Search Results (4)

The main purpose of this paper is to derive a new perturbation theory (PT) that has converging series. Such series arise in the nonlocal scalar quantum field theory (QFT) with fractional power potential. We construct a PT for the generating functional (GF) of complete Green functions (including disconnected parts of functions) $\mathcal{Z}\left[j\right]$ as well as for the GF of connected Green functions $\mathcal{G}\left[j\right]=ln\mathcal{Z}\left[j\right]$ in powers of coupling constant g. It has infrared (IR)-finite terms. We prove that the obtained series, which has the form of a grand canonical partition function (GCPF), is dominated by a convergent series—in other words, has majorant, which allows for expansion beyond the weak coupling g limit. Vacuum energy density in second order in g is calculated and researched for different types of Gaussian part $S_0\left[\varphi \right]$ of the action $S\left[\varphi \right]$. Further in the paper, using the polynomial expansion, the general calculable series for $\mathcal{G}\left[j\right]$ is derived. We provide, compare and research simplifications in cases of second-degree polynomial and hard-sphere gas (HSG) approximations. The developed formalism allows us to research the physical properties of the considered system across the entire range of coupling constant g, in particular, the vacuum energy density. Full article

Here, classical and quantum field theory of dipolar, axisymmetric quadrupolar and octupolar Bose gases is considered within a general approach. Dipole, axisymmetric quadrupole and octupole interaction potentials in the momentum representation are calculated. These results clearly demonstrate attraction and repulsion areas in corresponding gases. Then the Gross–Pitaevskii (GP) equation, which plays a key role in the present paper, is derived from the corresponding functional. The zoology of the form factors appearing in the GP equation is studied in details. The proper classes for the description of spatially non-uniform condensates form factors are chosen. In the Thomas–Fermi approximation a general solution of the GP equation with a quasilocal form factor is obtained. This solution has an interesting form in terms of a double rapidly converging series that universally includes all the interactions considered. Plots of condensate density functions for the exponential-trigonometric form factor are given. For the sake of completeness, in this paper we consider the GP equation with an optical lattice potential in the limit of small condensate densities. This limit does not distinguish between dipolar, quadrupolar and octupolar gases. An important analysis of the condensate stability, in other words the study of condensate excitations, is also performed in this paper. In the Gaussian approximation (from the Gross–Pitaevskii functional), a functional describing the perturbations of the condensate is derived in detail. This problem is an analog of the Bogolubov transformation used in the study of quantum Bose gases in operator formalism. For a probe wave function in the form of a plane wave, a spectrum of (Bogoliubov) excitations was obtained, from which an equation describing the threshold momentum for the emergence of instability was derived. An important result of this paper is the dependence of the threshold on the momentum of a stationary condensate. For completeness of the presentation, the approximating expression in the form of a rapidly converging series is obtained for the corresponding dependence, and plots of the corresponding series for the exponential-trigonometric form factor are given. Finally, in the conclusion a quantum hydrodynamic theory for dipolar, axisymmetric quadrupolar and octupolar gases is briefly presented, giving a clue to the experimental determination of the form factors. Full article

Nonlocal quantum field theory (QFT) of one-component scalar field $\phi$ in D-dimensional Euclidean spacetime is considered. The generating functional (GF) of complete Green functions $\mathcal{Z}$ as a functional of external source j, coupling constant g and spatial measure $d\mu$ is studied. An expression for GF $\mathcal{Z}$ in terms of the abstract integral over the primary field $\phi$ is given. An expression for GF $\mathcal{Z}$ in terms of integrals over the primary field and separable Hilbert space (HS) is obtained by means of a separable expansion of the free theory inverse propagator $\widehat{L}$ over the separable HS basis. The classification of functional integration measures $\mathcal{D}\left[\phi \right]$ is formulated, according to which trivial and two nontrivial versions of GF $\mathcal{Z}$ are obtained. Nontrivial versions of GF $\mathcal{Z}$ are expressed in terms of 1-norm and 0-norm, respectively. In the 1-norm case in terms of the original symbol for the product integral, the definition for the functional integration measure $\mathcal{D}\left[\phi \right]$ over the primary field is suggested. In the 0-norm case, the definition and the meaning of 0-norm are given in terms of the replica-functional Taylor series. The definition of the 0-norm generator $\Psi$ is suggested. Simple cases of sharp and smooth generators are considered. An alternative derivation of GF $\mathcal{Z}$ in terms of 0-norm is also given. All these definitions allow to calculate corresponding functional integrals over $\phi$ in quadratures. Expressions for GF $\mathcal{Z}$ in terms of integrals over the separable HS, aka the basis functions representation, with new integrands are obtained. For polynomial theories ${\phi }^{2n},n=2,3,4,\dots ,$ and for the nonpolynomial theory ${sinh}^4\phi$ , integrals over the separable HS in terms of a power series over the inverse coupling constant $1/\sqrt{g}$ for both norms (1-norm and 0-norm) are calculated. Thus, the strong coupling expansion in all theories considered is given. “Phase transitions” and critical values of model parameters are found numerically. A generalization of the theory to the case of the uncountable integral over HS is formulated—GF $\mathcal{Z}$ for an arbitrary QFT and the strong coupling expansion for the theory ${\phi }^4$ are derived. Finally a comparison of two GFs $\mathcal{Z}$ , one on the continuous lattice of functions and one obtained using the Parseval–Plancherel identity, is given. Full article

Nonlocal quantum theory of a one-component scalar field in D-dimensional Euclidean spacetime is studied in representations of $\mathcal{S}$ -matrix theory for both polynomial and nonpolynomial interaction Lagrangians. The theory is formulated on coupling constant g in the form of an infrared smooth function of argument x for space without boundary. Nonlocality is given by the evolution of a Gaussian propagator for the local free theory with ultraviolet form factors depending on ultraviolet length parameter l. By representation of the $\mathcal{S}$ -matrix in terms of abstract functional integral over a primary scalar field, the $\mathcal{S}$ form of a grand canonical partition function is found. By expression of $\mathcal{S}$ -matrix in terms of the partition function, representation for $\mathcal{S}$ in terms of basis functions is obtained. Derivations are given for a discrete case where basis functions are Hermite functions, and for a continuous case where basis functions are trigonometric functions. The obtained expressions for the $\mathcal{S}$ -matrix are investigated within the framework of variational principle based on Jensen inequality. Through the latter, the majorant of $\mathcal{S}$ (more precisely, of $-ln\mathcal{S}$ ) is constructed. Equations with separable kernels satisfied by variational function q are found and solved, yielding results for both polynomial theory ${\phi }^4$ (with suggestions for ${\phi }^6$ ) and nonpolynomial sine-Gordon theory. A new definition of the $\mathcal{S}$ -matrix is proposed to solve additional divergences which arise in application of Jensen inequality for the continuous case. Analytical results are obtained and numerically illustrated, with plots of variational functions q and corresponding majorants for the $\mathcal{S}$ -matrices of the theory. For simplicity of numerical calculation, the $D=1$ case is considered, and propagator for free theory G is in the form of Gaussian function typically in the Virton–Quark model, although the obtained analytical inferences are not, in principle, limited to these particular choices. Formulation for nonlocal QFT in momentum k space of extra dimensions with subsequent compactification into physical spacetime is discussed, alongside the compactification process. Full article