Scale transformations for renormalized field operators: Discussion of the soluble model

We discuss the operator formulation of the Zachariasen-Thirring model, describing the chain approximation to the propagator (the sum of three-particle massless bubbles) in massless λ⁴ theory. Such a model is formally scale-invariant and explicitly soluble. All intermediate steps of conventional renormalization procedure, regularization, introduction of appropriate counterterms, and cut-off free limit, are explicitly performed. In every step the scaling properties are discussed and respective dilatation currents are written down. After the proper choice of scale transformations for the renormalized field operator, we obtain the nonlocal dilatation current, defining the renormalized dilatation generator D_Λ^R(t). In the cut-off free limit Λ → ∞ the ET commutator of D_Λ^R(t) with renormalized field operators reproduces the Callan-Symanzik modification of “naive” canonical scale transformations. The renormalized scale transformations coincide in the cut-off free limit with renormalized dimensional transformations and define the exact symmetry of the renormalized theory.     

Spontaneous symmetry breaking in the present and early universe

Spontaneous symmetry breaking in λφ⁴ theory is formulated in terms of the operator φ², and in a manner which requires no specific expectation value to be assigned to φ. At the one-loop order of perturbation theory, a renormalized effective action for a field ζ, linearly related to φ², is obtained as a gradient expansion. Potential advantages of this formulation in applications to phase transitions in the early universe are discussed. They include the possibilities (i) of obtaining a well-defined semiclassical equation of motion, and (ii) of following the evolution of a field theory from an initial symmetrical high temperature state without the introduction, ad hoc, of regions in which 〈φ〉 ≠ 0.     

Field Equations in Quantum Electrodynamics

A formulation of quantum electrodynamics is presented, based on finite local field equations. These Dirac and Maxwell equations have the usual form except that the current operators f(x) and j^μ (x) are explicitly expressed as local limits of sums of non-local field products and suitable subtraction terms. These limits are shown to exist and to yield finite operators in the sense that the iterative solutions to the field equations are equivalent to conventional renormalized perturbation theory. The various invariance properties of the theory, including Lorentz invariance, gauge invariance, charge conjugation invariance, and renormalization invariance, are discussed and related directly to the field equations and current definitions. Initially only the general forms of the currents, based on dimensional arguments, are given. The electric current, for example, contains the (suitably defined) term :A³(x) :.The corresponding field equations are used to derive renormalized Dyson-Schwinger-type integral equations for the renormalized proper part functions ∑, II^μν, Λ^μ, and X^αβγδ (the four-photon vertex function), etc. Application of the boundary conditions ∑(p̀ = m) = ∑′(p̀ = m) = II(O) = II′(O) = II″(O) = Λ(p̀ = m, o) = X(O, O, O, O) = O completely specifies the current operators. Consistency is established by deriving the same equations from rigorous renormalization theory so that their iterative solutions are proved to reproduce the correct renormalized perturbation expansion. The electric current operator is exhibited in a manifestly gauge invariant form and in a form which is manifestly negative under charge conjugation. It is shown, in fact, that much of j^μ (x) can be determined directly from the requirements of gauge invariance and charge conjugation covariance, without recourse to the integral equations. It is suggested that equal time commutation relations can serve to similarly specify the rest of the current.     

On the interpretation of magnetic susceptibility data by means of a modified Curie-Weiss law

The analysis of the paramagnetic susceptibility of a substance by means of a modified Curie-Weiss law is discussed. It is shown that the experimental parameters x¹_o and C¹ should be renormalized in order to obtain physical quantities directly related to the crystal field groud state properties. Formulae are given to calculate the renormalized values of the parameters. The application to selected actinide compounds shows that the corrections may be significant for the interpretation of the magnetic data.     

How can we properly define the Q2 dependent parton distribution function in QCD?

We discuss how we can properly define the Q² dependent parton distribution functions in quantum chromodynamics within the framework of the operator product expansion and renormalization group techniques. It is proposed that the moments of the parton distribution functions at Q² should be defined as the hadronic expectation values of the twist-2 operators renormalized at Q². The integro-differential equations for the parton densities obtained by Altarelli and Parisi are reproduced in the leading logarithmic approximation. An application of our present formalism will be given in the case of a longitudinal structure function.     

The asymptotic behaviour of form factors

Similar to the method of S.A. Anikin and O.I. Zavialov a modified renormalization procedure is applied to the already renormalized current operator. This allows the derivation of an identity for the coefficient functions of this operator and the proof of new renormalization group equations. These equations are applied to the theory and the electromagnetic form factor of quarks in QCD. In the last case the one-loop approximation gives the leading behaviour exp(?clnlnQ²lnQ²) for large euclidean values of the momentum transfer Q² = ?q² at the external electromagnetic vertex. If the leading logarithmic terms (g²ln²Q²)ⁿ are extracted from this result, then the behaviour exp(?g²c′ln²Q²) is obtained. Whereas the first result is modified by contributions from two-loop calculations, the result concerning the leading logarithmic terms does not change.     

Improved renormalization group equations with dimensional regularization and the ε expansions

Due to the absence of dimensional cut-off parameters in the dimensional regularization scheme, vanishing of the renormalized mass of the scalar boson implies vanishing of its renormalized mass; thus the masses of both bosons and fermions in renormalizable field theories can be made finite by multiplicative mass renormalizations. The improved renormalization group equations in D dimensions are derived in such a way that both the large (or the small) momentum limits and the Wilson ? expansions can be uniformly treated for the fermion as well as the boson cases. We discuss the improved equations for φ₆³ theory, φ₄⁴ theory, quantumelectrodynamics, massive vector-gluon model, and non-Abelian guage theories incorporating fermions. For the latter three classes of theories, the gauge dependent problem of the coefficient functions in the improved renormalization group equations is discussed.     

Renormalization of the Lorentz-Abraham-Dirac equation for radiation reaction force in classical electrodynamics

Dirac’s analysis of radiation reaction force in classical electrodynamics suggested that a 4-momentum not collinear with 4-velocity could be introduced for a radiating electron. This would be equivalent to renormalization of the electron mass as an operator relating these 4-vectors. Dirac also pointed to an arbitrary choice made in deriving the Lorentz-Abraham-Dirac (LAD) equation. It was shown that renormalization substantially modifies the LAD equation under the additional requirement that the standard relativistic relation ℰ² = p ² c ² + m ² c ⁴ holds for the renormalized energy and momentum. The renormalized LAD equation is more rigorous than the LAD equation, because the drawbacks of the latter are eliminated, and is simpler than a well-known approximation of the LAD equation. The renormalized LAD equation appears to be better suited for numerical simulations of processes in ultrahigh-intensity laser-pulse fields.     

Wilsonian RG and redundant operators in nonrelativistic effective field theory

In a Wilsonian renormalization group (RG) analysis, redundant operators, which may be eliminated by using field redefinitions, emerge naturally. It is therefore important to include them. We consider a nonrelativistic effective theory (the so-called “pionless” nuclear effective field theory) as a concrete example and show that the off-shell amplitudes cannot be renormalized if the redundant operators are not included. The relation between the theories with and without such redundant operators is established in the low-energy expansion. We perform a Wilsonian RG analysis for the off-shell scattering amplitude in the theory with the redundant operator.     

Phase structure and critical behavior of lattice Φ4 theory at different temperatures

The Variational-Cumulant Expansion method is applied to investigate the phase structure and the critical behavior of lattice ?⁴ theory at two different temperatures The phase diagrams, the external current J as a function of the expectation value of the field operator $«ngle ?i _L»ngle $ are calculated to the third order analytically. The critical behaviors of the expectation value $«ngle ?i _L»ngle $, the renormalized mass m_R and $m_R/«ngle ?i _L»ngle $ are given and compared with the mean-field scaling laws with logarithmic scaling corrections. It is shown that, at a fixed bare coupling, the broken phase at a lower temperature might be restored at a higher temperature and a bound on the Higgs-boson mass at definite temperature can exist.     

Thermal Relaxation of a QED Cavity

We study repeated interactions of the quantized electromagnetic field in a cavity with single two-levels atoms. Using the Markovian nature of the resulting quantum evolution we study its large time asymptotics. We show that, whenever the atoms are distributed according to the canonical ensemble at temperature T>0 and some generic non-degeneracy condition is satisfied, the cavity field relaxes towards some invariant state. Under some more stringent non-resonance condition, this invariant state is thermal equilibrium at some renormalized temperature T ^*. Our result is non-perturbative in the strength of the atom-field coupling. The relaxation process is slow (non-exponential) due to the presence of infinitely many metastable states of the cavity field.     

Collision of Rydberg atom A** with atom B in the ground electronic state. Optical potential

A method for calculating the complex optical potential of slowly colliding Rydberg atom A^** and neutral atom B in the ground electronic state is suggested. The method is based on the asymptotic approach and the theory of multichannel quantum defects, which uses the formalism of renormalized Lippmann-Schwinger equations. The potential is introduced as the 〈q|V _opt|q〉 matrix element of the optical interaction operator, for which the integral equation is derived, and is calculated in the basis set of free particle wave functions |q〉. Fairly simple equations for the shift and broadening of the ionic term are obtained, and the principal characteristics of these equations are analyzed. By way of illustration, the optical potential of the Na^**(nl)+B systems, where B is a rare gas atom, is calculated.     

Gauge invariance and renormalization

The renormalization of Abelian and non-Abelian local gauge theories is discussed. It is recalled that whereas Abelian gauge theories are invariant to local c-number gauge transformations δA_μ(x) = ?_μ,…, with □Λ = 0, and to the operator gauge transformation δA_μ(x) = ?_μφ(x), …, δφ(x) = α^?1?·A(x), with □φ = 0, non-Abelian gauge theories are invariant only to the operator gauge transformations $\text{δA}{}_{\mathrm{\mu }}\text{(x) ～}\textcolor{red}{}{}_{\mathrm{\mu }}\text{C(x)}$, …, introduced by Becchi, Rouet and Stora, where

_μ is the covariant derivative matrix and C is the vector of ghost fields. The renormalization of these gauge transformation is discussed in a formal way, assuming that a gauge-invariant regularization is present. The naive renormalized local non-Abelian c-number gauge transformation $\text{δA}{}_{\mathrm{\mu }}\text{(x) = (Z}{}_1\text{/Z}{}_3\text{)gA}{}_{\mathrm{\mu }}\text{(x) ×}\text{Λ}\text{(x)+?}{}_{\mathrm{\mu }}\text{Λ}\text{(x)}$, …, is never a symmetry transformation and is never finite in perturbation theory. Only for $\text{Λ}\text{(x) = (Z}{}_3\text{/Z}{}_1\text{)L}$ with L finite constants or for $\text{Λ}\text{(x) = Ω}\text{z}\text{?}{}_3\text{C(x)}$ with Ω a finite constant does it become a finite symmetry transformation, where $\text{z}\text{?}{}_3$ is the ghost field renormalization constant. The renormalized non-Abelian Ward-Takahashi (Slavnov-Taylor) identities are consequences of the invariance of the renormalized gauge theory to this formation. It is also shown how the symmetry generators are renormalized, how photons appear as Goldstone bosons, how the (non-multiplicatively renormalizable) composite operator A_μ × C is renormalized, and how an Abelian c-number gauge symmetry may be reinstated in the exact solution of many asymptotically fr ee non-Abelian gauge theories.     

Renormalized vertex modification of Hall conductivity for dilute magnetic alloys

By means of the renormalized vertex procedure for the motion of Green's function developed by the authors, the vertex function of magnetic alloys, based on thes-d exchange interaction, is solved exactly and the corresponding Hall conductivity tensors are obtained. It is found that the value of the renormalized Hall conductivity is (1+h ²)^–1 times less than that before the renormalization (hereh is a reduced magnetic field). It is shown that the renormalized modification of the conductivity is very important in the cases with not too weak external magnetic field and slow relaxation time.     

Does a generic connection depend continuously on its curvature?

For a principal bundle with semi-simple structure group over a smooth four-dimensional base manifold, the set of connections (gauge potentials)A which are uniquely determined by their curvature (field or field strength)F is generic in the set of all potentials, endowed with the WhitneyC ^∞ topology. However, the operator taking each such fieldF to its potentialA is not continuous. Partial negative results are given concerning the existence of a smaller generic set on which this operator is continuous.     

An exact operator solution of the quantum Liouville field theory

Exact, closed form results are given expressing the quantum Liouville field theory in terms of a canonical free pseudoscalar field. The classical conformal transformation properties and a Bäcklund transformation of the Liouville model are briefly reviewed and then developed into explicit operator statements for the quantum theory. This development leads to exact expressions for the basic operator functions of the Liouville field: ?_μΦ, and e^nΦ. An operator product analysis is then used to construct the Liouville energy-momentum tensor operator, which is shown to be equal to that of a free pseudoscalar field. Dynamical consequences of this equivalence are discussed, including the relation between the Liouville and free field energy eigenstates. Liouville correlation functions are partially analyzed, and remaining open questions are discussed.     

Effect of electrodes on the properties of a thin ferroelectric film

The effect of various metal electrodes on the properties of thin ferroelectric films is considered using the phenomenological Ginzburg-Landau theory. The electric field produced by charges on electrodes is taken into account (with allowance for the screening of the charges in the metals) in the free energy functional and in the Euler-Lagrange equation for the film polarization. This equation is solved using the variational method, and the free energy functional is reduced to the conventional free energy with a renormalized coefficient of P ². This coefficient is dependent on the properties and thickness of the film and the properties of the electrode. Therefore, the physical characteristics of the size effect can be found by merely substituting the renormalized coefficient into the usual formulas from phenomenological theory. The calculations are shown to be in good agreement with the experimental data for Pt, Ir, IrO₂, and SrRuO₃.