Systems of covariance in quantum probability

Symmetry groups and systems of covariance are investigated in the framework of quantum probability theory. It is shown that a measurementX can be represented by a positive operator-valued measure on a sectorS of the amplitude space. Moreover, provides a generalized system of covariance for the generalized unitary representation of a symmetry group.     

Coherent states and induced representations

A generalization of the notion of coherent states is given. The following one-to-one correspondences are pointed out: (1) between covariant overcomplete systems of coherent states and a class of covariant semi-spectral measures; (2) between covariant semispectral measures and unitary irreducible subrepresentations of induced representations of Lie groups; (3) between unitary irreducible representations of Lie groups with covariant overcomplete systems of coherent states and unitary irreducible subrepresentations of induced representations, whose representation spaces are reproducing kernel Hilbert spaces.     

Induced representations and induced Hamiltonian actions

Given a Hamiltonian action of a closed subgroup Γ of a Lie group G, we construct a Hamiltonian action of G (the induced action). Our construction follows the well known scheme of inducing unitary representations due to Mackey. In order to bring out the similarity between these constructions we reformulate Mackey's scheme in terms of ⪡quantum reductions⪢.     

On the type and the standardform of induced UA representations

We derive to which of the three types an irreducible UA representation which is obtained with the procedure of generalised induction belongs. We analyse the question whether or not the irreducible induced UA representations are on standardform. The results hold as well for induced PUA representations.     

Linear covariance for nonlinear formal representations of the Poincaré group in 2+1 dimensions

A concept of linear covariance is defined for nonlinear formal representations of the Poincaré group. Then it is proved that the formal nonlinear representations previously built for 2+1 dimensions with irreducible unitary massless representations as free parts (cf. (1)) are nonlinearly equivalent to linearly covariant representations.     

Topologies induced by representations of the canonical commutation relations

This paper is concerned with continuity properties of representations of the canonical commutation relations, and is mainly devoted to a detailed discussion of the topologies induced on the test function spaces. The notion of closability of a representation of the canonical commutation relations is introduced and studied. We also discuss the strong continuity of functions of self-adjoint operators, and use bounded functions to define an analogue of the strong operator topology on the set of all self-adjoint operators.     

Classification of conformal representations induced from the maximal cuspidal parabolic

In the present paper we continue the project of systematic construction of invariant differential operators on the example of representations of the conformal algebra induced from the maximal cuspidal parabolic.     

Fractional squeezing-Hankel transform based on the induced entangled state representations

Based on the fact that the quantum mechanical version of Hankel transform kernel(the Bessel function) is just the transform between |q, r and(s, r′|, two induced entangled state representations are given, and working with them we derive fractional squeezing-Hankel transform(FrSHT) caused by the operator e~(-iα)(a_1~?a_2~?+a_1a_2)e~(-iπa_2~?a_2), which is an entangled fractional squeezing transform operator. The additive rule of the FrSHT can be explicitly proved.     

Commutators and lorentz covariance

The Lorentz transformation property of the right hand side of equal time commutators give conditions on Schwinger terms of commutators involving time derivatives. In particular, it is seen that current algebra relations imply special locality properties for the current divergence. Thus, an important element of the hypothesis of partially conserved currents is already contained in the current commutation rules.     

Non-linear equations and covariance

A discussion about the connection between non-linear wave equations covariant under the action of a Lie group on one hand, and the theory of non-linear representations of the covariance group developed elsewhere [1] on the other hand, is presented here.Applications to a class of relativistic non-linear wave-equations are then suggested.     

Canonical geometrodynamics and general covariance

By extending geometrodynamical phase space by embeddings and their conjugate momenta, one can homomorphically map the Lie algebra of space-time diffeomorphisms into the Poisson algebra of dynamical variables on the extended phase space.     

Lorentz covariance and kinetic charge

There is a one-to-one correspondence between inequivalent covariant displaced Fock representations of the free relativistic field and the 1-cohomology of the Poincaré group with coefficients in the 1-particle space.Representations with positive energy are obtained from cocycles with finite energy which have particle-like properties and are interpreted as condensed states of matter without a sharply defined mass.The 1-cohomology groups ofP ₊ are calculated. These are trivial in 3- or 4-dimensional space-time, or if the mass is non-zero. Non-trivial cocycles for subgroups lead to representations in whichP-invariance is spontaneously broken. We recoverP-invariance in a direct integral representation possessing a gauge group, and a superselection structure labelled by the velocities of the condensed states of matter which are the cocycles determining each irreducible component of the representation. A model in 4-dimensional space-time is constructed.     

Magnetic moments and general covariance

The principle of general covariance for deriving the passive equations is applied to the group of automorphisms of compact support of a principal bundle acting on the space of Cartan connections. Under an orbital constraint this yields the equations of motion postulated in [11] which have a symplectic character. When specialized to an affine connection these equations become the equations of interaction of spin and torsion studied in [5]. With the incorporation of a Higgs field, a forcing term is added to the equations which, in the case of an electromagnetic field, incorporate the effects of the magnetic moment. The Higgs field is of the character obtained by dimensional reduction.     

Lax representations and zero-curvature representations by the Kronecker product

It is shown that the Kronecker product can be applied to construct not only new Lax representations, but also new zero-curvature representations of integrable models. A characteristic difference between continuous and discrete zero-curvature equations is pointed out.     

Noisy covariance matrices and portfolio optimization

According to recent findings [#!bouchaud!#,#!stanley!#], empirical covariance matrices deduced from financial return series contain such a high amount of noise that, apart from a few large eigenvalues and the corresponding eigenvectors, their structure can essentially be regarded as random. In [#!bouchaud!#], e.g., it is reported that about 94% of the spectrum of these matrices can be fitted by that of a random matrix drawn from an appropriately chosen ensemble. In view of the fundamental role of covariance matrices in the theory of portfolio optimization as well as in industry-wide risk management practices, we analyze the possible implications of this effect. Simulation experiments with matrices having a structure such as described in [#!bouchaud!#,#!stanley!#] lead us to the conclusion that in the context of the classical portfolio problem (minimizing the portfolio variance under linear constraints) noise has relatively little effect. To leading order the solutions are determined by the stable, large eigenvalues, and the displacement of the solution (measured in variance) due to noise is rather small: depending on the size of the portfolio and on the length of the time series, it is of the order of 5 to 15%. The picture is completely different, however, if we attempt to minimize the variance under non-linear constraints, like those that arise e.g. in the problem of margin accounts or in international capital adequacy regulation. In these problems the presence of noise leads to a serious instability and a high degree of degeneracy of the solutions. Received 31 December 2001     

General covariance and open string interactions

It has been suggested that Witten's open string cubic interaction is independent of the background geometry. We investigate this suggestion in detail with a view to understanding the rôle of closed strings in Witten's open string field theory. The standard oscillator formulation of the vertex requires the use of a metric, if it is to be generally covariant, indicating that closed string interactions are required. However by changing to variables better suited to treating the midpoint, we show explicitly that it is possible to transform away all reference to the background metric. This supports the assertion that the cubic action (of Horowitz, Lykken, Rohm and Strominger) can also describe closed strings.     

Manifest gauge and Poincaré covariance

Manifest gauge invariance is known to be incompatible with manifest Poincaré covariance (Strocchi's theorem). By extending the notion of gauge invariance to that of gauge covariance, we circumvent that incompatibility, at least for free electromagnetic potentials. In the new formulation the potentials, A_G, for all permissible gauges G. act on a common Hilbert space. This formulation is shown to be inequivalent to the more conventional ones. (In particular, the Coulomb gauge is now inaccessible.) The abstract gauges G are represented by c-number potentials V_G, which play a central role in the theory. Even without interaction, they obey a field equation with a source, and thus they anticipate the existence of electric charges.