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Communications in Mathematical Physics

, Volume 8, Issue 4, pp 269–281 | Cite as

Eine scheinbare Abschwächung der Lokalitätsbedingung. II

  • H. -J. Borchers
  • K. Pohlmeyer
Article

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-y2|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):
$$X_1 ,...,X_{n - 1} X_1^i = \cdot \cdot \cdot = X_{n - 1}^i = 0i = 0,3$$
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 )⊂R4i=1,...,n−1 (in the euclidean topology ofR4) and c) a real number α>1 such that for all points (x):x1, ...,xn−1:x i U i (X r ) there are positive constantsC(n){(x)},h(n){(x)} with:
$$\left| {\left\langle {\left[ {A(x_1 )...A(x_{n - 1} ),A(x_n )} \right]} \right\rangle } \right|< C^{(n)} \left\{ {(x)} \right\}\exp \left\{ { - h^{(n)} \left\{ {(x)} \right\}r^\alpha } \right\}forx_n = \left( {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ r \\ \end{array} } \right),r > 1.$$
2. For any natural numbern there exist a) a configurationY(n):
$$Y_2 ,Y_3 ,...,Y_n Y_3^i = \cdot \cdot \cdot = Y_n^i = 0i = 0,3$$
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:Vi(Y i )⊂R4i=2, ...,n (in the euclidean topology ofR4) and c) a real number β>1 such that for all points (y):y2, ...,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)}\y2,y3, ...,y n totally space-like in the order 2, 3, ...,n and
$$\left| {\left\langle {\left[ {A(x_1 ),A(x_2 )} \right]A(y_3 )...A(y_n )} \right\rangle } \right|< C_{(n)} \left\{ {(y)} \right\}\exp \left\{ { - h_{(n)} \left\{ {(y)} \right\}r^\beta } \right\}$$
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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Literatur

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    Pohlmeyer, K.: Commun. math. Phys.7, 80 (1968).Google Scholar
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Copyright information

© Springer-Verlag 1968

Authors and Affiliations

  • H. -J. Borchers
    • 1
  • K. Pohlmeyer
    • 2
  1. 1.Institut für Theoretische Physik der Universität GöttingenGermany
  2. 2.II. Institut für Theoretische Physik der Universität HamburgGermany

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