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  • 1
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract We give a short proof for the decomposability of states on nuclear *-algebras into extremal states by using the integral decompositions of Choquet and the nuclear spectral theorem, recovering a recent result by Borchers and Yngvason. The decomposition of Wightman fields into irreducible fields is a special case of this. We also indicate a quick solution of the moment problem on nuclear spaces.
    Type of Medium: Electronic Resource
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  • 2
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract Recently, Nelson [2] has constructed relativistic fields from Euclidean fields which satisfy the Markoff and reflection property as well as an additional domain assumption. In this paper we replace the Markoff and reflection property by a weaker condition, a very simple positivity condition (“T-positivity”) which can be very easily expressed in terms of the expectation functionalE(f)=〈ω, exp {i φ (f)} ω〉. We show that the special role of the Markoff property in Nelson's approach is entirely due to features also shared byT-positivity. The role of Nelson's domain assumption (A′) in by-passing the difficulties with the paper of Osterwalder and Schrader [4] are made transparent, and possible ways to weaken this assumption are pointed out. If the conditions of [4] should turn out to be sufficient after all, (A′) can be replaced by a simple differentiability condition onE(τf). Our approach can also be applied to Fermi fields. The notion of Markoff and reflection property is discussed and shown to implyT-positivity.
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  • 3
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract We derive new correlation inequalities for even Ising ferromagnets whose interaction is invariant under some symmetry transformation and satisfies a growth condition. The recent results of Schrader [1] and Messager and Miracle-Sol [2] for the nearest neighbour (n.n.) Ising model reappear as a special case. In addition we obtain monotonicity of 〈σ0σ j 〉 under translation ofj perpendicular to diagonal hyperplanes and the inequality 〈σ0σ j 〉≧ $$\left\langle {\sigma _0 \sigma _{\left( {\sum {\left| {j_\nu } \right|,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{0} } } \right)} } \right\rangle $$ for n.n. and other interactions.
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  • 4
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract In a canonical field theory, the field Φ(f) and momentum π(g) are assumed defined for test functionsf andg which are elements of linear vector spaces and , respectively. Generally, the continuity of the map onto the unitary Weyl operatorsU(f),V(g) is taken as ray continuity, the barest minimum to recover the field operators as their generators, i.e.,U(f)=e iΦ(f) ,V(g)=e iπ(g) . This leaves open the question of whether any wider continuity properties follow and what form they would take. We show that much richer continuity properties do follow in a natural fashion for every cyclic representation of the canonical commutation relations. In particular, we show that the test function space may be taken as a metric space, that the space may be uniquely completed in this topology, and that the map into the unitary Weyl operators is strongly continuous in this topology. The topology induced by this metric is minimal in the sense that it is the weakest vector topology for which the mapsf→U(f),g→V(g) are strongly continuous. An expression for a suitable metric can easily be given in terms of a simple integral over a state on the Weyl operators.
    Type of Medium: Electronic Resource
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  • 5
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract We study mixing or spatial cluster properties and some of their consequences in classical lattice systems, in particular complete regularity and the weaker notion of strong mixing. Introducing the notion of reflection positivity as a generalization ofT-positivity of [1], we construct a generalized transfer matrixP and relate complete regularity to a spectral gap inP. It is shown that all reflection invariant Ising systems with n.n. and ferromagnetic n.n.n. interaction satisfy reflection positivity. For Ising ferromagnets with reflection positivity, exponential decay of the truncated 2-point function implies complete regularity. In particular, the 2-dimensional spin-1/2 Ising model is completely regular, except at the critical point. This complements a result of [2] that strong mixing fails at the critical point of this model and in this case verifies the suggestion of Jona-Lasinio [3] that critical behaviour should be linked with failure of strong mixing. We then show that strong mixing imposes severe restrictions on the possible form of limits of block spins. Strong mixing in each direction allows onlyindependent Gaussians as non-zero limit if the 2-point function exists; strong mixing in a single direction only will allow infinitely divisible distributions.
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  • 6
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract We generalize some notions of probability theory and theory of group representations to field theory and to states on the Borchers algebraS. It is shown that every field (relativistic and Euclidean, ...) can be decomposed into a countable number of prime fields and an infinitely divisible field. In terms of states this means that every state onS is a product of an infinitely divisible state and a countable number of prime states, and in this formulation it applies equally well to correlation functions of statistical mechanics and to moments of linear stochastic processes overS orD. Necessary and sufficient conditions for infinitely divisible states are given. It is shown that the fields of the ø 2 4 -theory are either prime or contain prime factors. Our results reduce the classification problem of Wightman and Euclidean fields to that of prime fields and infinitely divisible fields. It is pointed out that prime fields are relevant for a nontrivial scattering theory.
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  • 7
    ISSN: 1434-6036
    Keywords: 42.50 ; 32.90+a
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract We use the quantum jump method to study the photon statistics of a single laser-driven atom in the Λ configuration where both lower levels are strongly coupled to the common upper level. Under certain conditions we show that, for almost degenerate lower levels, light and dark periods occur which are similar to those of the well-known Dehmelt V system. Analytic results for their mean lengths and other statistical properties are given. For large separation of the lower levels we prove an interesting bunching property by the photons in the resonance fluorescence near the dark resonance. We propose a realistic system for which these effects may be observed.
    Type of Medium: Electronic Resource
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  • 8
    Electronic Resource
    Electronic Resource
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 26 (1985), S. 62-64 
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: Can the fields φ and π of a representation of the CCR's be written as φ=1/{φ1×1+1⊗φ2} and similarly for π, such that φi and πi satisfy the CCR's? What are the possible φi's and πi's? This is equivalent to a factorization of the corresponding generating functionals (scaled by 1/). Generalizing this question somewhat we show a noncommutative analog of Cramér's theorem of probability theory. If φ and π are Fock fields then so are φi, πi, i=1,2; similarly for quasifree representations of the CCR's. As an application we show that the fields of a representation of the CCR's whose generating functional differs from a Fock functional by a phase factor only are just shifted Fock fields.
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  • 9
    ISSN: 1572-9613
    Keywords: Operator Poisson process ; singular correlation functions ; noncommutative cumulant expansions ; partial summation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract Quantum shot noise consists of individual pulses which contribute time-dependent (operator) “potentials” toward a total potentialV(t). The averaged quantity 〈T exp ∫ t0 t dt′V(t′)〉 in general can no longer be calculated explicitly, in contrast to the classical case, and expansions are of interest. Noncommutative cumulant expansions are not directly applicable if the correlation functions ofV(t) have singularities, as happens in applications. It is shown here that these expansions, when applied to quantum shot noise, can be partially summed to give expansions in powers of the pulse densityυ. Three types of such expansions are established explicitly, and for two of them the derivation is direct. For one of them the first-order approximation is closely connected to the so-called unified theory of spectral-line broadening.
    Type of Medium: Electronic Resource
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  • 10
    ISSN: 1572-9613
    Keywords: Pressure broadening ; stochastic differential equations ; asymptotic decay ; N-particle limit
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract An atom in a gas or plasma experiences a random potential which gives rise to the so-called pressure broadening. The corresponding line shape is obtained in the usual two-level model by a trace operation from the Fourier transform of 〈T(t, 0)〉, the average of the time-development operator. Under certain technical assumptions it is rigorously shown by probabilistic techniques that 〈T(t, 0)〉 falls off faster thant −3+ε for any ε 〉 0, giving a continuous Fourier transform and line shape. An alternative expression is derived for the latter which explicitly displays its positivity and which is a limit over increasing perturber numbers. The latter generalizes results of von Waldenfels. Part I is preparatory for Part II, where a noncommutative cluster expansion is applied to the line-shape problem. Several open questions are pointed out which merit a rigorous investigation.
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