Abstract
If for a relativistic field theory the expectation values of the commutator (Ω|[A (x),A(y)]|Ω) vanish in space-like direction like exp {− const|(x-y 2|α/2#x007D; with α>1 for sufficiently many vectors Ω, it follows thatA(x) is a local field. Or more precisely:
For a hermitean, scalar, tempered fieldA(x) the locality axiom can be replaced by the following conditions
1. For any natural numbern there exist a) a configurationX(n):
with\(\left[ {\sum\limits_{i = 1}^{n - 2} {\lambda _i } (X_i^1 - X_{i + 1}^1 )} \right]^2 + \left[ {\sum\limits_{i = 1}^{n - 2} {\lambda _i } (X_i^2 - X_{i + 1}^2 )} \right]^2 > 0\) for all λ i ≧0i=1,...,n−2,\(\sum\limits_{i = 1}^{n - 2} {\lambda _i > 0} \), b) neighbourhoods of theX i 's:U i (X i )⊂R 4 i=1,...,n−1 (in the euclidean topology ofR 4) and c) a real number α>1 such that for all points (x):x 1, ...,x n−1:x i ∈U i (X r ) there are positive constantsC (n){(x)},h (n){(x)} with:
2. For any natural numbern there exist a) a configurationY(n):
with\(\left[ {\sum\limits_{i = 3}^{n - 1} {\mu _i (Y_i^1 - Y_{i{\text{ + 1}}}^{\text{1}} } )} \right]^2 + \left[ {\sum\limits_{i = 3}^{n - 1} {\mu _i (Y_i^2 - Y_{i{\text{ + 1}}}^{\text{2}} } )} \right]^2 > 0\) for all μ i ≧0,i=3, ...,n−1,\(\sum\limits_{i = 3}^{n - 1} {\mu _i > 0} \), b) neighbourhoods of theY i 's:V i(Y i )⊂R 4 i=2, ...,n (in the euclidean topology ofR 4) and c) a real number β>1 such that for all points (y):y 2, ...,y n y i ∈V i (Y i there are positive constantsC (n){(y)},h (n){(y)} and a real number γ(n){(y)∈a closed subset ofR−{0}−{1} with: γ(n){(y)}\y 2,y 3, ...,y n totally space-like in the order 2, 3, ...,n and
for\(x_1 = \gamma _{(n)} \left\{ {(y)} \right\}r\left( {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ 1 \\ \end{array} } \right),x_2 = y_2 - [1 - \gamma _{(n)} \{ (y)\} ]r\left( {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ 1 \\ \end{array} } \right)\) and for sufficiently large values ofr.
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Borchers, H.J., Pohlmeyer, K. Eine scheinbare Abschwächung der Lokalitätsbedingung. II. Commun.Math. Phys. 8, 269–281 (1968). https://doi.org/10.1007/BF01646268
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DOI: https://doi.org/10.1007/BF01646268