Physical unitarity in the Lagrangian Sp(2)-symmetric formalism

The structure of state vector space for a general (non-anomalous) gauge theory is studied within the Lagrangian version of the Sp(2)-symmetric quantization method. The physical S-matrix unitarity conditions are formulated. The general results are illustrated on the basis of simple gauge theory models.     

Basic Brackets of a 2D Model for the Hodge Theory Without its Canonical Conjugate Momenta

We deduce the canonical brackets for a two (1+1)-dimensional (2D) free Abelian 1-form gauge theory by exploiting the beauty and strength of the continuous symmetries of a Becchi-Rouet-Stora-Tyutin (BRST) invariant Lagrangian density that respects, in totality, six continuous symmetries. These symmetries entail upon this model to become a field theoretic example of Hodge theory. Taken together, these symmetries enforce the existence of exactly the same canonical brackets amongst the creation and annihilation operators that are found to exist within the standard canonical quantization scheme. These creation and annihilation operators appear in the normal mode expansion of the basic fields of this theory. In other words, we provide an alternative to the canonical method of quantization for our present model of Hodge theory where the continuous internal symmetries play a decisive role. We conjecture that our method of quantization is valid for a class of field theories that are tractable physical examples for the Hodge theory. This statement is true in any arbitrary dimension of spacetime.     

Are the Weyl and coherent state descriptions physically equivalent?

We investigate the consistency of coherent state quantization in regard to physical observations in the non-relativistic (or Galilean) regime. We compare this particular type of quantization of the complex plane with the canonical (Weyl) quantization and examine whether they are or not equivalent in their predictions. As far as only usual dynamical observables (position, momentum, energy, …) are concerned, the quantization through coherent states is proved to be a perfectly valid alternative. We successfully put to the test the validity of CS quantization in the case of data obtained from vibrational spectroscopy.     

Is the non-physical states conjecture valid?

The canonical quantization for N = 1 supergravity in the context of gravitational minisuperspace described by Gowdy T ³ and Bianchi class A cosmological models is analyzed in order to search for physical states. There are indeed physical states in the minisuperspace sector of the theory. This fact entails that the non-physical states conjecture has a restricted validity, and in consequence it cannot be considered a general result.     

Multiple quantization and the concept of information

The understanding of the meaning of quantization seems to be the main problem in understanding quantum structures. In this paper first the difference between quantized particle vs. radiation fields in the formalism of canonical quantization is discussed. Next von Weizsäcker's concept of multiple quantization which leads to an understanding of quantization as an iteration of probability theory is explained. Finally a connection between quantization and the idea of a general theory of information is considered. This brings together semantic information with the different levels of quantization and expresses the philosophical attitude of this paper concerning the interpretation of quantum theory.     

Spin-wave quantization in ferromagnetic nickel nanowires

The dynamical properties of uniform two-dimensional arrays of nickel nanowires have been investigated by inelastic light scattering. Multiple spin waves are observed that are in accordance with dipole-exchange theory predictions for the quantization of bulk spin waves. This first study of the spin-wave dynamics in ferromagnetic nanowire arrays reveals strong mode quantization effects and indications of a subtle magnetic interplay between nanowires. The results show that it is important to take proper account of these effects for the fundamental physics and future technological developments of magnetic nanowires.     

Time and a physical Hamiltonian for quantum gravity

We present a nonperturbative quantization of general relativity coupled to dust and other matter fields. The dust provides a natural time variable, leading to a physical Hamiltonian with spatial diffeomorphism symmetry. The surprising feature is that the Hamiltonian is not a square root. This property, together with the kinematical structure of loop quantum gravity, provides a complete theory of quantum gravity, and puts applications to cosmology, quantum gravitational collapse, and Hawking radiation within technical reach.     

Path integral quantization of field theories with second-class constraints

Faddeev's Hamiltonian path integral method for singular Lagrangians is generalized to the case when second-class constraints appear in the theory. The general formalism is then applied to several problems: quantization of the massive Yang-Mills field theory, light-cone quantization of the self-interacting scalar field theory, and quantization of a local field theory of magnetic monopoles.     

Continuous quantum hypothesis testing

I propose a general quantum hypothesis testing theory that enables one to test hypotheses about any aspect of a physical system, including its dynamics, based on a series of observations. For example, the hypotheses can be about the presence of a weak classical signal continuously coupled to a quantum sensor, or about competing quantum or classical models of the dynamics of a system. This generalization makes the theory useful for quantum detection and experimental tests of quantum mechanics in general. In the case of continuous measurements, the theory is significantly simplified to produce compact formulas for the likelihood ratio, the central quantity in statistical hypothesis testing. The likelihood ratio can then be computed efficiently in many cases of interest. Two potential applications of the theory, namely, quantum detection of a classical stochastic waveform and test of harmonic-oscillator energy quantization, are discussed.     

Fractional corresponding operator in quantum mechanics and applications: A uniform fractional Schrödinger equation in form and fractional quantization methods

In this paper we use Dirac function to construct a fractional operator called fractional corresponding operator, which is the general form of momentum corresponding operator. Then we give a judging theorem for this operator and with this judging theorem we prove that R–L, G–L, Caputo, Riesz fractional derivative operator and fractional derivative operator based on generalized functions, which are the most popular ones, coincide with the fractional corresponding operator. As a typical application, we use the fractional corresponding operator to construct a new fractional quantization scheme and then derive a uniform fractional Schrödinger equation in form. Additionally, we find that the five forms of fractional Schrödinger equation belong to the particular cases. As another main result of this paper, we use fractional corresponding operator to generalize fractional quantization scheme by using Lévy path integral and use it to derive the corresponding general form of fractional Schrödinger equation, which consequently proves that these two quantization schemes are equivalent. Meanwhile, relations between the theory in fractional quantum mechanics and that in classic quantum mechanics are also discussed. As a physical example, we consider a particle in an infinite potential well. We give its wave functions and energy spectrums in two ways and find that both results are the same.     

Essay: Initial Conditions for a Universe

In physical theories, boundary or initial conditions play the role of selecting special situations which can be described by a theory with its general laws. Cosmology has long been suspected to be different in that its fundamental theory should explain the fact that we can observe only one particular realization. This is not realized, however, in the classical formulation and in its conventional quantization; the situation is even worse due to the singularity problem. In recent years, a new formulation of quantum cosmology has been developed which is based on quantum geometry, a candidate for a theory of quantum gravity. Here, the dynamical law and initial conditions turn out to be linked intimately, in combination with a solution of the singularity problem.     

Phase transitions with several order parameters

The predictions of catastrophe theory for phase transitions involving more than one order parameter are given. These predictions are compared with those of other theories. For the simplest transition involving two order parameters it is found that there is a parameter which does not affect the topology of the phase diagram, which does affect certain angles in the diagram, and whose measured value will not depend on the scale of external physical variables. Comparison with renormalization theory predictions for this parameter leads to general observations on the relation of catastrophe theory and renormalization theory.     

Constraint algebra for interacting quantum systems

We consider relativistic constrained systems interacting with external fields. We provide physical arguments to support the idea that the quantum constraint algebra should be the same as in the free quantum case. For systems with ordering ambiguities this principle is essential to obtain a unique quantization. This is shown explicitly in the case of a relativistic spinning particle, where our assumption about the constraint algebra plus invariance under general coordinate transformations leads to a unique S-matrix.     

Canonical quantization theory of general singular QED system of Fermi field interaction with generally decomposed gauge potential

Quantization theory gives rise to transverse phonons for the traditional Coulomb gauge condition and to scalar and longitudinal photons for the Lorentz gauge condition. We describe a new approach to quantize the general singular QED system by decomposing a general gauge potential into two orthogonal components in general field theory, which preserves scalar and longitudinal photons. Using these two orthogonal components, we obtain an expansion of the gauge-invariant Lagrangian density, from which we deduce the two orthogonal canonical momenta conjugate to the two components of the gauge potential. We then obtain the canonical Hamiltonian in the phase space and deduce the inherent constraints. In terms of the naturally deduced gauge condition, the quantization results are exactly consistent with those in the traditional Coulomb gauge condition and superior to those in the Lorentz gauge condition. Moreover, we find that all the nonvanishing quantum commutators are permanently gauge-invariant. A system can only be measured in physical experiments when it is gauge-invariant. The vanishing longitudinal vector potential means that the gauge invariance of the general QED system cannot be retained. This is similar to the nucleon spin crisis dilemma, which is an example of a physical quantity that cannot be exactly measured experimentally. However, the theory here solves this dilemma by keeping the gauge invariance of the general QED system.     

The quantum theory of second class constraints: Kinematics

The problem of second class quantum constraints is here set up in the context ofC*-algebras, utilizing the connection with state conditions as given by the heuristic quantization rules. That is, a constraint set is said to be first class if all its members can satisfy the same state condition, and second class otherwise. Several heuristic models are examined, and they all agree with this definition. Given then a second class constraint set, we separate out its first class part as all those constraints which are compatible with the others, and we propose an algebraic construction for imposition of the constraints. This construction reduces to the normal one when the constraints are first class. Moreover, the physical automorphisms (assumed as conserving the constraints) will also respect this construction. The final physical algebra obtained is free of constraints, gauge invariant, unital, and with the right choice, simple. ThisC*-algebra also contains a factor algebra of the usual observables, i.e. the commutator algebra of the constraints. The general theory is applied to two examples—the elimination of a canonical pair from a boson field theory, as in the two dimensional anomalous chiral Schwinger model of Rajaraman [14], and the imposition of quadratic second class constraints on a linear boson field theory.     

Interference and interaction in Schrödinger's wave mechanics

Reminiscing on the fact that E. Schrödinger was rooted in the same physical tradition as M. Planck and A. Einstein, some aspects of his attitude to quantum mechanics are discussed. In particular, it is demonstrated that the quantum-mechanical paradoxes assumed by Einstein and Schrödinger should not exist, but that otherwise the epistemological problem of physical reality raised in this context by Einstein and Schrödinger is fundamental for our understanding of quantum theory. The nonexistence of such paradoxes just shows that quantum-mechanical effects are due to interference and not to interaction. This line of argument leads consequently to quantum field theories with second quantization, and accordingly quantum theory based both on Planck's constant h and on Democritus's atomism.     

Quantum Field Theory on Curved Backgrounds. I. The Euclidean Functional Integral

We give a mathematical construction of Euclidean quantum field theory on certain curved backgrounds. We focus on generalizing Osterwalder Schrader quantization, as these methods have proved useful to establish estimates for interacting fields on flat space-times. In this picture, a static Killing vector generates translations in Euclidean time, and the role of physical positivity is played by positivity under reflection of Euclidean time. We discuss the quantization of flows which correspond to classical space-time symmetries, and give a general set of conditions which imply that broad classes of operators in the classical picture give rise to well-defined operators on the quantum-field Hilbert space. In particular, Killing fields on spatial sections give rise to unitary groups on the quantum-field Hilbert space, and corresponding densely-defined self-adjoint generators. We construct the Schrödinger representation using a method which involves localizing certain integrals over the full manifold to integrals over a codimension-one submanifold. This method is called sharp-time localization, and implies reflection positivity.     

A new theory of elementary matter

This is the first of a series of articles that reviews and expands upon a new theory of elementary matter. This paper presents an exposition of the philosophical approach and its general implications. The ensuing explicit form of the mathematical expression of the theory and several applications in the atomic and elementary particle domains will be developed in the succeeding parts of this series.The theory is based on three axioms: the principle of general relativity, a generalized Mach principle, and a correspondence principle. The approach is basically a deterministic, relativistic field theory which fully incorporates the idea that any realistic physical system is in facta closed system, without separable parts. It is shown that the most primitive mathematical expression of this theory, following as anecessary consequence of its axioms, is in terms of a set of coupled nonlinear spinor field equations. Nevertheless, the exact formalism is constructed to asymptotically approach the quantum mechanical formalism for a many-particle system, in the limit of sufficiently small energy-momentum transfer among the components of the considered closed system. Thus, all of the mathematical predictions of nonrelativistic quantum mechanics are contained in this theory, as a mathematical approximation. However, predictions follow from the exact form of this theory (where energy-momentum transfer can be arbitrarily large) that are not contained in the quantum theory.