Anticommuting Field Research Articles

Finite-Size Effects in Non-neutral Two-Dimensional Coulomb Fluids

Thermodynamic potential of a neutral two-dimensional (2D) Cou\-lomb fluid, confined to a large domain with a smooth boundary, exhibits at any (inverse) temperature $\beta$ a logarithmic finite-size correction term whose universal prefactor depends only on the Euler number of the domain and the conformal anomaly number $c=-1$. A minimal free boson conformal field theory, which is equivalent to the 2D symmetric two-component plasma of elementary $\pm e$ charges at coupling constant $\Gamma=\beta e^2$, was studied in the past. It was shown that creating a non-neutrality by spreading out a charge $Q e$ at infinity modifies the anomaly number to $c(Q,\Gamma) = - 1 + 3\Gamma Q^2$. Here, we study the effect of non-neutrality on the finite-size expansion of the free energy for another Coulomb fluid, namely the 2D one-component plasma (jellium) composed of identical pointlike $e$-charges in a homogeneous background surface charge density. For the disk geometry of the confining domain we find that the non-neutrality induces the same change of the anomaly number in the finite-size expansion. We derive this result first at the free-fermion coupling $\Gamma\equiv\beta e^2=2$ and then, by using a mapping of the 2D one-component plasma onto an anticommuting field theory formulated on a chain, for an arbitrary coupling constant.

The quantum field as a quantum computer

It is supposed that at very small scales a quantum field is an infinite homogeneous quantum computer. On a quantum computer the information cannot propagate faster than c=a/τ, a and τ being the minimum space and time distances between gates, respectively. For one space dimension it is shown that the information flow satisfies a Dirac equation, with speed v=ζc and ζ=ζ(m) mass-dependent. For c the speed of light ζ−1 is a vacuum refraction index that increases monotonically from ζ−1(0)=1 to ζ−1(M)=∞, M being the Planck mass for 2a the Planck length. The Fermi anticommuting field can be entirely qubitized, i.e. it can be written in terms of local Pauli matrices and with the field interaction remaining local on qubits. Extensions to larger space dimensions are discussed.

FERMIONIC FIELDS IN THE FUNCTIONAL APPROACH TO CLASSICAL FIELD THEORY

In this paper, we present a formulation of the classical theory of fermionic (anticommuting) fields, which fits into the general framework proposed by Brunetti, Dütsch and Fredenhagen. It was inspired by the recent developments in perturbative algebraic quantum field theory and it also allows for a deeper structural understanding on the classical level. We propose a modification of this formalism that also allows to treat fermionic fields. In contrast to other formulations of classical theory of anticommuting variables, we do not introduce additional Grassman degrees of freedom. Instead the anticommutativity is introduced in a natural way on the level of functionals. Moreover, our construction incorporates the functional-analytic and topological aspects, which is usually neglected in the treatments of anticommuting fields. We also give an example of an interacting model where our framework can be applied.

Counter-ions at charged walls: Two-dimensional systems

We study equilibrium statistical mechanics of classical point counter-ions, formulated on 2D Euclidean space with logarithmic Coulomb interactions (infinite number of particles) or on the cylinder surface (finite particle numbers), in the vicinity of a single uniformly charged line (one single double layer), or between two such lines (interacting double layers). The weak-coupling Poisson-Boltzmann theory, which applies when the coupling constant [Formula: see text] is small, is briefly recapitulated (the coupling constant is defined as [Formula: see text] [Formula: see text] [Formula: see text] e (2) , where [Formula: see text] is the inverse temperature, and e the counter-ion charge). The opposite limit ( [Formula: see text] [Formula: see text] ∞ is treated by using a recent method based on an exact expansion around the ground-state Wigner crystal of counter-ions. These two limiting results are compared at intermediary values of the coupling constant [Formula: see text] = 2[Formula: see text] ([Formula: see text] = 1, 2, 3) , to exact results derived within a 1D lattice representation of 2D Coulomb systems in terms of anti-commuting field variables. The models (density profile, pressure) are solved exactly for any particles numbers N at [Formula: see text] = 2 and up to relatively large finite N at [Formula: see text] = 4 and 6. For the one-line geometry, the decay of the density profile at asymptotic distance from the line undergoes a fundamental change with respect to the mean-field behavior at [Formula: see text] = 6 . The like-charge attraction regime, possible for large [Formula: see text] but precluded at mean-field level, survives for [Formula: see text] = 4 and 6, but disappears at [Formula: see text] = 2 .

Nonlinear sigma model for a condensate composed of fermionic atoms

A nonlinear sigma model is derived for the time development of a Bose–Einstein condensate composed of fermionic atoms. Spontaneous symmetry breaking of a Sp ( 2 ) symmetry in a coherent state path integral with anticommuting fields yields Goldstone bosons in a Sp ( 2 ) ⧹ U ( 2 ) coset space. After a Hubbard–Stratonovich transformation from the anticommuting fields to a local self-energy matrix with anomalous terms, the assumed short-ranged attractive interaction reduces this symmetry to a SO ( 4 ) ⧹ U ( 2 ) coset space with only one complex Goldstone field for the singlet pairs of fermions. This bosonic field for the anomalous term of fermions is separated in a gradient expansion from the density terms. The U ( 2 ) invariant density terms are considered as a background field or unchanged interacting Fermi sea in the spontaneous symmetry breaking of the SO ( 4 ) invariant action and appear as coefficients of correlation functions in the nonlinear sigma model for the Goldstone boson. The time development of the condensate composed of fermionic atoms results in a modified Sine–Gordon equation.

Bosonization and the lattice Gross-Neveu model

We consider a lattice version of the bosonized Gross-Neveu model. It is explicitely chiral symmetric and its numerical simulation does not involve any anticommuting field. We study its non trivial 1/ N expansion up to the next-to-leading term comparing the results with explicit numerical simulations.

Functional methods in random classical field theory

A functional which generates N distinct point solutions to a classical wave equation with random coefficients in the presence of external sources is constructed. Statistical averaging over the random coefficients is then implemented using replica and/or anticommuting field techniques.

On some representations of the anticommutations relations

We study representations of the canonical anticommutation relations having the form: wherea(f),b*(f) and their adjoints are two basic anticommuting fields in a Fock Space. A complete determination of the type in terms of |K|=(K*K)1/2 and a sufficient condition for quasi-equivalence are given.

Field Theory of Equations with Many Masses

This paper develops the field theory of many mass equations with special attention to spin \textonehalf{} and the operator (1) of the author's previous paper on the irreducible volume character of events. The field is assumed to interact with the electromagnetic field which is introduced in a gauge-invariant way. General expressions for the charge-current four-vector and the symmetrical energy-momentum tensor are derived and are shown to satisfy the appropriate conservation theorems. According to a theorem of Leichter, the general solution is shown to be a superposition of nonorthogonal mass states which we designate as the root fields. Nevertheless, the physical quantities, such as the current four-vector, the energy-momentum tensor, etc., are shown to decompose into a sum over those of individual mass states but with an alternation of sign for consecutive roots. The Lagrangian takes the form of an alternating sum over the individual free-field Lagrangians for the mass states, plus the usual term $+(\frac{1}{c}){j}_{\ensuremath{\mu}}{A}_{\ensuremath{\mu}}$ for the interaction with the electromagnetic field. The matter field may be quantized by treating the root fields as independent anticommuting fields. The transformation to the interaction representation is obviously unaltered and the charge and mass renormalization may be treated following Schwinger. To the order of approximation in Schwinger II these renormalizations are not affected. It would seem that these methods of quantization, together with the usual treatment of the electromagnetic field, are at variance with the manifest nonlocal nature of the theory for the irreducible volume character of events.

Note on the Beta Interaction

It is shown that the most general Lorentz invariant interaction between four mutually anticommuting fields leads to matrix elements of the form $(S\ensuremath{-}T+P){f}_{1}(n, m)+(V\ensuremath{-}A){f}_{2}(n, m)+(S\ensuremath{-}A\ensuremath{-}P){f}_{3}(n, m).$ In the charge-exchange order, neutron and muon decay are compatible with ${f}_{2}={f}_{3}=0$; charged pion decay appears to correspond to ${f}_{1}={f}_{3}=0$. It is noted that the hypothesis of the same interaction between all fermions does not imply that ${f}_{1}$, ${f}_{2}$, and ${f}_{3}$ are numerical constants; it is possible that they are functions of new variables ($n, m$) needed to characterize the associated Feynman diagrams: $n$ indicates the number of particles produced in the reaction and $m$ the number of particles minus antiparticles. One can make a simple choice of the ${f}_{i}$ in such a way that the interaction is sometimes $S\ensuremath{-}T+P$, sometimes $V\ensuremath{-}A$, and never $S\ensuremath{-}A\ensuremath{-}P$.