Variational Calculations in Quantum Field Theory

The following sections are included:

• Preface

• Participants

• Contents

A field-theory version of the Rayleigh-Ritz variational method for bound states is presented, and applied to compute low-lying glueball masses. The method makes use of the exact path-integral representation of the vacuum state, as well as non-perturbative condensate contributions. The “gluon-chain” model of string formation, and possible applications of the variational method to a string-tension calculation, are also discussed.

Quantized SU(n) Yang-Mills fields are considered for n = 2, 3, and 4. Our calculations use the Hamiltonian formalism as applied to the continuum theory in a finite volume. We have examined states in the vacuum sector using variational methods in the Coulomb gauge. The results depend on the strength of the interaction constant. If this is big enough, the excited 0⁺⁺ states can be given a glueball interpretation.

The following sections are included:

• Introduction

• Sensitivity to High Frequencies

• Only Gaussian Trial States

• We Still Have To Do a Functional Integral

• Discussion

The following sections are included:

• Introduction

• QCD in Two Dimensions

• QCD in Four Dimensions: Gluons

• QCD in Four Dimensions: Quarks

• Conclusions

• Acknowledgements

• References

A Monte Carlo simulation of the exact path-integral representation of the Yang-Mills vacuum wavefunctional is carried out in three dimensions. The data for long-wavelength field configurations is accurately fit by the lattice version of

By insertion of Wilson lines into the path-integral representation it is also possible to measure the wavefunctional of a state containing heavy quark-antiquark charges. For large quark separations, this state represents the QCD string. Preliminary numerical results indicate the presence of constituent gluons in the confining string.

The following sections are included:

• Introduction

• The Variational Calculation

• The Lattice Corroboration [3,4]

• The Energy-Momentum Tensor's Trace Anomaly

• Outlook on the Future of QCD

• References

It is often stated that the only class of wavefunctionals Ψ[φ] in quantum field theory for which the energy expectation can be computed exactly is the Gaussians

(with K some positive definite operator) and their extension by shifting the field variable φ with a fixed c-number function, or multiplying the exponential by a polynomial in φ. The generalization we are proposing here is, in fact, also derived from the Gaussians; it involves a non-linear canonical transformation of the field operators. The basic idea is illustrated by the following quantum mechanical example…

I describe the calculation of pseudoscalar masses and decay coupling constants in QCD with three flavors, using a modified version of the Cornwall, Jackiw and Tom-boulis effective action for composite operators. Renormalization group corrections are taken into account. The effective potential depends on the standard parameters of QCD: Λ_QCD, m_u, m_d, m_s and on a further mass scale μ which discriminates between the infrared and the ultraviolet regime. The overall fit agrees quite well with the experimental data.

An equation of state is computed for a plasma of one flavour quarks interacting through some phenomenological potential in the Hartree approximation at zero temperature. Assuming that the confining potential is scalar and colour-independent, it is shown that the quarks undergo a first-order mass phase transition. In addition, due to the way screening is introduced, all the thermodynamic quantities computed are independent of the actual shape of the interquark potential. This equation of state is then generalized to a potential with scalar and vector components, Fock corrections are discussed and the case of a several quark flavour plasma is studied.

No doubt the man in the street believes that , is a trivial theory. This idea goes back to Landau and Pomeranchuk (for QED) [1] and Wilson [2]. Let us quote them: “We reach the fundamental conclusion that, from formal quantum electrodynamics, there apparently follows the equality of zero of the charge on the electron” and “The author has also studied the λ₀φ⁴ theory in four space-time dimensions. The solution of the renormalization group recursion formulas turns out to be more complicated than for the λ₀φ⁶ theory; also, the best evidence the author has at the moment on the λ₀φ⁴ theory suggests that the renormalized theory is always a free-field theory (in the no-cutoff limit) no matter how one lets λ₀ vary with the cutoff”…

The extension of the quantum-mechanical variational approximation to field theory is considered taking into account the restrictions imposed by renormalizability. It is found that the simplest variational ansatz for the states gives a well defined approximation to the effective potential in any model with only normal ordering divergences. For other super-renormalizable theories a new ansatz is proposed which also provides a well defined approximation.

The variational Approximation is generalized to a time-dependent approach capable of giving time-dependent Green's functions. This new Covariant Gaussian Approximation is represented, in full analogy with the classical approximation, as an initial truncation of the Dyson-Schwinger equations followed by functional differentiation of the effective action. The connection with the 1/N expansion for scalar theories is briefly discussed.

The covariant Gaussian approximation is applied to the O(2) invariant scalar theory and the U(l) Higgs model. It is found in this framework that neither Goldstone nor Higgs phenomena occur.

Variational Methods have been employed to study the phase structure of supersymmetric quantum field theories. O(N) symmetric vector models are treated in the large N limit. In this limit variational calculations give the exact solutions. A general framework is formulated for theories in d dimensions 2 ≤ d ≤ 3 and some examples are solved within this framework.

We use the variational basis function-expansion method to compute the one and two particle masses for the scalar field theory : λ(φ⁶ − φ⁴) + ½ μ²φ² : in 1+1 dimensions. The variational ansatz is a linear superposition of the known states of the scalar field theory defined by L₀ = ½∂_μφ∂^μφ − ½Ω²φ², where Ω, is a variational parameter. Results for the mass gap m₁ are obtained using a superposition of one-and three-particle states of L₀ as the variational trial state. Similarly, the two-particle bound state mass m₂ is computed using a superposition of two-and four-particle states of L₀. The new values of the mass gap and two-particle mass are used to recompute the two-particle mass fraction m₂/m₁ for various values of .

It would be of interest to study tightly bound systems in QED as it may shed some light on highly relativistic systems in QCD, although then the two-particle interaction is quite different. We therefore allow the fine structure constant to be a variable…

This is an attempt to translate the Feynman polaron path integral method into the language of a Hamiltonian formalism. The lowest energy and the effective mass expressions of Feynman are exactly reproduced in a Hamiltonian language. But we are not claiming that we have succeeded in obtaining a consistent Hamiltonian formalism which is equivalent to the path integral formalism. We have been able to express exactly what Feynman did in his method, in a more usual Hamiltonian language, term by term.

The following sections are included:

• Introduction

• Notation and Conventions

• Spherically Symmetric Potentials and Equations of Motion

• General Discussion of the Static Solutions

• Asymptotic Analysis

• Action and Energy

• References

A dense instanton-liquid model of Yang-Mills vacuum at zero and finite temperature is discussed by use of the Feynman variational principle. Importance of instanton-instanton interactions is emphasized.

The following sections are included:

• Introduction

• General Scope of Renormalization

• Regularization of Gauge Field Theories
  • Lattice Regularization
  • Regularization in the Coulomb gauge

• Many-Body Techniques for Yang-Mills Theory

• Aspects of Confinement

• References

The Landau gauge gluon propagator at finite temperature above and below the deconfinement transition is measured using lattice Monte Carlo simulation. The color electric and magnetic masses are determined. The most striking result of the calculation is that the time component of the gluon field appears to acquire a vacuum expected value in the deconfined region, corresponding to Higgs type spontaneous symmetry breaking.

The following sections are included:

• Introduction

• Systematic Variational Procedures
  • Lánczos Procedure for Upper Bounds
  • Weinstein-Stenger Procedure for Lower Bounds

• Some Two-Dimensional Examples
  • The O(3) σ-model
  • The Gross-Neveu model

• Four-Dimensional Gauge Theories
  • Solving the Compact Abelian Theory
  • Nonabelian Gauge Theory

• References

Systematic improvement schemes for solving finite size lattice systems variationally are reviewed. The Casimir-energy of conformally invariant two-dimensional field theories in restricted geometry is proposed as a simple but stringent test of the quality of the trial ground states. A generalized Lánczos-algorithm, proposed recently by Duncan and Roskies, leads to accurate conformal characteristics for the Z₂ and Z₃ symmetric quantum chains.

The t-expansion is a tool designed to improve on any variational calculation. We review its formulation and application to the solvable Heisenberg anti-ferromagnet model. This is used to decide on the approximants which we employ in order to obtain numerical estimates for physical quantities. The same technique is then applied to the SU(3) lattice gauge theory in 3+1 space time dimensions. Starting with strong-coupling wave-functions we investigate the pure glue sector. We review the calculations of the energy-density, the string-tension and several glueball masses. The scalar 0⁺⁺ glueball follows from the same calculation as the energy-density. The other glueballs which we study, with the J^PC quantum numbers of 1⁺⁻, 0⁻⁻, 1⁻⁻ and 2⁻⁻ are constructed by using wave-functions corresponding to appropriate representations of the cubic symmetry group. Estimates of physical masses are obtained from studies of scaling ratios.

A ground state of lattice SU(2) QCD with Kogut-Susskind fermions at finite chemical potential is studied by means of the t-expansion. A phase transition restoring both U(l) symmetry of the baryon number and the axial symmetry is predicted. This result is supported by a second order perturbative calculation.