Canonical Realization Research Articles

Covariant loops and strings in a positive definite Hilbert space

Relativistic loops and strings are defined in the conventional way as solutions of a one-dimensional wave equation with certain boundary conditions and satisfying the orthogonal gauge conditions. Conventional pseudo-Cartesian co-ordinates (rather than null-plane co-ordinates) are used. The creation and annihilation operator four-vector α † anda μ are required to be spacelike (orthogonal to the total momentumP μ), so that the resulting Fock spaceH + is positive definite. This requirement is shown to be mathematically consistent with Poincare invariance and to impose no additional physical constraints on the system. It can be implemented in a canonical realization of the Poincare algebra as a condition on state vectors, or in a noncanonical realization as an operator equation, as is done here. The spaceH + is further restricted by the Virasoro conditions to a physical subspace Φ which is of course also positive definite. In this way there arises no critical-dimension problem and Poincare invariance holds also in 3+1 dimensions. The energy and spin spectra are the same as usual, leading to linear Regge trajectories, except that there are no tachyons and no zero-mass states. The leading Regge trajectory has negative intercept.

Canonical realization of general time systems

The canonical realization of general time systems by dynamical systems is discussed employing the notation and definitions of Takahara and Mesarovic. Previous results are extended to the case of causal, but not necessarily stationary, systems and to stationary, strongly connected systems. Conditions are provided under which a time system is guaranteed to have a canonical representation from within a class of possible realizations.

Canonical realizations of classical lie algebras

We construct sets of canonical realizations for all classical Lie algebras (An,Bn,Cn,Dn). These realizations depend ond parameters,d=1, 2, 3,...,n; all Casimir operators are realized by multiples of identity. For most of the real forms of these algebras we give sets of realizations which are, moreover, in well-defined sense skew-Hermitian. Further we study extremal cases of the presented realizations. The realizations with minimal numbers of canonical pairs are discussed from the point of view of general results concerning minimal realizations. On the other hand, a connection is found between our “maximal” realizations ofAn and the Gel'fand-Kirillov Conjecture.

On the “near to minimal” canonical realizations of the lie algebraC n

It is proved that canonical realizations of the Lie algebraC n in the quotient division ringD 2(2n−2) of the Weyl algebraW 2(22−2) in 2n − 2 quantum canonical pairs are, in a definite sense, related to the standard minimal one inD 2n ⊂D 2(2n−2). Further, in any realization ofC n inW 2(2n-1) all Casimir operators are realized by multiples of identity element.

Identification in Canonical Form for Linear Time-Invariant Systems from Input-Output Data

An identification technique for determining a triple F, G, H in canonical form from the given input-output sequences has been presented. The proposed algorithm is the modification of Ackermann and Bucy technique for obtaining canonical realization from impulse response data.

Canonical realizations of the lie algebrasp(2n, R)

The generators of the Lie algebra of the symplectic groupsp(2n, R) are, recurrently, realized by means of polynomials in the quantum canonical variablesp i andq i. These realizations are skew-Hermitian, the Casimir operators are realized by constant multiples of identity elements, and, depending on the number of the canonical pairs used, they depend ond, d=1, 2, ...,n free real parameters.

Canonical realizations of the Poincaré group. II. Space–time description of two particles interacting at a distance, Newtonian-like equations of motion and approximately relativistic Lagrangian formulation

The physical meaning of the relativistic action-at-a-distance dynamics for two particles in a canonical framework is investigated on the basis of a general formalism introduced in previous works. Starting from the well-known prescription given by Bakamjian and Thomas in terms of ’’center-of-mass’’ (Q,P) and ’’internal’’ (ρ,π) canonical coordinates, we show how to construct physical, i.e., covariant, position vectors xτ (Q, P, ρ, π) (τ=1,2) which approach the free particle coordinates in the limit ρ→∞ for short range forces; this procedure is actually performed by means of a 1/c2 power expansion for any interaction potential U (ρ,π). In force of the zero-interaction theorem the physical coordinates, which do satisfy the world-line condition to any order in 1/c2, cannot play the role of canonical variables, i.e., the ’’localizability,’’ {xτi, xτj}=0 (τ=1,2), and the ’’causality’’ conditions {xτi, xτ′j}=0 (τ,τ′=1,2; τ≠τ′) cannot be simultaneously satisfied. It is possible, however, to satisfy the former set of equations to any order in 1/c2 by exploiting the arbitrariness lying in the definition of x1 and x2. By means of a suitable choice of a ’’gauge’’ for the internal variables, the remaining freedom is then shown to consist of the appearance of a single scalar function Λ (ρ,π). This function, entering the defining relations of x1, x2 in terms of the canonical variables Q, P, ρ, π, plays the role of an additional interaction potential which is effective for the space–time description of the particles in the interaction region, but does not affect the scattering properties of the system. On the other hand, assuming a static nonrelativistic limit of the canonical potential, U(0)=U(0)(ρ), the ’’causality’’ conditions are necessarily violated at the order of the radiation effects (1/c4). In terms of x1, x2, the equations of motion assume a Newtonian-like structure mτẍτ=Fτ[x1−x2,v1,v2] (τ=1,2), of the Currie type or a variety of manifestly covariant forms mτd2xμτ/ds2τ=Sμ νfν [x1(s1),x2(s2),u1(s1),u2(s2)], where Sμ ν is the Lorentz transformation which connects the laboratory frame with the Lorentz frame in which x1(s1) and x2(s2) are simultaneous. A final point is the derivation of the Newtonian-like equations of motion from a true Lagrangian variational principle δℱL [x1,x2,v1,v2]dt=0. It is shown in general that if U(0)(ρ) ≡0, this can be done only up to the post-Newtonian approximation, essentially because of the violation of the ’’causality’’ conditions at the order 1/c4. Then a general form of approximately relativistic Lagrangian for two particles is derived which actually contains all the examples quoted in the literature, among which the well-known Darwin–Breit and the Einstein–Infeld–Hoffmann Lagrangians. This investigation appears to disprove the widespread opinion according to which the zero-interaction theorem prevents the existence of invariant world lines and/or renders the relativity principle vacuous within a Hamiltonian framework.

Fundamental canonical realizations of connected Lie groups

An important class of transitive canonical realizations of connected Lie groups is studied by means of a general formalism. We give a simple method for the classification and the construction of these realizations.

Canonical realizations of the lie algebras gl( n, R) and sl( n, R) II. Casimir operators

The eigenvalues of Casimir operators in the canonical realization of gl( n, R) and sl( n, R) presented in the first part of this paper are expressed by concrete ploynomials in parameters characterizing the realization and some of their properties are derived.

A covariant hamiltonian formulation of electrodynamics in Coulomb gauge

Defining the electromagnetic potential in a dynamical sense and abandoning its traditionally assumed vectorial character, we have obtained an hamiltonian description of electrodynamics in Coulomb gauge. The surprising and spectacular feature of this new description is that it is relativistically covariant because we succeeded in building a canonical realization of the Lie algebra of the Poincaré group from the potential variables and because the Coulomb condition (i.e., τ · A = 0) can be imposed as a strong condition in this framework.

Minimal partial realizations in a canonical form

A canonical realization for a partial sequence of real, constant matrices is obtained by identifying the system invariants defined by Popov [11]. Further, the case where the resulting realization must also be stable is considered. The example used by Tether [5] is solved, and the results compared with those of Tether [5] and Ackermann [12].

Canonical realizations of the Poincaré group. I. General theory

The canonical realizations of the full Poincaré group in classical mechanics are studied by means of a general formalism introduced in preceding papers. The resulting classification displays significant analogies with the quantum one. The irreducible realizations, corresponding to positive, zero, and imaginary mass particles with or without spin, are discussed. Also the irreducible realizations of the homogeneous Lorentz group are classified. Particular attention is given to the nonirreducible realization describing a system of two free particles and to a discussion of the physical meaning of the ’’center-of-mass’’ and ’’internal’’ variables. It is seen that this formalism provides a most natural framework for the introduction of a direct interaction between the particles according to the well-known prescription given by Bakamjian and Thomas. Finally, some simple models of relativistic ’’rigid’’ systems are discussed.

A novel second-order canonical RC-active realization of high-pass notch filter

A high-pass notch filter with variable gain greater than unity is realized using a differential amplifier and only two capacitors that can be equal. The transmission zero and pole frequency are completely independent of the amplifier dc gain. Design formulas, sensitivity analysis, and measure and trim tuning process are presented. An example of a 60-Hz rejection filter is discussed, the response of which is within the required specifications even with a \pm 5-percent capacitance variation. Two modifications of the original structure can be used to realize the same filter, extending the available range of the gain to cover all requirements.

Minimal Bases of Rational Vector Spaces, with Applications to Multivariable Linear Systems

A minimal basis of a vector space V of n-tuples of rational functions is defined as a polynomial basis such that the sum of the degrees of the basis n-tuples is minimum. Conditions for a matrix G to represent a minimal basis are derived. By imposing additional conditions on G we arrive at a minimal basis for V that is unique. We show how minimal bases can be used to factor a transfer function matrix G in the form $G = ND^{ - 1} $, where N and D are polynomial matrices that display the controllability indices of G and its controller canonical realization. Transfer function matrices G solving equations of the form $PG = Q$ are also obtained by this method; applications to the problem of finding minimal order inverse systems are given. Previous applications to convolutional coding theory are noted. This range of applications suggests that minimal basis ideas will be useful throughout the theory of multivariable linear systems. A restatement of these ideas in the language of valuation theory is given in an Ap...

$H^2 $-Functions and Infinite-Dimensional Realization Theory

In this paper the realization question for infinite-dimensional linear systems is examined for both bounded and unbounded operators. In addition to obtaining realizability criteria covering the basic cases, we discuss the relationship between canonical realizations of the same system. What one finds is that the set of transfer functions which are realizable by triples $(A,b,c)$ with A bounded is related in a close way to the space of complex functions analytic and square integrable on the disk $| {s < 1} |$, and that the set of transfer functions which are realizable by triples $(A,b,c)$ with A unbounded but generating a strongly continuous semigroup is related in a close way to functions analytic and square integrable on a half-plane. This relation makes possible a deeper study between the transfer function and the models which realize it. Some examples illustrate the results and their applications.

A minimal canonical realization algorithm for impulse response matrix using moments

An algorithm based on moments of impulse response matrix is proposed which will yield an irreducible canonical form for linear time-invariant multivariable systems. The proposed method is also computationally simpler than the existing techniques.

A canonical realization of active RC filters†

A canonical structure of an RC active filter is presented which can realize any transfer function. It consists of a fixed passive part and an active part that is computed from two triangular systems of linear equations. The solution vectors are the gain values of the active part and can be implemented as input resistances of summing amplifiers. The equations are generated systematically from the given passive part and the required transfer function. The realization requires at most n isolation amplifiers and two operational amplifiers to obtain any transfer function of degree n. An extra amplifier is added to minimize the sensitivity.

The jaumann structure in wave‐digital filters

AbstractThis paper deals with an extension of the wave‐digital filter structures introduced by Fettweis.A new structure is introduced, viz, the digital translation of the Jaumann structure. The Jaumann structure is a well‐known canonical realization of analogue, symmetric, grounded lossless two‐port networks. With this structure, transfer functions with complex transmission zeros can be obtained in a wave‐digital realization, which is not possible by means of a wave‐digital ladder filter. Two examples are given, one of which has been implemented using TTL digital integrated circuits showing excellent properties.

A Note on Easily Testable Realizations for Logic Functions

It is shown that at most, n + 3 tests are required to detect any single stuck-at fault in an AND gate or a single faulty EXCLUSIVE OR (EOR) gate in a Reed-Muller canonical form realization of a switching function.

Canonical lowpass prototype network for symmetric coupled-resonator bandpass filters

A lowpass prototype network that represents a canonical realisation for symmetric coupled-resonator bandpass filters is presented. A synthesis procedure for the prototype is outlined.