# (p.173) Appendix The Separating Corollary and the Axiomatic Proof of the Spin–Statistics Theorem

# (p.173) Appendix The Separating Corollary and the Axiomatic Proof of the Spin–Statistics Theorem

In the Wightman axiomatic proof of the spin–statistics theorem outlined in Section 1.2.1 of Chapter 1, one shows that if the wrong spin–statistics connection is assumed, then the fields are identically zero. This requires a “separating corollary,” which plays an important role in the discussion of axiomatic NQFTs in Chapter 3. The goal of this appendix is to provide an account of this corollary and the role it plays in the axiomatic proof.

In the Wightman axiomatic treatment of RQFTs, one can associate to every bounded region $\mathcal{O}$ of Minkowski spacetime, the polynomial algebra of operators, bounded and unbounded, smeared with test functions with support in $\mathcal{O}$. This algebra is supposed to provide candidates for local measurements confined to $\mathcal{O}$. In the following I will restrict attention to von Neumann algebras consisting of all such bounded operators. One can show that Wightman axiom W4iii (the *spectrum condition* (SC); see Table 3.1 in Section 3.2.1) entails that the vacuum state is *cyclic* for any local (von Neumann) algebra of operators $\mathfrak{R}(\mathcal{O})$ associated with a spatiotemporal region $\mathcal{O}$ of spacetime. This means that the set of states $\left\{\varphi |0;\varphi \in \Re (\mathcal{O})\right\}$ generated by acting on the vacuum with any member of $\mathfrak{R}(\mathcal{O})$ is dense in $\mathcal{H}$.^{1} This is the Reeh–Schlieder theorem (see, e.g., Streater and Wightman, 1964: 138). One next notes the following general result in the theory of operator algebras (see, e.g., Bratelli and Robinson, 1987: 85):

General resultAny cyclic vector for a von Neumann algebra $\mathfrak{R}$ is separating for its commutant ${\mathfrak{R}}^{{}^{\prime}}$.

The commutant ${\mathfrak{R}}^{{}^{\prime}}$ consists of operators that commute with all operators in $\mathfrak{R}$. Wightman axiom W3 (*relativistic local commutativity* (LC); see Table 3.1 in Section 3.2.1) entails that $\mathfrak{R}(\mathcal{O}{)}^{{}^{\prime}}\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}\mathfrak{R}({\mathcal{O}}^{\prime})$, where ${\mathcal{O}}^{{}^{\prime}}$ is the *causal complement* of $\mathcal{O}$, defined as the set of all points *causally separated* (in this context, spacelike separated) from points in $\mathcal{O}$. In words: the commutant of a local algebra associated with a spacetime region $\mathcal{O}$ of Minkowski spacetime is the local algebra associated with the causal complement ${\mathcal{O}}^{\prime}$ of $\mathcal{O}$. Thus, provided the causal complement of any region is non-empty, the general result entails that the vacuum is separating
(p.174)
for any local algebra in Minkowski spacetime. We thus have the following *separating corollary* (Theorem 4-3 in Streater and Wightman, 1964: 139):

Separating corollaryLet $\mathfrak{R}(\mathcal{O})$ be a local algebra of operators associated with an open region $\mathcal{O}$ of spacetime. Suppose (i) the vacuum is cyclic for $\mathfrak{R}(\mathcal{O})$; (ii) the causal complement ${\mathcal{O}}^{\prime}$ is non-empty; and (iii) relativistic local commutativity (W3) holds. Then the vacuum is

separatingfor $\mathfrak{R}(\mathcal{O})$.

Separability of the vacuum means that, given any bounded region $\mathcal{O}$ of Minkowski spacetime, and any operator *ϕ* associated with $\mathcal{O}$, if *ϕ* annihilates the vacuum, $\varphi |0\u3009=0$, then it is identically zero, $\varphi =0$.

The axiomatic spin–statistics theorem (Streater and Wightman, 1964: 150) demonstrates that if the wrong spin–statistics connection is imposed on the fields *ϕ*(*x*), *ϕ*^{†}(*x*), then, provided LC holds, the fields vanish. To obtain this result, we first encode statistics in the fields by assuming the *statistics–locality connection* (StLC) from Section 1.1.2, i.e., we assume bosonic fields commute and fermionic fields anti-commute (Streater and Wightman, 1964: 147). We next assume the *wrong* spin–statistics connection, i.e., we assume half-integer spin fields are bosonic and integer spin fields are fermionic. This entails half-integer-spin fields commute and integer-spin fields anti-commute at spacelike separated distances. Thus,

This was called the *wrong spin–locality connection* (NSpLC) in Section 2.1.4. We now want to show that NSpLC entails that the fields annihilate the vacuum. We can then appeal to the separating corollary to argue that the fields must therefore vanish.

Note first that the vacuum expectation values of the fields are boundary values of complex Wightman functions in the following sense,

where $\mathrm{\zeta}=\mathrm{\xi}-i\mathrm{\eta}$, with $\mathrm{\xi},\mathrm{\eta}\in \text{}\mathbb{R}$, $\mathrm{\xi}=x\phantom{\rule{thinmathspace}{0ex}}-\phantom{\rule{thinmathspace}{0ex}}y$, and *W*, $\stackrel{\u02c6}{W}$ are in general distinct complex Wightman functions of a single argument. The relations (NSpLC) induce the following relations between the corresponding Wightman functions,

(p.175) The PT condition of Section 1.2.1 entails that $\stackrel{\u02c6}{W}$ satisfies

for *ℱ* = (number of conjugate spinor indices in *ϕϕ*^{†}) = (number of indices in *ϕ*) = even/odd if *ϕ* is integer/half-odd integer spin. So equation (A.2) is equivalent to

Combining equations (A.1) and (A.3) yields

In the boundary limit *η* → 0, this corresponds to $\u30080|\varphi (x){\varphi}^{{}^{\u2020}}(y)\text{}|0\u3009+\u30080|{\varphi}^{{}^{\u2020}}(-y)\text{}\varphi (-x)\text{}|\text{}\text{}0\u3009=0$, and this entails $\varphi (x)|0\u3009=0$. The separating corollary then entails that *ϕ*(*x*) must vanish. Thus the axiomatic spin–statistics theorem takes the schematic form (where NSSC denotes the wrong spin–statistics connection):

(i) $[\mathrm{S}\mathrm{t}\mathrm{L}\mathrm{C}\phantom{\rule{thinmathspace}{0ex}}\mathrm{\&}\phantom{\rule{thinmathspace}{0ex}}\mathrm{N}\mathrm{S}\mathrm{S}\mathrm{C}]\Rightarrow \mathrm{N}\mathrm{S}\mathrm{p}\mathrm{L}\mathrm{C}$

(ii) [(PT of Wightman functions) & NSpLC] ⇒ (fields annihilate the vacuum)

(iii) [LC & (fields annihilate the vacuum)] ⇒ (fields vanish)

where (iii) is the separating corollary (under the assumptions of a cyclic vacuum state and non-empty causal complements). Now note that a sufficient condition for PT invariance of Wightman functions is that the fields satisfy restricted Lorentz invariance (RLI) and the SC. We can use this result and combine (ii) and (iii) to obtain:

(ii

^{′}) [RLI & SC & LC & NSpLC] ⇒ (fields vanish)

Since StLC entails LC, (i) and (ii^{′}) can be combined to produce

which is equivalent to

To obtain the form of the axiomatic spin–statistics theorem in Section 1.2.1 of Chapter 1, we have to assume that ~NSSC is equivalent to SSC (i.e., the failure of the wrong spin–statistics connection is just the spin–statistics connection). This essentially is the assumption that “… one puts aside the possibility of laws of (p.176) statistics other than Bose–Einstein or Fermi–Dirac,” as Streater and Wightman (1964: 147) state. We thus obtain

Note that entailment (i) above is not explicitly mentioned by Streater and Wightman (1964: 150). While they explicitly adopt StLC (1964: 147), they identity the “wrong connection of spin with statistics” with the wrong spin–locality connection NSpLC (1964: 150). Greenberg (1998: 146) points out that NSpLC “does not relate directly to particle statistics,” but then goes on to claim “… for that reason this theorem should not be called the spin–statistics theorem.” This appears to miss the implicit entailment (i) in Streater and Wightman’s argument.

## Notes:

(1.) This is to be distinguished from the cyclicity axiom (W4ii) (Table 3.2, Section 3.2.1), which is the requirement that the vacuum state be cyclic for the “global” algebra $\mathfrak{R}$, i.e., the von Neumann algebra of all operators defined on $\mathcal{H}$, as opposed to a local algebra $\mathfrak{R}(\mathcal{O})$ of operators defined only on test functions with support in $\mathcal{O}$.