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cs/0005010
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Note that MATH is anti-monotonic in its first argument, that is, MATH implies MATH, and monotonic in its second argument. Fix a program MATH, a stable model MATH of MATH, and a set of literals MATH such that MATH agrees with MATH. Define MATH and MATH . Let MATH be the least fixed point of MATH. Since MATH agrees with MATH, MATH and MATH. Hence, for varying MATH, the least fixed point of MATH, which is equal to the least fixed point of MATH, is a subset of MATH by REF . In other words, MATH.
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cs/0005010
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Assume that MATH covers MATH and that MATH. Then, MATH and MATH for every MATH, since MATH. Thus, MATH is the least fixed point of MATH from which we infer, by REF , that MATH is a stable model of MATH.
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cs/0005010
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Note that for every atom MATH, the list MATH is traversed at most twice and the lists MATH and MATH are traversed at most once. For every rule MATH, the list MATH is traversed at most twice. To be precise, the list MATH is only traversed in the procedure MATH when MATH or MATH is being set and this can only happen once. The same holds for the lists MATH and MATH. Moreover, the only other place where the list MATH is examined is in the function MATH and there too only once, when MATH is decremented to one. The list MATH is only traversed in MATH and MATH. These functions are, for each rule, called at most once from the procedure MATH. The function MATH is also called from MATH, but then MATH is true and the list is gone through if MATH, which only happens once. Finally, MATH is called from MATH, but only once when MATH.
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cs/0005010
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Note that for every atom MATH, the list MATH is traversed at most twice and the lists MATH is traversed at most once.
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cs/0005010
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Let MATH be the set of atoms that appear as not-atoms in some cycles or that appear in the heads of some choice rules of MATH. Define MATH and MATH . If MATH, then MATH does not appear in the head of a choice rule nor does it appear as a not-atom in MATH . Hence, MATH for any MATH. Now, for any MATH and for any MATH, MATH if and only if MATH. Let MATH and MATH be two stable models of MATH for which MATH holds. If MATH, then MATH implies MATH . Thus, MATH by symmetry. Consequently, MATH implies MATH.
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cs/0005010
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Let MATH. Assume that MATH. Then, there exists an atom MATH and a rule MATH that is not a choice rule such that MATH . Since otherwise MATH, MATH, and consequently MATH by REF . As the atoms in MATH can not appear as not-atoms in the body of MATH, MATH for MATH and MATH. Hence, MATH since MATH is monotonic in its second argument. But then, MATH which is a contradiction. Thus, MATH covers MATH.
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cs/0005010
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We show that completeness is not lost when backjumping is introduced. Since backtracking is chronological, and therefore exhaustive, for a depth of recursion smaller than MATH, we only have to consider the case MATH. Assume that there is no stable model agreeing with MATH and that when MATH returns false, MATH returns true. Furthermore, assume that MATH is a stable model agreeing with the set MATH and that the set covers MATH. Then, MATH returns true for MATH since there is no stable model agreeing with MATH. Take MATH such that MATH or MATH, and let MATH be the smallest number for which MATH returns true and MATH returns false. Then, MATH returns true by REF, IREF, and EREF. Hence, by REF there is no stable model that agrees with MATH and this contradicts the existence of MATH. Thus, there is no stable model agreeing with MATH. We complete the proof by induction on the size of MATH . If MATH covers MATH, if the function MATH returns false and if MATH, then there is no stable model agreeing with MATH. Let MATH, let MATH return false and let MATH. Then, MATH returns false for MATH and MATH, and by induction there is no stable model agreeing with MATH. If MATH returns true, then by the above there is no stable model agreeing with MATH either. If, on the other hand, MATH returns false, then MATH returns false and there is again by induction no stable model that agrees with MATH. Therefore, there is no stable model that agrees with MATH.
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cs/0005010
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Given MATH we can check whether MATH using the oracle MATH a linear number of time. Namely, the language MATH is decided by the function CASE: MATH each atom MATH in MATH in order, most significant first there exists a stable model of MATH that agrees with MATH the least significant atom is in MATH return true return false . Hence, MATH. Deciding the least significant atom of the lexicographically largest satisfying assignment of a Boolean formula is MATH-complete CITE. Thus, MATH is also MATH-complete.
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cs/0005010
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We begin by showing that the problem is in MATH. Assume without loss of generality that all weights are positive. If MATH is the minimize statement MATH then define MATH . Furthermore, if MATH is a set of atoms, then define MATH . Since the oracle MATH is in NP, the procedure MATH in REF shows that the problem of finding an optimal stable model of a logic program is in MATH. The procedure begins with a binary search for the weight of the optimal model. It then constructs the model using a linear number of oracle calls. We prove that the problem is MATH-hard by noticing that the problem NAME Sat can be reduced to Opt. If we are given a set of clauses, each with an integer weight, then NAME Sat is the problem of finding the truth assignment that satisfies a set of clauses with the greatest total weight. The problem NAME Sat is MATH-complete CITE. For each clause MATH with weight MATH, create the rule MATH . For each atom MATH that appears in a clause create the rule MATH. Finally, add the maximize statement MATH . The maximal stable model of this program is the truth assignment of greatest total weight.
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cs/0005010
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Let MATH. Then, MATH . Consequently, MATH . In addition, MATH implies MATH . As MATH implies MATH (we can assume MATH), MATH . It follows that MATH . Hence, MATH and therefore MATH . Thus, MATH . Let MATH. Then, MATH . Hence, MATH is a fixed point of MATH and MATH . Now, MATH as MATH and we have proved that the well-founded model of MATH is a subset of MATH.
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cs/0005010
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Consider the program MATH and assume that MATH is a modular mapping. Then, MATH is unsatisfiable as MATH has no stable models. It follows that also MATH is unsatisfiable. But this implies that MATH has no stable models, which is clearly not the case. Hence, MATH is not modular.
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cs/0005010
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Define MATH . Then, MATH implies MATH, which in turn implies MATH by the monotonicity of MATH. Hence, MATH, and consequently MATH . Now, MATH implies MATH, which by the definition of MATH implies MATH. Thus, MATH. Moreover, for any fixed point MATH, MATH and hence MATH by definition.
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cs/0005012
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CASE: Let MATH stem from MATH. This implies MATH and MATH. For each MATH and MATH, MATH, this implies MATH and hence MATH and MATH are isomorphic. CASE: REF hold by construction. Obviously, MATH stems from MATH and from REF it follows that each interpretation MATH stemming from MATH is isomorphic to MATH, hence REF holds. CASE: Since MATH stems from MATH, MATH implies that MATH is admissible with respect to MATH. If MATH is admissible with respect to MATH, then there is an interpretation MATH stemming from MATH with MATH. Since MATH is isomorphic to MATH, this implies MATH.
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cs/0005012
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For the only if-direction let MATH be an interpretation with MATH and MATH. From REF it follows that the canonical witness MATH is a witness for MATH that is admissible with respect to MATH. For the if-direction let MATH be an witness for MATH that is admissible with respect to MATH. This implies that there is an interpretation MATH stemming from MATH with MATH. For each interpretation MATH that stems from MATH, it holds that MATH due to REF .
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cs/0005012
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MATH is a witness, hence there is an interpretation MATH stemming from MATH. From REF and the fact that MATH satisfies the properties stated in REF it follows that, for each MATH, MATH . Hence, MATH and MATH is admissible with respect to MATH.
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cs/0005012
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The if-direction follows from the definition of ``correct absorption". For the only if-direction, let MATH be a concept and MATH a witness for MATH that is admissible with respect to MATH. This implies the existence of an interpretation MATH stemming from MATH such that MATH and MATH. Since MATH we have MATH and hence the canonical witness MATH is an unfolded witness for MATH.
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cs/0005012
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Trivially, MATH holds. Given an unfolded witness MATH, we have to show that there is an interpretation MATH stemming from MATH with MATH. We fix an arbitrary linearisation MATH of the ``uses" partial order on the atomic concept names appearing on the left-hand sides of axioms in MATH such that, if MATH uses MATH, then MATH and the defining concept for MATH is MATH. For some interpretation MATH, atomic concept MATH, and set MATH, we denote the interpretation that maps MATH to MATH and agrees with MATH on all other atomic concepts and roles by MATH. For MATH, we define MATH in an iterative process starting from an arbitrary interpretation MATH stemming from MATH and setting MATH . Since, for each MATH there is exactly one axiom in MATH, each step in this process is well-defined. Also, since MATH may only restrict the interpretation of atomic roles, MATH for each MATH. For MATH it can be shown that MATH is an interpretation stemming from MATH with MATH. First we prove inductively that, for MATH, MATH stems from MATH. We have already required MATH to stem from MATH. Assume the claim was proved for MATH and MATH does not stem from MATH. Then there must be some MATH such that either REF MATH but MATH or REF MATH but MATH (since we assume MATH to stem from MATH and MATH is the only atomic concept whose interpretation changes from MATH to MATH). The two cases can be handled dually: CASE: From MATH it follows that MATH, because MATH is unfolded. Since MATH stems from MATH and MATH is a witness, REF implies MATH. But this implies MATH, which is a contradiction. CASE: From MATH it follows that MATH because MATH is unfolded. Since MATH stems from MATH and MATH is an witness, REF implies MATH. Since MATH this implies MATH, which is a contradiction. Together this implies that MATH also stems from MATH. To show that MATH we show inductively that MATH for each MATH. This is obviously true for MATH. The interpretation of MATH may not depend on the interpretation of MATH because otherwise REF would imply that MATH uses itself. Hence MATH and, by construction, MATH. Assume there is some MATH such that MATH. Since MATH and only the interpretation of MATH has changed from MATH to MATH, MATH must hold because of REF . But this implies that MATH occurs in MATH and hence MATH uses MATH which contradicts MATH. Thus, we have MATH for each MATH and hence MATH.
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cs/0005012
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In both cases, MATH holds trivially. CASE: Let MATH be a concept and MATH be an unfolded witness for MATH with respect to the absorption MATH. This implies that MATH is unfolded with respect to the (smaller) absorption MATH. Since MATH is a correct absorption, there is an interpretation MATH stemming from MATH with MATH. Assume MATH. Then, without loss of generality, there is an axiom MATH such that there exists a MATH. Since MATH is unfolded, we have MATH and hence REF implies MATH, a contradiction. Hence MATH and MATH is admissible with respect to MATH. CASE: Let MATH be a concept and MATH be an unfolded witness for MATH with respect to the absorption MATH. From MATH we define a new witness MATH for MATH by setting MATH, MATH, and definig MATH to be the function that, for every MATH, maps MATH to the set MATH . It is easy to see that MATH is indeed a witness for MATH and that MATH is also unfolded with respect to the absorption MATH. This implies that MATH is also unfolded with respect to the (smaller) absorption MATH. Since MATH is a correct absorption of MATH, there exists an interpretation MATH stemming from MATH such that MATH. We will show that MATH also holds. Assume MATH, then there is an axiom MATH and a MATH such that MATH but MATH. By construction of MATH, MATH implies MATH because otherwise MATH would hold in contradiction to REF . Then, since MATH is unfolded, MATH, which, again by REF , implies MATH, a contradiction. Hence, we have shown that there exists an interpretation MATH stemming from MATH such that MATH. By construction of MATH, any interpretation stemming from MATH also stems from MATH, hence MATH is admissible with respect to MATH.
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cs/0005018
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By the fact that MATH is MATH-closed and REF , it is sufficient to prove that for all positive literals MATH, all NAME of MATH are finite. Let us consider an atom MATH. If MATH is defined in MATH, then the thesis trivially holds by hypothesis. If MATH is defined in MATH, MATH is bounded with respect to MATH by hypothesis and thus MATH is defined. The proof proceeds by induction on MATH. Base. Let MATH. In this case, by acceptability of MATH, there are no clauses in MATH whose head unifies with MATH and whose body is non-empty. Hence, the thesis holds. Induction step. Let MATH. It is sufficient to prove that for all direct descendants MATH in the NAME of MATH, if MATH is a computed answer for MATH then all NAME of MATH are finite. Let MATH be a clause of MATH such that MATH. Let MATH and for all MATH, let MATH and MATH be a substitution such that MATH is a computed answer of MATH in MATH. We distinguish two cases. If MATH is defined in MATH then the thesis follows by hypothesis. Suppose that MATH is defined in MATH. We prove that MATH is bounded and MATH. The thesis will follow by the induction hypothesis. Let MATH be a substitution such that MATH is ground. By soundness of CITE, there exists MATH such that MATH and MATH is a ground instance of MATH and MATH. Therefore MATH . Since MATH is bounded, we can conclude that MATH is bounded and also that MATH.
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cs/0005018
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Let MATH be a query strongly bounded with respect to MATH and MATH. We prove the theorem by induction on MATH. Base. Let MATH. This case follows immediately by REF , where MATH, MATH is empty and MATH is the class of strongly bounded queries with respect to MATH and MATH, and the fact that a strongly bounded atom is also bounded. Induction step. Let MATH. Also this case follows by REF , where MATH, MATH and MATH is the class of strongly bounded queries with respect to MATH and MATH. In fact, CASE: MATH is acceptable with respect to MATH and MATH; CASE: for all queries MATH, all NAME of MATH are finite, by REF and the inductive hypothesis; CASE: for all atoms MATH, if MATH is defined in MATH then MATH is bounded with respect to MATH, by definition of strong boundedness.
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cs/0005018
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It is sufficient to extend the proof in CITE by showing that if a query MATH is well-moded and MATH is ground then both MATH and MATH are well-moded. This follows immediately by definition of well-modedness. If MATH is non-ground then the query above has no descendant.
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cs/0005018
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Similarly to the case of well-moded programs, to extend the result to general programs it is sufficient to show that if a query MATH is well-typed then both MATH and MATH are well-typed. In fact, by REF , MATH is well-typed and by REF , if the first literal in a well-typed query is negative, then it is not used to deduce well-typedness of the rest of the query.
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cs/0005018
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Let MATH be the class of well-moded queries of MATH. By REF , MATH is MATH-closed. Moreover CASE: MATH is acceptable with respect to a moded level mapping MATH and MATH, by hypothesis; CASE: for all well-moded queries MATH, all NAME of MATH are finite, by hypothesis; CASE: for all well-moded atoms MATH, if MATH is defined in MATH then MATH is bounded with respect to MATH, by REF , since MATH is a moded level mapping. Hence by REF we get the thesis.
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cs/0005018
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Let MATH be the class of well-typed queries of MATH. By REF , MATH is MATH-closed. Moreover CASE: MATH is acceptable with respect to a level mapping MATH and MATH, by hypothesis; CASE: for all well-typed queries MATH, all NAME of MATH are finite, by hypothesis; CASE: for all well-typed atoms MATH, if MATH is defined in MATH then MATH is bounded with respect to MATH, by hypothesis. Hence by REF we have the thesis.
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cs/0005018
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It is a consequence of REF and (the proof of) REF.
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cs/0005018
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Immediate by the definitions of semi-acceptability and strongly boundedness, since we are considering a finest decomposition.
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cs/0005018
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Since we are considering a finest decomposition of MATH, by REF , MATH is acceptable with respect to MATH, while MATH is acceptable with respect to MATH such that if MATH is defined in MATH then MATH else MATH. By REF all NAME of MATH are bounded with respect to MATH and MATH. By definition of boundedness, for all NAME MATH of MATH, MATH is bounded with respect to MATH. By definition of MATH, for all atoms MATH bounded with respect to MATH we have that: if MATH is defined in MATH then MATH is bounded with respect to MATH, while if MATH is defined in MATH then MATH is bounded with respect to MATH and hence with respect to MATH (since MATH). Hence the thesis follows.
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cs/0005026
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Suppose we have a MATH-bit length message and a perfect (non predictable) random encryption key with the same length. It is easy to see that we can establish a bijective relation between the bits of the message and the bits of the key. Let us suppose that the MATH-th bit of the encryption key could change the state of the MATH-th bit of the message; as a result, there are no statistical methods that any potential attacker could hope to use to infer detectable patterns in the message and that could allow the prediction of the final state of the bits in the message from the ciphertext, what is a simple consequence of Information Theory.
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cs/0005026
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We have a MATH-bit length ciphertext MATH obtained using an exclusive-or bitwise operator as described in subsection REFEF (here MATH where MATH denotes the MATH-th bit of the message and MATH labels the MATH-th bit of the encryption key). It is possible to hide an arbitrary message MATH if we define the key as: MATH where MATH.
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cs/0005026
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Suppose we have a ciphertext MATH provided with the message as digital signature. These ciphertext hides a codeword (a random bitmap) MATH and a message field digest MATH that depends on the codeword itself. It is easy to see that we have MATH possible codewords and a message field digest for each codeword field. Applying REF we can find a key for each pair MATH that allows us to obtain the ciphertext MATH.
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gr-qc/0005031
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Let MATH. REF gives that MATH is an open neighbourhood of MATH in MATH. Hence, by the definition of MATH, there exists a smooth timelike curve MATH of MATH having a future endpoint at MATH in MATH. One has MATH by REF . Since MATH is a smooth timelike curve of MATH with a future endpoint at MATH one therefore has MATH by the definition of MATH. There follows MATH. A similar argument gives MATH which implies MATH. Hence one has MATH and similarly MATH. This establishes REF . Let MATH be a future-directed null geodesic generator of MATH and let MATH. Let MATH for MATH and let MATH. One has MATH. By REF and the definition of MATH there exists a smooth future-directed timelike curve MATH of MATH with a future endpoint at MATH. Indeed there exists a sequence of smooth future-directed timelike curves MATH of MATH converging pointwise to MATH in MATH, each with a future endpoint at MATH. Since the smooth timelike curves MATH of MATH converge pointwise to MATH in MATH one has that MATH is a causal curve of MATH with a future endpoint at MATH. Since REF gives MATH and since MATH is a null hypersurface of MATH it follows that MATH is a null geodesic generating segment of MATH. Since MATH and MATH were arbitrary in MATH it follows that MATH is a null geodesic generating segment of MATH. Clearly MATH cannot have a future endpoint MATH otherwise MATH would have a future endpoint at MATH and so would not be a generator of MATH, contrary to hypothesis. Similarly MATH cannot have a past endpoint in MATH. Hence MATH is a null geodesic generator of MATH. The corresponding result for MATH is similar. This establishes REF .
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gr-qc/0005031
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The time reverse of REF gives that MATH cannot be contained entirely in MATH. Let MATH and let MATH. One then has MATH. Let MATH. The set MATH is compact in MATH by REF and MATH is compact in MATH. So by REF the set MATH is a slice of MATH. The set MATH is contained in MATH which does not intersect MATH. Hence MATH is contained in MATH. The set MATH is a slice of MATH. Suppose there exists a point MATH. Then there exists a causal curve MATH of MATH from MATH to MATH, and a timelike curve MATH of MATH from MATH to MATH. Let MATH. Then MATH lies in the topological boundary of MATH in MATH and one has MATH for all MATH. For each MATH the set MATH is non-achronal in MATH and contained in MATH. So, by the asymptotic chronological consistency of MATH, one has that MATH is non-achronal in MATH for each MATH. Since MATH is a timelike curve of MATH from MATH to MATH one thus has MATH for all MATH. REF gives that MATH is closed in MATH. Hence one has MATH for all MATH. But since MATH lies in the topological boundary of MATH in MATH the point MATH must lie in the topological boundary of MATH in MATH. This is impossible because MATH is contained in MATH which is open in MATH. One thus has MATH. Now let MATH be a slice of MATH lying strictly to the past of MATH along the generators of MATH. One has MATH. Hence a point of MATH lies in MATH iff it lies on a causal curve of MATH in MATH from MATH to MATH. Moreover, since REF gives that MATH and MATH are both closed in MATH, the set MATH is closed in MATH. Let MATH. One has MATH. Hence a point of MATH lies in MATH iff it lies on a causal curve of MATH in MATH from MATH to MATH. One thus has MATH. Since MATH has been shown to be closed in MATH it follows that MATH is closed in MATH.
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gr-qc/0005031
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One may assume MATH otherwise one may redefine MATH as MATH. The Lemma then gives that there exists a slice MATH of MATH such that MATH.
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gr-qc/0005031
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It suffices to assume MATH since one may otherwise redefine MATH as MATH. The set MATH is open in MATH and MATH is a slice of MATH. By the time reverse of REF one has that MATH is compact in MATH and such that MATH. The time reverse of REF gives MATH whereby one has MATH. Hence one has MATH. Let MATH be the compact set MATH or, if this is empty, let MATH be any non-empty compact set of MATH. The set MATH is a slice of MATH. Since MATH does not intersect MATH one has MATH. Let MATH be as in the statement of REF and let MATH. Let MATH be a slice of MATH lying strictly to the past of both MATH and MATH along the generators of MATH. One then has MATH and MATH. The set MATH is a slice of MATH. Let MATH and let MATH be a future-directed, past endless timelike curve of MATH having a future endpoint at MATH. Let MATH be the maximal segment of MATH to MATH in MATH. Then MATH is a timelike curve of MATH to MATH. One clearly has MATH. In order to show that MATH cuts MATH it therefore suffices to show that MATH cuts MATH. Suppose first that MATH is past endless in MATH. The time reverse of REF gives that MATH cannot contain all of MATH, whilst the time reverse of REF gives MATH. Hence MATH cuts MATH. Since MATH is past endless and timelike in MATH it follows that MATH cuts MATH. Because MATH has a future endpoint at MATH one thus has that MATH cuts both MATH and MATH and so cuts MATH. Now suppose that MATH has a past endpoint MATH in MATH. The point MATH must lie in the topological boundary of the open set MATH in MATH otherwise it would lie in MATH, in which case MATH would be a past endpoint to MATH in MATH and MATH would be past extendible in MATH. Because MATH is open in MATH the set MATH cannot contain MATH. Since MATH is closed in MATH the set MATH is an open neighbourhood of MATH in MATH and so is cut by MATH. In view of MATH one thus has that MATH cuts MATH. Since MATH has a future endpoint at MATH it follows that MATH cuts both MATH and MATH and so cuts MATH. Since MATH cuts MATH and therefore cuts MATH one has that MATH cuts MATH. Hence MATH cuts MATH.
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gr-qc/0005031
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Let MATH be the associated ASE space-time and MATH an ASE asymptote of MATH. If MATH was a timelike curve of MATH from MATH to MATH then, because MATH is a null hypersurface of MATH, one would have MATH so there would exist a timelike curve MATH of MATH such that MATH. But then MATH would be a timelike curve of MATH from MATH to MATH. This would contradict the achronality of MATH in MATH. Hence MATH is achronal in MATH. Suppose there exists MATH. Since MATH is partially future asymptotically predictable from MATH there exists a slice MATH of MATH such that MATH. In the case MATH, every past endless timelike curve of MATH to MATH would cut MATH, and so MATH would be an open neighbourhood of MATH and so would intersect MATH. This is impossible since MATH is achronal in MATH. So suppose MATH. There exists MATH lying strictly to the past of MATH on the null geodesic generator of MATH through MATH. In view of MATH, every past endless timelike curve of MATH to MATH must cut MATH. But then MATH is an open neighbourhood in MATH of MATH and therefore of MATH and so must intersect MATH. So again one has a contradiction to the achronality of MATH. One thus has MATH. Suppose there exists MATH. Let MATH be a slice of MATH such that MATH. By REF there exists a slice MATH of MATH such that MATH for MATH and MATH. Since MATH is partially future asymptotically predictable from MATH one may assume that MATH is taken sufficiently far to the past in MATH to give MATH. Let MATH. By REF one has MATH. Hence there is a timelike curve MATH of MATH from MATH to some point MATH. In view of MATH one has that MATH is a timelike curve of MATH from MATH to MATH. The set MATH cannot intersect MATH otherwise MATH would be an open neighbourhood of MATH and so MATH would not be achronal in MATH. Since the past endless null geodesic generating segment of MATH to MATH clearly does not cut MATH it follows that MATH does not intersect MATH. But MATH is a neighbourhood of MATH in MATH so one has a contradiction. One now has MATH. Since MATH is a diffeomorphism onto its image it follows that MATH is relatively closed in MATH. Hence MATH is closed in MATH. One has MATH because MATH is closed in MATH and does not intersect MATH. And, because MATH is a conformal isometry onto its image, one has MATH. Hence MATH is empty.
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gr-qc/0005031
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It suffices to assume MATH since one may otherwise redefine MATH as MATH. Let MATH be a slice of MATH and let MATH be a slice of MATH lying strictly to the past of MATH along the null geodesic generators of MATH. Let MATH and MATH. By REF there exists a slice MATH of MATH such that MATH and MATH for MATH. REF gives MATH which, since MATH lies strictly to the future of MATH along the null geodesic generators of MATH, implies that MATH is a neighbourhood of MATH in MATH. REF gives MATH whereby one has that MATH is a neighbourhood of MATH in MATH. Thus there exists an open neighbourhood MATH of MATH in MATH. The set MATH is an open neighbourhood of MATH in MATH. In view of REF one may, by passing to a subset of MATH if necessary, assume MATH. By passing to a further subset of MATH if necessary, one may arrange that each point of MATH is a future endpoint of a timelike curve of MATH in MATH from MATH. REF gives that there exists a slice MATH of MATH such that every past endless timelike curve of MATH to MATH cuts MATH. One may assume that MATH lies strictly to the past of MATH along the null geodesic generators of MATH. Since MATH is partially future asymptotically predictable from MATH there exists a slice MATH of MATH lying strictly to the past of MATH along the generators of MATH such that MATH. Suppose there exists a timelike curve MATH of MATH from MATH to MATH. In view of MATH the set MATH cannot intersect MATH. So, because every past endless timelike curve of MATH to MATH cuts MATH, there exists MATH such that MATH. By the construction of MATH there exists MATH. If MATH did not cut MATH one could concatenate the past endless null geodesic generating segment of MATH to MATH, a timelike curve in MATH from MATH to MATH, and the segment MATH of MATH from MATH to MATH to obtain a past endless causal curve MATH of MATH to MATH which did not cut MATH. For an open neighbourhood MATH of MATH in MATH there would exist MATH such that MATH. But then MATH would be an open neighbourhood of MATH in MATH not intersecting MATH, which gives a contradiction. Thus MATH must cut MATH and indeed there must exist MATH such that MATH. One now has that MATH is a timelike curve of MATH from MATH to MATH. Hence MATH intersects MATH and so intersects MATH. This contradicts the achronality of MATH in MATH. Hence there can be no timelike curve of MATH from MATH to MATH. One thus has MATH. Suppose there exists MATH. Since MATH is relatively open in MATH one can construct an open neighbourhood MATH of MATH in MATH such that every point of MATH is a past endpoint of a timelike curve of MATH in MATH to MATH. Then MATH intersects MATH and so MATH intersects MATH, which is impossible. Hence MATH does not intersect MATH. There follows MATH.
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gr-qc/0005031
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One may, by passing to a subset of MATH if necessary, assume MATH. Suppose, for the purpose of obtaining a contradiction, that MATH is non-empty. Then MATH is non-empty and so is MATH. By REF there exists a slice MATH of MATH such that MATH. Since MATH is a non-empty proper subset of MATH it follows that MATH is a non-empty proper subset of MATH. Hence there exists MATH. There exists a null geodesic generator MATH of MATH to MATH having either a past endpoint in MATH or no past endpoint in MATH. In the former case MATH could not be a null geodesic generating segment of MATH because it would have a past endpoint in MATH. In the latter case MATH could not be a null geodesic generating segment of MATH because it would then cut MATH. Hence one has MATH. Suppose MATH were past endless in MATH. Then MATH would be a past endless causal curve of MATH in MATH. Let MATH lie strictly to the future of MATH along the null geodesic generator of MATH though MATH. Then one could deform MATH to the future in MATH so as to give a past endless timelike curve MATH of MATH to MATH in MATH. Clearly MATH could not intersect MATH because MATH is achronal in MATH. Since MATH is timelike curve of MATH to MATH in MATH it follows that there would exist a neighbourhood of MATH in MATH that did not intersect MATH. This would be contrary to the future asymptotic predictability of MATH from MATH. Thus MATH must have a past endpoint in MATH at MATH. Consequently there exists a null geodesic MATH of MATH such that MATH is the unique maximal segment of MATH in MATH. For each MATH, every open neighbourhood of MATH in MATH intersects both MATH and MATH. One thus has MATH and hence that MATH is a null geodesic generator of MATH. Since MATH has a past endpoint at MATH in MATH it follows that MATH has a past endpoint at MATH in MATH. Since MATH has a future endpoint at MATH in MATH it follows that MATH is future endless and future complete in MATH. One may assume that MATH is an affine future-directed null geodesic of MATH of the form MATH. Let MATH and MATH be null normal fields to MATH along a relative open neighbourhood MATH of MATH in MATH, normalised such that MATH, with MATH. The induced metric on MATH is given by MATH, whilst MATH and MATH are null second fundamental forms of MATH along MATH. By the definition of a closed trapped surface one has MATH and MATH along MATH. The vector field MATH along MATH defines a congruence of future endless affine null geodesics of MATH from MATH with tangents that coincide with MATH along MATH. Let MATH also denote the tangents to these null geodesics. Each tangent vector to MATH at MATH may be NAME propagated along MATH with respect to MATH to yield a vector field MATH along MATH. From vanishing torsion one has MATH and hence that MATH satisfies the defining equation MATH for a NAME field along MATH. Note that MATH is orthogonal to MATH at MATH and satisfies MATH along MATH, and so is orthogonal to MATH along MATH. One may parallelly propagate the vector MATH along the integral curves of MATH and so define MATH along these curves. Then MATH is a projection operator such that MATH. One thus has that MATH satisfies MATH for MATH now defined all along MATH. One also has MATH. The expansion and shear tensors of the vector fields MATH along MATH may be expressed as MATH and MATH respectively, where MATH is the scalar expansion. Then, defining MATH, one has that MATH satisfies the NAME equation MATH . By means of REF one thus has MATH for all MATH such that MATH. Hence there exists MATH such that MATH. Thus MATH is conjugate to MATH at MATH and so there exists a NAME field MATH along MATH which is non-zero and tangent to MATH at MATH and such that MATH. One may, by an adaptation of the technique of HE REF , use the vector field MATH to construct a timelike curve of MATH from MATH to MATH for any MATH. But this is impossible because MATH is a generator of MATH. This establishes the required contradiction.
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gr-qc/0005050
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CASE: Suppose MATH is a photon surface. Let MATH and let MATH be null. There exists an affine null geodesic MATH of MATH such that MATH. One has MATH along MATH. At MATH this gives MATH. CASE: Let MATH. By REF one has MATH null MATH. Let MATH be an orthonormal basis for MATH with MATH timelike and MATH, MATH spacelike. Any null MATH, normalized such that MATH, has components MATH with respect to MATH for some MATH. A calculation gives MATH . This must vanish for all MATH. One thus has MATH and MATH. Since MATH is trace-free one must also have MATH. There follows MATH. CASE: For any curve in MATH with null tangent MATH one has MATH where MATH denotes covariant differentiation in MATH with respect to MATH. The second term on the right of MATH vanishes by hypothesis. If MATH is tangent to an affine null geodesic of MATH then the term on the left of MATH also vanishes and so MATH is tangent to an affine null geodesic of MATH. CASE: Let MATH and let MATH be null. Let MATH be an affine null geodesic of MATH such that MATH. Then, by REF , MATH is an affine null geodesic of MATH such that MATH, MATH.
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gr-qc/0005050
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Let MATH be the induced Lorentzian MATH-metric on MATH and, for each MATH, let MATH be the induced Riemannian MATH-metric on MATH. The expansion of MATH in MATH is given by MATH where the covariant derivative is that of MATH. Since MATH is both shear-free and vorticity-free in MATH, the NAME equation for MATH in MATH assumes the form MATH where MATH is the NAME tensor of MATH. From first principles one has MATH where MATH is the second fundamental form of each MATH in MATH. Since MATH is shear-free and vorticity free in MATH one has MATH. The second fundamental form of MATH admits the canonical decomposition MATH. REF therefore give MATH which combine to yield MATH . One may now substitute for the second term on the right of REF to obtain MATH . From first principles one has MATH, and the NAME theorem gives MATH. Substituting for MATH and MATH in REF one obtains MATH . This agrees with REF iff MATH. Construct, for the tangent bundle MATH of MATH, an orthonormal basis field of the form MATH, with MATH and MATH unit spacelike. With respect to this basis one has MATH . By spherical symmetry the vector field MATH must be proportional to MATH. Hence one has MATH. The vanishing of MATH is thus equivalent to the vanishing of MATH. One has MATH iff MATH is a photon surface.
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gr-qc/0005050
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By spherical symmetry, and since MATH is unit timelike, the vector field MATH must be proportional to MATH. Hence it suffices to show that MATH is a photon surface iff along MATH one has MATH or equivalently MATH . Construct for MATH a local orthonormal basis field of the form MATH. With respect to this basis field the components of MATH form a diagonal matrix with MATH . REF is thus equivalent to MATH which is in turn equivalent to MATH. In view of the trace-free property of MATH, REF is thus equivalent to MATH. From REF one has that MATH holds along MATH iff MATH is a photon surface.
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gr-qc/0005050
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Note that the unit future-directed timelike tangent field MATH along MATH in REF is proportional to the restriction to MATH of the Killing field MATH. Suppose first that there exists a MATH-invariant MATH-sphere MATH such that REF holds. The quantities MATH, MATH and MATH remain constant as MATH is mapped along the flow lines of the Killing field MATH. So they also remain constant as they are mapped along the flow lines of MATH. Hence REF holds with the term on the left and the first term on the right both zero. Thus, by REF , MATH is a photon surface of MATH. By REF is MATH-invariant. For the converse, suppose that MATH is a MATH-invariant photon surface of MATH. Then REF holds for every MATH-invariant MATH-sphere MATH. Since MATH induces groups of local isometries, the area MATH of MATH is independent of the parameter MATH. Hence the term on the left and the first term on the right of REF both vanish and one obtains REF .
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gr-qc/0005050
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Since MATH is both spherically symmetric and static, the surface MATH is of the form MATH . The unit spacelike normal to MATH is therefore given by MATH for MATH . The second fundamental form of MATH is given by MATH . The vector fields MATH form an orthonormal frame field along MATH, with MATH and MATH unit spacelike and MATH unit timelike. One has MATH . REF holds iff MATH is proportional to MATH and hence iff MATH . This is equivalent to MATH which is in turn equivalent to REF .
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gr-qc/0005050
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Fix MATH and let MATH be the outward future-directed null normal field along each MATH, MATH, normalized such that MATH, where MATH is the outward radial unit tangent to MATH . Since MATH is parallelly propagated along each of the geodesic integral curves of MATH, one has that MATH is a well-defined, nowhere-zero null vector field along MATH. For MATH the vector field MATH has the form MATH for MATH . The expansion of MATH is given by MATH . The condition that MATH is marginally outer trapped therefore implies MATH . The non-negativity of MATH and MATH gives, by means of REF , that MATH and MATH are non-decreasing functions of MATH. Thus REF holds iff at least one of MATH holds. Suppose the first of REF fails. Then the second must hold and one has MATH . From the boundedness of MATH and MATH on MATH one has, by means of REF , that MATH is finite. This is incompatible with the second of REF . Hence the first of REF must hold. Let MATH be the left side of REF . By the non-negativity of MATH and the first of REF one has MATH. By REF and l'Hôpital's rule one has MATH and hence MATH. Hence there exists some MATH such that MATH. The hypersurface MATH is a MATH-invariant photon surface of MATH.
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gr-qc/0005050
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By REF with REF one has MATH. By REF one therefore has MATH. The left side of REF is thus bounded from below by MATH. This is positive by REF . The left side of REF is therefore non-vanishing for all MATH.
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gr-qc/0005050
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Let MATH be the left side of REF . Since MATH and MATH are MATH, piecewise MATH functions of MATH, one has by REF that MATH is a MATH, piecewise MATH function of MATH. The function MATH is then a piecewise MATH function of MATH which, by means of of the NAME REF , is given by MATH . For MATH such that MATH REF reduces to MATH . From REF one has MATH, whence by REF one has MATH. Thus REF gives MATH. By REF and l'Hôpital's rule one has MATH and hence MATH. Since it has been established that MATH is negative for all MATH such that MATH, one must therefore have MATH for all MATH. Hence the space-time can contain no MATH-invariant timelike photon surfaces.
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hep-th/0005002
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Let MATH be a state of ghost number MATH and MATH-invariant (MATH). Assuming that the filtration is bounded, we write MATH where MATH. Then, MATH . Each parenthesis vanishes separately since they carry different MATH. So, MATH. The MATH-cohomology is trivial by assumption, thus MATH. But then MATH, which is cohomologous to MATH, has no MATH piece. By induction, we can eliminate all MATH, so MATH; MATH is actually MATH-exact.
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hep-th/0005002
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FGZ's filtration is originally given for the MATH flat spacetime as MATH . The filtration itself does not require MATH; this filtration can be naturally used for MATH, replacing MATH with MATH. Then, the modified filtration assigns the following degrees to the operators: MATH . The operator MATH satisfies REF and the degree of each term in MATH is non-negative. Because the eigenvalue of MATH is bounded below from REF , the total number of oscillators for a given mass level is bounded. Thus, the degree for the states is bounded for each mass level. Note that the unitarity of the compact CFT MATH is essential for the filtration to be bounded. The degree zero part of MATH is given by MATH . We break MATH as follows: MATH . The NAME spaces MATH, MATH and MATH are decomposed according to the ghost number MATH: MATH . From REF , the differentials act as follows: MATH and MATH. Thus, MATH and MATH are complexes with differentials MATH and MATH. Note that MATH is the differential for MATH as well as for MATH. Then, the NAME formula REF relates the cohomology group of MATH to those of MATH and MATH: MATH . Later we will prove the following lemma: MATH if MATH and MATH. Then, REF reduces to MATH which leads to MATH for MATH because MATH. The cohomology group MATH is not exactly what we want. However, NAME and NAME have shown that MATH . See pages REF of Ref. CITE. Thus, MATH . We will later prove the NAME duality theorem, MATH REF . Therefore, MATH . This is the vanishing theorem for MATH.
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hep-th/0005002
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The number of states of the NAME space MATH and that of the NAME module MATH are the same for a given level MATH. Thus, the NAME module furnishes a basis of the NAME space if all the states in a highest weight representation, MATH are linearly independent, where MATH. This can be shown using the NAME determinant. Consider the matrix of inner products for the states at level MATH: MATH . The NAME determinant is then given by MATH where MATH is a positive constant and the multiplicity of the roots, MATH, is the partition of MATH. The zeros of the NAME determinant are at MATH where MATH . For MATH, MATH so that MATH. Thus, the states MATH are linearly independent if MATH.
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hep-th/0005002
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Using REF , a state MATH can be written as MATH where MATH and MATH. Note that the states in MATH all have nonpositive ghost number: MATH. We define a new filtration degree MATH as MATH which corresponds to MATH . The algebra then determines MATH (for MATH) from the assignment. The operator MATH satisfies REF and the degree of each term in MATH is non-negative. The degree zero part of MATH is given by MATH . Since we want a bounded filtration, break up MATH according to MATH eigenvalue MATH : MATH where MATH . Note that MATH is finite dimensional since MATH satisfies MATH . Thus, the above filtration is bounded for each MATH. We first consider the MATH-cohomology on MATH for each MATH. Define an operator MATH such as MATH where MATH means that the term is missing (When MATH, MATH). Then, it is straightforward to show that MATH . The operator MATH is called a homotopy operator for MATH. Its significance is that the MATH-cohomology is trivial except for MATH. If MATH is closed, then MATH . Thus, a closed state MATH is actually an exact state if MATH. Therefore, the MATH-cohomology is trivial if MATH since MATH. And now, again using REF , the MATH-cohomology MATH is trivial if MATH. Because MATH, we can define MATH . Furthermore, the isomorphism MATH can be established. Consequently, MATH if MATH.
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hep-th/0005002
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At each mass level, states MATH in MATH are classified into two kinds of representations: BRST singlets MATH and BRST doublets MATH, where MATH. The ghost number of MATH is the ghost number of MATH plus REF. Therefore, MATH causes these pairs of states to cancel in the index and only the singlets contribute: MATH . We have used the vanishing theorem on the last line.
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hep-th/0005002
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At a given mass level, the matrix of inner products among MATH takes the form MATH . We have used MATH, MATH and MATH. If MATH were degenerate, there would be a state MATH which is orthogonal to all states in MATH. Thus, the matrix MATH should be nondegenerate. (Similarly, the matrix MATH should be nondegenerate as well.) So, a change of basis MATH sets MATH. Finally, going to a basis, MATH the inner product MATH becomes block-diagonal: MATH . Therefore, BRST doublets again make no net contribution: MATH . This proves REF .
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hep-th/0005002
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We prove REF by explicitly calculating the both sides. In order to calculate the left-hand side of REF , take an orthonormal basis of definite MATH, MATH [the basis REF ], MATH and an orthonormal basis of MATH. Then, MATH. Similarly, for the right-hand side, take an orthonormal basis of definite MATH, MATH, MATH and an orthonormal basis of MATH. From these relations, the left-hand side of REF becomes MATH . The right-hand side becomes MATH . This proves REF .
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hep-th/0005023
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The cohomology groups with coefficient in the constant sheaf MATH on homotopic paracompact topological spaces are isomorphic CITE. If MATH is a vector bundle, its base MATH is a strong deformation retract of the infinite-order jet space MATH. To show this, let us consider the map MATH . A glance at the transition functions REF shows that, given in the coordinate form, this map is well-defined if MATH is a vector bundle. It is a desired homotopy from MATH to the base MATH which is identified with its image under the global zero section of the vector bundle MATH.
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hep-th/0005023
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Local exactness of a vertical complex on a coordinate chart MATH, MATH on MATH follows from a version of the NAME lemma with parameters (see, for example, CITE). We have the the corresponding homotopy operator MATH where MATH. Since MATH is a vector bundle, it is readily observed that this homotopy operator is globally defined on MATH, and so is the exterior form MATH.
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hep-th/0005057
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MATH being a bounded NAME set, the same is true of its closure MATH, so that, due to compactness and convexity of MATH, there exists a number MATH for which the relation MATH is satisfied, where MATH denotes the sum MATH with MATH terms. The spectrum condition then entails: MATH . Note, that in the derivation of this result compactness of MATH is needed to ensure that the distance between MATH and MATH is positive; other shapes of MATH are possible as long as convexity and the condition MATH are preserved, e. g. wedges in MATH. For arbitrary MATH all the operators MATH, MATH, belong to MATH whilst their product MATH is an element of MATH, hence by REF MATH . Now, CITE states that for any MATH and any MATH where MATH is the orthogonal projection onto the intersection of the kernels of MATH-fold products MATH for arbitrary MATH, MATH is compact and the supremum extends over all unit vectors MATH. According to REF MATH if we take MATH, so that the following estimate, uniform in MATH, is a consequence of REF combined with almost locality of MATH (compare REF): MATH . The positive operators MATH thus constitute an increasing net which is bounded in MATH. According to CITE this net has a least upper bound in MATH, which is its MATH-strong limit MATH and satisfies MATH . For MATH this is the desired estimate REF. The MATH-weak topology of MATH is induced by the positive normal functionals of the space MATH, so that integrability of MATH in the MATH-weak topology is implied by integrability of the functions MATH for any MATH. Now, given any compact subset MATH of MATH, there holds the estimate MATH and, as a consequence of the Monotone Convergence Theorem CITE, the functions MATH indeed turn out to be integrable for any MATH. Thus the integral of the mapping MATH with respect to the MATH-weak topology exists (compare CITE) and, through an application of NAME 's Dominated Convergence Theorem CITE, is seen to be the MATH-weak limit of the net of operators MATH which coincides with the MATH-strong limit MATH established above. Formally MATH which is the last of the above assertions.
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hep-th/0005057
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CASE: By partition of unity (compare CITE), applied to elements of MATH which have arbitrary energy-momentum transfer in MATH, any MATH can be written as a finite sum MATH where the MATH belong to MATH and the operators MATH have energy-momentum transfer in compact and convex subsets MATH of MATH. Since MATH we infer MATH so that by REF the increasing net MATH turns out to be bounded, having a least upper bound in MATH that is its MATH-strong limit MATH. Making again use of the above order relation for MATH one arrives at MATH for any MATH and any MATH, where the right-hand side of this relation is integrable as was shown in the proof of REF . Then the reasoning applied there establishes the MATH-weak integrability of MATH together with the relation MATH CASE: Consider MATH with MATH. By polarization MATH where MATH for MATH, and according to REF MATH converges MATH-strongly to MATH . Now, let MATH be a normal functional on MATH. By polar decomposition (compare CITE) there exist a partial isometry MATH and a positive normal functional MATH subject to the relation MATH, such that MATH, allowing for the following estimate (MATH arbitrary): MATH where we made use of the fact that MATH for any MATH. Now, from the first part of this Proposition we infer that it is possible to integrate the above expression over all of MATH to get for any MATH the estimate MATH . Note, that the normal functionals MATH and the MATH-weak integrals commute due to CITE. Taking the infimum with respect to MATH one finally arrives at MATH . This relation is valid for any normal functional in MATH, so that the MATH-weak integrability of MATH is established, the relation MATH being an immediate consequence (compare the proof of REF ). Another fact implied by REF is the estimate MATH . Since any MATH is a linear combination of operators of the form MATH, the above relations REF through REF are easily generalized to establish the second part of the Proposition.
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hep-th/0005057
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For the MATH-seminorms on MATH the assertion follows from the order relation for operators in MATH. Let MATH belong to the left ideal MATH, then MATH which by REF has the consequence MATH . This relation extends by continuity of the seminorms to all of MATH. In case of the MATH-topologies, note that for any NAME set MATH the functional MATH, defined through MATH, belongs to MATH if MATH does. From this we infer, since moreover MATH implies MATH, that MATH for any MATH and thus, by REF, that MATH, a relation which by continuity of the seminorms is likewise valid for any operator in the completion MATH.
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hep-th/0005057
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CASE: Note, that we can define a linear subspace MATH of MATH consisting of all those operators MATH which fulfill MATH for any bounded NAME set MATH. On this space the mappings MATH act as seminorms whose restrictions to MATH coincide with MATH (compare the Remark following REF ). Now let MATH be arbitrary. Given a bounded NAME set MATH we can then find a sequence MATH in MATH satisfying MATH . The second equation implies MATH so that NAME 's Dominated Convergence Theorem can be applied to get for any functional MATH and any compact MATH . According to REF each term in the sequence on the right-hand side is majorized by the corresponding MATH and this sequence in turn converges to MATH by assumption, so that in passing from MATH to MATH and to the supremum over all MATH we get MATH . This final estimate shows, by arbitrariness of MATH and the selected MATH, that MATH is a subspace of MATH and, from MATH, it eventually follows that for all these MATH and MATH CASE: The proof of the second part follows the same lines of thought. We introduce the subspace MATH consisting of operators MATH satisfying MATH for any bounded NAME set MATH and furnish it with the locally convex topology defined by the seminorms MATH. An arbitrary MATH is, for given MATH, approximated by a sequence MATH with respect to the norm and the MATH-topology. As above one has MATH and infers MATH . This establishes, by arbitraryness of MATH and MATH, that MATH, and the equation MATH implies that for these MATH and MATH .
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hep-th/0005057
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CASE: For any MATH and arbitrary MATH the relation MATH leads to the estimate MATH for any MATH and thus, by REF and the notation of the proof of REF , to MATH . This shows that MATH belongs to MATH and at the same time that the seminorm MATH (on MATH) can be replaced by MATH to yield REF. CASE: Let MATH be a normal functional on MATH with MATH. By polar decomposition there exist a partial isometry MATH and a positive normal functional MATH with MATH such that MATH. Then MATH for any MATH and the method used in the proof of REF can be applied to get, in analogy to REF, MATH where we made use of REF. According to the notation introduced in the proof of REF this result expressed in terms of the seminorm MATH on MATH reads MATH from which we infer, as in the first part of the present proof, not only that MATH is an element of MATH but also that MATH can be substituted by MATH to give REF.
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hep-th/0005057
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CASE: According to REF , we have for any MATH whereas the reverse inequality is a consequence of REF . This proves the assertion. CASE: Note, that MATH is invariant under the operation of taking adjoints defined by MATH with MATH, MATH, for any linear functional MATH on MATH. Thus MATH for any MATH (compare the proof of REF ), which is sufficient to establish both of the assertions.
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hep-th/0005057
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MATH as well as its intersection MATH with the positive cone MATH are invariant under the mapping MATH defined by MATH for any unitary operator MATH and any linear functional MATH on MATH. CASE: Now, MATH for any MATH. This implies MATH for any MATH and any MATH, henceforth MATH . Therefore the introductory remark in combination with REF yields for any MATH: MATH which, as in the proof of REF , establishes the assertions. CASE: The same argument applies to the seminorm MATH, so that for MATH .
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hep-th/0005057
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CASE: Note, that continuity of the mapping MATH with respect to the locally convex space MATH is equivalent to its continuity with respect to each of the topologizing seminorms MATH. Let the NAME subset MATH of MATH be arbitrary but fixed. We shall first consider the special point MATH and restrict attention to an operator MATH having energy-momentum transfer MATH which, under transformations from a sufficiently small neighbourhood MATH of the neutral element MATH, stays bounded in a compact and convex subset MATH of MATH. This means that all operators MATH, MATH, have energy-momentum transfer in the common set MATH and relation REF applies to the differences MATH yielding MATH . An estimate for the integrand on the right-hand side can be based on relation REF, requiring suitable approximating nets of local operators for MATH. Given MATH there exists a neighbourhood MATH of MATH such that MATH for MATH, and if MATH is an approximating net of local operators for MATH, then MATH for any MATH and MATH. Now MATH holds for any MATH, so that the operators MATH, MATH, constitute the large radius part of approximating nets for each of MATH, MATH, subject to the bound MATH which is independent of MATH. Then, according to the remark following REF , there exist approximating nets MATH for the almost local operators MATH that fulfill the estimates MATH and, for MATH, MATH, where in view of REF the second inequality amounts to MATH for suitable MATH. Making use of relation REF in the same remark we arrive at MATH for any MATH, where MATH and MATH denote the characteristic functions pertaining to the compact ball of radius MATH in MATH and its complement, respectively. The above relation REF holds for any MATH, and its right-hand side turns out to be an integrable majorizing function for the mapping MATH irrespective of MATH, if MATH. Another consequence of REF is that the function REF converges pointwise to REF on MATH in the limit MATH due to strong continuity of the automorphism group MATH. Therefore we can apply NAME 's Dominated Convergence Theorem to the integral on the right-hand side of REF, evaluated for any sequence MATH approaching MATH and infer MATH . Since MATH as a topological space satisfies the first axiom of countability, this suffices to establish continuity of the mapping MATH in MATH with respect to the MATH-topology. An arbitrary operator MATH can be represented as MATH where MATH comply with the above assumptions on MATH and the operators MATH belong to the quasi-local algebra MATH for any MATH. According to REF we have MATH where the right-hand side vanishes in the limit MATH due to the preceding result and strong continuity of the group MATH. Thus the mapping MATH turns out to be continuous in MATH with respect to MATH for arbitrary MATH. The restriction to the specific point MATH is inessential in the last step since for arbitrary MATH one has MATH explicitly showing that continuity of MATH in MATH is equivalent to continuity of MATH in MATH with respect to any of the seminorms MATH, where MATH belongs to MATH. CASE: Continuity of a mapping with values in the locally convex space MATH is equivalent to its continuity with respect to all seminorms MATH. The problem at hand thus reduces to the one already solved in the first part, if one takes into account the shape of general elements of MATH according to REF . CASE: According to REF we have to show that for any vacuum annihilation operator MATH the mapping MATH is differentiable in the locally convex space MATH and has derivatives coinciding with those existing in the uniform topology by assumption. Let MATH be given and consider the local chart MATH around MATH. Due to the presupposed differentiability of the mapping MATH with respect to the uniform topology the corresponding residual term at MATH with respect to MATH is given by MATH using the notation MATH for elements of MATH, and satisfies the limit condition MATH . To prove the assertion it has to be shown that REF stays true when the norm is replaced by any of the seminorms MATH. Now, according to the Mean Value REF , we have for small MATH where the integral is to be understood with respect to the norm topology of MATH. Thus the residual term REF can be re-written as MATH where in the last REF is used to represent the linear operator MATH in terms of partial derivatives of MATH which can be expressed by means of analytic functions MATH on MATH and NAME transformed derivatives MATH of MATH (MATH is the dimension of the NAME group). As a consequence of the first statement of this proposition, the integrand on the right-hand side is continuous with respect to all seminorms MATH, so that the integral exists in the complete locally convex space MATH. By CITE this leads to the following estimate for the residual term MATH where evidently the right-hand side vanishes in the limit MATH. Thus REF for differentiability of mappings with values in a locally convex space is fulfilled, and according to the counterpart REF the derivatives of MATH with respect to both the uniform and locally convex topologies on MATH coincide.
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hep-th/0005057
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CASE: By REF is an integrable majorizing function for the integrand of REF, so MATH exists as a NAME integral in MATH. The same holds true for the integrals constructed by use of an approximating net MATH for the almost local operator MATH: MATH . Due to compactness of MATH, these operators belong to the local algebras MATH pertaining to standard diamonds in MATH which have each a MATH-dimensional basis of radius MATH where MATH and MATH are suitable positive constants. Now, MATH so that we arrive at the estimate (MATH is the measure of the compact set MATH) MATH which holds for any MATH. Due to almost locality of MATH, the right-hand side vanishes in the limit of large MATH, so that the operator MATH itself turns out to be almost local: MATH with approximating net MATH. Let MATH denote the energy-momentum transfer of the vacuum annihilation operator MATH, then, by the NAME Theorem CITE, the following equation is valid for any MATH . In the special case MATH, MATH the NAME transform of MATH, the inner integrals on the right-hand side vanish for any MATH so that we infer MATH which shows that the energy-momentum transfer of MATH is contained in the compact subset MATH of MATH. Therefore MATH is indeed a vacuum annihilation operator from MATH. Finally, infinite differentiability with respect to the uniform topology of the mapping MATH has to be established. By REF is infinitely often differentiable with respect to the NAME group, which implies that likewise all the operators MATH belong to MATH for any MATH. Their residual terms at MATH with respect to the canonical coordinates MATH of the first kind, as introduced in CITE, can, using the notation MATH for the transformations in MATH, be expressed by MATH where the last equation stems from an application of the Mean Value REF , which holds true for small MATH. By REF the term MATH on the second line is continuous in MATH, so that it is possible to multiply REF with the function MATH and subsequently integrate over its compact support MATH. Taking into account that each of the automorphisms MATH is uniformly continuous, thus commuting with NAME integrals, this yields MATH which has the shape of a residual term for MATH at MATH. Now, the operator-norm of MATH can be estimated according to REF by MATH which, due to continuity of MATH and MATH with respect to the operator-norm topology, is majorized on the compact set MATH by a constant MATH. As a consequence of the last equation in REF we then get for any MATH and small MATH the bound MATH which is integrable over MATH by assumption; restricting furthermore attention to sequences MATH converging to MATH, we see that the left-hand side of REF converges pointwise to MATH. With this information at hand it is possible to apply NAME 's Dominated Convergence Theorem CITE to the left-hand side of REF to get MATH which is sufficient to establish REF for differentiability of the mapping MATH at MATH. The linear operator defining the corresponding derivative is according to the right-hand side of REF in connection with REF given by MATH where MATH are functions from MATH with compact support MATH. Since MATH is invariant under differentiation we conclude from the first two paragraphs of the present proof and the above considerations that the partial derivatives MATH are again almost local vacuum annihilation operators which belong to MATH. Thus by induction, repeatedly using these methods, MATH is seen to be an element of MATH with almost local derivatives of any order, i. CASE: MATH. CASE: By REF the mappings MATH and MATH are continuous with respect to the uniform topology and all the MATH- and MATH-topologies, respectively, staying bounded on the compact set MATH. This implies their measurability in the locally convex spaces MATH and MATH together with the fact that their product with the integrable function MATH is majorized in each of the norm and seminorm topologies by a multiple of MATH. As a consequence the integrals MATH and MATH exist in the complete locally convex spaces MATH and MATH, respectively, and REF is an immediate upshot CITE.
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By translation invariance of the norm MATH as well as of the seminorms MATH (compare REF ) the (measurable) integrand on the right-hand side of REF is majorized by the functions MATH and MATH for any bounded NAME set MATH. These are NAME and therefore MATH exists as a unique element of MATH, satisfying the claimed estimates REF. Next, we consider an arbitrary function MATH. By NAME 's Theorem CITE and translation invariance of NAME measure MATH where the term in brackets on the right-hand side of the last equation is the convolution product MATH of MATH and MATH. Its NAME transform MATH is given by MATH (compare CITE), so that this function vanishes if MATH and MATH have disjoint supports. Therefore MATH entails MATH and this shows that the NAME transform of MATH has support in MATH, which henceforth contains the energy-momentum transfer of MATH.
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hep-th/0005057
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First we consider the special case of two elements MATH and MATH in MATH having energy-momentum transfer in compact and convex subsets MATH and MATH of MATH, respectively, with the additional property that MATH and MATH lie in the complement of MATH, too. According to REF MATH and we are left with the task to investigate for large MATH the behaviour of the functions MATH and MATH. Since the arguments of both terms belong to MATH, having energy-momentum transfer in the compact and convex subsets MATH and MATH of MATH, we can apply REF in connection with REF to get the estimate (for the second term) MATH . Let MATH and MATH be approximating nets for MATH and MATH, respectively, satisfying MATH and MATH. Then the elements MATH constitute the large radius part of approximating nets for the almost local operators MATH, MATH, subject to the estimate MATH for any MATH with suitable MATH. Now, as suggested by the remark following REF , there exist approximating nets MATH, MATH, with MATH and MATH, so that, according to REF, the integrand of REF is bounded by MATH which implies MATH . Evaluation of the integrals on the right-hand side yields (for MATH) polynomials of degree MATH in MATH, so that, due to the decay properties of MATH, there exists a uniform bound MATH . The same reasoning applies to the term MATH, thus establishing the asserted rapid decrease for the mapping MATH, according to relation REF. In the general case of almost local elements MATH and MATH one has, by REF MATH and rapid decay is an immediate consequence of almost locality for all terms but the second one on the right-hand side of this inequality. Using suitable decompositions of MATH and MATH in terms of elements of MATH complying pairwise with the special properties exploited in the previous paragraph, the remaining problem of decrease of the mapping MATH reduces to the case that has already been solved above, thus completing the proof.
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Due to the assumed continuity of MATH, the assertions follow from REF in connection with REF .
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Due to the assumed continuity of the functional MATH, there exists a bounded NAME set MATH such that, according to REF in connection with REF , for any MATH there holds the inequality MATH . Therefore the linear operator MATH turns out to be continuous, so that the assertion follows by an application of REF from the result of REF , stating that the mappings MATH are differentiable for any MATH (compare the remark of that place).
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REF state that MATH exist in the complete locally convex spaces MATH and MATH, respectively. Now, the functional MATH, which lies in MATH according to the remark following REF , is linear and continuous with respect to MATH and, by REF , also with respect to both MATH. Therefore it commutes with the locally convex integrals CITE, which proves the assertion. The annexed estimates are a further simple application of the results contained in REF .
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Let MATH be an arbitrary compact subset of MATH and note that MATH . Thus, according to the construction of MATH, MATH belongs to the algebra of counters and exists furthermore as an integral in the locally convex space MATH. Therefore the functional MATH can be interchanged with the integral CITE to give MATH . Application of REF then leads to the estimate MATH where we made use of the positivity of MATH. The above inequality survives in the limit MATH and the convergence of the right-hand side to a finite real number establishes the integrability of the function MATH as a consequence of the Monotone Convergence Theorem CITE. In view of REF, one finally arrives at the asserted bound MATH .
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First, we re-write the argument MATH, commuting the operators MATH and MATH, to get MATH . This implies MATH where the first term on the right-hand side is evidently integrable over MATH, due to almost locality of the operators encompassed by the commutator. For MATH we have the estimate MATH . The second term can be estimated by use of the NAME inequality applied to the positive functional MATH: MATH . Integration of the first term on the right-hand side is possible according to the previous REF and gives MATH where MATH is any bounded NAME set containing the sum of MATH and the energy-momentum transfer MATH of MATH. Concerning the second term on the right of REF, we get, upon commuting MATH and MATH to the interior, MATH where again use was made of the positivity of MATH. The rapid decay of commutators of almost local operators with respect to the MATH-seminorm established in REF of Subsection REF can be combined with REF to show integrability over MATH: MATH which holds for any bounded NAME set MATH, where MATH denotes the energy-momentum transfer of MATH. By REF, the left-hand side of REF turns out to be integrable, and a bound for this integral is proportional to MATH. In connection with REF this establishes the assertion for a suitable constant MATH that can be deduced from relations REF.
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As implied by REF , MATH is a continuous mapping on MATH with respect to the MATH-topology, hence it is uniformly continuous on any compact set MATH. This means that to MATH there exists MATH such that MATH and MATH imply MATH where MATH denotes the MATH-dimensional volume of MATH. Consequently, under the above assumption on MATH and MATH, we infer from REF MATH . By compactness of MATH, there exist finitely many elements MATH such that the MATH-balls around these points cover all of MATH; moreover, since MATH is the weak limit of the net MATH, we can find MATH such that MATH implies MATH for any MATH. Now, for MATH and MATH, MATH and, selecting for MATH an appropriate MATH in a distance less than MATH, we can put the above results together to get the estimate MATH which holds for any MATH and MATH. Thus weak (i. e. pointwise) convergence of the net MATH is indeed uniform convergence on compact subsets of MATH. Upon integration over MATH we arrive at MATH . Now, by REF so that to MATH there exists a compact subset MATH satisfying MATH . Then, as a consequence of REF, we get for any MATH . Combining REF with REF for the compact set MATH yields for MATH (note, that MATH only depends on MATH) MATH . By arbitrariness of MATH this proves the possibility to interchange integration and the limit with respect to MATH as asserted in REF.
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If a function MATH belongs to the space MATH, then the operator MATH lies in MATH, according to REF , and has energy-momentum transfer in MATH, the support of the NAME transform of MATH. If this happens to satisfy MATH, we infer MATH and henceforth, by REF , MATH. Since MATH is assumed to belong to MATH, REF results in MATH which, according to the preceding considerations, entails MATH . Now, let MATH be an arbitrary function from MATH with MATH, MATH, then MATH, so that REF is fulfilled for any function of this kind, proving the assertion.
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Taking into account the fact that the NAME measure on MATH is invariant under translations, one can express MATH for any finite time MATH and any given MATH by the following integral MATH . Next, we want to evaluate MATH which, according to the respective limits of MATH-integration, can be split into a sum of three integrals to be estimated separately: MATH both MATH and MATH contribute to the third integral MATH where we used the abbreviation MATH for the interval of MATH-integration. Setting (for MATH large enough) MATH we finally arrive at the estimate MATH . The net MATH approximates MATH uniformly on compact subsets of MATH in the limit of large MATH, i. e. given MATH and MATH there exists a positive number MATH such that MATH implies MATH for any MATH with MATH. On the other hand, given MATH there exists MATH such that MATH for any MATH and any MATH. Combining these results with the special properties of MATH, i. e. uniform continuity on compact balls in MATH and approximate constancy at infinity, we infer that for large MATH the term MATH falls below any given positive bound. Therefore the right-hand side of REF vanishes with MATH since MATH exceeds any positive value in this limit. Now, let MATH be the weak limit of the subnet MATH, i. CASE: MATH then there holds for any MATH, any MATH and any MATH the subsequent inequality MATH . By the reasoning of the preceding paragraph and the above condition for subnet convergence, all three terms on the right-hand side vanish with respect to the directed set MATH, since in this limit MATH. As a result the intermediate term has to be equal to MATH, thereby establishing translation invariance of MATH.
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Consider the functional MATH at finite time MATH. Applying to the absolute value of its defining REF the NAME inequality with respect to the inner product (MATH large enough) MATH of square-integrable functions MATH and MATH depending on the time variable MATH, one gets in the special case of MATH the estimate MATH . Now, let MATH be a compact subset of MATH; then, by positivity of the functional MATH, CITE together with the NAME Theorem CITE leads for arbitrary MATH to MATH which is preserved in the limit MATH, which exists on account of the assumed integrability of the mapping MATH. On commuting MATH and the integrals one arrives at MATH and the combination of REF gives MATH . We want to replace the term MATH by the norm MATH and, to do so, define the function MATH, which is a non-negative element of MATH as is MATH itself. Then for any MATH there holds the equation MATH . Next, consider for an arbitrary function MATH the following inequality, based on an application of NAME 's Theorem and the reasoning of REF, MATH where we made use of the coordinate transformation MATH followed by the transformation MATH and introduced the abbreviations MATH as well as MATH for the interval of MATH-integration. Similar to the proof of REF , the expression MATH is seen to vanish for all MATH in the limit of large MATH, so that by NAME 's Dominated Convergence Theorem the left-hand side of REF converges to REF. This reasoning in particular applies to the functions MATH as well as MATH and thus to the third and fourth term on the right of REF . On the other hand, substitution of MATH by MATH in the integral of REF likewise gives a non-negative result for all times MATH. Combining all these informations and specializing to a subnet MATH approximating MATH or MATH, one arrives at the following version of REF, valid for asymptotic times: MATH . Making use of REF , this result can be expressed in terms of the functional MATH to yield MATH .
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CASE: The proof of the various properties stated in the Theorem is readily carried out, once the NAME has been realized. CASE: Since a particle weight satisfies the NAME inequality its null space MATH turns out to be a left ideal in MATH (and hence in MATH). The defining sesquilinear form endows the quotient space of MATH by MATH with a pre-Hilbert space structure; its completion MATH contains by construction the range of the canonical homomorphism MATH as a dense subspace. MATH and MATH being left ideals in MATH, the definition MATH makes sense on the range of MATH and can be extended to all of MATH due to the estimate MATH which is founded on the fact that the particle weight is a non-negative sesquilinear form and the operator MATH is positive. Since MATH is unital, this yields a non-zero, non-degenerate representation of the quasi-local algebra on the NAME space MATH. CASE: The norm on MATH induces a seminorm on MATH via the linear mapping MATH and this coincides with MATH as defined for particle weights. Therefore the asserted continuity of the mapping MATH is an immediate consequence of the respective property in REF . CASE: By construction, the canonical homomorphisms MATH and MATH coincide and furthermore MATH, so that the assumption of MATH-differentiability is self-evident. CASE: The existence of a strongly continuous unitary representation of space-time translations in MATH is a direct consequence of translation invariance of the particle weight MATH and its continuity under NAME transformations with respect to MATH. NAME 's Theorem (compare CITE and CITE) connects the spectrum of its generator MATH to the support of the NAME transform of MATH in REF by virtue of the relation MATH which holds for any MATH and any MATH. To clarify this fact, note, that the projection-valued measure MATH corresponding to MATH is regular, i. CASE: MATH is for any NAME set MATH the strong limit of the net MATH. For each compact MATH consider an infinitely often differentiable function MATH with support in MATH that envelops the characteristic function for MATH (compare CITE): MATH. According to the assumption of REF the left-hand side of REF vanishes for any MATH of the above kind, and this means that all the bounded operators MATH equal MATH not only on the dense subspace spanned by vectors MATH, MATH, but on all of MATH. Due to the fact that MATH majorizes MATH, this in turn implies MATH and thus, by arbitrariness of MATH in connection with regularity, the desired relation MATH. CASE: The reversion of the above arguments in order to establish that the scalar product on MATH possesses the characteristics of a particle weight is self-evident.
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CASE: Due to continuity of the particle weight MATH with respect to NAME transformations as claimed in REF , the integrand of REF can be estimated with respect to the seminorm MATH induced on MATH, which gives the NAME function MATH. Therefore the integral in question indeed exists in the completion of the locally convex space MATH not only with respect to the norm topology but also with respect to the seminorm MATH. Furthermore the corresponding NAME of MATH implies that MATH coincides with MATH for any MATH, a relation which extends to the respective completions (compare CITE) thus resulting in REF is then an immediate consequence, again on grounds of continuity under NAME transformations. CASE: According to REF , the particle weight MATH is invariant under space-time translations and so is the seminorm MATH. Therefore the integrand of REF is majorized by the NAME function MATH, so that the respective integral exists in the completion of MATH. The first equation of REF arises from the same arguments that were already applied above, whereas the second one is then a consequence of NAME 's Theorem (compare REF). Again on the ground of translation invariance, the estimate REF is an immediate conclusion from REF.
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The energy-momentum transfer of an operator MATH can be stated in terms of the support properties of the NAME transform of the mapping MATH considered as an operator-valued distribution (compare the remark following REF ). For the operator MATH this has the consequence that MATH if MATH is any NAME function with MATH. In this case we have, by an application of REF , MATH . Upon insertion of REF into the formulation REF of NAME 's Theorem, the reasoning applied in the proof of REF yields the assertion.
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To establish this result we follow in the main the strategy of the proof of REF . Applied to the problem at hand in terms of MATH, this yields initially the estimate MATH for any MATH. The first term on the right-hand side turns out to be majorized by MATH in view of the fact that the particle weight is invariant under translations and that the representation MATH is continuous. As the operators involved are almost local without exception, the norm of the commutator taking part in this expression decreases rapidly, thus rendering it integrable. The second term requires a closer inspection. One has MATH again by translation invariance of the particle weight in the last estimate. Now, a substitute of REF has to be sought for, which was applied in the proof of REF to get an estimate for REF, corresponding to the right-hand side of REF. Note, that MATH has the same energy-momentum transfer with respect to the unitary group MATH as the operator MATH has regarding the underlying positive energy representation, and that, according to REF , MATH and MATH belong to the spectral subspaces pertaining to the compact sets MATH and MATH. As in addition the spectrum of MATH is restricted to a displaced forward light cone, all of the arguments given in the proofs of REF also apply to the representation MATH, so that e. CASE: MATH is seen to exist in the MATH-topology on MATH. For this term we thus have MATH . The same holds true for the other expression on the right-hand side of REF, which shows that MATH is an integrable function, too. Altogether, we have thus established the Cluster Property for particle weights.
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Let MATH denote the NAME of the particle weight MATH with associated spectral measure MATH for the generator MATH of the intrinsic space-time translations. For the time being, suppose that MATH is an open bounded NAME set in MATH. Let furthermore MATH be an arbitrary element of MATH and MATH. We are interested in an estimate of the term MATH. Note, that the spectral measure is regular, so that MATH is the strong limit of the net MATH. As MATH is assumed to be open, there exists for each compact subset MATH of MATH an infinitely often differentiable function MATH with MATH that fits between the corresponding characteristic functions CITE: MATH. Thus the respective operators are subject to the relation MATH from which we infer that for arbitrary MATH . By density of all the vectors MATH in MATH, it is thereby established that MATH which implies for the scalar product in to be considered here MATH . Since MATH is the NAME transform of a rapidly decreasing function MATH, which therefore belongs to the space MATH, REF can be applied to yield for the right-hand side of REF MATH where, following the remark pertaining to REF , the ultimate expression is based on the fact that MATH as a consequence of REF in connection with REF . The approximating functionals MATH for MATH in the form REF with a non-negative function MATH allow, through an application of CITE, for the following estimate of their integrand: MATH . Here the spectral projections MATH pertaining to the NAME set MATH, which is both bounded and open, could be introduced, since, according to REF , the energy-momentum transfer of MATH is contained in MATH by construction. An immediate consequence of the above relation is MATH which extends to the limit functional MATH: MATH . Insertion of this result into REF yields MATH and in the limit MATH, in compliance with REF, MATH . Passing to the supremum with respect to all MATH such that MATH (these constitute a dense subset of the unit ball in MATH), we get through an application of CITE MATH . This establishes the defining REF for MATH-boundedness with MATH in the case of an open bounded NAME set MATH. But this is not an essential restriction, since an arbitrary bounded NAME set MATH is contained in the open set MATH, MATH, consisting of all those points MATH for which MATH. Since MATH is likewise a bounded NAME set, we get MATH as an immediate consequence of REF, where MATH. This covers the general case and thereby proves MATH-boundedness for the asymptotic functionals MATH.
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With the definitions MATH and MATH, MATH, where the latter obviously leaves invariant MATH and is such that the corresponding spectral measure turns out to be MATH for any NAME set MATH, all features of the restricted MATH-particle weight are readily checked on the grounds of REF .
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The presuppositions of this theorem meet the requirements for an application of CITE. This supplies us with CASE: a standard NAME space MATH, CASE: a bounded positive measure MATH on MATH, CASE: a MATH-measurable field MATH on MATH consisting of irreducible representations MATH of the MATH-algebra MATH on the NAME spaces MATH, CASE: and an isomorphism (a linear isometry) MATH from MATH onto the direct integral of these NAME spaces, such that MATH transforms MATH into the direct integral of the representations MATH according to MATH and the maximal abelian NAME algebra MATH can be identified with the algebra of diagonalisable operators via MATH with an appropriate function MATH. At first sight, the different statements of CITE listed above seem to cover almost all of the assertions of the present REF , but one must not forget that the disintegration is to be expressed in terms of a field of restricted MATH-particle weights. So we are left with the task to establish their defining properties in the representations MATH supplied by the standard disintegration theory. In accomplishing this assignment, one has to see to it that simultaneously relation REF is to be satisfied, which means that one is faced with the following problem: In general the isomorphism MATH connects a given vector MATH not with a unique vector field MATH but rather with an equivalence class of such fields, characterized by the fact that its elements differ pairwise at most on MATH-null sets. In contrast to this, REF associates the vector field MATH with MATH for any MATH, leaving no room for any ambiguity. In particular, the algebraic relations prevailing in the set MATH which carry over to MATH have to be observed in defining each of the mappings MATH which are characteristic of a restricted MATH-particle weight. The contents of the theorem quoted above, important as they are, can therefore only serve as the starting point for the constructions carried out below, in the course of which again and again MATH-null sets have to be removed from MATH to secure definiteness of the remaining components in the disintegration of a given vector. In doing so, one has to be cautious not to apply this procedure uncountably many times; for, otherwise, by accident the standard NAME space MATH arising in the end could happen to be itself a MATH-null set. Then, if MATH denotes the restriction of MATH to this set, one would have MATH, in contradiction to the disintegration REF of the non-zero representation MATH. CASE: As indicated above, our first task in view of REF will be to establish the existence of MATH-linear mappings MATH from MATH onto a dense subset MATH of each of the component NAME spaces supplied by CITE with the property MATH . Now, by relation REF, there exists to each MATH an equivalence class of vector fields on MATH which corresponds to the element MATH in MATH. The assumed MATH-linearity of the mapping MATH carries first of all over to these equivalence classes, but, upon selection of a single representative from each class, it turns out that every algebraic relation in question is fulfilled in all components of the representatives involved, possibly apart from those pertaining to a MATH-null set. So, if we pick out one representative of the vector MATH for every MATH in the numerable set MATH and designate it as MATH, all of the countably many relations that constitute MATH-linearity are satisfied for MATH-almost all of the components of these representatives. They can thus be taken to define the linear mappings of the form REF for all MATH in a NAME subset MATH of MATH, which is left by the procedure of dismissing an appropriate MATH-null set for each algebraic relation to be satisfied. The same reasoning can be applied to the disintegration of vectors of the form MATH with MATH and MATH. Again with REF in mind, the number of relations REF to be satisfied is countable, so that in view of relation REF the mere removal of an appropriate MATH-null set from MATH leaves only those indices MATH behind, for which the mappings MATH indeed have the desired REF . In this way we have implemented by hand the first defining property of restricted MATH-particle weights in the representations MATH for MATH-almost all indices MATH. The only thing that remains to be done in this connection is to show that MATH is a dense subset MATH in MATH. But, according to CITE, the fact that the set MATH is total in MATH by assumption implies that the corresponding property holds for MATH-almost all MATH in the disintegration. Thus there exists a non-null NAME set MATH, such that the corresponding mappings MATH, MATH, have this property, too. In this way all of the characteristics presented in the first item of REF are fulfilled for MATH by the mappings REF constructed above, and additionally we have MATH CASE: In the next step, the mappings MATH have to be extended to the set MATH of all NAME transforms of operators from MATH in such a way that the counterpart of REF is continuous. In the present notation this is the mapping MATH . At this point the special selection of MATH as consisting of compactly regularized vacuum annihilation operators comes into play, and also the invariance of this set under transformations MATH will be of importance. Great care has to be taken in these investigations based on the differentiability properties of the operators in question, that not uncountably many conditions are imposed on the mappings MATH, since anew not all of them will share the claimed extension property, but only a MATH-null subset of MATH shall get lost on the way. To start with, note that the NAME group MATH can be covered by a sequence of open sets MATH with compact closures MATH, MATH, contained in corresponding open charts MATH with the additional property that the sets MATH are convex (e. g. consider the translates of the canonical coordinates MATH around MATH to all elements of MATH and take suitable open subsets thereof). Select one of these compacta, say MATH, and fix an element MATH, which by assumption is given as a compactly supported regularization of an element MATH: MATH where MATH is an infinitely often differentiable function on MATH with compact support MATH in the NAME group MATH. According to REF the mapping MATH commutes with this integral so that the vector MATH in MATH takes on the shape MATH . The same equation holds for the NAME transforms of the operator MATH as well, so that invariance of the NAME measure on MATH implies for any MATH the equations MATH . The derivatives of the mapping MATH, the domain of MATH restricted to the neighbourhood MATH in MATH, are thus explicitly seen to be expressible in terms of derivatives of the functions MATH . So, let MATH and MATH be a pair of NAME transformations lying in the common neighbourhood MATH; then the following equation results from an application of the Mean Value REF to the MATH-differentiable mapping MATH (compare REF ): MATH where MATH and MATH belong to the compact and convex set MATH. Now, the vector MATH defines a positive functional on the algebra MATH, and we want to show that this vector functional can be majorized by a positive normal functional in MATH. To establish this fact, note, that the integrals in REF exist in the uniform topology of MATH, so that they commute with every bounded linear operator MATH. Hence MATH . This equation is invariant with respect to an exchange of MATH and MATH. In the case of a positive operator MATH the following relation holds for arbitrary vectors MATH and MATH in MATH: MATH which, applied to the integrand of REF and to that resulting from an interchange of MATH and MATH, yields MATH upon execution of a trivial integration over MATH and MATH, respectively. As in REF we can pass to the following representation for the integrand on the right-hand side of REF: MATH . The derivatives which show up in REF depend by construction continuously on the parameters MATH and MATH, MATH and MATH as well as MATH and MATH, so that their absolute values, taken on the compact domains MATH, MATH and MATH, respectively, are bounded by MATH for all MATH with a suitable non-negative constant MATH. Hence the non-negative matrix element in REF can be estimated by MATH which is independent of MATH, so that insertion into REF yields MATH . Since the positive operator MATH can be written as MATH for suitable MATH, the integrand on the right-hand side allows for the following estimate, making use of the relation between the geometric and the arithmetic mean of two non-negative numbers: MATH . As a consequence of this inequality entered into REF, one integration over MATH can be carried out on its right-hand side for each resulting term of the sum, so that finally MATH where the last integral can be viewed as a positive normal functional on MATH in the variable MATH, as announced at the beginning of this paragraph. Now, let MATH be a measurable subset of MATH then, according to REF, it corresponds via the associated characteristic function MATH to a projection MATH in the selected maximal abelian NAME algebra MATH. If MATH in turn denotes the orthogonal projection from MATH onto the NAME space MATH, we can define MATH as a positive operator in MATH, which is therefore subject to REF. This relation can then be re-written for MATH in terms of the restricted MATH-particle weight MATH: MATH where now the integral on the right-hand side defines a positive normal functional on the NAME algebra MATH through MATH . Specializing to NAME transformations MATH and MATH from the countable subgroup MATH, the unique disintegration of the vector MATH occurring on the left-hand side of REF is already explicitly given by REF for all MATH so that MATH . On the other hand, the positive normal functional MATH of REF is easily seen by CITE in connection with REF to correspond to a unique integrable field MATH of positive normal functionals on the NAME algebras MATH in the direct integral decomposition of MATH. Explicitly, MATH for any MATH with an appropriate function MATH. The above relation stays true, if we replace MATH by MATH, since both differ at most by a MATH-null set. So, in view of relations REF through REF can for any measurable subset MATH of MATH corresponding to the orthogonal projection MATH be expressed in terms of integrals according to MATH . Due to arbitrariness of MATH, we then infer, making use of elementary results of integration theory CITE, that for MATH-almost all MATH there holds the estimate MATH where we replaced the points MATH and MATH from the space MATH of coordinates for MATH by their pre-images MATH and MATH from MATH. The important thing to notice at this point is that, apart from the factor MATH, the terms on the right-hand side of REF hinge upon the operator MATH (determining the function MATH as well as its support MATH) and on the neighbourhood MATH with compact closure MATH containing MATH. Therefore this estimate also holds for any other pair of NAME transformations in MATH with the same MATH-dependent factor; of course, in each of the resulting countably many relations one possibly loses a further MATH-null subset of MATH. The reasoning leading up to this point can then be applied to any combination of an operator in the numerable selection MATH with an open set from the countable cover of MATH to produce in each REF relation of the form of REF for the respective NAME transformations in MATH. Simultaneously, the domain of indices MATH, for which all of these inequalities are valid, shrinks to an appropriate MATH-measurable non-null subset MATH of MATH. Consider now an arbitrary element MATH, which belongs to at least one of the open sets MATH from the covering of the NAME group already used above. By density of MATH in MATH, the transformation MATH can be approximated by a sequence MATH. This is a NAME sequence in the initial topology of MATH, so that relation REF implies that for each MATH the corresponding sequences MATH likewise have the NAME property with respect to the NAME space norms. Their limits in each of the spaces MATH, MATH, thus exist and are obviously independent of the approximating sequence of NAME transformations from MATH. Therefore, we can write MATH a result that holds for arbitrary MATH as long as MATH is taken from the non-null set MATH. According to CITE, which lays down the notion of measurability for vector fields, the mapping MATH that arises as the pointwise limit of measurable vector fields on MATH, is itself measurable with respect to the restriction of MATH to this subset of MATH and turns out to be a representative of the vector MATH (compare CITE, and note that we can neglect the null set missing in MATH compared to MATH). The question now is, if the limits MATH, constructed by the above method for arbitrary operators MATH and any transformation MATH, can unambiguously be identified for all MATH in MATH with vectors MATH, which satisfy a relation of the form REF. One of the situations, in which an inconsistency possibly arises, is the appearance of two different representations for a single element MATH: MATH where MATH, and MATH. In this case the pair of operators is connected by the NAME transformation MATH, which belongs to the subgroup MATH of MATH according to the constructions of REF. Therefore MATH which implies that MATH for any sequence MATH approximating MATH. But then the transformations on the right-hand side of the last equation constitute another sequence in MATH, which in this case tends to MATH in the limit MATH. As a consequence of the independence of the limits REF from the selected sequence in MATH, we could define MATH . The only problem that is still left open with respect to an unequivocal definition of vectors of the form MATH, MATH, occurs when the vacuum annihilation operator MATH happens to be an element of MATH, so that its components in the NAME spaces MATH have already been fixed in the initial step. But, as MATH is a numerable set, such a coincidence will be encountered at most countably often, so that relation REF indeed turns out to be the unique definition of MATH for all MATH, such that the relation MATH is satisfied, where again MATH is a MATH-measurable subset which differs from MATH only by a null set. The MATH-vectors corresponding to elements of MATH that arise as NAME transforms of MATH are defined according to REF, in particular by the relations REF. As a result, when MATH and MATH are closely neighbouring elements of MATH so that their products with MATH lie in the common open neighbourhood MATH, we get the following estimate, which is a direct consequence of the above constructions inserted into relation REF and which holds for any MATH: MATH . This shows that the continuity property with respect to generic NAME transformations as expressed in REF is fulfilled by all the extended mappings MATH introduced above for arbitrary MATH. CASE: The last property of restricted MATH-particle weights to be established is the existence of unitary groups MATH in the representations MATH which satisfy the relations REF. To construct them we first consider one element MATH of the countable space MATH together with a single space-time translation MATH in the numerable dense subgroup MATH of MATH. By REF , the NAME algebra MATH is contained in the commutant of MATH, which means that for any measurable subset MATH of MATH with associated orthogonal projection MATH there holds the equation MATH . Since this result is valid for arbitrary measurable sets MATH, we infer by CITE that for MATH-almost all MATH the vectors are subject to the relation MATH A corresponding equation can be derived for any other of the countable number of combinations of elements in MATH and MATH, so that REF is true in all of these cases when the domain of MATH is restricted to the MATH-measurable subset MATH, which again differs from MATH only by a null set. On MATH we can then define for arbitrary MATH the mappings MATH which are indeed determined unambiguously according to REF. By the same relation they are norm-preserving and, moreover, turn out to be MATH-linear operators on the countable spaces MATH. We want to extend the definition given by REF in two respects: All space-time translations MATH should be permissible, and all vectors of MATH are to belong to the domain of the resulting operators. Now, let MATH be an arbitrary element of MATH, i. CASE: MATH and consider MATH approximated by the sequence MATH. Then, by REF in connection with REF , the translates by MATH and MATH of the vectors MATH are for MATH subject to the following relation: MATH . Since MATH is a subset of MATH, we have relation REF at our disposal; moreover, by assumption, the group of automorphisms MATH is strongly continuous. As a result, the sequences MATH both have the NAME property and are thus convergent as well as bounded in the respective norm topologies. Applied to the elements of MATH and MATH appearing in the representation REF of MATH this has the consequence, that the right-hand side of REF can be made arbitrarily small for all pairs MATH exceeding a certain number. The terms on the left-hand side of this inequality thus turn out to be part of NAME sequences MATH which converge in the NAME spaces MATH. Since a renewed application of the above arguments shows that the arising limits are independent of the approximating sequence in MATH, the following relation unambiguously defines the mappings MATH for arbitrary MATH and MATH: MATH . Again these mappings act as MATH-linear operators on the spaces MATH and preserve the NAME space norm. As a consequence they can, by the standard procedure used for completions of uniform spaces, be continuously extended in a unique fashion to all of the NAME space on condition that their countable domain constitutes a dense subset of MATH; but this is the case as MATH is contained in MATH. Changing the notation from MATH to MATH for these extensions, their definition on arbitrary vectors MATH then reads for any MATH and this definition is again independent of the selected sequence. For any MATH and any MATH the vector field MATH, which is the pointwise limit of a sequence of measurable vector fields by REF and hence itself measurable CITE, corresponds to MATH, the equivalent limit in MATH (where we neglect the difference between MATH and MATH which is of measure MATH): MATH . Having defined the family of mappings MATH for MATH, we now have to check that they obey REF. First of all, note that, as an immediate consequence of the way in which they were introduced, these mappings are MATH-linear and norm-preserving. Another property that is readily checked by use of the relations REF in connection with the estimates arising from REF with MATH replacing MATH is the fact that for arbitrary MATH . From this we infer that, as MATH, each operator MATH has the inverse MATH and thus proves to be an isometric isomorphism of MATH. Hence, in accordance with REF, the set MATH indeed turns out to be a unitary group in MATH. The strong continuity of this group is easily seen: Consider the operator MATH as defined in REF and two sequences MATH, MATH in MATH converging to MATH and MATH, respectively. Then REF stays valid if we replace the translations MATH by MATH in each case. In compliance with REF it is then possible to pass to the limit, which results in the obvious estimate MATH . This explicit inequality shows that the right-hand side can be made arbitrarily small for all MATH in an appropriate neighbourhood of MATH; for the first term this is brought about by the strong continuity of the automorphism group MATH, whereas for the second term it is a consequence of relation REF. The defining condition for strong continuity is therefore satisfied for vectors in the dense subset MATH. If now an arbitrary vector MATH is considered, we can expand the difference MATH by introducing the corresponding translates of any element MATH and, making use of the property of norm-preservation of the unitaries, arrive at MATH . The right-hand side of this inequality can again be made smaller than any given bound by first choosing a suitable element MATH from a small neighbourhood of MATH and then, in dependence on this selected vector MATH but irrelevant for the statement, selecting an appropriate neighbourhood of translations MATH around MATH as implied by REF. Thereby we have established strong continuity of the unitary group MATH. Before considering the support of the spectral measure MATH associated with this strongly continuous unitary group, we mention a result on the interchange of integrations with respect to the NAME measure on MATH and the bounded positive measure MATH on MATH, which proves to be necessary as NAME 's Theorem does not apply to the situation in question. Let MATH be a continuous bounded function in MATH, then MATH is an integrable mapping for any MATH and MATH. Moreover, it is NAME integrable over any compact MATH-dimensional interval MATH, and this integral is the limit of a NAME sequence (compare CITE): MATH where MATH denotes the MATH-th subdivision of MATH, MATH are the NAME measures of these sets, and MATH are corresponding intermediate points. The sums on the right-hand side of this equation turn out to be MATH-measurable when their dependence on MATH is taken into account, and so is the limit on the left-hand side. Moreover this property is preserved in passing to the limit MATH, so that MATH is MATH-measurable and, in addition, integrable since MATH . The counterpart of REF is valid in MATH, too, and, if MATH denotes a measurable subset of MATH with associated orthogonal projection MATH, this equation reads MATH . Then REF in connection with REF imply MATH where, in the second equation, use was made of NAME 's Dominated Convergence Theorem in view of the fact that the integrable function MATH majorizes both sides of REF. Relation REF again stays true in passing to the limit MATH: MATH which is the announced result on the commutability of integrations in the present context. The support of the spectral measure MATH associated with the generators MATH of the unitary group MATH can now be investigated by use of the method applied in the proof of the fourth item of the first part of REF . Note, that the complement of the closed set MATH can be covered by an increasing sequence MATH of compact subsets (take e. g. the intersection of the compact ball of radius MATH with the complement of the open MATH-neighbourhood of MATH). To each of these sets one can find an infinitely often differentiable function MATH with support in MATH that has the property MATH. As before, let MATH be a measurable subset of MATH with associated orthogonal projection MATH, then, by assumption on the spectral support of the unitary group implementing space-time translations in the underlying particle weight, we have MATH for any MATH and any pair of vectors MATH and MATH with MATH, and this, according to REF, implies MATH . By arbitrariness of MATH in the last expression, we conclude once more that for MATH-almost all MATH . Eventually, if we want this equation to hold for any element of the countable set of triples MATH, a non-null set MATH is left, and REF stays valid for the remaining MATH even if the special vectors MATH and MATH are replaced by arbitrary ones. NAME 's Theorem then implies (compare REF) that MATH and therefore, by the order relation inherent in the definition of MATH, we have MATH for any MATH. As the spectral measure MATH is regular, one can pass to the limit MATH and thereby arrives at the desired result MATH . The defining REF in connection with REF implies that for arbitrary operators MATH and MATH and for any translation MATH one can write MATH . Since the vectors MATH, MATH, constitute a dense subset of MATH for MATH we infer from this equation that MATH an equation that readily extends to all translations MATH in MATH and, by uniform density of MATH in MATH, also to any operator MATH in the MATH-algebra MATH: MATH . This proves the counterpart of REF. The action of the group MATH on MATH according to REF is an immediate consequence of the defining relations REF in connection with REF and the continuity property as expressed by REF. In the present setting we thus have MATH . Now, let MATH be an arbitrary element of MATH having energy-momentum transfer MATH. Defined as the support of the NAME transform of an operator-valued distribution (compare the Remark following REF ), MATH is a closed NAME set, so that the arguments given in the preceding paragraph can again be applied when MATH replaces MATH and MATH and the functions MATH now correspond to an increasing sequence of compact sets MATH which constitute a cover of MATH. As a result we arrive at the equivalent of REF, so that MATH holds for MATH-almost all MATH even if the index MATH is allowed to run through all natural numbers. As in the preceding paragraph we then conclude that for all of these MATH and all MATH one has MATH and hence MATH. According to the regularity of the spectral measure MATH, passage to the limit with respect to MATH yields the equation MATH. By countability, this last result is valid for arbitrary MATH if a MATH-measurable non-null set MATH is appropriately selected, from which the indices MATH are to be taken. The complementary statement thus constitutes a restricted version of the counterpart of REF: MATH . Now, let MATH be an arbitrary element of MATH, then the energy-momentum transfer of its NAME transform by MATH, i. e. of the operator MATH, is given by MATH, so that, according to REF, MATH . This result can be applied to investigate the case of a generic element from MATH. For arbitrary MATH approximated by the sequence MATH we have, by virtue of the relevant parts of REF, MATH and NAME 's Dominated Convergence Theorem in connection with NAME 's Theorem yields for any function MATH and any index MATH . In the limit of large MATH one finds the energy-momentum transfer MATH of MATH in a small MATH-neighbourhood of MATH. Therefore, in view of REF, the right-hand side of REF vanishes for all MATH exceeding a certain bound MATH if MATH is chosen in such a way that MATH. The NAME transform of the distribution MATH is thus seen to be supported by MATH from which we infer MATH which is the equivalent of REF for arbitrary operators in MATH. REF are readily generalized, making use of the order structure of spectral projections reflecting the inclusion relation of NAME subsets of MATH. If MATH denotes the set of operators from MATH having energy-momentum transfer in the NAME set MATH, then MATH and thus the counterpart of REF is established for the remaining indices MATH from the non-null subset MATH of MATH. The above construction has supplied us with a measurable subset MATH of the standard NAME space MATH, that was introduced at the outset, emerging from an application of CITE. MATH is a non-null set, differing from MATH only by a MATH-null set. Moreover it is itself a standard NAME space (compare the definition in CITE), and we shall denote the restriction of the measure MATH to it by MATH; MATH is again a bounded positive measure. Moreover, and this has been the central aim of the previous investigations, the field MATH indeed consists of MATH-particle weights. What remains to be done now is a verification of the properties listed in REF. CASE: Arising as the restriction to a measurable subset in MATH of a field of irreducible representations, the family of representations MATH is obviously MATH-measurable and its components inherit the feature of irreducibility. CASE: As MATH and MATH only differ by a MATH-null set, one has MATH and the relations REF can be reformulated, using the right-hand side of REF and an isomorphism MATH, which is the composition of MATH with the isometry that implements the above unitary equivalence. As an immediate consequence of REF we then get the equivalence assertion of REF. Moreover, by REF, the operator MATH connects the vector fields MATH with vectors MATH for MATH in the desired way as expressed in REF. CASE: REF is the mere reformulation of REF in terms of MATH and MATH. CASE: According to the argument preceding REF, the mappings MATH, with MATH restricted to MATH and MATH as well as MATH taken from MATH, are measurable for all vectors MATH and MATH in the dense subsets MATH, and this suffices, by CITE, to establish measurability of the field MATH for arbitrary MATH. Moreover, this is a bounded field of operators, so that it defines a bounded operator on MATH which is given by REF as an immediate consequence of REF, bearing in mind that MATH and MATH only differ by a MATH-null set. The demonstration of REF on the other hand is less straightforward. Assume first of all that the NAME set MATH in question is open. Then we can make use of the regularity of spectral measures and construct, according to CITE, a sequence of compact subsets MATH as well as of infinitely often differentiable functions MATH with support in MATH such that MATH and furthermore MATH for any MATH. The discussion on page REF f. - with MATH replacing MATH and MATH instead of MATH - demonstrates, making use of NAME 's Theorem, that the sequence appearing on the right-hand side consists of MATH-measurable functions of MATH, so that we infer that its limit MATH is MATH-measurable, too. Another application of NAME 's Theorem in connection with REF in terms of MATH then shows that MATH and, as NAME 's Dominated Convergence Theorem allows for the passage to the limit function under the last integral, we get according to REF MATH . This formula, as yet valid only for open NAME sets MATH, is readily generalized to closed NAME sets and then, since by regularity the spectral measure of an arbitrary NAME set is approximated by a sequence in terms of compact subsets of it, to any NAME set. By polarization and the fact that the ket vectors with entries from MATH are dense in MATH and MATH, respectively, we first conclude with CITE that all the fields MATH are measurable for arbitrary NAME sets MATH and then pass to the aspired REF from REF. CASE: The unitary operators MATH, MATH, defined in REF belong to the NAME algebra MATH, according to REF, and are thus diagonalisable in the form MATH . According to the construction of these operators, we can re-express this result in terms of the canonical unitary group of REF: MATH . The definition MATH then provides the asserted canonical choice of a strongly continuous unitary group on each NAME space MATH. Its spectral properties are derived from those of the canonical group MATH by the methods that have already been used repeatedly above. Possibly a further MATH-null subset of MATH gets lost by this procedure. This finishes the proof of the assertions of REF .
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CASE: MATH-boundedness of the particle weight MATH means, according to REF , that to any bounded NAME set MATH there exist another such set MATH containing MATH and an appropriate positive constant MATH, so that the estimate MATH holds for any MATH. Then a finite cover of MATH by sets of diameter less than a given MATH (which exists on account of the hypothesis of precompactness) induces a corresponding cover of MATH, which is composed of sets with a diameter smaller than MATH as REF shows. This establishes totally boundedness of the set MATH . By arbitrariness of MATH as well as of the bounded region MATH, the representation MATH is thus seen to satisfy REF and NAME in the sense that the mappings MATH are altogether precompact. CASE: According to the construction of MATH from MATH explained in the proof of REF , both of these representations are related by the inequality MATH which holds for any MATH. Then MATH-boundedness of the underlying particle weight again implies the existence of a bounded NAME set MATH such that MATH . This relation replaces REF in the proof of the first part, so that we conclude that indeed MATH inherits the precompactness properties of the underlying quantum field theory in the sense that all the sets MATH are totally bounded for any MATH whenever MATH is an arbitrary bounded NAME set and MATH is one of the countably many localization regions from MATH. Again that is sufficient to establish the fact that the NAME - NAME Compactness Condition is satisfied in the restricted setting for Local Quantum Physics introduced in REF.
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Select a dense sequence MATH in the norm-separable MATH-algebra MATH and consider the countable set of compact balls MATH of radius MATH in MATH. The corresponding operators MATH are decomposable according to REF : MATH and CITE tells us that the respective norms are related in compliance with the equation MATH . With regard to all possible combinations of operators MATH and compact balls MATH we thus infer that there exists a measurable non-null subset MATH of MATH such that for all MATH and all indices MATH and MATH the estimate MATH holds. Now, let MATH be an arbitrary bounded NAME set which is thus contained in a compact ball MATH and note that, by continuity of the representations MATH and MATH, REF extends to arbitrary operators MATH. Therefore MATH and this implies, according to REF, the existence of a bounded NAME set MATH such that MATH . The arguments given in the proof of REF can then again be applied to the present situation to show that for MATH the irreducible representations MATH altogether meet the requirements of the NAME - NAME Compactness Condition.
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CASE: Let MATH be a bounded NAME set and suppose that MATH is a normal functional on MATH. Then the same applies to the functional MATH, and therefore the mapping MATH is continuous with respect to the relative MATH-weak topology of MATH. Now, according to the Compactness Condition, MATH maps the unit ball MATH of the local MATH-algebra MATH onto the relatively compact set MATH. The restriction of MATH to MATH is now obviously continuous with respect to the relative MATH-weak topologies, but this result can be tightened up in the following sense: The relative MATH-weak topology, being NAME and coarser than the relative norm topology, and the relative uniform topology itself coincide on the compact norm closure of MATH due to a conclusion of general topology CITE. Therefore MATH is still continuous on MATH when its image is furnished with the norm topology instead. Now, suppose that MATH is an arbitrary bounded NAME set and let MATH be another bounded NAME set with the property that REF is satisfied. Then the linear mapping MATH is well-defined and continuous with respect to the uniform topologies of both domain and image. Therefore, as a consequence of the previous paragraph, we infer that the composition of this map with the restriction of MATH to MATH is continuous, when MATH is endowed with the MATH-weak topology whereas the range carries the relative norm topology. The resulting map is explicitly determined as the restriction to MATH of MATH . If MATH denotes a MATH-weakly continuous functional on MATH, the same is true regarding MATH for any bounded NAME set MATH, and moreover, due to strong continuity of the spectral measure, MATH is the uniform limit of the net of functionals MATH for MATH. Given a MATH-weakly convergent net MATH with limit MATH, we conclude from the discussion in the preceding paragraph that MATH . Therefore, by means of the estimate MATH it is easily seen that, upon selection of a suitable bounded NAME set MATH, the right-hand side can be made smaller than any given bound for MATH with an appropriate index MATH. This being true for any MATH-weakly continuous functional MATH on MATH and arbitrary nets MATH in MATH converging to MATH with respect to the MATH-weak topology of MATH, we have thus established that the restrictions of the representation MATH to each of the unit balls MATH are MATH-weakly continuous. According to CITE this assertion extends to the entire local MATH-algebra MATH, so that MATH indeed turns out to be locally normal. CASE: The arguments given above in the case of MATH can be transferred literally to the representations MATH and MATH, MATH, in view of the relations REF established in the proofs of REF , which substitute REF used in the first part. The evident modifications to be applied include the restriction to local algebras MATH where MATH is a member of the countable family MATH. CASE: Complementary to the statements of the second part, CITE exhibits that the representations MATH and MATH, MATH, allow for unique MATH-weakly continuous extensions MATH and MATH onto the weak closures MATH CITE of the local algebras, which, due to the NAME Theorem CITE, coincide with the strong closures and thus, by the very construction of MATH, MATH, expounded in REF, contain the corresponding local MATH-algebras MATH of the underlying quantum field theory. Now, due to the net structure of MATH, the quasi-local MATH-algebra MATH is its MATH-inductive limit, i. e. the norm closure of the MATH-algebra MATH. As the representations MATH and MATH, MATH, are altogether uniformly continuous on this MATH-algebra CITE, they can in a unique way be continuously extended to its completion MATH CITE, and these extensions, again denoted MATH and MATH, respectively, are easily seen to be compatible with the algebraic structure of MATH. MATH and MATH are thus representations of this quasi-local algebra, evidently irreducible in the case of MATH, and moreover locally normal, since, by construction, they are MATH-weakly continuous when restricted to local algebras MATH pertaining to the countable subclass of regions MATH, and an arbitrary local algebra MATH is contained in at least one of these. The statement on uniqueness of the extensions is then an immediate consequence of the fact that they are uniquely determined by the property of being MATH-weakly continuous on the local MATH-algebras MATH. Regarding the disintegration of operators MATH for arbitrary MATH, note that any operator MATH is the MATH-weak limit of a sequence MATH in MATH with MATH. For nets in MATH this statement is a consequence of NAME 's Density Theorem CITE in connection with CITE and the various relations between the different locally convex topologies on MATH. The specialization to sequences is justified by CITE in view of the separability of MATH. The operators MATH define fundamental sequences of measurable vector fields MATH (compare CITE) and, as the operators MATH are decomposable, all the functions MATH are measurable for arbitrary MATH. The same is valid for the pointwise limit of this sequence CITE MATH and that suffices, according to CITE, to demonstrate that MATH is a measurable field of operators. As the sequence MATH converges MATH-weakly to MATH by assumption and since, moreover, MATH is finite and the family of operators MATH is bounded by MATH for any MATH, we conclude with NAME 's Dominated Convergence Theorem applied to the decompositions of MATH with respect to MATH (which differs from MATH only by a null set), that MATH . If MATH denotes the isometry which implements the unitary equivalence MATH and has all the properties of the operator MATH introduced in REF , then, by density of the set MATH in MATH, we infer from REF that MATH . This relation has been established under the presupposition that MATH belongs to some local MATH-algebra MATH. Now, it is possible to reapply the above reasoning in the case of an arbitrary element MATH of the quasi-local algebra MATH, which can be approximated uniformly by a sequence MATH from MATH. In this way, REF is extended to all of MATH so that we end up with the final equation MATH demonstrating that indeed MATH .
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In a first step it will be shown that, setting MATH for any compact subset MATH of MATH, the following estimate is in force for arbitrary bounded NAME sets MATH: MATH with a suitable constant MATH and an appropriate bounded NAME set MATH. If MATH denotes a state on MATH which is induced by a vector MATH we can immediately adopt the inequalities of CITE to get MATH . The integrand of the second term on the right-hand side is subject to the relation MATH with MATH denoting the closure of MATH. Upon insertion into REF, removal of the resulting common factor MATH on both sides and passing to the supremum with respect to all unit vectors MATH, we get MATH where use is made of the fact that all the vectors MATH belong to the subspace MATH for arbitrary MATH and MATH. The preparatory estimate REF is now established by complete induction on MATH, where this natural number is defined in dependence on the sets MATH and MATH through the condition MATH (compare the proof of REF on page REF). For MATH we have, according to the spectrum condition, MATH so that REF is trivially fulfilled since its left-hand side vanishes. Now assume that the condition MATH is valid, which, stated another way, means that the intersection of MATH with the complement of MATH is empty. As MATH is a bounded NAME set we can apply the induction hypothesis for MATH, i. CASE: REF with MATH replaced by MATH, to infer that there exists a bounded NAME set MATH which satisfies MATH for an appropriate constant MATH. This estimate inserted into REF amounts to MATH from which to conclude the validity of REF with suitable constant MATH and proper bounded NAME set MATH is an obvious task. Now, having established REF, we can specialize it to MATH and pass to the limit MATH as in the proof of REF , noting that MATH and that, due to almost locality of MATH, the integral on the right-hand side can be extended over all of MATH. As a result, in view of REF , one arrives at the desired REF , where the formulation chosen makes explicit its dependence on MATH and MATH.
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We have to calculate the integral on the right-hand side of REF and, to do so, it is split into two parts according to MATH or MATH with an abitrary radius MATH which is held fixed for the moment. For large MATH we use the estimate REF for the integrand and get in terms of the norm MATH: MATH . Accordingly, the respective integral is subject to the inequality MATH . The integrand for small MATH is evaluated observing the spectral projections arising on the right-hand side of REF, which are abbreviated as MATH with closure MATH. This leads to MATH where MATH - the inclusion of MATH into this definition being required at the very end of the present argumentation. The corresponding integral satisfies the inequality MATH . The integrals remaining in REF are known from calculus (compare CITE): MATH where the factor MATH is defined via the MATH-function as MATH . Collecting the results from REF one gets for the complete integral MATH with suitable MATH-dependent factors. So far the value of MATH has been left open. To get the concise REF we deliberately choose MATH so that REF simplifies to MATH . Inserting the square root of REF into this estimate and carrying the result over to REF, we finally arrive at REF, where MATH has been included in the definition of the coefficients. Note, that the above argument is independent of the occurrence of MATH or MATH, for in this case MATH, so that REF is trivially fulfilled. This consequence is immediate from the norm property of MATH. As to the second of the above conditions, it turns out to be important that we have included MATH into the definition of MATH. Thence the named assumption implies MATH, and MATH is a result of REF .
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hep-th/0005057
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Let MATH denote the closed MATH-ball, MATH, in MATH with respect to MATH. By REF , the condition MATH, MATH, implies MATH, stating a property of uniform approximation. This means that, given MATH, there exists a radius MATH, take e. CASE: MATH, such that to any MATH we can find a local operator MATH with MATH . Again according to REF , one also has MATH, so that the collection of local operators just introduced belongs to the closed ball of radius MATH in MATH. Now, the NAME - NAME Compactness Condition ensues that there exists a finite number of operators MATH, MATH, in this ball such that any MATH satisfies the condition MATH for at least one MATH. Combining this with REF, we see that for any MATH there exists a suitable operator MATH with MATH . It is an immediate consequence that finitely many elements from MATH can be selected, serving as centres of MATH-balls which cover the set MATH . By arbitrariness of MATH, we have thus established precompactness of the mapping MATH in the sense of the Proposition.
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hep-th/0005057
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Let MATH and MATH be arbitrary distinct points in MATH. We shall assume MATH and want to show that MATH. Define MATH and consider one of the seminorms MATH topologizing MATH. There are two possibilities: MATH . Depending on the actual situation we define an interval MATH, choosing MATH, MATH in REF and MATH, MATH in REF . Independent of this selection is the estimate MATH . The same procedure can then be applied to the interval MATH, to the resulting interval MATH and so on. In this way a sequence of intervals MATH is constructed, which is decreasing with respect to the inclusion relation: MATH. Furthermore the lengths are explicitly known as MATH and the estimate REF can be generalized to MATH . There exists exactly one point MATH belonging to all intervals of this sequence and by REF, so that for MATH in a small REF-neighbourhood MATH the increment of MATH at MATH is represented by MATH with a mapping MATH satisfying MATH . Hence, given MATH, there exists MATH such that MATH and MATH are majorized by MATH for MATH. According to REF this implies MATH where we made use of the length formula for the interval MATH. From REF one then infers MATH so that, by arbitrariness of MATH and MATH together with the separation property of the seminorms, we see that MATH. This relation holds for any MATH, MATH and extends by the supposed continuity of MATH to all of MATH, establishing MATH as stated.
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hep-th/0005057
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By CITE MATH is a well-defined MATH-valued mapping on the compact interval MATH. For MATH and MATH satisfying MATH we have MATH hence MATH . Now by assumption, MATH is continuous on the compact interval MATH of integration for any of the defining seminorms MATH of MATH, and, according to CITE, one has for any MATH the estimate MATH where the right-hand side vanishes in the limit MATH. Thus REF corresponds to the representation REF in terms of the increment MATH with a residual term MATH satisfying REF. This proves differentiability of MATH on MATH along with relation REF.
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hep-th/0005057
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Given MATH and MATH as above we define two mappings MATH and MATH on the compact interval MATH to MATH respectively MATH through MATH . From REF we infer MATH for any MATH. This implies, according to REF , that the mapping MATH is constant on the interval MATH (Note, that MATH as well as MATH are continuous.). Hence MATH which is just REF re-written.
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hep-th/0005057
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CASE: If MATH is continuously differentiable the mappings MATH are continuous for any MATH; furthermore REF correspond for each MATH exactly to REF setting MATH, so that the first part of the statement is almost trivial. CASE: Let all the partial derivatives of MATH exist as continuous mappings MATH, then, for small MATH, we have through an application of the Mean Value NAME REF for any MATH . Due to continuity of the mappings MATH, the second term on the right-hand side multiplied with MATH can be estimated by MATH where the last expression of the above inequality is seen to vanish in the limit MATH by assumption. Thus REF in connection with REF establishes continuous differentiability of the mapping MATH with MATH .
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hep-th/0005057
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For MATH and sufficiently small MATH consider the following expression which involves two increments of MATH: MATH . By assumption on the existence and continuity of the mixed derivatives we can apply the Mean Value REF twice to the above expression: One can consider the increments with respect to MATH and apply the Mean Value Theorem to them first and afterwards to the resulting integrand which takes on the form of an increment with respect to MATH, or one carries out the same procedure with the roles of MATH and MATH interchanged. Upon division by MATH this yields the integrals MATH for any MATH. Specializing to sequences MATH and MATH in this set which converge to MATH, it is a consequence of NAME 's Dominated Convergence Theorem (compare CITE) that for MATH the left-hand side converges to MATH whereas the right-hand side approaches MATH in the locally convex topology of MATH. Since this topology separates the elements of MATH, we conclude that these limits coincide and get the assertion by arbitrariness of MATH.
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hep-th/0005057
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CASE: To prove the non-trivial part, suppose that MATH is arbitrary but fixed. Then MATH, MATH, is a local chart around MATH with MATH. According to the definition of MATH we have MATH and, since the automorphisms are norm-preserving, the assumed differentiability of the mapping MATH at MATH carries over to MATH which by REF means that MATH is differentiable at MATH: MATH . In view of REF this relation can be re-written with respect to an arbitrary local chart MATH on MATH containing MATH: MATH where the matrix elements of MATH are analytic in MATH. Since the automorphisms are norm-preserving and act stongly continuous on MATH, it is evident that application of the above operator to any vector MATH yields a continuous mapping on MATH to MATH, thus establishing continuous differentiability of MATH on MATH as stated. CASE: Let MATH be arbitrary and consider the local chart MATH, MATH, around MATH with inverse MATH. Then MATH so that the assumed differentiablity of MATH at MATH and thus, according to the first part, at MATH with respect to the local chart MATH implies differentiability of MATH at MATH. By an application of REF we have MATH where the matrix elements of MATH are analytic in MATH. This in turn can, again by use of REF, be generalized to any MATH lying in the local chart MATH: MATH an expression which is obviously continuous in both variables MATH and MATH when applied to an arbitrary element MATH of MATH.
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hep-th/0005057
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By assumption on MATH (relations REF) in connection with linearity of MATH, the increment of MATH at MATH allows for the representation MATH where MATH for any seminorm MATH, MATH. But, due to continuity of MATH, there exist to any seminorm MATH on MATH a finite number of seminorms MATH on MATH, MATH, and a positive constant MATH such that for any MATH and therefore MATH . This is just the formulation of REF for MATH and thus proves, according to REF, differentiability of this mapping at MATH together with REF. The remainder of the assertion is a trivial consequence.
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hep-th/0005057
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Let MATH be a dense sequence of non-zero vectors in MATH and let MATH denote the NAME algebra generated by MATH. According to NAME 's Density Theorem, MATH coincides with the strong closure MATH of the algebra MATH, which by assumption acts non-degenerately on MATH (compare CITE, CITE). First we assume the existence of a separating vector for MATH, which is thus cyclic for MATH CITE. Then any normal functional on MATH is of the form MATH with MATH CITE. Choose operators MATH satisfying MATH which is possible due to NAME 's Density Theorem CITE. Let MATH denote the norm-separable MATH-algebra generated by the unit element MATH together with all the operators MATH, MATH, and select a normal functional MATH on MATH with the properties MATH and MATH. Without loss of generality we can assume MATH. To any MATH there exist vectors MATH from the dense sequence in MATH rendering MATH and MATH small enough so that MATH . Making use of REF we then get the estimate MATH which in connection with REF implies MATH . By arbitraryness of MATH we infer MATH in contradiction to the assumption that MATH be normalized. Thus, MATH implies MATH, i. e. any normal functional on MATH annihilating MATH annihilates MATH as well. Now, since the MATH-algebra MATH acts non-degenerately on MATH, NAME 's Density Theorem tells us that its strong and MATH-weak closures coincide with MATH, and this in turn is equal to the NAME algebra MATH; for the existence of an element MATH not contained in MATH would, by the NAME, imply a MATH-weakly continuous (normal) functional that vanishes on MATH but not on MATH in contradiction to the above result. Now suppose that there does not exist a separating vector for the NAME algebra MATH. Then the sequence MATH is such a vector for the NAME algebra MATH, where MATH denotes the identity representation of MATH in MATH. The result of the preceding paragraph thus applies to the MATH-algebra MATH of operators on the separable NAME space MATH which is weakly dense in MATH: MATH. We infer that there exists a norm-separable MATH-subalgebra MATH of MATH including its unit MATH, which is strongly dense in MATH. Now, MATH is a faithful representation of MATH on MATH and its inverse MATH is a faithful representation of MATH on MATH which is continuous with respect to the strong topologies of MATH and MATH. Therefore MATH is a norm-separable MATH-subalgebra of MATH, containing the unit element MATH and lying strongly dense in MATH.
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hep-th/0005103
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CASE: ` REF ': Since MATH is a group homomorphism this is trivial. CASE: ` REF ': Assume that MATH is free but MATH. Thus MATH, in contradiction to the NAME isomorphism REF .
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hep-th/0005273
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Assume first that MATH is real, and consider the functional MATH for MATH. Then MATH where MATH and MATH was used. Hence MATH is invariant as well, and REF follows using uniqueness of the integral (up to normalization). For MATH, we define MATH with the star structure REF . Using MATH and MATH, an analogous calculation shows that MATH is invariant under the action of MATH, which again implies REF .
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hep-th/0005273
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For MATH, we have MATH and hermiticity is immediate. For MATH, consider MATH . NAME follows using REF : MATH . Using REF for MATH, it is not difficult to see that they are also positive - definite.
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hep-th/0005273
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Using MATH (which follows from REF ) and MATH, it is easy to check that MATH for all MATH, and REF follows. REF follows immediately from MATH, and To see REF , one needs the well - known relation MATH, as well as MATH; the latter follows from the quasitriangularity of MATH. The commutation relations among the MATH are obtained as in CITE by observing MATH using the commutation relations MATH, as well as REF . The remaining relations can be checked similarly.
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math-ph/0005010
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Though MATH-modules MATH fail to be MATH-modules CITE, one can use the fact that the sheaves MATH are projections MATH of sheaves of MATH-modules. Let MATH be a locally finite open covering of MATH and MATH the associated partition of unity. For any open subset MATH and any section MATH of the sheaf MATH over MATH, let us put MATH. Then, MATH provide a family of endomorphisms of the sheaf MATH, required for MATH to be fine. NAME MATH of MATH also yield the MATH-module endomorphisms MATH of the sheaves MATH. They possess the properties required for MATH to be a fine sheaf. Indeed, for each MATH, there is a closed set MATH such that MATH is zero outside this set, while the sum MATH is the identity morphism.
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