Enlarged Hilbert Space Research Articles

Magnetic states at short distances

The magnetic interactions between a fermion and an antifermion of opposite electric or color charges in the 1S0 and 3P0 states are very attractive and singular near the origin and may allow the formation of new bound and resonance states at short distances. In the two body Dirac equations formulated in constraint dynamics, the short-distance attraction for these states for point particles leads to a quasipotential that behaves near the origin as -\alpha^2/r^2. Both 1S0 and 3P0 states admit two types of eigenstates with drastically different behaviors for the radial wave function u=r\psi. One type of states, with u growing as r^{1+\sqrt(1-4*\alpha^2)/2} at small r, will be called usual states. The other type of states with u growing as r^{(1-\sqrt(1-4*\alpha^2))/2} will be called peculiar states. Both of the usual and peculiar eigenstates have admissible behaviors at short distances. The usual bound 1S0 states possess attributes the same as those one usually encounters in QED and QCD. In contrast, the peculiar bound 1S0 states, yet to be observed, have distinctly different bound state properties and scattering phase shifts. For the 3P0 states, the usual solutions lead to the standard bound state energies and no resonance, but resonances have been found for the peculiar states whose energies depend on the description of the internal structure of the charges, the mass of the constituent, and the coupling constant. The existence of both usual and peculiar eigenstates in the same system leads to the non-self-adjoint property of the mass operator and two non-orthogonal complete sets. The mass operator can be made self-adjoint with a single complete set of admissible states by introducing a new peculiarity quantum number and an enlarged Hilbert space.

Purification of mixed states with closed timelike curve is not possible

In ordinary quantum theory, any mixed state can be purified in an enlarged Hilbert space by adding an ancillary system. The purified state does not depend on the state of any extraneous system with which the mixed state is going to interact or on the physical interaction. Here, we prove that it is not possible to purify a mixed state that traverses a closed timelike curve (CTC) and is allowed to interact in a consistent way with a causality-respecting (CR) quantum system in the same manner. In other words, if a CTC quantum system with a mixed state has to undergo an arbitrary interaction with an arbitrary CR system then it has to be in a ``proper'' mixture, as it cannot be regarded as a subsystem of a pure entangled state. Thus, in general for arbitrary interactions between CR and CTC systems, there is no universal purification by embedding in a larger Hilbert space for mixed states with CTCs. This shows that in quantum theory with CTCs there can exist proper and improper mixtures.

Projective Hilbert space structures at exceptional points

A non-Hermitian complex symmetric 2 × 2-matrix toy model is used to study projective Hilbert space structures in the vicinity of exceptional points (EPs). The bi-orthogonal eigenvectors of a diagonalizable matrix are Puiseux-expanded in terms of the root vectors at the EP. It is shown that the apparent contradiction between the two incompatible normalization conditions with finite and singular behaviour in the EP-limit can be resolved by projectively extending the original Hilbert space. The complementary normalization conditions correspond then to two different affine charts of this enlarged projective Hilbert space. Geometric phase and phase-jump behaviour are analysed, and the usefulness of the phase rigidity as measure for the distance to EP configurations is demonstrated. Finally, EP-related aspects of -symmetrically extended quantum mechanics are discussed and a conjecture concerning the quantum brachistochrone problem is formulated.

Upper Quantum Lyapunov Exponent and Anosov Relations for Quantum Systems Driven by a Classical Flow

We generalize the definition of quantum Anosov properties and the related Lyapunov exponents to the case of quantum systems driven by a classical flow, i.e. skew-product systems. We show that the skew Anosov properties can be interpreted as regular Anosov properties in an enlarged Hilbert space, in the framework of a generalized Floquet theory. This extension allows us to describe the hyperbolicity properties of almost-periodic quantum parametric oscillators and we show that their upper Lyapunov exponents are positive and equal to the Lyapunov exponent of the corresponding classical parametric oscillators. As second example, we show that the configurational quantum cat system satisfies quantum Anosov properties.

Simplifying Schmidt number witnesses via higher-dimensional embeddings

We apply the generalised concept of witness operators to arbitrary convex sets, and review the criteria for the optimisation of these general witnesses. We then define an embedding of state vectors and operators into a higher-dimensional Hilbert space. This embedding leads to a connection between any Schmidt number witness in the original Hilbert space and a witness for Schmidt number two (i.e. the most general entanglement witness) in the appropriate enlarged Hilbert space. Using this relation we arrive at a conceptually simple method for the construction of Schmidt number witnesses in bipartite systems.

Fermion-number violation in regularizations that preserve fermion-number symmetry

There exist both continuum and lattice regularizations of gauge theories with fermions which preserve chiral U(1) invariance (``fermion number''). Such regularizations necessarily break gauge invariance but, in a covariant gauge, one recovers gauge invariance to all orders in perturbation theory by including suitable counterterms. At the nonperturbative level, an apparent conflict then arises between the chiral U(1) symmetry of the regularized theory and the existence of 't Hooft vertices in the renormalized theory. The only possible resolution of the paradox is that the chiral U(1) symmetry is broken spontaneously in the enlarged Hilbert space of the covariantly gauge-fixed theory. The corresponding Goldstone pole is unphysical. The theory must therefore be defined by introducing a small fermion-mass term that breaks explicitly the chiral U(1) invariance and is sent to zero after the infinite-volume limit has been taken. Using this careful definition (and a lattice regularization) for the calculation of correlation functions in the one-instanton sector, we show that the 't Hooft vertices are recovered as expected.

Dual state vector of nonlinear coherent state and its application in complex P-representation

We introduce the dual vector | z,f (n)* of a single-mode nonlinear coherent state (NCS) by using the contour integral form of δ-function, and the explicit form of | z,f (n)* is obtained in enlarged Hilbert space. The nonlinear Fock states and their contour representation are deduced. A new completeness relation composed of | z,f (n)* and NCS in contour integral form is derived and its application in constructing a complex P-representation of density operators is shown.

A unified approach to Aharonov-Bohm, Aharonov-Casher and which-path experiments

A unified approach to Aharonov-Bohm, Aharonov-Casher and which-path experiments is presented, using an enlarged Hilbert space. This Hilbert space contains quasi-periodic Aharonov-Bohm wavefunctions R(x+2) = R(x)exp(i) with various values of . Thus it can describe which-path Aharonov-Bohm experiments where the phase is uncertain due to decoherence that occurs as a result of the observation of the paths of the electric charges. The same Hilbert space contains quasi-periodic Aharonov-Casher wavefunctions which describe magnetic flux tubes winding around an electric charge and which are related through a Fourier transform to the Aharonov-Bohm wavefunctions. The duality between these two phenomena is discussed. The decoherence occuring in which-path experiments is studied quantitatively. Magnetic and electric superselection rules, appropriate for the Aharonov-Bohm and Aharonov-Casher experiments correspondingly, are also discussed.

Decomposition of the electromagnetic field in lossless inhomogeneous dispersive dielectrics.

The interaction of the electromagnetic field with lossless Lorentz oscillators modeling the inhomogeneous dielectrics is diagonalized in a closed Fabry-P\'erot resonator using the generalized polariton transformation. Resulting dispersion relations coincide with the classical ones obtained by the solution of the wave equation. The corresponding decomposition is, however, orthogonal and complete in the enlarged Hilbert space. \textcopyright{} 1996 The American Physical Society.

Relativistic field formulation of simultaneous quantum measurement

The simultaneous measurement of Dirac field operators is formulated in analogy to the work of von Neumann and Arthurs-Kelly. Meter fields are coupled to the system field with a relativistically invariant bilinear interaction. Measurement of vacuum meter field expectation values provides for the simultaneous measurement of noncommuting system components. It is shown that two meter coupling allows for a simultaneous minimum in the variance of the subsequent meter measurements. A pseudoscalar self-interaction of the Dirac field is shown to allow simultaneous measurement of positive energy field operators with negative energy meters. The simultaneous measurement ofn noncommuting field operators is obtained by coupling the system ton fermionic fields. Also, in this paper the related concept of mutual simultaneous measurement is developed. This requires that any operators in the enlarged Hilbert space are measurable by the remaining fields as meters. System embedding into a larger Hilbert space results in added noise due to the zero point motion of the meter fields. By the negentropy principle of Brillouin, the added noise is equivalent to entropy. A criterion determining the interaction among fields is that the averaged added noise in the components of each quantum field is minimized. This criterion defines an optimum fermionic mass matrix through the determination of the entangling interaction.

Ordering of "ladder" operators, the Wigner function for number and phase, and the enlarged Hilbert space

A suitable ordering of phase exponential operators has been compared with the antinormal ordering of the annihilation and creation operators of a single mode optical field. The extended Wigner function for number and phase in the enlarged Hilbert space has been used for the derivation of the Wigner function for number and phase in the original Hilbert space.

FROM QUANTUM FIELDS TO LOCAL VON NEUMANN ALGEBRAS

The subject of the paper is an old problem of the general theory of quantized fields: When can the unbounded operators of a Wightman field theory be associated with local algebras of bounded operators in the sense of Haag? The paper reviews and extends previous work on this question, stressing its connections with a noncommutive generalization of the classical Hamburger moment problem. Necessary and sufficient conditions for the existence of a local net of von Neumann algebras corresponding to a given Wightman field are formulated in terms of strengthened versions of the usual positivity property of Wightman functionals. The possibility that the local net has to be defined in an enlarged Hilbert space cannot be ruled out in general. Under additional hypotheses, e.g., if the field operators obey certain energy bounds, such an extension of the Hilbert space is not necessary, however. In these cases a fairly simple condition for the existence of a local net can be given involving the concept of “central positivity” introduced by Powers. The analysis presented here applies to translationally covariant fields with an arbitrary number of components, whereas Lorentz covariance is not needed. The paper contains also a brief discussion of an approach to noncommutative moment problems due to Dubois-Violette, and concludes with some remarks on modular theory for algebras of unbounded operators.

Floquet spectrum for two-level systems in quasiperiodic time-dependent fields

We study the time evolution ofN-level quantum systems under quasiperiodic time-dependent perturbations. The problem is formulated in terms of the spectral properties of a quasienergy operator defined in an enlarged Hilbert space, or equivalently of a generalized Floquet operator. We discuss criteria for the appearance of pure point as well as continuous spectrum, corresponding respectively to stable quasiperiodic dynamics and to unstable chaotic behavior. We discuss two types of mechanisms that lead to instability. The first one is due to near resonances, while the second one is of topological nature and can be present for arbitrary ratios between the frequencies of the perturbation. We treat explicitly an example of this type. The stability of the pure point spectrum under small perturbations is proven using KAM techniques.

Local nets and self-adjoint extensions of quantum field operators

We consider Wightman fields having the property that some closed extensions of the field operators generate locally commuting von Neumann algebras. We show that for such fields the hermitian field operators have self-adjoint extensions, possibly in an enlarged Hilbert space, such that bounded functions of the self-adjoint operators commute locally.

Variational results as saddle-point approximations: The Anderson impurity model.

The slave-boson functional-integral approach to the Hubbard model by Kotliar and Ruckenstein is applied to the spin-degenerate Anderson impurity model. At T=0 a simple saddle-point approximation to the functional integral is shown to be equivalent to a special case of a variational ansatz presented by the author in the 1970's. The question of how to choose the freedom in the form of the hopping Hamiltonian in the enlarged Hilbert space including the slave bosons is discussed.