On structure of the algebra of field operators

Borchers, H. J.

doi:10.1007/bf02745645

Cited by 204 publications

(196 citation statements)

References 7 publications

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“…2 ' I conjug1tion on P(JYJ) (and hence the action of L 0 of the Hilbert space H) is uniquely determined by the condition (14) (d) The mapping J-cp{J} is such that if{/"' (16) for all X E p(R 1 ) and all Y E (J(R 2 ).…”

Section: Basic Assumptions; Discussion Of Notationmentioning

confidence: 99%

On the duality condition for a Hermitian scalar field

Bisognano

Wichmann

1975

Journal of Mathematical Physics

523

638

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A general Hermitian scalar field, assumed to be an operator−valued tempered distribution, is considered. A theorem which relates certain complex Lorentz transformations to the TCP transformation is stated and proved. With reference to this theorem, duality conditions are considered, and it is shown that such conditions hold under various physically reasonable assumptions about the field. A theorem analogous to Borchers’ theorem on relatively local fields is stated and proved. Local internal symmetries are discussed, and it is shown that any such symmetry commutes with the Poincaré

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Section: Basic Assumptions; Discussion Of Notationmentioning

confidence: 99%

On the duality condition for a Hermitian scalar field

Bisognano

Wichmann

1975

Journal of Mathematical Physics

523

638

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“…It will be convenient to have a more explicit description of J c^ d n and the duals d' n~y -}r tnm To this end we generalize somewhat a construction called "space-like symmetrization" in [1]. Define first an equivalence relation between points in R d ' n : …”

Section: And ^Afa^i(t) Implies A^^t) For All I Also L(t) Resp I(t)mentioning

confidence: 99%

“…These are precisely the points X = (x l9 ••-, x n ) 9 where (x i -x j ) 2 = Q for some pairs ij with a i} .=-}-l } the sign of the other (Xi-Xj) 2 and if a =|= a ', then ^ n W C ^^ for some a /7 . 3) £(*>:= u ^ is a closed regular set in the sense of [15], p. 98-100, for all k. Because of 3) we can now decompose r (1) :…”

Section: Proposition (I) F^y C>n Ifandonlyiff=0 (Ii) the Mappings Dmentioning

confidence: 99%

“…The decomposition is unique up to terms with support in U ^H^c^. Suppose /e 9" is such that supp \ We can then write /=#+/* with supp gf}B (1) =0 and supp h contained in the open set Note that T (1) -At each step the decomposition can be done in such a way k that an equicontinuous set of distributions is decomposed into equicontinuous sets. This follows from the proofs of the decomposition theorems pp.…”

Section: Proposition (I) F^y C>n Ifandonlyiff=0 (Ii) the Mappings Dmentioning

confidence: 99%

“…The locality ideal in Borchers 9 tensor algebra [1,2] is the two-sided ideal generated by commutators of test functions with space-like separated supports. Its importance comes from the fact that quantum fields satisfying the requirement of local commutativity can be regarded as Hilbert space representations of the tensor algebra annihilating this ideal.…”

Section: §L Introductionmentioning

confidence: 99%

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On the Locality Ideal In the Algebra of Test Functions for Quantum Fields

Yngvason

1984

Publ. Res. Inst. Math. Sci.

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Some basic properties of the locality ideal in Borchers's tensor algebra are established. It is shown that the ideal is a prime ideal and that the corresponding quotient algebra has a faithful Hilbert space representation. A topology is determined for which the positive cone in the quotient algebra is normal, and it is shown that every w-point distribution satisfying the locality condition is a linear combination of positive functional which also satisfy that condition. §L IntroductionThe locality ideal in Borchers 9 tensor algebra [1, 2] is the two-sided ideal generated by commutators of test functions with space-like separated supports. Its importance comes from the fact that quantum fields satisfying the requirement of local commutativity can be regarded as Hilbert space representations of the tensor algebra annihilating this ideal. Equivalently, such fields define representations of the corresponding quotient algebra. Among other things it will be shown that the states on this algebra separate points, and consequently that it has a faithful Hilbert space representation. For the tensor algebra itself this was first shown in [3]. The question whether this is also true for the quotient algebra was posed in [2], but has remained unsettled till now.The present paper is a sequel to [4, 5 3 6], and we refer to these papers and also to [2] for definitions and further references. Borchers 9 tensor algebra will be denoted by ^; it is the completed tensor algebra over the Schwartz space tf (R d ). The locality ideal 3* c is generated by elements of the form f®g - §®f with /, g^ff(R d ) satisfying the condition f(x)g(y)=Q whenever (x-y) e R d is not space-like. The method used for the investigation of 3 f e and the

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Conservation of estimates in quantum field theory

Bratteli¹

1972

Comm Pure Appl Math

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On structure of the algebra of field operators

Cited by 204 publications

References 7 publications

On the duality condition for a Hermitian scalar field

On the duality condition for a Hermitian scalar field

On the Locality Ideal In the Algebra of Test Functions for Quantum Fields

Conservation of estimates in quantum field theory

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