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The Hall-Wightman Theorem in
Relativistic Quantum Field Theory

The Hall-Wightman Theorem in Quantum Field Theory is a seemingly trivial result that has profound consequences and provides the basis for a rigorous proof of the Spin-Statistics Theorem and consequently the PCT Theorem. Previous attempts to prove the Spin-Statistics Theorem assumed the PCT Theorem or some truncated version of it.

The development of the Hall-Wightman Theorem started with the thesis of D. Hall at Princeton University in 1956. Arthur Strong Wightman then wrote an article entitled "Quantum Field Theory in Terms of Vacuum Expectation Values." He then co-authored with D. Hall in 1958 an article entiitled "A Theorem on Invariant Analytical Functions with Applications to Relativistic Quantum Field Theory," in the Danish journal Matematisk-fysiske Meddelelser udgivet af Det Kongelige Danske Videnskabernes Selskab (Mathematical and Physical Studies published by the Royal Danish Academy of Sciences). This contains what later became known as the Hall-Wightman Theorem.

Hall-Wightman Theorem states that

The vacuum expected value of the product of two fields at two points is analystically continuable to separation distances of the two points including zero.

In symbols, if x and y are the two points in 3D space or 4D spacetime of the field φ(z) and the vacuum state in Dirac notation is |0>: then

<0|φ(x)φ(y)|0> = F(2)(x−y)

where F(2)(z) is Hall and Wightman's notation for a function with two arguments.

(To be continued.)

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