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Register a new userMirror channel eigenvectors of the $d$-dimensional fishnets
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We present a basis of eigenvectors for the graph building operators acting along the mirror channel of planar fishnet Feynman integrals in $d$-dimensions. The eigenvectors of a fishnet lattice of length $L$ depend on a set of $L$ quantum numbers $(u_k,l_k)$, each associated with the rapidity and bound-state index of a lattice excitation. Each excitation is a particle in $(1+1)$-dimensions with $O(d)$ internal symmetry, and the wave-functions are formally constructed with a set of creation/annihilation operators that satisfy the corresponding Zamolodchikovs-Faddeev algebra. These properties are proved via the representation - new to our knowledge - of the matrix elements of the fused R-matrix with $O(d)$ symmetry as integral operators on the functions of two spacetime points. The spectral decomposition of a fishnet integral we achieved can be applied to the computation of Basso-Dixon integrals in higher dimensions.
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The Casimir effect for rectangular boxes has been studied for several decades. But there are still some points unclear. Recently, there are new developments related to this topic, including the demonstration of the equivalence of the regularization methods and the clarification of the ambiguity in the regularization of the temperature-dependent free energy. Also, the interesting quantum spring was raised stemming from the topological Casimir effect of the helix boundary conditions. We review these developments together with the general derivation of the Casimir energy of the $p$-dimensional cavity in ($D+1$)-dimensional spacetime, paying special attention to the sign of the Casimir force in a cavity with unequal edges. In addition, we also review the Casimir piston, which is a configuration related to rectangular cavity.
In this paper we consider a conformal invariant chain of $L$ sites in the unitary irreducible representations of the group $SO(1,5)$. The $k$-th site of the chain is defined by a scaling dimension $Delta_k$ and spin numbers $frac{ell_k}{2}$, $frac{dot{ell}_k}{2}$. The model with open and fixed boundaries is shown to be integrable at the quantum level and its spectrum and eigenfunctions are obtained by separation of variables. The transfer matrices of the chain are graph-builder operators for the spinning and inhomogeneous generalization of squared-lattice fishnet integrals on the disk. As such, their eigenfunctions are used to diagonalize the mirror channel of the the Feynman diagrams of Fishnet conformal field theories. The separated variables are interpreted as momentum and bound-state index of the $textit{mirror excitations}$ of the lattice: particles with $SO(4)$ internal symmetry that scatter according to an integrable factorized $mathcal{S}$-matrix in $(1+1)$ dimensions.
We reconsider the thermal scalar Casimir effect for $p$-dimensional rectangular cavity inside $D+1$-dimensional Minkowski space-time. We derive rigorously the regularization of the temperature-dependent part of the free energy by making use of the Abel-Plana formula repeatedly and get the explicit expression of the terms to be subtracted. In the cases of $D$=3, $p$=1 and $D$=3, $p$=3, we precisely recover the results of parallel plates and three-dimensional box in the literature. Furthermore, for $D>p$ and $D=p$ cases with periodic, Dirichlet and Neumann boundary conditions, we give the explicit expressions of the Casimir free energy in both low temperature (small separations) and high temperature (large separations) regimes, through which the asymptotic behavior of the free energy changing with temperature and the side length is easy to see. We find that for $D>p$, with the side length going to infinity, the Casimir free energy tends to positive or negative constants or zero, depending on the boundary conditions. But for $D=p$, the leading term of the Casimir free energy for all three boundary conditions is a logarithmic function of the side length. We also discuss the thermal Casimir force changing with temperature and the side length in different cases and find with the side length going to infinity the force always tends to zero for different boundary conditions regardless of $D>p$ or $D=p$. The Casimir free energy and force at high temperature limit behave asymptotically alike in that they are proportional to the temperature, be they positive (repulsive) or negative (attractive) in different cases. Our study may be helpful in providing a comprehensive and complete understanding of this old problem.
We investigate the existence and properties of a double asymptotic expansion in $1/N^{2}$ and $1/sqrt{D}$ in $mathrm{U}(N)timesmathrm{O}(D)$ invariant Hermitian multi-matrix models, where the $Ntimes N$ matrices transform in the vector representation of $mathrm{O}(D)$. The crucial point is to prove the existence of an upper bound $eta(h)$ on the maximum power $D^{1+eta(h)}$ of $D$ that can appear for the contribution at a given order $N^{2-2h}$ in the large $N$ expansion. We conjecture that $eta(h)=h$ in a large class of models. In the case of traceless Hermitian matrices with the quartic tetrahedral interaction, we are able to prove that $eta(h)leq 2h$; the sharper bound $eta(h)=h$ is proven for a complex bipartite version of the model, with no need to impose a tracelessness condition. We also prove that $eta(h)=h$ for the Hermitian model with the sextic wheel interaction, again with no need to impose a tracelessness condition.
We develop the formalism of quantum mechanics on three dimensional fuzzy space and solve the Schrodinger equation for a free particle, finite and infinite fuzzy wells. We show that all results reduce to the appropriate commutative limits. A high energy cut-off is found for the free particle spectrum, which also results in the modification of the high energy dispersion relation. An ultra-violet/infra-red duality is manifest in the free particle spectrum. The finite well also has an upper bound on the possible energy eigenvalues. The phase shifts due to scattering around the finite fuzzy potential well have been calculated.
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