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Part II, Chapter 6

The Spin-Statistics Theorem

Why bosons commute and fermions anticommute: a deep theorem of nature

🔗Course Connections

📚Prerequisites

QFT Part II

Dirac Quantization

Fermionic anticommutators

→

QFT Part I

Lorentz Invariance

Relativistic transformation properties

→

▶️

Video Lecture

Lecture 19: Spin-Statistics Theorem - MIT 8.323

The deep connection between spin and quantum statistics (MIT QFT Course)

💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.

6.1 Statement of the Theorem

🎯 Spin-Statistics Theorem

In any local, Lorentz invariant quantum field theory with positive energy:

Integer spin (s = 0, 1, 2, ...) fields MUST obey Bose-Einstein statistics(commutators)
Half-integer spin (s = 1/2, 3/2, 5/2, ...) fields MUST obey Fermi-Dirac statistics (anticommutators)

This is not a choice—it's forced by consistency of the theory!

💡Why This Matters

The spin-statistics theorem explains fundamental facts about our universe:

Why electrons (spin 1/2) obey the Pauli exclusion principle
Why photons (spin 1) can occupy the same state (lasers!)
Why atomic structure exists (electron shells fill up)
Why chemistry works (valence electrons can't all drop to ground state)
Why matter is stable (fermions resist compression)

6.2 Heuristic Argument

We'll give a simplified version of the proof. The full rigorous proof (Pauli, 1940; Lüders & Zumino, 1958) uses advanced techniques from axiomatic QFT.

Step 1: Locality (Microcausality)

For spacelike separated points (x - y)² < 0, observables must commute:

$$[\hat{\mathcal{O}}(x), \hat{\mathcal{O}}(y)] = 0 \quad \text{for spacelike } (x-y)$$

This ensures causality: measurements at x cannot instantly affect measurements at y if they're spacelike separated (no faster-than-light signaling).

Step 2: Fields as Observables

For integer spin fields (scalars, vectors), the field itself can be an observable. Therefore, we need:

$$[\hat{\phi}(x), \hat{\phi}(y)] = 0 \quad \text{for spacelike } (x-y)$$

This is satisfied if [â_k, â^†_q] = ... (commutators). ✓

Step 3: Spinors Are Not Observable!

For half-integer spin fields (spinors like ψ), the field itself is notan observable—it transforms with a sign under 2π rotation!

The observable is the bilinear ψ̄ψ (or current j^μ = ψ̄γ^μψ). We need:

$$[\hat{\bar{\psi}}(x)\hat{\psi}(x), \hat{\bar{\psi}}(y)\hat{\psi}(y)] = 0 \quad \text{for spacelike } (x-y)$$

Let's check if anticommutators for ψ give us this:

\begin{align*} [\bar{\psi}(x)\psi(x), \bar{\psi}(y)\psi(y)] &= \bar{\psi}(x)\psi(x)\bar{\psi}(y)\psi(y) - \bar{\psi}(y)\psi(y)\bar{\psi}(x)\psi(x) \\ &= \bar{\psi}(x)[\psi(x)\bar{\psi}(y)]\psi(y) - \bar{\psi}(y)[\psi(y)\bar{\psi}(x)]\psi(x) \\ &= \bar{\psi}(x)[-\bar{\psi}(y)\psi(x)]\psi(y) - \bar{\psi}(y)[-\bar{\psi}(x)\psi(y)]\psi(x) \\ &\quad \text{(using anticommutators)}\\ &= 0 \quad \text{for spacelike separation!} \end{align*}

The minus signs from anticommutation make everything cancel! If we had used commutators, this would fail. ✓

Step 4: Connection to Spin

The key mathematical fact (from representation theory of the Lorentz group):

Integer spin: Tensor representations → fields are real/complex numbers
Half-integer spin: Spinor representations → fields change sign under 2π rotation

This sign change under 2π rotation is precisely what forces us to use anticommutators for spinors!

6.3 What Happens If You Violate Spin-Statistics?

❌ Scenario 1: Bosonic Quantization of Spin-1/2

If we tried to quantize the Dirac field with commutators [ψ̂, ψ̂^†]:

Negative norm states: Some states would have ⟨ψ|ψ⟩ < 0 (ghost states!)
Violated causality: [ψ̄ψ(x), ψ̄ψ(y)] ≠ 0 for spacelike separation
Unstable vacuum: Hamiltonian unbounded from below (infinite negative energy)

❌ Scenario 2: Fermionic Quantization of Spin-0

If we tried to quantize a scalar field with anticommutators {φ̂, φ̂^†}:

Wrong propagator: Would get wrong analytical structure
Violated Lorentz invariance: Spinless object can't have fermionic statistics
Mathematical inconsistency: No consistent S-matrix

6.4 Examples in Nature

Integer Spin Bosons

Photon (γ): Spin 1, mediates EM force
Gluons (g): Spin 1, mediate strong force
W±, Z bosons: Spin 1, weak force
Higgs boson: Spin 0, mass generation
Graviton (hypothetical): Spin 2, gravity
Pions (π): Spin 0, nuclear force

All obey Bose-Einstein statistics!

Half-Integer Spin Fermions

Electron (e⁻): Spin 1/2, matter
Quarks (u,d,s,c,b,t): Spin 1/2, hadrons
Neutrinos (νₑ,νᵤ,ντ): Spin 1/2, weak int.
Muon (μ⁻), Tau (τ⁻): Spin 1/2, leptons
Proton, Neutron: Spin 1/2, nuclei
Gravitino (hypothetical): Spin 3/2, SUSY

All obey Fermi-Dirac statistics!

No exceptions have ever been found! Every particle discovered obeys spin-statistics.

6.5 Experimental Tests

The spin-statistics theorem has been tested to extremely high precision:

Pauli Exclusion Violation Tests

If electrons violated Pauli exclusion, atomic transitions forbidden by exclusion would occur. Experiments look for X-rays from "impossible" transitions in conductors.

Result: No violations found. Pauli exclusion principle holds to better than 1 part in 10²⁶!

Photon Statistics Tests

Lasers rely on bosonic stimulated emission (many photons in same state). If photons were fermions, lasers wouldn't work!

Result: Lasers work. Photons are definitely bosons.

6.6 Summary Table

Spin-Statistics Connection

How spin determines quantum statistics

Aspect	Integer Spin (Bosons)	Half-Integer Spin (Fermions)
Examples	Photons (s=1), Higgs (s=0), gravitons (s=2)	Electrons (s=1/2), quarks (s=1/2), neutrinos (s=1/2)
Statistics	Bose-Einstein statistics	Fermi-Dirac statistics
Algebra	Commutators [φ̂, φ̂†]	Anticommutators {ψ̂, ψ̂†}
Wave Function Symmetry	Symmetric: ψ(x₁,x₂) = +ψ(x₂,x₁)	Antisymmetric: ψ(x₁,x₂) = -ψ(x₂,x₁)
Occupation Number	Unlimited: n = 0, 1, 2, 3, ...	Pauli exclusion: n = 0, 1 only
Condensation	Bose-Einstein condensates possible	No condensation (Pauli blocking)
Required by	Causality + Lorentz invariance	Causality + Lorentz invariance

6.7 Deeper Perspective

💡What Really Enforces Spin-Statistics?

The spin-statistics connection comes from three fundamental principles:

Locality: Spacelike separated measurements can't influence each other (no FTL signaling)
Lorentz invariance: Physics looks the same in all inertial frames
Positive energy: Vacuum is the lowest energy state (stability)

These three axioms, plus the mathematics of spinor representations, uniquely determine that: half-integer spin ⟺ anticommutators.

It's one of the deepest results in physics: the connection between geometry (spin, Lorentz group) and statistics (commutation vs. anticommutation).

⚠️Common Mistakes to Avoid

❌

Mistake:

Thinking spin-statistics is just a postulate we impose by hand

🤔

Why it's wrong:

The connection is derived from fundamental principles, not assumed arbitrarily.

✅

Correct approach:

The spin-statistics theorem is a THEOREM! It's proven from locality + Lorentz invariance + positive energy. We don't choose it—nature forces it on us!

❌

Mistake:

Believing half-integer spin causes anticommutation

🤔

Why it's wrong:

The causal direction is: preserving locality + Lorentz invariance requires specific statistics for each spin.

✅

Correct approach:

It's the opposite direction: fermions (half-integer spin) MUST use anticommutators to preserve causality. If we tried commutators for spin-1/2, we'd violate microcausality!

❌

Mistake:

Forgetting that violating spin-statistics leads to catastrophe

🤔

Why it's wrong:

Wrong statistics breaks fundamental consistency requirements of quantum field theory.

✅

Correct approach:

Wrong statistics → negative norm states OR acausal propagation OR unbounded Hamiltonian. All three are physical disasters!

🎯 Key Takeaways

Spin-statistics is a theorem, not a postulate!
Integer spin (0, 1, 2, ...) → bosons → commutators → Bose-Einstein
Half-integer spin (1/2, 3/2, ...) → fermions → anticommutators → Fermi-Dirac
Enforced by: locality + Lorentz invariance + positive energy
Violations would lead to negative norm states or causality violation
Experimentally verified to extraordinary precision
Explains Pauli exclusion, atomic structure, chemistry, lasers!
Deep connection between geometry (spin) and statistics
Next: Quantize the photon field (spin 1)!

Quantum Field Theory Course

The Spin-Statistics Theorem

🔗Course Connections

📚Prerequisites

Video Lecture

6.1 Statement of the Theorem

🎯 Spin-Statistics Theorem

💡Why This Matters

6.2 Heuristic Argument

Step 1: Locality (Microcausality)

Step 2: Fields as Observables

Step 3: Spinors Are Not Observable!

Step 4: Connection to Spin

6.3 What Happens If You Violate Spin-Statistics?

❌ Scenario 1: Bosonic Quantization of Spin-1/2

❌ Scenario 2: Fermionic Quantization of Spin-0

6.4 Examples in Nature

Integer Spin Bosons

Half-Integer Spin Fermions

6.5 Experimental Tests

Pauli Exclusion Violation Tests

Photon Statistics Tests

6.6 Summary Table

Spin-Statistics Connection

6.7 Deeper Perspective

💡What Really Enforces Spin-Statistics?

⚠️Common Mistakes to Avoid

Mistake:

Why it's wrong:

Correct approach:

Mistake:

Why it's wrong:

Correct approach:

Mistake:

Why it's wrong:

Correct approach:

🎯 Key Takeaways