On the energy-momentum tensor in gauge field theories

The energy-momentum tensor in spontaneously broken non-Abelian gauge field theories is studied. The motivation is to show that recent results on the finiteness and gauge independence of S-matrix elements in gauge theories extends to observable amplitudes for transitions in a gravitational field. Path integral methods and dimensional regularization are used throughout. Green's functions Γ_μν^(j)(q; p₁,…,p_j) involving the energy-momentum tensor and j particle fields are proved finite to all orders in perturbation theory to zero and first order in q, and finite to one loop order for general q. Amputated Green's functions of the energy momentum tensor are proved to be gauge independent on mass shell.     

The energy-momentum tensor in scalar and gauge field theories

A previous study of the energy-momentum tensor in ?⁴ theory and spontaneously broken non-Abelian gauge field theories is extended here to show finiteness to all orders in perturbation theory. Divergences of Green's functions Γ_μν^(j) (q; p₁, …, p_j) involving the energy-momentum tensor θ_μν and j particle fields are removed by counterterms of the ordinary Lagrangian plus a renormalization of the coefficient of the Callan-Coleman-Jackiw improvement term in θ_μν. Physically the extra renormalization means that the mean square “mass radius” of elementary spin zero particles must be specified from experiment.     

Renormalized Normal Product and equations of motion of Φ4-coupling

The notion of a Renormalized Normal Product (RNP) in Euclidean space of 1 ≤ r ≤ 4 dimensions is introduced for a Φ⁴-model in a nonperturbative approach. The essential ingredients used for this purpose are the composite operators defined in perturbation theory and the renormalized G-convolution product constructed in the axiomatic field theory framework in Euclidean momentum space. Convergent equations of motion for the connected Green's functions are established where the interaction term is represented by the RNP. The corresponding renormalization constants are defined as boundary values of the RNP by imposing “physical” renormalization conditions. In the special case of 2-dimensions it is proved that these equations conserve analyticity and algebraic properties (in complex Minkowski space of 2-momenta) coming from the first principles of general local field theory, together with properties of asymptotic behaviour at infinity (in Euclidean space of 2-momenta).     

Strong coupling limit of the gφ4 theory

The strong coupling limit of the gφ⁴ theory in the framework of the path integral formalism. An expansion of the Green's functions in negative powers of the coupling constant is obtained; at each order the dependence on the external momenta is of polynomial type. A renormalization procedure is proposed; the asymptotic behaviour of the Callan-Symanzik β function is studied and the existence of a stable ultraviolet fixed point is established.     

An eikonal perturbation theory having applications in hadron dynamics. II. Massive quantum electrodynamics

An eikonal perturbation theory (EPT), derived in previous work for a superrenormalizable coupling, is here developed for massive quantum electrodynamics (MQED) involving scalar or spinor matter fields minimally coupled to neutral massive vector gluons. After summarizing the functional method, we present the EPT for the external field problem. In agreement with results known within ordinary perturbation theory (OPT) in the eikonal approximation (EA), from an exact eikonal equation derived here we show that the EPT for the external field problem provides an excellent approximation method for Green's functions at large momenta. We then discuss some general features of the EPT for MQED, and show that it leads to a renormalizable approximation method. Our approach is then illustrated by deriving explicit expressions for various renormalized Green's functions in lowest order EPT. We also discuss some asymptotic properties of such Green's functions and indicate how to proceed with calculations in higher orders. As in our previous work, we again find that the renormalization procedure in EPT bears close resemblance to the one for OPT. Contrary to what happens with the EA, the inclusion of self-interactions and of other field-theoretic effects does not spoil the virtues of the EPT as a far better high-momenta approximation than the OPT. As a typical example, if s is an energy parameter and g the coupling constant with $\text{g}{}^2\text{4π}\text{< 1}$, OPT to order g²ⁿ often fails to be a good approximation as soon as $\text{(}\text{g}{}^2\text{4π}\text{)}\text{ln}\text{s ～ 1}$, while in such cases EPT to order g²ⁿ is still a good approximation as long as $\text{(}\text{g}{}^2\text{4π}\text{)}{}^{\mathrm{n+1}}\text{ln}\text{s < 1}$. We also find that the EPT is superior to the EA in that, contrary to the EA, it provides a step-by-step rigorous and renormalizable iterative approximation method which can account for self-interactions and other field-theoretic effects. We emphasize that the EPT is much simpler and more general than other explicit approximate summation methods of classes of OPT Feynman graphs.In field theory, we consider the use of the EPT as a generalization of the EA for discussing, e.g. high-energy behaviors in MQED as well as infrared divergence and bound-state problems in the limit of massless gluons. It is also suggested that, in view of its nice field-theoretic and high-energy properties, the EPT for MQED might provide a useful laboratory where ideas and problems in hadron dynamics could be meaningfully investigated within a Lagrangian field theory.     

Higher order field equations II; Limit properties of green's functions

In a previous paper wave propagation was studied according to a sixth-order partial differential equation involving a complex mass M. The corresponding Yang-Feldman integral equations (indicated as SM-YF-equations), were formulated using modified Green's functions G^M_R(x) and G^M_A(x), which then incorporate the partial differential equation together with certain boundary conditions. In this paper certain limit properties of these modified Green's functions are derived: (a) It is shown that for |M| → ∞ the Green's functions G^M_R(x) and G^M_A(x) approach the Green's functions Δ_R(x) and Δ_A(x) of the corresponding KG-equation (Klein-Gordon equation). (b) It is further shown that the asymptotic behaviour of G^M_A(x) and G^M_A(x) is the same as of Δ_R(x) and Δ_A(x) - and also the same as for D_R(x) and D_A(x) for t→ ± ∞, where D_R and D_A are the Green n's functions for the KG-equation with mass zero. It is essential to take limits in the sense of distribution theory in both cases (a) and (b). The property (b) indicates that the wave propagation properties of the SM-YF-equations, the KG-equation with finite mass and the KG-equation with mass zero are closely related in an asymptotic sense.     

Legendre transforms and r-particle irreducibility in quantum field theory: The formalism for r = 1, 2

We analyze the first and second Legendre transforms Γ^(r) (r = 1, 2) of the generating functional G for connected Green's functions in Euclidean boson field theories. By using Spencer's idea of t-lines we define and prove irreducibility properties independently of perturbation theory. In particular we prove that Γ^(r) generates r-irreducible vertex functions, r-irreducible expectations and r-field projectors; moreover, Γ⁽²⁾ generates (generalized) Bethe-Salpeter kernels with 2-cluster-irreducibility properties.     

A fixed point for quantum gravity

Using the 1/N expansion a fixed point of the renormalization group is found for quantized gravitational theories which is non-trivial in all dimensions, d, including four. Using the fixed point it is shown how Einstein's theory can be renormalized for 3<d<4. In four dimensions the pure Einstein theory does not exist, but the R + C_μναβ² theory does. It is shown how gravitational theories whose quantum lagrangians are scale invariant may be renormalized such that the scale invariance is broken only by the choice of the critical renormalization group trajectory. A comparison is made with the renormalization of four-fermion and Yukawa theories in 4?? dimensions which suggests that quantum gravity might exist in four dimensions even if those theories do not.     

Non-linear σ-model and CP(n−1) at 2 + ε dimensions

Two systems, O(n) non-linear σ-model and CP^(n?1), are studied in the light of Elitzur's theorem, on the disappearance of infrared singularities at two dimensions. The consequences of the theorem are expressed in dimensional regularization, and issues like the proper analytic continuation to d = 2 + ε, the peculiarities of momentum-space Green functions near d = 2 and their renormalization, and the exponentiation of Green functions are clarified.The analysis is applied to compute the renormalization constants, and the gauge-invariant critical exponent η associated with the wave function of CP^(n?1) at one order higher than previously done. Finally, we conjecture on a possible connection between infrared finiteness and renormalizability.     

A fixed point approach to the Green's functions of Φ24: Local existence of the Borel transforms

We prove the existence at small μ of a unique solution of the system $\text{B =}\text{M}\text{(B)}$ to be satisfied by the set of Borel transformed euclidean Green's functions of a Φ₂⁴ theory, μ being the Borel conjugate variable of the coupling λ. As a byproduct, a new proof of convergence of the Borel transformed perturbative series is obtained.     

On the definition of renormalized field products

We study the redefinition of the field products appearing in a Lagrangian and its equations of motion in a Normal Product framework. We propose a method of defining these products, which give the finite Green's functions, in such a way that the canonical derivation of the equations of motion is preserved. This involves the use of the Wilson Expansion in a Dimensionally Regularized form. As an example a ?⁴, ?³, field theory in four dimensions is fully redefined to the 1-loop level.     

Functional approach to the parametric derivatives of renormalized green's functions with multiple insertions

Functional derivation of formulae for parametric derivatives of renormalized Green's functions with multiple insertions of composite operators is presented. An application to the derivative with respect to the ultraviolet cutoff in the Pauli–Villars regularized ?⁴₄ theory is given.     

Spin autocorrelation of Ni in K2Mn0.975Ni0.025F4 studied by nuclear spin-lattice relaxation

From the nuclear spin-lattice relaxation of the out-of-layer ¹⁹F nuclei in magnetic fields perpendicular to the c-axis the low-frequency component of the autocorrelation function 〈S^z(t)S^z(O)〉 of Ni in ordered K₂Mn_0.975Ni_0.025F₄ is found to be substantially reduced relative to the Mn host. The experimental rates vs temperature are in accord with those for relaxation involving two spin excitations calculated with local Green's functions.     

Summation of perturbation series in the generalized Hubbard model

The cumulant expansion is proposed for the summation of perturbation series in the generalized Hubbard model, which considers strongly correlated d-electrons hybridized with nearly free conduction electrons. It is shown, that summation of the principle series for the Green's functions leads to the Kondo renormalization of d-level hybridized with s-band.     

An effective lagrangian with no U(1) problem in CPn−1 models and QCD

We write an effective lagrangian which gives two-point Green's functions satisfying the anomalous Ward identities for the U(1) axial vector current with a singlet particle that has a non-vanishing mass in the chiral limit. We show that the mechanism that has been postulated by Witten and Veneziano for solving the U(1) problem in the framework of the 1/n expansion in QCD is fully active in the two-dimensional CP^n? model where the 1/n expansion can be explicitly performed.     

Infra-red asymptotic behaviour of the one-fermion green's function in a scalar model with isospin

In a theory where massive fermions interact with a massless scalar field of isospin 1, the behaviour of the one-fermion Green's function is found to differ from the free Green's function by a factor $\text{(1 ? (2g}{}^2\text{/π}{}^2\text{)1}\text{n}\text{m|x ? y|)}{}^{\mathrm{<mtext>?</mtext><mtext>3</mtext><mtext>8</mtext>}}$, in the limit of large separation |x ? y|.     

Quantum field theory in the infinite temperature limit

The T = ∞ limit for renormalizable 4-dimensional Euclidean QFT is considered. A general argument is presented in three examples: φ³, QED, QCD. Using an expansion of the Green's functions generating functional, it is shown at T = ∞ quantum dynamics generally becomes 3 dimensional. All superficially divergent diagrams survive at T = ∞ and ensure renormalization of effective dynamics. The correction to naive dimensional reduction is studied; appearance of “electric” masses in QED and QCD is shown to be the result of such a correction. A curious symmetry of the generating functional in QED and QCD, its implications and breaking by the thermal corrections of heavy modes are discussed. Presence of the symmetry implies survival of some fermion modes at T = ∞.     

Scale invariance in the thirring model with SU(n) symmetry

It is shown that the condition for scale invariance of the Green's functions is satisfied for two sequences of the SU(n) coupling g_v. For all these solutions the c-number Schwinger term in the commutator of the SU(n) currents has a negative values different from the free field value.     

Generating functional approach to the Green's functions diagrammatic technique for spin and pseudospin lattice systems: I. General theory

The theory of the irreducible many-point Green's functions, describing spin and pseudospin lattice systems, is formulated with the help of the generating functional approach. The diagrammatic technique for the generating functional is also developed. Special attention is paid to the construction and summation of the diagrammatic series for the one- and two-point Green's functions. Closed formulae for the one-point Green's function and the generalized Vaks-Larkin- Pikin equation are obtained. The $\text{1}\text{z}$ expansion scheme near the critical temperature of the order-disorder phase transition, is discussed, where z denotes the effective number of nearest- neighbours for a given site in a crystal lattice.