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Journal of Mathematical Physics

Gauge transformations and normal states of the CCR

M. Fannes, André F. Verbeure

Published Oct 1, 1975• 7 citations
Publisher

Abstract

Let ω be any normal state on the CCR-algebra, and τχ the gauge transformation corresponding to the continuous character χ on the test function space as an Abelian group; denote by K the set of these characters; then we prove that the linear hull of {ω0τχ‖χ-K} is norm dense in all normal states. It is also proved that the theorem is in general false if we take the convex hull.

DOI
10.1063/1.522423

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Let ω be any normal state on the CCR-algebra, and τχ the gauge transformation corresponding to the continuous character χ on the test function space as an Abelian group; denote by K the set of these characters; then we prove that the linear hull of {ω0τχ‖χ-K} is norm dense in all normal states. It is also proved that the theorem is in general false if we take the convex hull.
Abstract The set of states on CCR(H)\textrm{CCR}({{\mathcal {H}}})CCR(H) CCR ( H ) , the CCR algebra of a separable Hilbert space H{{\mathcal {H}}}H H , is here looked at as a natural object to obtain a non-commutative version of Freedman’s theorem for unitarily invariant stochastic processes. In this regard, we provide a complete description of the compact convex set of states of CCR(H)\textrm{CCR}({{\mathcal {H}}})CCR(H) CCR ( H ) that are invariant under the action of all automorphisms induced in second quantization by unitaries of H{{\mathcal {H}}}H H . We prove that this set is a Bauer simplex, whose extreme states are either the canonical trace of the CCR algebra or Gaussian states with variance at least 1.
In this paper we prove some theorems about the n-representability problem for reduced density operators. The first theorem (Theorem 6) sharpens a theorem proved by Garrod and Percus. Let Tnp be the set of all n-representable p-density operators. Then a density operator Dp belongs to Tnp¯ (the bar indicates the closure with respect to a certain topology) if and only if Tr (DpBp) ≥ 0 for all bounded self-adjoint p-particle operators Bp, such that their n-expansion (pn)ΓpnBp≡ ∑ i1<…<ipBp(i1…ip)is a positive operator in n-particle space. Moreover, it is shown that Tnp¯ is the closed convex hull of the exposed points of Tnp of finite one-rank (Theorem 9). A more practical version of this theorem may be formulated in the following manner (cf. Theorem 8). Consider the set γp of subspaces of the n-particle space, occurring as an eigenspace to the deepest eigenvalue of a bounded n-particle operator which is the n expansion of some p-particle operator. Choose from every element of γp one (and only one) vector (function) and form the corresponding reduced p-particle operator. Tnp¯ is the closed convex hull of all these p-density operators (cf. Theorem 9). For p = 1, this theorem reduces to Coleman's theorem about the n representability of the 1 matrix.
The basis-independent approach to canonical commutation relations (CCR's), which allows arbitrary test-function spaces for smearing the field operators, is a generalization of the basis-dependent approach, in which the fields are smeared with an orthonormal system (or finite linear combinations thereof) to obtain an infinite set of qk and pk. Using recent results on continuity properties of representations of the CCR's, we show that every representation of the basis-independent type in a separable Hilbert space can be obtained by continuous extension of a suitable representation of the basis-dependent type where properties like irreducibility, cyclicity, etc., remain unaffected. In this sense, both approaches are equivalent, and the classification problem for CCR's is reduced from the simultaneous consideration of all representations for all possible test-function spaces to those for a single one (up to isomorphism).
Let {o^,..., a) n ] and {<TI,..., a n ] be finite sets of normal states over the bounded linear operators acting over some infinite-dimensional separable Hilbert-space.In the paper mutually equivalent sets of necessary and sufficient conditions are derived that there exists a completely positive, unity preserving linear map Ttransforming 0-1,..., a n simultaneously into o)i..., o) n : a) k = a k ° T,k = l,... 9 n.Of particular interest is this 6 '/z-tuple-problem" in case of pairs of density operators (normal states), i.e. for n=2.In this situation possible connections with the notion of "generalized transition probability" are analyzed, and at least in case of normal states over the bounded operators a characterization of functionals of this type is proposed and applied. § lo Main Results, ExamplesLet jtf be a unital C*-algebra ? with unit 7, topological dual jtf*, and group of unitary elements <%(j&).To each Fe ^(j/) let us associate a bounded linear operator T v over the dual by T ¥ (a))(X) = co(VXV*) = co v (X), for all XEJ^ and any COG jtf*.On the set of bounded linear operators ^(j/*) over j/* we may introduce the weak operator topology (w-topology) which is characterized by the system of semi-norms {q^xi coe.s/*,XEJ/} given by q (0 ,x(T) = \T(a>)(X)\, with Te^(X*).Then, by % u (<xf} the weak closure of the convex hull of the operators T v will be denoted: ^M(j^) = conv{T F : VE<%(JZ?)} W .In this paper as an essential result the following theorem is proved (proofs of most of the assertions of this part are given In § §3-5): LI Ttoeoremo Let 3? be an infinite-dimensional separable complex
A detailed analysis of and a general decomposition theorem for in general unbounded symplectic transformations on an arbitrary complex pre-Hilbert space (one–boson test function space) are given. The structure of strongly continuous symplectic groups on such spaces is determined. The connection between quadratic Hamiltonians, Bogoliubov transformations, and symplectic transformations is discussed in the Fock representation, and their relevance for squeezing operations in quantum optics is pointed out. The results for this rather general class of transformations are proved in a self-contained fashion.