirac field, from which the general \((-1)^{2s}\) pattern follows.

Why it matters

The spin-statistics connection is the reason the periodic table exists, the reason electrons in a white dwarf resist gravitational collapse, and the reason lasers and Bose–Einstein condensates are possible. It converts an experimentally observed rule — half-integer spin particles obey the Pauli exclusion principle, integer spin particles do not — into a theorem forced by special relativity and quantum mechanics together.

It also demonstrates that relativity and quantum theory are not independently consistent: neither microcausality nor energy positivity alone selects the statistics, but demanding both at once leaves exactly one option per spin. Non-relativistic quantum mechanics cannot prove it; the light cone is essential.

Assumptions

Natural units \(\hbar=c=1\) and Minkowski metric \(\eta=\mathrm{diag}(+,-,-,-)\).Only bookkeeping; restoring \(\hbar,c\) rescales the Pauli–Jordan function but changes no conclusion. Free (non-interacting) fields with the standard mode expansions.With interactions the argument runs on asymptotic in/out fields and the LSZ formalism; the connection survives but the proof is heavier. Positive energy: single-particle states have \(E_p=+\sqrt{\mathbf{p}^2+m^2}\), so only \(e^{-iE_pt}\) appears as the positive-frequency part.If negative-energy modes are kept as physical, the Hamiltonian has no lower bound and the vacuum is unstable. Microcausality: observables built from the fields commute at spacelike separation, \([\mathcal{O}(x),\mathcal{O}(y)]=0\) for \((x-y)^2<0\).Drop it and spacelike-separated measurements interfere, allowing acausal signalling; the theorem then has no premise and either statistics is formally allowed. The fields transform in finite-dimensional representations of the proper orthochronous Lorentz group \(SO^+(1,3)\), labelled by spin \(s\).In these representations a rotation by \(2\pi\) multiplies the field by \((-1)^{2s}\); this sign is the engine of the whole result. Without a well-defined spin the classification collapses.

Derivation

\[ \phi(x)=\int\frac{d^3p}{(2\pi)^3\,2E_p}\left(a_{\mathbf p}\,e^{-ip\cdot x}+a_{\mathbf p}^{\dagger}\,e^{+ip\cdot x}\right),\qquad [a_{\mathbf p},a_{\mathbf q}^{\dagger}]=(2\pi)^3\,2E_p\,\delta^3(\mathbf p-\mathbf q) \]

Import the real scalar mode expansion from scalar-field-mode-expansion-fock-space; \(p\cdot x=E_p t-\mathbf p\cdot\mathbf x\) on shell. A

\[ [\phi(x),\phi(y)]=\int\frac{d^3p}{(2\pi)^3\,2E_p}\left(e^{-ip\cdot(x-y)}-e^{+ip\cdot(x-y)}\right)\;\equiv\; i\,\Delta(x-y) \]

Insert the expansion; only the \([a,a^{\dagger}]\) and \([a^{\dagger},a]\) cross terms survive, and the delta functions collapse one momentum integral. This defines the Pauli–Jordan function \(\Delta\). A

\[ \Delta(-x)=-\Delta(x),\qquad \Delta(\Lambda x)=\Delta(x)\ \ \forall\,\Lambda\in SO^+(1,3) \]

Each exponential is Lorentz invariant with the invariant measure \(d^3p/2E_p\); swapping the two terms is \(x\to -x\), so \(\Delta\) is odd and Lorentz scalar. B

\[ (x-y)^2<0\ \Rightarrow\ \exists\,\Lambda:\ \Lambda(x-y)=-(x-y)\ \Rightarrow\ \Delta(x-y)=\Delta(-(x-y))=-\Delta(x-y)=0 \]

For spacelike separation \(x-y\) and \(-(x-y)\) lie in the same orbit of the proper orthochronous group (a rotation reaches \(-\)of a spacelike vector without leaving the branch). Invariance plus oddness force zero. Hence \([\phi(x),\phi(y)]=0\) outside the light cone. C

\[ \{\phi(x),\phi(y)\}_{\text{trial}}=\int\frac{d^3p}{(2\pi)^3\,2E_p}\left(e^{-ip\cdot(x-y)}+e^{+ip\cdot(x-y)}\right)\;\equiv\;\Delta_1(x-y) \]

Now quantize the same scalar with anticommutators \(\{a,a^{\dagger}\}=(2\pi)^3 2E_p\delta^3\). The relative sign flips: the two terms now add. B

\[ \Delta_1(-x)=+\Delta_1(x),\qquad \Delta_1(x-y)\big|_{(x-y)^2<0}\neq 0 \]

\(\Delta_1\) is even, so the argument of Step 4 gives \(\Delta_1=\Delta_1\), no cancellation; explicitly \(\Delta_1\) is the invariant \(\tfrac{m}{4\pi^2\sqrt{-x^2}}K_1(m\sqrt{-x^2})\), positive for spacelike \(x\). A spin-0 field with anticommutators violates microcausality. Integer spin must use commutators. C

\[ \psi(x)=\int\frac{d^3p}{(2\pi)^3\,2E_p}\sum_{s}\left(a^{s}_{\mathbf p}\,u^{s}(p)\,e^{-ip\cdot x}+b^{s\dagger}_{\mathbf p}\,v^{s}(p)\,e^{+ip\cdot x}\right) \]

Import the Dirac expansion from dirac-equation-from-lagrangian, with spin sums \(\sum_s u^s\bar u^s=\gamma^\mu p_\mu+m\) and \(\sum_s v^s\bar v^s=\gamma^\mu p_\mu-m\). A

\[ \{\psi_a(x),\bar\psi_b(y)\}=\int\frac{d^3p}{(2\pi)^3\,2E_p}\Big[(\gamma^\mu p_\mu+m)_{ab}\,e^{-ip\cdot(x-y)}+(\gamma^\mu p_\mu-m)_{ab}\,e^{+ip\cdot(x-y)}\Big] \]

Anticommutate; \(\{a^s,a^{r\dagger}\}\) and \(\{b^s,b^{r\dagger}\}=(2\pi)^3 2E_p\delta^{sr}\delta^3\) collapse the sums to the two completeness relations. A

\[ \{\psi_a(x),\bar\psi_b(y)\}=\left(i\gamma^\mu\partial_\mu^{(x)}+m\right)_{ab}\,i\,\Delta(x-y)\;\xrightarrow{(x-y)^2<0}\;0 \]

Use \(\gamma^\mu p_\mu\,e^{\mp ip\cdot x}=\pm i\gamma^\mu\partial_\mu e^{\mp ip\cdot x}\) to pull the Dirac operator out; what remains is exactly the scalar \(i\Delta\) of Step 2, which vanishes spacelike, and a derivative of a function that is identically zero on the open spacelike region is zero. Anticommutators give a causal Dirac field. C

\[ [\psi_a(x),\bar\psi_b(y)]_{\text{trial}}=\left(i\gamma^\mu\partial_\mu^{(x)}+m\right)_{ab}\,\Delta_1(x-y)\;\neq\;0\ \ \text{for }(x-y)^2<0 \]

Quantizing Dirac with commutators flips the relative sign of the two terms in Step 8, replacing \(\Delta\) by the even \(\Delta_1\), which does not vanish spacelike. Spin-\(\tfrac12\) with commutators is acausal. C

\[ H=\int\frac{d^3p}{(2\pi)^3}\,E_p\sum_{s}\Big(a^{s\dagger}_{\mathbf p}a^{s}_{\mathbf p}\;\underbrace{-\;b^{s}_{\mathbf p}b^{s\dagger}_{\mathbf p}}_{\text{normal order}}\Big) \]

Compute the Dirac Hamiltonian directly. The antiparticle term appears with the opposite sign — this is the crux. B

\[ \{b,b^{\dagger}\}\!:\ -b\,b^{\dagger}=+b^{\dagger}b-\text{const}\ \Rightarrow\ H\ge 0;\qquad [b,b^{\dagger}]\!:\ -b\,b^{\dagger}=-b^{\dagger}b-\text{const}\ \Rightarrow\ H\ \text{unbounded below} \]

Only anticommutators turn the dangerous \(-bb^\dagger\) into a positive number operator; commutators would let antiparticle occupation lower the energy without limit, destroying the vacuum. Energy positivity independently selects Fermi statistics for spin \(\tfrac12\). C

Result

\[ \boxed{\ \text{spin } s\in\mathbb{Z}:\ [\,\phi(x),\phi(y)\,]=0\ \ (x{-}y)^2<0\qquad\Longleftrightarrow\qquad \text{spin } s\in\mathbb{Z}+\tfrac12:\ \{\psi(x),\bar\psi(y)\}=0\ \ (x{-}y)^2<0\ } \]

Reading. Whether the causal spacelike-vanishing bilinear is the commutator or the anticommutator is fixed entirely by the spin. The \((-1)^{2s}\) sign a field acquires under a \(2\pi\) rotation (and equivalently under the \(x\to -x\) exchange of the two mode terms) is exactly cancelled only when the graded bracket matches the spin: symmetric bracket for integer spin, antisymmetric for half-integer. Energy positivity gives the same verdict, so the two independent requirements agree.

Units check. In natural units \([\phi]=1\) (mass\(^1\)), so \([\phi,\phi]\) and \(\Delta\) carry mass dimension \(2\); the measure \(d^3p/2E_p\) has dimension \(3-1=2\), matching. For Dirac \([\psi]=\tfrac32\), so \(\{\psi,\bar\psi\}\) has dimension \(3\); the operator \(i\gamma^\mu\partial_\mu+m\) adds dimension \(1\) to the dimension-\(2\) object \(\Delta\), giving \(3\). Consistent.

Limiting cases

Massless limit \(m\to0\): the mass term in Step 9 drops but \(\{\psi,\bar\psi\}=i\gamma^\mu\partial_\mu\,i\Delta\) still vanishes spacelike; statistics are mass-independent, as required (photons are bosons at every energy).
Non-relativistic limit \(c\to\infty\): the light cone opens to the whole of space, microcausality becomes vacuous, and the theorem loses its premise — statistics must then be postulated, which is why non-relativistic QM cannot derive Pauli exclusion.
General spin \(s\): a rank-\(2s\) tensor/spinor field carries \(2s\) Lorentz indices, contributing \((-1)^{2s}\) under the exchange; the cancellation reproduces commutators for even and anticommutators for odd \(2s\).
Equal times, \(t_x=t_y\), \(\mathbf x\neq\mathbf y\): a special spacelike case — \([\phi,\dot\phi]\) gives the canonical \(i\delta^3(\mathbf x-\mathbf y)\) while \([\phi,\phi]=0\), the familiar equal-time algebra.

Breaks when

Lorentz invariance is broken (a preferred frame, a lattice cutoff at finite spacing, or non-commutative spacetime): Step 3–4 fail because \(x-y\) and \(-(x-y)\) are no longer connected by an invariance, so a spacelike bracket need not vanish and "wrong-statistics" fields become formally consistent.
The energy is not bounded below (ghost fields, higher-derivative theories with negative-norm modes, or keeping negative-energy solutions): Step 11–12 no longer discriminate, and the theorem's second pillar collapses.
Reduced spatial dimensions (2+1D): the rotation group is abelian and \(2\pi\) rotations can give an arbitrary phase \(e^{i\theta}\), not just \(\pm1\); anyons with fractional statistics evade the integer/half-integer dichotomy entirely.
Indefinite-metric or gauge sectors: for gauge fields quantized covariantly (Gupta–Bleuler), the naive field commutators are not directly observable and the theorem must be phrased on gauge-invariant bilinears; applied naively it appears to "break."

Failure modes

Claiming the theorem is about wavefunction symmetry. The primitive statement is about field (anti)commutators and microcausality; multi-particle wavefunction (anti)symmetry is a consequence, not the premise.
Trying to prove it non-relativistically. Students invoke "identical particles" alone; without the light cone there is nothing to force the sign, and both statistics are consistent in Galilean QM.
Sign-of-metric and \(p\cdot x\) slips. Getting \(p\cdot x=E_pt+\mathbf p\!\cdot\!\mathbf x\) wrong swaps positive- and negative-frequency parts and appears to reverse the conclusion.
Forgetting the antiparticle term's sign in \(H\). Missing the \(-bb^{\dagger}\) in Step 11 hides the energy-positivity argument and makes anticommutators look optional.
Using \(\{\psi,\psi\}\) instead of \(\{\psi,\bar\psi\}\). The causal object for the Dirac field is the field with its Dirac conjugate; \(\{\psi_a,\psi_b\}\) vanishes for a different (spin-sum) reason and confuses the microcausality check.
Assuming \(\Delta_1\) vanishes spacelike "by symmetry." It is even and manifestly nonzero (a Bessel function), which is precisely why the wrong statistics fail.

Discussion

The heart of the theorem is a single sign. Exchanging the two mode terms in a field bilinear is the operation \(x\to -x\); a field of spin \(s\) carries \(2s\) Lorentz indices and so acquires \((-1)^{2s}\) under that exchange. The graded bracket contributes its own sign, \(-1\) for a commutator and \(+1\) for an anticommutator (as seen in Steps 2 vs 5). The spacelike bracket vanishes only when these two signs cancel, which pins commutators to integer and anticommutators to half-integer spin. Everything else — Bessel functions, spin sums, Hamiltonians — is machinery around that observation.

Two logically independent arguments converge here. Microcausality (Steps 4, 6, 9, 10) is a statement about the algebra of observables and the causal structure of spacetime. Energy positivity (Steps 11–12) is a statement about the spectrum and the stability of the vacuum. That both single out the same statistics is a deep consistency check on relativistic quantum field theory: had they disagreed, no local, causal, stable quantum field theory of that spin could exist. In fact for some spins that is what happens — a spin-\(\tfrac12\) field forced to be bosonic is both acausal and unstable — which is why nature contains no such particles.

The most rigorous modern proofs discard mode expansions entirely. In the Wightman axiomatic framework, one analytically continues the two-point function \(W(x-y)=\langle 0|\phi(x)\phi(y)|0\rangle\) into complex spacetime; the domains of holomorphy fixed by spectral positivity (energy bounded below) and the transformation law under the complexified Lorentz group \(SL(2,\mathbb{C})\) force a definite relation between \(W(x-y)\) and \(W(y-x)\) at the "Jost points" — real spacelike separations reachable by complex Lorentz transformations. Demanding that the correct bilinear vanish there yields the connection with full generality, independent of the free-field assumption, and exposes the theorem's true content as an interplay of complex analysis, the spectrum condition, and the topology of the Lorentz group.

Common misconceptions. The Pauli exclusion principle is not an extra postulate bolted onto quantum mechanics; it is the low-energy shadow of field anticommutation. Conversely, Bose condensation is not particles "wanting to be together" but the absence of any exclusion. And the theorem does not say fermions "are" antisymmetric wavefunctions by definition — antisymmetry is derived, and could in principle have failed, which is exactly what makes the result a theorem rather than a convention.

Worked examples

Two identical spin-\(\tfrac12\) electrons in a 1D infinite well of width \(L\): Fermi statistics forbids both from sharing the same spatial and spin state. Single-level energies \(E_n=\dfrac{n^2\pi^2\hbar^2}{2m_eL^2}\).

Symbols first: the spin-singlet (antisymmetric spin) allows a symmetric spatial state so both electrons occupy \(n=1\), giving \(2E_1\); the spin-triplet (symmetric spin) requires an antisymmetric spatial state, the lowest being \(n=1,2\), giving \(E_1+E_2=5E_1\). The energy gap \(3E_1\) is a direct fingerprint of anticommutation. A

\[ E_1=\frac{\pi^2(1.055\times10^{-34}\,\text{J s})^2}{2(9.11\times10^{-31}\,\text{kg})(1.0\times10^{-9}\,\text{m})^2}=6.03\times10^{-20}\,\text{J}=0.377\,\text{eV} \]

Insert \(L=1.0\,\)nm and the electron mass. A

\[ E_{\text{singlet}}=2E_1=0.75\ \text{eV},\qquad E_{\text{triplet}}=5E_1=1.88\ \text{eV},\qquad \Delta E=3E_1=1.13\ \text{eV} \]

Reading. Had electrons been bosons, both configurations would sit at \(2E_1\) and the \(1.13\,\)eV splitting — measurable as an optical line — would not exist. The exclusion is a consequence of the theorem, not an input.

Units check. \(\hbar^2/(m L^2)\) has units \((\text{J s})^2/(\text{kg}\,\text{m}^2)=\text{J}\). Correct.

Electron degeneracy pressure in a carbon white dwarf: because electrons are fermions, they fill momentum states up to a Fermi momentum \(p_F=\hbar(3\pi^2 n_e)^{1/3}\), where \(n_e=\rho/(\mu_e m_u)\) with \(\mu_e=2\) nucleons per electron.

Symbols first: the exclusion principle (anticommutation) forces a filled Fermi sea whose pressure, not thermal motion, supports the star. Evaluate \(n_e\), then \(p_F\), then the Fermi energy \(E_F=p_F^2/2m_e\). B

\[ n_e=\frac{1.0\times10^{9}\,\text{kg m}^{-3}}{2(1.66\times10^{-27}\,\text{kg})}=3.0\times10^{35}\,\text{m}^{-3},\qquad p_F=(1.055\times10^{-34})\big(3\pi^2\cdot3.0\times10^{35}\big)^{1/3} \]

Take \(\rho=10^{9}\,\)kg m\(^{-3}\). Then \(3\pi^2n_e=8.9\times10^{36}\,\)m\(^{-3}\), cube root \(2.07\times10^{12}\,\)m\(^{-1}\). B

\[ p_F=2.18\times10^{-22}\,\text{kg m s}^{-1}\quad\Big(\tfrac{p_F}{m_ec}=0.80,\ \text{mildly relativistic}\Big),\qquad E_F=\frac{p_F^2}{2m_e}=2.6\times10^{-14}\,\text{J}=0.16\,\text{MeV} \]

Compute \(p_F\), compare to \(m_ec=2.73\times10^{-22}\,\)kg m s\(^{-1}\), then the (non-relativistic estimate of the) Fermi energy. B

\[ E_F\approx0.16\ \text{MeV}\ \gg\ k_BT_{\text{thermal}}\ (\sim\!\text{keV}) \]

Reading. The Fermi energy vastly exceeds the thermal energy, so the electrons are fully degenerate: the pressure holding up the white dwarf is a pure consequence of anticommutation. Boson "electrons" would all fall into \(p=0\), exert no such pressure, and no white dwarf could exist.

Units check. \(p_F=\hbar\,n_e^{1/3}\) has units \((\text{J s})(\text{m}^{-1})=\text{kg m s}^{-1}\); \(p_F^2/m_e\) has units \((\text{kg m s}^{-1})^2/\text{kg}=\text{J}\). Correct.

Problems

Show explicitly that \(\Delta(x)\) defined in Step 2 is odd, \(\Delta(-x)=-\Delta(x)\), directly from its integral representation (do not invoke Lorentz invariance).
Solution
Substitute \(x\to -x\) in \(i\Delta(x)=\int\frac{d^3p}{(2\pi)^3 2E_p}(e^{-ip\cdot x}-e^{+ip\cdot x})\): the exponents swap, giving \(\int\frac{d^3p}{(2\pi)^3 2E_p}(e^{+ip\cdot x}-e^{-ip\cdot x})=-i\Delta(x)\). Hence \(\Delta(-x)=-\Delta(x)\). The measure \(d^3p/2E_p\) is unchanged because it depends only on \(|\mathbf p|\).
For the Dirac field, verify the dimensional consistency of Step 9 in natural units by assigning mass dimensions to \(\psi\), \(\Delta\), and \(i\gamma^\mu\partial_\mu+m\).
Solution
From the kinetic term \(\bar\psi\,i\gamma^\mu\partial_\mu\psi\) in a 4D Lagrangian of dimension \(4\): \(2[\psi]+1=4\Rightarrow[\psi]=\tfrac32\). So \(\{\psi,\bar\psi\}\) has dimension \(3\). \(\Delta\) has dimension \(2\) (Problem-independent: it equals \([\phi,\phi]\) with \([\phi]=1\)). The operator \(i\gamma^\mu\partial_\mu+m\) has dimension \(1\). Then \(1+2=3\), matching \([\{\psi,\bar\psi\}]=3\).
Compute the lowest three total energies (in units of \(E_1\)) for three identical spin-\(\tfrac12\) fermions in the 1D box of Worked Example 1, ignoring spin degeneracy beyond the exclusion it enforces, assuming each spatial level \(n\) holds at most two electrons.
Solution
Each level \(n\) holds \(2\) electrons (spin up/down). Three electrons: fill \(n=1\) (two) and \(n=2\) (one). Ground total \(=2E_1+E_2=2E_1+4E_1=6E_1\). First excited: promote the \(n=2\) electron to \(n=3\): \(2E_1+9E_1=11E_1\); or promote one \(n=1\) electron to \(n=2\) (now doubly filled) giving \(E_1+2E_2=E_1+8E_1=9E_1\). So the ordered lowest three totals are \(6E_1,\ 9E_1,\ 11E_1\).
Using the white-dwarf numbers of Worked Example 2, estimate the degeneracy pressure via \(P\approx\tfrac{2}{5}n_eE_F\) and comment on its order of magnitude.
Solution
\(P\approx\tfrac25 n_e E_F=0.4\times(3.0\times10^{35}\,\text{m}^{-3})(2.6\times10^{-14}\,\text{J})=3.1\times10^{21}\,\text{Pa}\). Units: \(\text{m}^{-3}\cdot\text{J}=\text{J m}^{-3}=\text{Pa}\). This \(\sim3\times10^{21}\,\)Pa dwarfs any laboratory pressure and is the outward force balancing gravity in the star — a macroscopic manifestation of fermion anticommutation.
Explain, with the sign bookkeeping, why a spin-0 field quantized with anticommutators fails both microcausality and energy positivity, whereas the spin-\(\tfrac12\) Dirac field with commutators also fails both. What general rule does this illustrate?
Solution
Spin-0, anticommutators: the field-exchange sign is \((-1)^{2s}=+1\) and the bracket sign is \(+1\), so they do not cancel — Step 5–6 give the even \(\Delta_1\neq0\) spacelike (microcausality fails); and the scalar Hamiltonian \(\int E_p(a^\dagger a+b^\dagger b)\) requires \([a,a^\dagger]=+\) to be positive-normed, so anticommutation also spoils the spectrum/norm. Spin-\(\tfrac12\), commutators: exchange sign \((-1)^{2s}=-1\), bracket sign \(-1\), product \(+1\) — no cancellation, so Step 10 gives a nonzero spacelike bracket (microcausality fails), and Step 12 shows \(H\) unbounded below (energy positivity fails). General rule: the spacelike bracket vanishes and the energy is bounded below iff the graded-bracket sign matches \((-1)^{2s}\), i.e. commutators for integer and anticommutators for half-integer spin.