Heisenberg Field Research Articles

Asymptotic expansion of the commutator of heisenberg fields with respect to finite-dimensional irreducible representations of the Lorentz group

Asymptotic expansion of the commutator of heisenberg fields with respect to finite-dimensional irreducible representations of the Lorentz group

The N-quantum approximation in the Bronzan–Lee extended-source static model

The Bronzan–Lee model of a soluble field theory with vertex function is analyzed from the viewpoint of the N-quantum approximation. Determination of the Heisenberg field expansions in the single-source one- and two-meson sectors of the model yields three scattering amplitudes, a production amplitude, and the renormalization constants in agreement with other derivations. The solutions to one algebraic and three linear integral equations, the mathematical complexity of which is reminiscent of the Tamm–Dancoff eigenvalue approach in the Lee model with two nontrivial sources, are required for this purpose. Finite results compatible with the conventional renormalization program are obtained from the field expansions and the equations of motion by prescribing that each normal-ordered product of in-fields in an expansion have the same quantum numbers as its Heisenberg field and, so that all relevant terms are included for a sector, all possible combinations of in-fields consistent with the quantum numbers of the sector must occur. In general, this recipe is substituted in place of power counting, which evidently is inappropriate in higher sectors of the Lee model. The investigation of one- and two-meson exchange interactions of two sources in the Lee model via the N-quantum approximation has been carried out, and similar work is contemplated in charged scalar theory and the classic Chew–Low model.

Relation between projective geometry and group contraction in spontaneously broken symmetry theories

By using projective geometry arguments, it is shown that in a spontaneously broken symmetry theory the linearization procedure by which one passes from non-linear Heisenberg fields equations to the linear free fields equations leads to a group contraction mechanism: the symmetry group that appears in observations is a group contraction of the dynamical invariance group.

Consistent canonical quantization of a model field theory with Schwinger term

Canonical quantization of a finite two-dimensional model field theory is consistently carried out, in spite of the occurrence of a non-canonical Schwinger term. Heisenberg field operators are defined perturbatively by means of the dimensionally regularized Feynman chronological product, while their equal-time commutators may be related to free-field equal-time commutators through the use of the non-covariant Dyson chronological product. A concrete agreement (including the axial anomaly) is thus found between the quantum Lagrangian and Hamiltonian formulations.

Bound states and solitons in the Gross-Neveu model

The results for the spectrum of bound states and of solitons first deduced by Dashen, Hasslacher, and Neveu for a model of interacting fermions by techniques of functional integration are obtained here by methods based on Heisenberg field mechanics analogous to those applied previously to models of self-interacting bosons. The method of solution is suggested by a simplified physical picture of the bound states: These are computed in a Hartree approximation in which the self-consistent potential is a sum of contributions from the fermions (and antifermions) occupying orbitals in the conventional many-body picture and from the vacuum fluctuations of single-closed-loop type. In the same approximation the self-consistent field generated by the heavy soliton is a result of the vacuum fluctuations alone. As the main new technical contribution, we deduce and solve directly equations determining the self-consistent fields as well as the amplitudes ("wave functions") from which these are constructed. We comment on the degeneracy of the heavy soliton state.

Unstable particles in a self-consistent field theory

We indicate how unstable particles can be introduced into the self-consistent field theory formulation of Umezawa where the equal-time commutation relations for Heisenberg fields are derived and not assumed. The Lee model is used to illustrate the results.

Equal-Time Commutation Relations for Heisenberg Fields in the Lee Model

>~ 4 > has been successfully applied to problems in both high energy physics5> and the area of many-body problems, particularly in superconductivity 6> and ferromagnetism. 7> In this paper we apply this method to the solvable Dirac­ Lee model which, unlike the pair model considered in Ref. 3), has mass and coupling constant renormalizations. The equal-time commutation relations among Heisenberg fields are derived and not assumed. Further we show that the wave­ function renormalization constant is determined from microcausality. This result sheds light on the question why in Ref. 3) the existence of the~ bound state was found to be closely connected with microcausality. The self-consistent method exploits the duality between basic fields and observed particles concretely exhibited in the presence of interactions. The basic Heisenberg fields obey nonlinear field equations which reflect the laws of nature, while the physical particles are the ones that appear in observations. These physical fields obey free field equations and define the physical Fock space. The field equations for the Heisenberg fields are to be regarded as operator equations in this physical Fock space. The Heisenberg field is related to the physical field by means of a mapping known as the dynamical map.4> The set ?f physical fields is taken to form an irreducible operator ring. Therefore, any operator of the Fock space can be written as a sum of normal products of physical flelds. The ability of this method to predict new particles rests on this postulate. The asymptotic condition may tell us that the physical fields are not complete. The completion of this set of fields then requires the existence of other particles at the level of observation and should be incorporated. 8>·8> The success of this method therefore rests on the proper choice of the set of physical fields.

The Goldstone theorem and dynamical rearrangement of symmetry in the path-integral formalism

The spontaneous breakdown of symmetry is investigated in the path-integral formalism. The generating functional is modified by the addition of a limiting “ϵ term” which fixes the direction of the breaking. The method is illustrated by means of several models with Abelian symmetries. The Goldstone theorem is a result of Ward-Takahashi identities in which the ϵ term plays a decisive role. The problem of how the original symmetry transformation of the Heisenberg fields manifests itself at the level of observable (physical) particles, expressed in terms of the in-field operators φ in ( χ), is investigated. It is proved that, if spontaneous breakdown is present, the in-field transformation φ in ( χ) → φ in ( χ) + α ( χ), with α (χ) satisfying the same field equation as φ in ( χ) (Boson transformation), is an invariant transformation (i.e. leaves the equation of motion of the Heisenberg fields unchanged). It is also found that, when spontaneous breaking occurs, the invariant transformations of the Heisenberg and in-fields are different (dynamical rearrangement of symmetry). The form of the invariant in-field transformation is established for several models.

Relativistic quantum theory for charged spinless particles in external vector fields

Guided by a diagonalized form of the classical field-energy we construct a time-dependent canonical pair of Schrödinger fields Φ t (x) and Π t (x) which diagonalizes the field-HamiltonianH t . These Schrödinger fields in general belong to inequivalent representations of the canonical commutation relations for differentt's. The Heisenberg field is constructed by solving the Heisenberg equation of motion and its time-evolution turns out to be governed by a unitary operator, i.e. the Heisenberg fields at different times are unitarily equivalent. Scattering theory (including eventual incoming and/or outgoing bound-states) is finally constructed.

Quasisecular Clothing Transformation for theφ4Model of Quantum Field Theory

The first-order renormalized Heisenberg field for the ${\ensuremath{\varphi}}^{4}$ model of quantum field theory, calculated by the quasisecular perturbation method, is used to construct a clothing transformation. The clothing transformation gives the Fourier amplitude of the renormalized field from that of the bare field, except for the quasisecular terms. The properties of the clothing transformation are examined in the cases of one, two, and three space dimensions. The connection with the strange representations of the canonical commutation relations is discussed.

Algebraic Solution for the Källén-Pauli State Vectors in the Vθ Sector by a Congruence Transformation

From the representation of Bolsterli of the Vθ sector physical states in a basis having singular integral equations of the separable type, a congruence transformation carries the representation into the Källén-Pauli Heisenberg field components. This solves the Källén-Pauli singular integral equation algebraically. The resulting state vectors are proven explicitly to be both complete and orthonormal and furnish a new Möller wave matrix.

Quadratic Fermion Interaction Hamiltonian

The interaction Hamiltonian λ∫:ψ̄(0)(x)ψ(0)(x):g(x)dsx,g(x)∈S(Rs) is studied. An ultraviolet cutoff is introduced. We remove this cutoff, and take the limit g → 1 in S(Rs), by working with the Heisenberg fields. The limiting fields are well defined on the Fock space associated with the bare mass m0. In the limit we get a new representation of the canonical anticommutation relations which is given by a (generalized) Bogoliubov transformation. The new representation is not always unitarily equivalent to the bare mass Fock representations.

Derivation of Equal-Time Commutation Relations in a Self-Consistent Quantum Field Theory

A self-consistent quantum field theory is proposed wherein the field equations for the Heisenberg fields are postulated. The Heisenberg fields are required to be linear combinations of normal products of the physical free fields, and the coefficients of the expansion are determined self-consistently. No a priori knowledge of the equal-time commutation relations among the Heisenberg fields is assumed. Using a solvable model, it is shown that local microscopic causality requires the existence of a bound state.

Gauge invariance and the charge operator in the Heisenberg picture

The transformations of the renormalized interpolating fields generated by the gauge transformation of the free asymptotic fields restrict the interpolating fields to satisfying certain commutation rules. These commutation rules, in turn, give a non-phenomenological charge operator for the interpolating Heisenberg fields. Redundant states and the eigenvalue problem with the indefinite metric are discussed.

Theory of the raman effect in metals: D. L. Mills, A. A. Maradudin, and E. Burstein. Dept. of Physics, Univ. of California, Irvine, Calif.

Canonical quantization of a finite two-dimensional model field theory is consistently carried out, in spite of the occurrence of a non-canonical Schwinger term. Heisenberg field operators are defined perturbatively by means of the dimensionally regularized Feynman chronological product, while their equal-time commutators may be related to free-field equal-time commutators through the use of the non-covariant Dyson chronological product. A concrete agreement (including the axial anomaly) is thus found between the quantum Lagrangian and Hamiltonian formulations.

Asymptotic states in the quantum field theory

The purpose of the present work is to c a r r y out the H a a g Ruelte p r o g r a m [1, 2] within the f r a m e work of the Bogolyubov approach to the quantum field theory, when it is based not on the Heisenberg field but on the S-matr ix, which depends on the intensity of switching on interaction. The analysis will be p e r formed wholly within the f ramework of the vers ion of the theory* discussed in [3].

On interacting local quantum fields describing many masses and spins

We investigate the possible restrictions of locality on interacting quantum fields, which create or destroy many-particle states with different masses and spins. First we look at the vacuum expectation value of the commutator (or anticommutator) of two Heisenberg fields. Demanding that these fields transform as representations of the homogeneous Lorentz group and using theCPT-theorem we prove that locality gives no relations between the different spectral functions, and the relations between the masses as functions of the spin are the same as those found by Abarbanel and Frishman for free fields. Then with the help of the Yang-Feldman equations we look at the problem in perturbation theory. We find that demanding locality at each order in perturbation theory in a manner consistent with the general results found above imposes restrictions on the possible forms of these interactions.

On equal-time commutation relations of renormalized currents in perturbation theory I

Equal-time current commutation relations are considered in renormalizable field theories. Renormalized currents are obtained by means of solutions of the Yang-Feldman equations for Heisenberg field operators in perturbation theory. For the computation of matrix elements of current commutators we apply Jost-Lehmann-Dyson type techniques. The equal time limit is taken with the help of symmetrical time-smearing functions which interpolate the δ-function. Our methods avoid any cut-off procedure and lead therefore to unambiguous results. In order to avoid spin complications, our general methods are applied to trilinear resp. quadrilinear couplings of isoscalar and isovector spin 0-mesons in first order perturbation theory. We find that the zero-space components of the current-commutator matrix elements behave for small time separationT like ln(T) grad x δ(X–Y).

Field-Theory Calculations in Broken Symmetries

A method based on field theory is developed for calculating corrections to broken-symmetry sum rules. The Gell-Mann-Okubo formulas for the bare masses of the baryons and mesons are found from equal-time commutators in the current algebra by using the canonical commutation relations for unrenormalized Heisenberg field operators. The corrections to the Gell-Mann-Okubo formula for the physical masses are calculated from the self-masses in perturbation theory. It is conjectured that the logarithmic infinity occurring in the second-order perturbation-theory result for the baryon self-mass cancels in the mass sum rule, and the corrections can be evaluated. For pure $D$ coupling, the correction in second-order perturbation theory reduces the discrepancy in the baryon-mass sum rule from 12 to 7 MeV.

Geometric Theory of Neutrinos

A theory of neutrinos is constructed within the present scheme of geometrodynamics. First, a special class of $c$-number Heisenberg fields is considered and a plane-wave-like solution of the covariant neutrino equation is obtained in the two-component limit. It is shown that the special class of neutrinos forms a Rainich null geometry. Conversely, a geometry suitable for the special neutrinodynamics is distinguished from general null geometries by imposing the condition that the null eigenvector of geometry is the gradient of the neutrino complexion. Finally, the geometrical parallelism between gravito-electrodynamics and gravito-neutrinodynamics is discussed.