Hilbert Space Of Physical States Research Articles

Theory of Response to Perturbations in Non-Hermitian Systems Using Five-Hilbert-Space Reformulation of Unitary Quantum Mechanics.

Non-Hermitian quantum-Hamiltonian-candidate combination of a non-Hermitian unperturbed operator with an arbitrary “small” non-Hermitian perturbation is given a mathematically consistent unitary-evolution interpretation. The formalism generalizes the conventional constructive Rayleigh–Schrödinger perturbation expansion technique. It is sufficiently general to take into account the well known formal ambiguity of reconstruction of the correct physical Hilbert space of states. The possibility of removal of the ambiguity via a complete, irreducible set of observables is also discussed.

Entanglement entropy in lattice theories with Abelian gauge groups

We revisit the issue of the geometrical separability of the Hilbert space of physical states on lattice Abelian theories in the context of entanglement entropy. We discuss the conditions under which vectors in the Hilbert space, as well as the gauge-invariant algebra, admit a tensor product decomposition with a geometrical interpretation. With the exception of pure gauge lattices with periodic boundary conditions which contain topological degrees of freedom, we show that the Hilbert space is geometrically separable.

Admissible perturbations and false instabilities inPT-symmetric quantum systems

In ${\cal PT}-$symmetric quantum mechanics one of the most characteristic mathematical features of the formalism is the explicit Hamiltonian-dependence of the physical Hilbert space of states ${\cal H}={\cal H}(H)$. Some of the most important physical consequences are discussed, with emphasis on the dynamical regime in which the system is close to the quantum phase transition. Consistent perturbation treatment of such a regime is proposed. An illustrative application of the innovated perturbation theory to a non-Hermitian but ${\cal PT}-$symmetric user-friendly family of $J-$parametric "discrete anharmonic" quantum Hamiltonians $H=H(\vec{\lambda})$ is given. The models are shown to admit the standard probabilistic interpretation if and only if the parameters remain compatible with the reality of the spectrum, $\vec{\lambda} \in {\cal D}^{(physical)}$. In contradiction to the conventional wisdom the systems are shown stable with respect to the admissible perturbations lying inside the domain ${\cal D}^{(physical)}$. This observation holds even in the immediate vicinity of the phase-transition boundaries $\partial {\cal D}^{(physical)}$.

Symmetric scalar constraint for loop quantum gravity

In the framework of loop quantum gravity, we define a new Hilbert space of states which are solutions of a large number of components of the diffeomorphism constraint. On this Hilbert space, using the methods of Thiemann, we obtain a family of gravitational scalar constraints. They preserve the Hilbert space for every choice of lapse function. Thus adjointness and commutator properties of the constraint can be investigated in a straightforward manner. We show how the space of solutions of the symmetrized constraint can be defined by spectral decomposition, and the Hilbert space of physical states by subsequently fully implementing the diffeomorphism constraint. The relationship of the solutions to those resulting from a proposal for a symmetric constraint operator by Thiemann remains to be elucidated.

PT -symmetric quantum state discrimination

The objective of this paper is to explain and elucidate the formalism of quantum mechanics by applying it to a well-known problem in conventional Hermitian quantum mechanics, namely the problem of state discrimination. Suppose that a system is known to be in one of two quantum states, | ψ 1 〉 or | ψ 2 〉. If these states are not orthogonal, then the requirement of unitarity forbids the possibility of discriminating between these two states with one measurement; that is, determining with one measurement what state the system is in. In conventional quantum mechanics, there is a strategy in which successful state discrimination can be achieved with a single measurement but only with a success probability p that is less than unity. In this paper, the state-discrimination problem is examined in the context of quantum mechanics and the approach is based on the fact that a non-Hermitian -symmetric Hamiltonian determines the inner product that is appropriate for the Hilbert space of physical states. It is shown that it is always possible to choose this inner product so that the two states | ψ 1 〉 and | ψ 2 〉 are orthogonal. Using quantum mechanics, one cannot achieve a better state discrimination than in ordinary quantum mechanics, but one can instead perform a simulated quantum state discrimination, in which with a single measurement a perfect state discrimination is realized with probability p .

A discrete, unitary, causal theory of quantum gravity

A discrete model of Lorentzian quantum gravity is proposed. The theory is completely background free, containing no reference to absolute space, time, or simultaneity. The states at one slice of time are networks in which each vertex is labelled with two arrows, which point along an adjacent edge, or to the vertex itself. The dynamics is specified by a set of unitary replacement rules, which causally propagate the local degrees of freedom. The inner product between any two states is given by a sum over histories. Assuming it converges (or can be Abel resummed), this inner product is proven to be Hermitian and fully gauge-degenerate under spacetime diffeomorphisms. At least for states with a finite past, the inner product is also positive. This allows a Hilbert space of physical states to be constructed.

Quantum Gowdy model within the new loop quantum cosmology improved dynamics

The linearly polarized Gowdy T3 model can be regarded as compact Bianchi I cosmologies with inhomogeneous modes allowed to travel in one direction. We study a hybrid quantization of this model that combines the loop quantization of the Bianchi I background, adopting the improved dynamics scheme put forward by Ashtekar and Wilson-Ewing, with a Fock quantization for the inhomogeneities. The Hamiltonian constraint operator provides a resolution of the cosmological singularity and superselects separable sectors. We analyze the complicated structure of these sectors. In any of them the Hamiltonian constraint provides an evolution equation with respect to the volume of the associated Bianchi I universe, with a well posed initial value problem. This fact allows us to construct the Hilbert space of physical states and to show that we recover the standard quantum field theory for the inhomogeneities.

Inhomogeneous loop quantum cosmology: Hybrid quantization of the Gowdy model

The Gowdy cosmologies provide a suitable arena to further develop loop quantum cosmology, allowing the presence of inhomogeneities. For the particular case of Gowdy spacetimes with the spatial topology of a three-torus and a content of linearly polarized gravitational waves, we detail a hybrid quantum theory in which we combine a loop quantization of the degrees of freedom that parametrize the subfamily of homogeneous solutions, which represent Bianchi I spacetimes, and a Fock quantization of the inhomogeneities. Two different theories are constructed and compared, corresponding to two different schemes for the quantization of the Bianchi I model within the improved dynamics formalism of loop quantum cosmology. One of these schemes has been recently put forward by Ashtekar and Wilson-Ewing. We address several issues, including the quantum resolution of the cosmological singularity, the structure of the superselection sectors in the quantum system, or the construction of the Hilbert space of physical states.

Complete set of inner products for a discrete PT-symmetric square-well Hamiltonian

A discrete N-point Runge–Kutta version H(N)(λ) of one of the simplest non-Hermitian square-well Hamiltonians with real spectrum is studied. Its possible Hermitizations mediated by nontrivial (often called “non-Dirac”) metrics Θ≠I are considered as a source of nonequivalent standard probabilistic interpretations of this quantum model. A complete set of these alternative, multiparametric metrics Θ=Θ(a,b,…)(N)(λ) defining all the eligible Hamiltonian-dependent representations of the physical Hilbert space of states is constructed, in closed form, for any coupling λ∊(−1,1) and for any matrix dimension N.

Sturm-Schrödinger equations: Formula for metric

Sturm-Schrödinger equations Hψ = EWψ with H ≠ H † and W≠ = W † ≠ = I are considered, with a weak point of the theory lying in the purely numerical matrix-inversion form of the double-series definition of the necessary metric operator Θ in the physical Hilbert space of states [M Znojil, J. Phys. A: Math. Theor. 41, 215304 (2008]. This shortcoming is removed here via an amended, single-series definition of Θ.

Entanglement entropy in lattice gauge theories

We report on the recent progress in theoretical and numerical studies of entanglement entropy in lattice gauge theories. It is shown that the concept of quantum entanglement between gauge fields in two complementary regions of space can only be introduced if the Hilbert space of physical states is extended in a certain way. In the extended Hilbert space, the entanglement entropy can be partially interpreted as the classical Shannon entropy of the flux of the gauge fields through the boundary between the two regions. Such an extension leads to a reduction procedure which can easily be implemented in lattice simulations by constructing lattices with special topology. This enables us to measure the entanglement entropy in lattice Monte-Carlo simulations. On the simplest example of Z2 lattice gauge theory in dimensions we demonstrate the relation between the entanglement entropy and the classical entropy of the field flux. For lattice gauge theory in four dimensions, we find a signature of non-analytic dependence of the entanglement entropy on the size of the region. We also comment on the holographic interpretation of the entanglement entropy.

Entanglement entropy in gauge theories and the holographic principle for electric strings

We consider quantum entanglement between gauge fields in some region of space A and its complement B. It is argued that the Hilbert space of physical states of gauge theories cannot be decomposed into a direct product H A ⊗ H B of Hilbert spaces of states localized in A and B. The reason is that elementary excitations in gauge theories—electric strings—are associated with closed loops rather than points in space, and there are closed loops which belong both to A and B. Direct product structure and hence the reduction procedure with respect to the fields in B can only be defined if the Hilbert space of physical states is extended by including the states of electric strings which can open on the boundary of A. The positions of string endpoints on this boundary are the additional degrees of freedom which also contribute to the entanglement entropy. We explicitly demonstrate this for the confining and deconfining phases of the three-dimensional Z 2 lattice gauge theory both numerically and using a simple trial ground state wave function. The entanglement entropy appears to be saturated almost completely by the entropy of string endpoints, thus reminding of a “holographic principle” in quantum gravity and AdS/CFT correspondence.

Time-dependent version of crypto-Hermitian quantum theory

For many quantum models an apparent non-Hermiticity of observables just corresponds to their hidden Hermiticity in another, physical Hilbert space. For these models we show that the existence of observables which are manifestly time-dependent may require the use of a manifestly time-dependent representation of the physical Hilbert space of states.

Scattering theory with localized non-Hermiticities

In the context of the recent interest in solvable models of scattering mediated by non-Hermitian Hamiltonians (cf. H. F. Jones, Phys. Rev. D 76, 125003 (2007)) we show that the well-known variability of the ad hoc choice of the metric $\ensuremath{\Theta}$ which defines the physical Hilbert space of states can help us to clarify several apparent paradoxes. We argue that with a suitable $\ensuremath{\Theta}$, a fully plausible physical picture of the scattering can be recovered. Quantitatively, our new recipe is illustrated on an exactly solvable toy model.

Identification of observables in quantum toboggans

Schrödinger equations for ‘quantum toboggans’ with real energies are given the generalized eigenvalue-problem form Hψ = EWψ, where H ≠ H† and W ≠ W† ≠ I. The consistent probabilistic interpretation of these models is provided. The new double-series formula for the necessary ad hoc metric Θ = Θ(H, W) is derived which defines the acceptable inner products in the physical Hilbert space of states. The formula degenerates to the usual single series in the non-tobogganic trivial-weight limit W → I.

On the Role of the Normalization Factors κn and of the Pseudo-Metric P ≠ P† in Crypto-Hermitian Quantum Models

Among P-pseudo-Hermitian Hamiltonians H = P 1 H † P with real spectra, the weakly pseudo-Hermitian ones (i.e., those employing non-self-adjoint P 6 P † ) form a remarkable subfamily. We list some reasons why it deserves a special attention. In par- ticular we show that whenever P 6 P † , the current involutive operator of charge C gets complemented by a nonequivalent alternative involutive quasiparity operator Q. We show how, in this language, the standard quantum mechanics is restored via the two alternative inner products in the physical Hilbert space of states, with h 1 |PQ| 2i = h 1 |CP | 2i.

BRST quantization of noncommutative gauge theories

In this paper, the BRST symmetry transformation is presented for the noncommutative U(N) gauge theory. The nilpotency of the charge associated to this symmetry is then proved. As a consequence for the space-like non-commutativity parameter, the Hilbert space of physical states is determined by the cohomology space of the BRST operator as in the commutative case. Further, the unitarity of the S-matrix elements projected onto the subspace of physical states is deduced.

Quantum dynamics of the polarized GowdyT3model

The polarized Gowdy ${\mathbf{T}}^{3}$ vacuum spacetimes are characterized, modulo gauge, by a ``point particle'' degree of freedom and a function \ensuremath{\varphi} that satisfies a linear field equation and a nonlinear constraint. The quantum Gowdy model has been defined by using a representation for \ensuremath{\varphi} on a Fock space F. Using this quantum model, it has recently been shown that the dynamical evolution determined by the linear field equation for \ensuremath{\varphi} is not unitarily implemented on F. In this paper, (1) we derive the classical and quantum model using the ``covariant phase space'' formalism, (2) we show that time evolution is not unitarily implemented even on the physical Hilbert space of states $\mathcal{H}\ensuremath{\subset}\mathcal{F}$ defined by the quantum constraint, and (3) we show that the spatially smeared canonical coordinates and momenta as well as the time-dependent Hamiltonian for \ensuremath{\varphi} are well-defined, self-adjoint operators for all time, admitting the usual probability interpretation despite the lack of unitary dynamics.

Bulk versus boundary dynamics in anti–de Sitter spacetime

We investigate the details of the bulk-boundary correspondence in Lorentzian signature anti--de Sitter space. Operators in the boundary theory couple to sources identified with the boundary values of non-normalizable bulk modes. Such modes do not fluctuate and provide classical backgrounds on which bulk excitations propagate. Normalizable modes in the bulk arise as a set of saddlepoints of the action for a fixed boundary condition. They fluctuate and describe the Hilbert space of physical states. We provide an explicit, complete set of both types of modes for free scalar fields in global and Poincar\'e coordinates. For ${\mathrm{AdS}}_{3},$ the normalizable and non-normalizable modes originate in the possible representations of the isometry group $\mathrm{SL}(2,R{)}_{L}\ifmmode\times\else\texttimes\fi{}\mathrm{SL}(2,R{)}_{R}$ for a field of given mass. We discuss the group properties of mode solutions in both global and Poincar\'e coordinates and their relation to different expansions of operators on the cylinder and on the plane. Finally, we discuss the extent to which the boundary theory is a useful description of the bulk spacetime.

A combinatorial approach to diffeomorphism invariant quantum gauge theories

Quantum gauge theory in the connection representation uses functions of holonomies as configuration observables. Physical observables (gauge and diffeomorphism invariant) are represented in the Hilbert space of physical states; physical states are gauge and diffeomorphism invariant distributions on the space of functions of the holonomies of the edges of a certain family of graphs. Then a family of graphs embedded in the space manifold (satisfying certain properties) induces a representation of the algebra of physical observables. We construct a quantum model from the set of piecewise linear graphs on a piecewise linear manifold, and another manifestly combinatorial model from graphs defined on a sequence of increasingly refined simplicial complexes. Even though the two models are different at the kinematical level, they provide unitarily equivalent representations of the algebra of physical observables in separable Hilbert spaces of physical states (their s-knot basis is countable). Hence, the combinatorial framework is compatible with the usual interpretation of quantum field theory.