On semipositive definiteness of 2n-degree forms

The aim of this paper is to present necessary and sufficient conditions for the semipositive definiteness of 2n-degree forms. These conditions allow to verify whether a given map Λ: $\text{A}$ → $\text{A}$ (where $\text{A}$ is the algebra of linear operators on the n-dimensional Hilbert space $\text{H}$) is a semipositive map.     

Eigenfunction expansions of operators admitting separation of an infinite number of variables

Let A_k be a self-adjoint operator in a Hilbert space H_k (k = 1, 2, …) and let $\text{L}$ be an operator of the form $\text{L}\text{= A}{}_{\mathrm{r}}\text{? 1 ? 1 ? … + 1 ? A}{}_2\text{? 1 ? 1 ? … + …}$ acting in the infinite tensor product $\text{?}\text{k=1}\text{∞}\text{H}{}_{\mathrm{k}}$. We construct the spectral theory of these operators. In particular, the expansion is generalized eigenvectors of this operator is constructed using the eigenvectors of the operators A_k.     

On the entropy of systems undergoing time evolution

The aim of the present paper is to discuss the behaviour of entropy of physical systems undergoing time evolution. We discuss the case of an infinite-dimensional separable Hilbert space $\text{H}$ where the entropy of a microstate ? is given by the formula s(?) = ?Tr(?ln?). Information about a physical system is given by the mean values Tr(?A_i) = m, i = 1, …, N, of N self-adjoint (not necessarily bounded) operators A_i.     

A mathematical interpretation of Dirac's formalism part b: generalized eigenfunctions in trajectory spaces

Starting with a Hilbert space $\text{L}{}_2\text{(}\text{R}\text{,μ)}$ we introduce the dense subspace $\text{R}\text{(}\text{L}{}_2\text{(}\text{R}\text{,μ))}$ where $\text{R}$ is a positive self-adjoint Hilbert–Schmidt operator on $\text{L}$₂(R,μ). For the space $\text{R}\text{(}\text{L}{}_2\text{(}\text{R}\text{,μ))}$ a measure-theoretical Sobolev lemma is proved. The results for the spaces of type $\text{R}\text{(}\text{L}{}_2\text{(}\text{R}\text{,μ))}$ are applied to nuclear analyticity spaces $\text{S}{}_{\mathrm{<mtext>X,</mtext><mtext>A</mtext>}}\text{=}\text{?}\text{t>0}\text{e}{}^{\mathrm{<mtext>-t</mtext><mtext>A</mtext>}}\text{(X)}$, where e^?t$\text{A}$ is a Hilbert–Schmidt operator on the Hilbert space X for each t>0. We solve the so-called generalized eigenvalue problem for a general self-adjoint operator $\text{P}$ in X.     

Semi-simple real subalgebras of non-compact semi-simple real Lie algebras IV

The investigation of the problem of embedding a semi-simple real Lie algebra $\text{L}$′ in a non-compact semi-simple real Lie algebra $\text{L}$ is extended to the case in which at least one of the real Lie algebras has a semi-simple complex extension, which consists of the direct sum of two simple complex Lie algebras. Detailed procedures are given, which together with those given previously, allow the construction of all embeddings of $\text{L}$′ in $\text{L}$ when their complex extensions are A₁, B₁, C₁, D₁ or a direct sum of any two of these. The procedures are illustrated by considering examples corresponding to complex Lie algebra embeddings A₁?(A₂⊕A₂), (A₁⊕A₁)?(A₂⊕A₂), (A₁⊕A₁)?A₃, (A₁⊕A₁)??(A₃⊕A₃) and (A₁⊕A₁)?(A₃⊕A₂). Because of its physical significanc embeddings of SL(2,C) in simple and semi-simple real Lie algebras are studied in detail.     

Positive cones in enveloping algebras

The concept of strong ordering on enveloping algebras of finite-dimensional Lie algebras is introduced and studied as a generalization of the corresponding notion for the commutative polynomial algebra. A linear functional f on an enveloping algebra $\text{E}$ (G) is called strongly positive if f(x) ? 0 for all x ? $\text{E}$(G) which are mapped on positive operators for all G-integrable irreducible representations of $\text{E}$(G). We prove that for each real connected Lie group G ≠ R₁ there are positive, not strongly positive, linear functionals on $\text{E}$(G). A non-commutative problem of moments is defined. It has a solution iff the corresponding linear functional is strongly positive.     

The uniform and the strong topology on realizations of the algebra of polynomials

It is shown that under quite general assumptions on the operators A₁,…,A_n (unbounded, symmetric) and on the domain $\text{D}$ on the realization $\text{P}\text{(A}{}_1\text{,…,A}{}_{\mathrm{n}}\text{)}$ of the algebra of polynomials $\text{P}\text{(x}{}_1\text{,…,x}{}_{\mathrm{n}}\text{)}$, the strongest locally convex topology τ_st coincides with the uniform topology τ_$\text{D}$ as well as with the strong operator topology τ_s. In the case n = 2 some conditions are given, under which these general assumptions are fulfilled.     

Decoherence Functionals for von Neumann Quantum Histories: Boundedness and Countable Additivity

Gell–Mann and Hartle have proposed a significant generalisation of quantum theory in which decoherence functionals perform a key role. Verifying a conjecture of Isham–Linden–Schreckenberg, the author analysed the structure of bounded, finitely additive, decoherence functionals for a general von Neumann algebra A (where A has no Type I₂ direct summand). Isham et al. had already given a penetrating analysis for the situation where A is finite dimensional. The assumption of countable additivity for a decoherence functional may seem more plausible, physically, than that of boundedness. The results of this note are obtained much more generally but, when specialised to L(H), the algebra of all bounded linear operators on a separable Hilbert space H, give: Let d be a countably additive decoherence functional defined on all pairs of projections in L(H). If H is infinite dimensional then d must be bounded. By contrast, when H is finite dimensional, unbounded (countably additive) decoherence functionals always exist for L(A). Received: 6 December 1996 / Accepted: 18 May 1997     

On invariant states and the commutant of a group of quasi-free automorphisms of the car algebra

We study primary states of the CAR algebra which are left invariant under quasi-free automorphisms α_U corresponding to unitaries U of a von Neumann algebra $\text{M}$ on the one-particle Hilbert space, and show that they are quasi-free states ?_A corresponding to self-adjoint operators A in $\text{M}$′ with 0 ? A ? 1, under the assumption that $\text{M}$ does not contain any finite type Ifactor direct summands. Next we study automorphisms of the CAR algebra which commute with α_U for U in a von Neumann algebra $\text{M}$ and show that they are quasi-free automorphisms α_U with U in $\text{M}$′ under the same assumption on $\text{M}$ as above. Finally by using the latter result we obtain a generalization of a theorem of Hugenholtz and Kadison [3].     

Perturbation theory for the effective interaction in nuclei

Starting from a decomposition of the Hamiltonian H(x) of the nuclear many-body problem in the form H(x) = H₀ + xV, where H₀ is a shell-model Hamiltonian, V the residual interaction, and x a strength parameter, we introduce a general effective interaction W(x) describing the interaction of nucleons within a shell, and the associated effective operators $\text{A}\text{?}\text{(x)}$. We display some properties of these operators. From a particular choice of W(x) we obtain the expressions introduced earlier by several authors. The convergence of the expansions for W(x) and $\text{A}\text{?}\text{(x)}$ in powers of x is investigated. It is shown that W(x) and $\text{A}\text{?}\text{(x)}$ are holomorphic in a domain of the complex x-plane including the point x = 0. With the help of a generalization of the von Neumann-Wigner noncrossing rule, we exhibit the nature of the common singularity of W(x) and $\text{A}\text{?}\text{(x)}$ which is closest to the origin and thus defines the radius r₀ of convergence of the expansions of W and $\text{A}\text{?}$. It is shown that r₀ is unaffected by the cancellation of unlinked diagrams. A criterion of consistency is established, which shows that most of the practical calculations of W lead to results which are inconsistent with the definition of W.     

On the spaces of positive and negative frequency solutions of the Klein-Gordon equation in curved space-times

In a space-time $\text{(V}{}_{\mathrm{n}}\text{×}\text{R}\text{;g)}$ with V_n closed (n ≠ 2) satisfying certain global conditions, we can write the Klein-Gordon equation, relative to a suitable class of atlases, in the evolution form du/dt = T^-1(t)u, on Sobolev spaces K^l(V_n) = H^l(V_n) × H^l?1(V_n), where the spectrum of T^-1(t) is imaginary. Following papers by T. Kato and J. Kisyński we prove the existence of the evolution operator for this equation. The space $\text{K}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{(V}{}_{\mathrm{n}}\text{)}$ has a natural strongly-symplectic structure ω. We determine the explicit form of complex-structure-positive operators of this structure. We prove that any two such operators, say J₁, J₂, are symplectically equivalent, (i.e. there is a symplectic transformation S such that J₂ = SJ₁S^-1). Spaces of positive and negative frequency solutions are then unique modulo symplectic equivalence. Each operator J determines a regular kernel on space-time which satisfies the properties of the kernel postulated by A. Lichnérowich in his program of quantization of fields in curved space-times. We carry out explicit calculations in the case of Robertson-Walker space-times. If an additional condition is satisfied by the given space-time, a unique complex-structure-positive operator can be selected in a natural way. This condition is satisfied by globally stationary space-times.     

Dirac dynamics on stochastic phase spaces for spin 1/2 particles

The Foldy-Wouthuysen representation of the dynamics of a free spin $\text{1}\text{2}$ particle is formulated in a Hilbert space $\text{H}$(Γ) of spinor-valued functions over Γ-space. $\text{H}$(Γ) carries a reducible Wigner-type representation of the Poincaré group. The transition to the Dirac representation in a new bispinor Hilbert space $\text{K}$(Γ) is effected by means of a generalized inverse Foldy-Wouthuysen transformation. Explicit expressions are derived for the resolution generators η of invariant subspaces $\text{K}$±(Γ_η) carrying irreducible representations of the resulting representations of the Poincaré group. The formalism is recast in a manifestly covariant form and the Dirac equation on $\text{H}$(Γ_s) with minimal coupling to a four-potential is examined. It is shown that the resulting external field theory is gauge-invariant and relativistically covariant.     

Semi-simple real subal gebras of non-compact semi-simple real lie algebras V

The investigation of the problem of embedding a semi-simple real Lie algebra $\text{L}$ in a non-compact semi-simple real Lie algebra $\text{L}$ is extended to the case when $\text{L}$ and/or $\text{L}$ is exceptional. Matrix representations for all the exceptional Lie algebras are calculated. Detailed procedures are given, which, together with those given in previous papers, allow the construction of all embeddings of $\text{L}$ in $\text{L}$, when their complex extensions are A₁, B₁, C₁, D₁, E₆, E₇, E₈, F₄, G₂ or a direct sum of any two of there. The procedures are illustrated by examples, including all real semi-simple Lie subalgebras of real forms of G₂ and sub-algebras of real forms of F₄ whose complex extensions are B₄ or A₁ (representation (16) + (9)). Because of its physical significance, all embeddings of SL (2, C) in real forms of F₄ and E₆ are given. Many of these are new results.     

The p-approximations and inverse problems of expansion and contraction of n-fermion operators

The orthogonal projectors from the space $\text{L}$(n) of n-fermion operators onto its sub-spaces $\text{L}$_p(n) consisting of p-reducible elements of $\text{L}$(n>), as well as those from $\text{L}$(n) onto $\text{L}$_p(n)?$\text{L}$_p?1(n) are constructed. Using the above projectors the inverse problems of contraction and expansion are solved.     

A positive energy dynamics and scattering theory for directly interacting relativistic particles

Starting from the tensor product of N irreducible positive energy representations of the Poincaré group describing N free relativistic particles with arbitrary spins and positive masses, we construct an interacting positive energy representation by modifying the total 4-momentum operator. We first make a transformation to a Hilbert space on which the free total 4-momentum operator equals the product of a dimensionless center-of-mass 4-vector ($\text{(|k|}{}^2\text{+ 1)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$, k) and a free “reduced Hamiltonian” H_r⁰, which is a positive operator acting only on internal variables, and then replace H_r⁰ by an interacting reduced Hamiltonian H_r = H_r⁰ + V, where V commutes with the Lorentz group and is such that H_r is a positive operator. The resulting product form is shown to imply that the wave operators interwine the free and interacting representations so that the S-operator is Lorentz invariant. From a physical point of view the scheme is related to the framework first introduced by Bakamjian and Thomas, in which the Hamiltonian and boost generators are modified, but the above procedure makes a mathematically rigorous discussion much simpler. In the spin-zero case we introduce a natural generalization of the pair potentials of nonrelativistic N-particle Schrödinger theory to the present relativistic setting, study its scattering theory, and point out some problems that do not have analogs at the nonrelativistic level. In the $\text{spin}\text{-}\text{1}\text{2}$ case we propose, inspired by the Dirac equation, explicit reduced Hamiltonians to describe atomic energy levels and present arguments making plausible that their eigenvalues are in closer agreement with the experimental data than their nonrelativistic counterparts. We also consider extensions to arbitrary spin and, in the $\text{spin}\text{-}\text{1}\text{2}$ case, coupling of a quantized radiation field. In view of eventual applications to “completely integrable” one-dimensional field theories the case of one space dimension is studied as well, both in quantum mechanics and in classical mechanics.     

On observability of N-level quantum systems

The purpose of this paper is to discuss necessary and sufficient conditions for observability of N-level quantum systems. We assume that the information about a physical system is given by the mean values Tr(?(t_j)A_i) = m_Ai(t_j), of n self-adjointoperators A₁,…,A_n on $\text{H}$ at some instants t₁ < t₂ <…<t_s. The question of theminimal number n of operators A₁,…,A_n (physical quantities $\text{A}{}_1\text{, …,}\text{A}{}_{\mathrm{n}}$) for which the quantum system $\text{S}$ is (A₁,…,A_n)-observableis discussed.     

Quasi-multiplication and 1-algebras

Let $\text{B}$ be a ¹-algebra with identity. With $\text{B}$ we associate a quasi-algebra Q($\text{B}$) consisting of sequences whose entries are elements of $\text{B}$. For A,B in $\text{B}$ we give general expansions pertaining to (A+B)ⁿ, e^A+B and e^Ae^B. We also discuss the case where $\text{B}$ is the ¹-algebra generate creation and annihilation operators. Another example deals with the ¹-algebra of field operators.     

Relativistic hamiltonian equations for any spin

We discuss 2(2J + 1)-component Poincaré-invariant Hamiltonian theories that describe free particles of definite mass and spin and that are subject to the conditions (a) every observable $\text{O}$ is either Hermitian or pseudo-Hermitian (i.e., O = ?₃O⁺?₃) and (b) the theory is invariant under the discrete symmetries. Our treatment is based on the Heisenberg equations of motion and on the Lie algebra of the Poincaré group. Explicit formulas are found for the generators of this algebra, including the Hamiltonian H, and all relations between the operators Γ and H that are both necessary and sufficient for $\text{K =}\text{1}\text{2}\text{[x}\text{, H]}{}_{\mathrm{+}}\text{+ Γ}$ to generate Lorentz boosts are found. To illustrate the utility of our results, we apply them to obtain explicit generalizations of the Dirac equation to any spin, by requiring that Γ = 0, and of the Sakata-Taketani spin-0 and spin-1 equations to any spin, by requiring that $\text{Γ = ??}{}_3\text{(}\text{1}\text{2m}\text{)}\text{S × p}$.     

Irreducible gauge theory of a consolidated Salam-Weinberg model

We derive the Salam-Weinberg model by gauging an internal simple supergroup $\text{SU}\text{(}\text{2}\text{1}\text{)}$. The theory uniquely assigns the correct SU(2)_L ? U(1) eigenvalues for all leptons, fixes θ_W = 30°, generates the W^±_σ, Z⁰_σ and A_σ together with the Higgs-Goldstone $\text{I}{}_{\mathrm{<mtext>L</mtext>}}\text{=}\text{1}\text{2}$ scalar multiplets as gauge fields, and imposes the standard spontaneous breakdown of SU(2)_L ? U(1). The masses of intermediate bosons and fermions are directly generated by $\text{SU}\text{(}\text{2}\text{1}\text{)}$ universality, which also fixes the Higgs field coupling.