Jost Points, Edge-of-the-Wedge, and Locality

Jost points are real configurations lying inside the extended analytic domain of Wightman functions. At such configurations, locality supplies equality between differently ordered boundary values, and the edge-of-the-wedge theorem promotes equality on an open real set to a common holomorphic continuation. The mechanism is powerful but local: it requires a genuine Jost region and matching distributional boundary values.

Required background. Wightman fields, domains, and axioms supplies local commutativity, and tube domains and complex Lorentz covariance supplies the analytic functions to be joined.

Helpful background. Distributional kernels on manifolds clarifies boundary values, while microcausality and relativistic compatibility gives the physical meaning of local exchange.

Real points of the extended tube

Let $\xi_j=x_j-x_{j+1}$ be successive differences. A real configuration is a Jost configuration when every nonzero combination with nonnegative coefficients is spacelike:

$$\left(\sum_{j=1}^{n-1}\lambda_j\xi_j\right)^2<0 \quad\text{for all}\quad \lambda_j\geq0, \quad \sum_j\lambda_j>0.$$

Equivalently, the closed convex cone generated by the successive differences contains only spacelike vectors apart from the origin. This criterion is stronger than asking only that each adjacent difference be spacelike. It characterizes the relevant real points of the extended tube; see Streater and Wightman 2016, §§ 2-4 and 4-1, pp. 63–73 and 134–136.

Jost configurations form an open set. That openness matters: the identity and edge-of-the-wedge theorems cannot be triggered by equality at an isolated real configuration.

Locality joins orderings

Consider two scalar Wightman orderings that differ by exchanging adjacent fields:

$$W_n(\ldots,x_j,x_{j+1},\ldots), \qquad W_n(\ldots,x_{j+1},x_j,\ldots).$$

They begin as boundary values of holomorphic functions on differently permuted tubes. If $x_j-x_{j+1}$ is spacelike, local commutativity makes their distributional boundary values equal on every open test-function neighborhood that remains spacelike. For fermionic fields, the equality includes the statistics sign and the field labels are exchanged.

The edge-of-the-wedge theorem says, in the relevant distributional form, that holomorphic functions on two wedges with a common real edge and equal distributional boundary values on an open subset are restrictions of one holomorphic function on a connected neighborhood of the union. The needed several-complex-variable statement is given in Streater and Wightman 2016, § 2-5, pp. 74–83. Repeating adjacent exchanges at Jost points relates larger classes of orderings and provides the analytic core of the Jost proofs of CPT and spin–statistics.

The proof has three logically separate inputs: spectral support creates tubes; complex Lorentz covariance makes Jost points accessible from them; locality identifies boundary values. Dropping any one step breaks this particular continuation argument.

A scalar three-point configuration

The application connects directly to scalar two- and three-point functions, where the conformal form is developed.

For three scalar fields, take

$$\xi_1=(0,a,0,0), \qquad \xi_2=(0,0,b,0), \qquad a,b\neq0.$$

Every nonzero nonnegative combination has square $-(\lambda_1a)^2-(\lambda_2b)^2<0$, so this is a Jost point. In a conformal scalar three-point function, the orderings $W_3(x_1,x_2,x_3)$ and $W_3(x_2,x_1,x_3)$ therefore have equal spacelike boundary values for identical bosonic fields. Edge-of-the-wedge joins their tube functions near this real region. What is continued is the distribution with its boundary prescription, not an unqualified pointwise power law; this distinction is important when Lorentzian conformal correlators acquire branch cuts.

Now choose successive differences that are individually spacelike but have a future-timelike positive sum. The point fails the convex-cone criterion and need not lie in the Jost region. Alternatively, suppose two formal analytic expressions agree at one real point but no open set of distributional boundary values is shared. Neither case satisfies the edge-of-the-wedge hypotheses. These are not technical nuisances; they block the theorem.

An independent check of the displayed example is immediate in the $(+---)$ convention: the time component of every positive combination vanishes, while at least one spatial component is nonzero. Thus no hidden null boundary is included.

Converse and failure boundaries

Local commutativity implies the stated boundary equalities, but equality of one selected correlator does not reconstruct operator locality. To infer locality from vacuum distributions one needs the relevant hierarchy of exchange identities and vacuum cyclicity; equality only at Jost points is commonly called weak local commutativity and has a special role in the CPT theorem.

Edge-of-the-wedge also does not erase singularities or establish a maximal analytic domain. It gives a continuation into a neighborhood/envelope determined by its hypotheses. Claims about global sheets, crossing, or singularity locations require additional analysis.

Exercises

Give two spacelike successive differences in $1+1$ dimensions whose positive sum is timelike, showing that pairwise spacelikeness is not the Jost criterion.

Solution

Take $\xi_1=(1,2)$ and $\xi_2=(1,-2)$. Each has square $1-4=-3<0$, but $\xi_1+\xi_2=(2,0)$ has square $4>0$. Hence the cone generated by $\xi_1,\xi_2$ contains a timelike vector, so the configuration is not Jost.

References

• Hall, David, and Arthur S. Wightman. 1957. “A Theorem on Invariant Analytic Functions with Applications to Relativistic Quantum Field Theory.” Matematisk-fysiske Meddelelser, Det Kongelige Danske Videnskabernes Selskab 31 (5): 1–41. Catalog record.
• Streater, Raymond F., and Arthur S. Wightman. 2016. PCT, Spin and Statistics, and All That. Princeton Landmarks in Physics. Princeton University Press. DOI.