arXiv subjects

Władysław Wachowski

Publications and source records attributed to Władysław Wachowski.

5 recordsLinked to original sources

arXiv2026

Schwinger--DeWitt expansion for the heat kernel of nonminimal operators in causal theories

We suggest a systematic calculational scheme for heat kernels of covariant nonminimal operators in causal theories whose characteristic surfaces are null with respect to a generic metric. The calculational formalism is based on a pseudodifferential operator calculus which allows one to build a linear operator map from the heat kernel of the minimal operator to the nonminimal one. This map is realized as a local expansion in powers of spacetime curvature, dimensional background fields, and their covariant derivatives with the coefficients -- the functions of the Synge world function and its derivatives. Finiteness of these functions, determined by multiple proper time integrals, is achieved by a special subtraction procedure which is an important part of the calculational scheme. We illustrate this technique on the examples of the vector Proca model and the vector field operator with a nondegenerate principal symbol. We also discuss smoothness properties of heat kernels of nonminimal operators in connection with the nondegenerate nature of their operator symbols.

hep-th↗

arXiv2026

New "metric-affine-like" generalization of Yang-Mills theory

We suggest a new generalization of the $\mathrm{U}(n)$ Yang-Mills theory obtained by relaxing the condition of covariant constancy of the Hermitian form in the fibers, $\nabla_a g_{αβ'} \ne 0$. This theory is a simpler analogue of the metric-affine gravity with $\nabla_a g_{bc} \ne 0$. In our case, connection $\nabla_a$ and Hermitian form $g_{αβ'}$ are two independent variables so total curvature and total potential are no longer anti-Hermitian matrices: thus, along with the standard YM potential $\boldsymbol{A}_a$ and field strength tensor $\boldsymbol{F}_{ab}$, it contains non-trivially interacting fields $\boldsymbol{B}_a$, $\boldsymbol{h}$, and $\boldsymbol{G}_{ab}$, $\boldsymbol{N}_a$, forming a non-Abelian generalization of Stückelberg theory. Due to the spontaneous symmetry breaking $\mathrm{GL}(n,\mathbb{C}) \to \mathrm{U}(n)$, these new fields can be made massive, and the limit $M\to\infty$ restores the standard YM theory.

hep-th↗

arXiv2026

"Metric-affine-like" generalization of YM (mal-YM): detailed classical consideration

We consider the ``metric-affine-like'' generalization of the Yang-Mills theory (mal-YM) which we first proposed earlier. In this model, the connection is no longer assumed to be compatible with the Hermitian form in the fibers. As a consequence, along with the usual YM potential $\boldsymbol{A}_a$ and the field strength tensor $\boldsymbol{F}_{ab}$, it contains non-trivially interacting fields $\boldsymbol{B}_a$, $\boldsymbol{h}$, and $\boldsymbol{G}_{ab}$, $\boldsymbol{N}_a$, forming a non-Abelian generalization of Stückelberg theory. Due to the spontaneous symmetry breaking $GL(n,\mathbb{C}) \to U(n)$, these new fields can be made massive and the limit $M\to\infty$ restores the standard YM theory. We perform a detailed analysis of this theory on the classical level. We discuss in detail geometric motivation for the model, field transformations, gauge symmetry and its spontaneous breaking, action, equations of motion, Noether identities, gauge fixing, and other issues.

hep-th↗

arXiv2025

Functorial properties of Schwinger-DeWitt expansion and Mellin-Barnes representation

We consider integral kernels for functions $f(\hat F)$ of a minimal second-order differential operator $\hat F(\nabla)$ on a curved spacetime. We show that they can be expanded in a functional series, analogous to the DeWitt expansion for the heat kernel, by integrating the latter term-by-term. This procedure leads to a separation of two types of data: all information about the bundle geometry and the operator $\hat F(\nabla)$ is still contained in the standard HaMiDeW coefficients $\hat a_k[F | x,x']$ (we call this property ``off-diagonal functoriality''), while information about the function $f$ is encoded in some new scalar functions $\mathbb{B}_α[f | σ]$ and $\mathbb{W}_α[f | σ, m^2]$, which we call basis and complete massive kernels, respectively. These objects are calculated for operator functions of the form $\exp(-τ\hat F^ν)/(\hat F^μ+ λ)$ as multiple Mellin--Barnes integrals. The article also discusses subtle issues such as the validity of the term-by-term integration, the regularization of IR divergent integrals, and the physical interpretation of the resulting expansions.

hep-th↗

arXiv2024

Commutator technique for the heat kernel of minimal higher derivative operators

We suggest a new technique of the asymptotic heat kernel expansion for minimal higher derivative operators of a generic $2M$-th order, $F(\nabla)=(-\Box)^M+\cdots$, in the background field formalism of gauge theories and quantum gravity. This technique represents the conversion of the recently suggested Fourier integral method of generalized exponential functions [Phys. Rev. D105, 065013 (2022), arXiv:2112.03062] into the commutator algebra of special differential operators, which allows one to express expansion coefficients for $F(\nabla)$ in terms of the Schwinger-DeWitt coefficients of a minimal second order operator $H(\nabla)$. This procedure is based on special functorial properties of the formalism including the Mellin-Barnes representation of the complex operator power $H^M(\nabla)$ and naturally leads to the origin of generalized exponential functions without directly appealing to the Fourier integral method. The algorithm is essentially more straightforward than the Fourier method and consists of three steps ready for computer codification by symbolic manipulation programs. They begin with the decomposition of the operator into a power of some minimal second order operator $H(\nabla)$ and its lower derivative "perturbation part" $W(\nabla)$, $F(\nabla)=H^M(\nabla)+W(\nabla)$, followed by considering their multiple nested commutators. The second step is the construction of special local differential operators -- the perturbation theory in powers of the lower derivative part $W(\nabla)$. The final step is the so-called procedure of their {\em syngification} consisting in a special modification of the covariant derivative monomials in these operators by Synge world function $σ(x,x')$ with their subsequent action on the HaMiDeW coefficients of $H(\nabla)$.

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