Commutation Relations Of Operators Research Articles

From shockwaves to the gravitational memory effect

We study the relationship between shockwave geometries and the gravitational memory effect in four-dimensional asymptotically flat spacetime. In particular, we show the ’t Hooft commutation relations of shockwave operators are equivalent to the commutation relation between soft and Goldstone modes parametrizing a sector of the gravitational phase space. We demonstrate this equivalence via a diffeomorphism that takes a shockwave metric to a metric whose transverse traceless component is the gravitational memory. The shockwave momentum in ’t Hooft’s analysis is related to the soft graviton mode, which is responsible for the memory effect, while the shift in the shockwave position is related to the Goldstone mode. This equivalence opens new directions to utilize the gravitational memory effect to explore the observational implications of shockwave geometries in flat space.

Embarking on the path to quantum field theory

The harmonic oscillator represents as a unique tool in modern physics since it is not only exactly solvable but also relates many phenomena. This paper introduces the relation between Quantum Field Theory and other physical theories and discusses how harmonic oscillator behave in Quantum Mechanics and Quantum Field Theory by the quantization of Klein-Gordon Field. To this end, this paper applies energy-momentum equation given by special relativity to Schrdingers Equation to get the Klein-Gordon Field, and then uses canonical method to quantize the field, which is called the second quantization of the field. The comparison is made between the result quantized field, the quantum harmonic oscillator and classical harmonic oscillator. The similarity of two physical systems is demonstrate by the comparison between commutation relations of operators. The similar commutation relations then help people to interpret quantized Klein-Gordon Field as a system that contain a quantized harmonic oscillator in each space coordinate.

A generalization of the canonical commutation relation and Heisenberg uncertainty principle for the orbital operators

Abstract In this paper, the orbits of observable physical quantities of position and momentum operators at the states of quantum Hilbert spaces are created. Also the Hilbert space of finite orbits and the Fréchet–Hilbert space of all orbits are created and the orbital operators corresponding to these observable operators in these spaces of orbits are defined and studied. We call this process the orbitization, and the obtained model the orbital quantum mechanics. The orbitization process is compared with the quantization process. The generalization of well-known canonical commutation relations for orbital operators corresponding to the position and momentum operators is established. Also, the Heisenberg uncertainty principle for the orbital operators is proved and the question of achieving equality in the Heisenberg inequality is considered.

Carrollian approach to 1 + 3D flat holography

The isomorphism between the (extended) BMS4 algebra and the 1 + 2D Carrollian conformal algebra hints towards a co-dimension one formalism of flat holography with the field theory residing on the null-boundary of the asymptotically flat space-time enjoying a 1 + 2D Carrollian conformal symmetry. Motivated by this fact, we study the general symmetry properties of a source-less 1 + 2D Carrollian CFT, adopting a purely field-theoretic approach. After deriving the position-space Ward identities, we show how the 1 + 3D bulk super-translation and the super-rotation memory effects emerge from them, manifested by the presence of a temporal step-function factor in the same. Temporal-Fourier transforming these memory effect equations, we directly reach the bulk null-momentum-space leading and sub-leading soft graviton theorems. Along the way, we construct six Carrollian fields {S}_0^{pm } , {S}_1^{pm } , T and overline{T} corresponding to these soft graviton fields and the Celestial stress-tensors, purely in terms of the Carrollian stress-tensor components. The 2D Celestial shadow-relations and the null-state conditions arise as two natural byproducts of these constructions. We then show that those six fields consist of the modes that implement the super-rotations and a subset of the super-translations on the quantum fields. The temporal step-function allows us to relate the operator product expansions (OPEs) with the operator commutation relations via a complex contour integral prescription. We deduce that not all of those six fields can be taken together to form consistent OPEs. So choosing {S}_0^{+} , {S}_1^{+} and T as the local fields, we form their mutual OPEs using only the OPE-commutativity property, under two general assumptions. The symmetry algebra manifest in these holomorphic-sector OPEs is then shown to be Vir overset{wedge }{ltimes overline{textrm{sl}left(2,{mathbb{R}}right)}} with an abelian ideal.

Quantum counterfactuality with identical particles

Quantum self-interference enables the counterfactual transmission of information, whereby the transmitted bits involve no particles traveling through the channel. In this work, we show how counterfactuality can be realized even when the self-interference is replaced by interference between identical particles. Interestingly, the facet of indistinguishability called forth here is associated with first-order coherence, and is different from the usual notion of indistinguishability associated with the (anti-)commutation relations of mode operators. From an experimental perspective, the simplest implementation of the proposed idea can be realized by slight modifications to existing protocols for differential-phase-shift quantum key distribution or interaction-free measurement.

Intrinsic approach to 1 + 1D Carrollian Conformal Field Theory

The 3D Bondi-Metzner-Sachs (BMS3) algebra that is the asymptotic symmetry algebra at null infinity of the 1 + 2D asymptotically flat space-time is isomorphic to the 1 + 1D Carrollian conformal algebra. Building on this connection, various preexisting results in the BMS3-invariant field theories are reconsidered in light of a purely Carrollian perspective in this paper. In direct analogy to the covariant transformation laws of the Lorentzian tensors, the flat Carrollian multiplets are defined and their conformal transformation properties are established. A first-principle derivation of the Ward identities in a 1 + 1D Carrollian conformal field theory (CCFT) is presented. This derivation introduces the use of the complex contour-integrals (over the space-variable) that provide a strong analytic handle to CCFT. The temporal step-function factors appearing in these Ward identities enable the translation of the operator product expansions (OPEs) into the language of the operator commutation relations and vice versa, via a contour-integral prescription. Motivated by the properties of these step-functions, the iϵ-forms of the Ward identities and OPEs are proposed that permit for the hassle-free use of the algebraic properties of the latter. Finally, utilizing the computational techniques developed, it is shown that the modes of the quantum energy-momentum tensor operator generate the centrally extended version of the infinite-dimensional 1 + 1D Carrollian conformal algebra.

Explicit Calculation of Structural Commutation Relations for Stochastic and Dynamical Graph Grammar Rule Operators in Biological Morphodynamics.

Many emergent, non-fundamental models of complex systems can be described naturally by the temporal evolution of spatial structures with some nontrivial discretized topology, such as a graph with suitable parameter vectors labeling its vertices. For example, the cytoskeleton of a single cell, such as the cortical microtubule network in a plant cell or the actin filaments in a synapse, comprises many interconnected polymers whose topology is naturally graph-like and dynamic. The same can be said for cells connected dynamically in a developing tissue. There is a mathematical framework suitable for expressing such emergent dynamics, "stochastic parameterized graph grammars," composed of a collection of the graph- and parameter-altering rules, each of which has a time-evolution operator that suitably moves probability. These rule-level operators form an operator algebra, much like particle creation/annihilation operators or Lie group generators. Here, we present an explicit and constructive calculation, in terms of elementary basis operators and standard component notation, of what turns out to be a general combinatorial expression for the operator algebra that reduces products and, therefore, commutators of graph grammar rule operators to equivalent integer-weighted sums of such operators. We show how these results extend to "dynamical graph grammars," which include rules that bear local differential equation dynamics for some continuous-valued parameters. Commutators of such time-evolution operators have analytic uses, including deriving efficient simulation algorithms and approximations and estimating their errors. The resulting formalism is complementary to spatial models in the form of partial differential equations or stochastic reaction-diffusion processes. We discuss the potential application of this framework to the remodeling dynamics of the microtubule cytoskeleton in cortical microtubule networks relevant to plant development and of the actin cytoskeleton in, for example, a growing or shrinking synaptic spine head. Both cytoskeletal systems underlie biological morphodynamics.

Weak equivalence principle in quantum space

Owing to the development of String Theory and Quantum Gravity, studies of quantized spaces described by deformed commutation relations for operators of coordinates and operators of momenta have received much attention. In this paper, the implementation of the weak equivalence principle is examined in the quantized spaces described by different types of deformed algebras, among them the noncommutative algebra of canonical type, Lie type, and the nonlinear deformed algebra with an arbitrary function of deformation depending on momenta. It is shown that the deformation of commutation relations leads to the mass-dependence of motion of a particle (a composite system) in a gravitational field, and, hence, to violation of the weak equivalence principle. We conclude that this principle is recovered in quantized spaces if one considers the parameters of the deformed algebras to be different for different particles (bodies) and to be determined by their masses.

On the light-ray algebra in conformal field theories

We analyze the commutation relations of light-ray operators in conformal field theories. We first establish the algebra of light-ray operators built out of higher spin currents in free CFTs and find explicit expressions for the corresponding structure constants. The resulting algebras are remarkably similar to the generalized Zamolodchikov’s W∞ algebra in a two-dimensional conformal field theory. We then compute the commutator of generalized energy flow operators in a generic, interacting CFTs in d > 2. We show that it receives contribution from the energy flow operator itself, as well as from the light-ray operators built out of scalar primary operators of dimension ∆ ≤ d − 2, that are present in the OPE of two stress-energy tensors. Commutators of light-ray operators considered in the present paper lead to CFT sum rules which generalize the superconvergence relations and naturally connect to the dispersive sum rules, both of which have been studied recently.

Statistical model describing Bose-Einstein and Fermi-Dirac statistics.

Beginning with the assumption that the quantum state of a many-particle system is a functional on the internal space of the particles, a unified quantum statistics, which may interpolate smoothly between Bose-Einstein and Fermi-Dirac statistics, is proposed. The commutation relations for such particle creation and annihilation operators are derived, and statistical partition function and thermodynamical properties of an ideal gas of the particles are investigated. Besides the Bose-Einstein condensation, another kind of particle condensation may appear.

Parton interpretation and twist-4 parton distributions

Through explicit examples we show that twist-4 parton distributions have no parton interpretation in the sense that parton or partons inside a hadron can carry the momentum fraction x of the hadron with x>1 or x<−1. The studied twist-4 parton distributions of collinear factorization are power-divergent for |x|>1. The corresponding transverse momentum dependent parton distributions have finite contributions for |x|>1 and they have hence no parton interpretation. Parton distributions are defined with products of field operators. In the case of twist-4, different orderings give different results. The differences are determined by commutation relations of field operators. This has been explicitly checked in the studied case. The implications of our results are discussed.

Geometric constraints on two-electron reduced density matrices

For many-electron systems, the second-order reduced density matrix (2-RDM) provides sufficient information for characterizing their properties of interests in physics and chemistry, ranging from total energy, magnetism, quantum correlation and entanglement to long-range orders. Theoretical prediction of the structural properties of 2-RDM is an essential endeavor in quantum chemistry, condensed matter physics and, more recently, in quantum computation. Since 1960s, enormous progresses have been made in developing RDM-based electronic structure theories and their large-scale computational applications in predicting molecular structure and mechanical, electrical and optical properties of various materials. However, for strongly correlated systems, such as high-temperature superconductors, transition-metal-based biological catalysts and complex chemical bonds near dissociation limit, accurate approximation is still out of reach by currently most sophisticated approaches. This limitation highlights the elusive structural feature of 2-RDM that determines quantum correlation in many-electron system. Here, we present a set of constraints on 2-RDM based on the basic geometric property of Hilbert space and the commutation relations of operators. Numerical examples are provided to demonstrate the pronounced violation of these constraints by the variational 2-RDMs. It is shown that, for a strongly correlated model system, the constraint violation may be responsible for a considerable portion of the variational error in ground state energy. Our findings provide new insights into the structural subtlety of many-electron 2-RDMs.

Analytical treatments for magnetic multipole moment integrals

This work presents a new method for the analytical evaluation of magnetic multipole moment integrals (MMMIs) using Slater type orbitals (STOs), the Fourier transform method and some translation equations. These integrals are one type of the molecular integrals and their solutions are important for better understanding of properties of molecular structure. In this work, magnetic multipole moment operators are translated to another atomic center and the x-, y-, and z- components of MMMIs are obtained in terms of overlap integrals. Moreover, by using ladder operators, the matrix representations and commutation relations of magnetic dipole moment operators have been obtained successfully.

Properties of states on Weyl algebra with variable multiplication law

We consider possible quantum effects for infinite systems implied by variations of the multiplication law in the algebra of observables. Using the algebraic formulation of quantum theory, we study the behavior of states ω under changes in the defining relations of the canonical commutation relations (CCR-algebra). These defining relations of the multiplication law depend explicitly on the symplectic form σ, which encodes commutation relations of canonical field operators. We consider the change in this form given by simple rescaling of σ by a positive parameter h. We analyze to what extent changes in h preserve the original state space (this gives restrictions on the admissible changes in the scaling parameter h) and which properties have original quantum states ω as states on the new algebra. We answer such questions for the quasi-free states. We show that any universally invariant state can be interpreted as a convex combination of Fock states with different values of constant h. The second important class of states we study is the KMS-state; here, the rescaling alters in a nontrivial way the relevant dynamics. We also show that it is possible to go beyond the limits restricting the changes in h, but then one has to restrict the CCR-algebra to a subalgebra.

Gravitational domain wall and stability with some symmetry algebra

In this paper, first, we will try to introduce the gravitational domain wall as a physical system. In the second step, we also introduce the Hun differential equation as a mathematical tools. We factorize the known Heun’s equation as form of operators [Formula: see text], [Formula: see text] and [Formula: see text]. Then we compare the differential equation of gravitational domain wall with corresponding Hun equation. In that case the above-mentioned operators can be obtained for the gravitational system by the comparing process. Finally, we employ such operators and achieve the corresponding symmetry algebra with the usual commutation relation of operators to each other. Here, by having such operators, we investigate the stability of system.

Quantum challenge-response identification using single qubit unitary operators

In challenge-response authentication, the claimants are required to prove that they are legitimate users while minimizing the exposure of their confidential information. We propose a quantum challenge-response identification protocol using single qubit unitary operators that utilizes the mechanism of challenge-response, unlike most existing protocols that use (one-time) password authentication. We utilized the commutation relations of single qubit unitary operators to design this protocol that ensures the security of confidential information during identification in a quantum computing environment. For security analysis, we estimated the average fidelity of two quantum states after performing operations with the unitary operators based on the random numbers generated by two users. Subsequently, we evaluated the amount of information acquired by an eavesdropper to show that the proposed protocol is robust against impersonation and intercept-measurement attacks. Finally, we described the advantages of the protocol by comparing it with the existing quantum identification protocols.

Fluctuation-dissipation theorem and fundamental photon commutation relations in lossy nanostructures using quasinormal modes

We provide theory and formal insight on the Green function quantization method for absorptive and dispersive spatial-inhomogeneous media in the context of dielectric media. We show that a fundamental Green function identity, which appears, e.g., in the fundamental commutation relation of the electromagnetic fields, is also valid in the limit of non-absorbing media. We also demonstrate how the zero-point field fluctuations yields a non-vanishing surface term in configurations without absorption, when using a more formal procedure of the Green function quantization method. We then apply the presented method to a recently developed theory of photon quantization using quasinormal modes [Franke et al., Phys. Rev. Lett. 122, 213901 (2019)] for finite nanostructures embedded in a lossless background medium. We discuss the strict dielectric limit of the commutation relations of the quasinormal mode operators and present different methods to obtain them, connected to the radiative loss for non-absorptive but open resonators. We show exemplary calculations of a fully three-dimensional photonic crystal beam cavity, including the lossless limit, which supports a single quasinormal mode and discuss the limits of the commutation relation for vanishing damping (no material loss and no radiative loss).

Electromagnetic field quantization near a dielectric slab and spontaneous emission rate determination

The Green function approach of the electromagnetic field quantization is used to quantize the electromagnetic field near dispersive and dissipative dielectric slabs. The dielectric function of the slab is an arbitrary complex function of frequency-satisfying Kramers–Kronig relations. For the atoms with the electric dipole moment perpendicular and parallel to the slab, the vector potential operator commutation relations are used to drive the spontaneous emission rate of an atom via Fermi’s golden rule. By choosing a Lorentz model for the dielectric function of the slab, the typical spatial variations of the spontaneous emission rates near the slab are shown.

Commutation relations of vertex operators for Uq(sl^(M|N))

We consider commutation relations and invertibility relations of vertex operators for the quantum affine superalgebra Uq(sl^(M|N)) by using bosonization. We show that vertex operators give a representation of the graded Zamolodchikov-Faddeev algebra by direct computation. Invertibility relations of type-II vertex operators for N &amp;gt; M are very similar to those of type-I for M &amp;gt; N.

Real Gentile Statistical Systems: N-Annulenes.

Gentile statistics, which is famous for its advantages in dealing with composite particle systems, is a kind of fractional statistics. Lately, researchers are concentrating on finding real systems that obey Gentile statistics. Five years ago, we discovered that the cyclic hydrocarbon polyenes called N-annulenes in the Hückel model had certain correspondence with Gentile oscillators. According to their rotation and dihedral symmetry, these two systems had the same energy levels and partition functions. In this paper, we give further discussions. We discuss the transformations of wave functions between N-annulenes and Gentile oscillators, the creation and annihilation operators in site pictures of N-annulenes, the coherent state, and the mathematical proofs of the intermediate commutation relations of operators. All of our works prove that N-annulenes in the Hückel model are real Gentile statistical systems and offer a new algebraic method to deal with the problems of electron-electron and electron-phonon interactions.