Related papers: Algebraic Varieties in Second Quantization

We develop algebraic geometry for coupled cluster (CC) theory of quantum many-body systems. The high-dimensional eigenvalue problems that encode the electronic Schr\"odinger equation are approximated by a hierarchy of polynomial systems at…

Well defined quantum field theory (QFT) for the electroweak force including quantum electrodynamics (QED) and the weak force is obtained by considering natural unitary representations of a group $K\subset U(2,2)$, where $K$ is locally…

Both algebras, Clifford and Grassmann, offer "basis vectors" for describing the internal degrees of freedom of fermions. The oddness of the "basis vectors", transferred to the creation operators, which are tensor products of the finite…

We develop and numerically analyze an algebraic-geometric framework for spin-adapted coupled-cluster (CC) theory. Since the electronic Hamiltonian is SU(2)-invariant, physically relevant quantum states lie in the spin singlet sector. We…

We determine the number of complex solutions to a nonlinear eigenvalue problem on the Grassmannian in its Pl\"ucker embedding. This is motivated by quantum chemistry, where it represents the truncation to single electrons in coupled cluster…

We present in Part II the description of the internal degrees of freedom of fermions by the superposition of odd products of the Clifford algebra elements, either $\gamma^a$'s or $\tilde{\gamma}^a$'s, which determine with their oddness the…

The first part of this thesis deals with certain properties of the quantum symmetric and exterior algebras of Type 1 representations of $U_q(g)$ defined by Berenstein and Zwicknagl. We define a notion of a commutative algebra object in a…

We extend recent algebro-geometric results for coupled cluster theory of quantum many-body systems to the truncation varieties arising from the doubles approximation (CCD), focusing on the first genuinely nonlinear doubles regime of four…

Application of the so-called refined algebraic quantization scheme for constrained systems to the relativistic particle provides an inner product that defines a unique Fock representation for a scalar field in curved space-time. The…

We construct a cluster algebra structure within the quantum cohomology ring of a quiver variety associated with an $A$-type quiver. Specifically, let $Fl:=Fl(N_1,\ldots,N_{n+1})$ denote a partial flag variety of length $n$, and…

As a first numerical application of the light-front coupled-cluster (LFCC) method, we consider the odd-parity massive eigenstate of $\phi_{1+1}^4$ theory. The eigenstate is built as a Fock-state expansion in light-front quantization, where…

We introduce a family of cluster algebras of infinite rank associated with root systems of type $A$, $D$, $E$. We show that suitable completions of these cluster algebras are isomorphic to the Grothendieck rings of the categories…

The Dirac theory implies the existence of an internal vector space, in addition to spin space. Using Dirac's coupling of variables in internal space to those in physical space, we construct a new configuration structure for particles in the…

We construct the affine version of the Fomin-Kirillov algebra, called the affine FK algebra, to investigate the combinatorics of affine Schubert calculus for type $A$. We introduce Murnaghan-Nakayama elements and Dunkl elements in the…

This work completes the construction of purely algebraic version of the theory of non-linear quantum chemistry methods. It is shown that at the heart of these methods there lie certain algebras close in their definition to the well-known…

Let $G$ be a simple, simply connected, simply laced algebraic group. We construct a monoidal category of representations of the quantum affine algebra $U_q(\widehat{\mathfrak{g}})$ whose Grothendieck ring contains a cluster algebra with…

Cluster varieties are geometric objects that have recently found applications in several areas of mathematics and mathematical physics. This thesis studies the geometry of a large class of cluster varieties associated to compact oriented…

We formulate a theory of generalized Fock spaces which underlies the different forms of quantum statistics such as ``infinite'', Bose-Einstein and Fermi-Dirac statistics. Single-indexed systems as well as multi-indexed systems that cannot…

Motivated by a recent conjecture by Hernandez and Leclerc [arXiv:0903.1452], we embed a Fomin-Zelevinsky cluster algebra [arXiv:math/0104151] into the Grothendieck ring R of the category of representations of quantum loop algebras U_q(Lg)…

This work provides a quantum-computing-first derivation of the Unitary Coupled Cluster ansatz, showing that its structure emerges naturally from fermionic algebra under unitary constraints. By explicitly connecting second quantization,…