Representation of group generators by Boson or Fermion operators application to spin perturbation theory

The generator coordinate method approach to the dynamic group representation is discussed in general. In various cases, either in group space or in coset space, representations of the dynamic group can readily be obtained with the generator coordinate method. Boson representation is just one form of the generator coordinate method approach to the dynamic group representation. Various representations of the dynamic group are described by the generator coordinate method approach to the dynamic group representation in a unified way. Not only is the algebraic structure of generators preserved, but also conditions imposed by (a) the Pauli principle, (b) symmetry properties, (c) dynamic properties, and (d) other concrete properties of nuclear systems are well incorporated in these representations, so that the original fermion representation is faithfully realized. The generator coordinate method approach to the dynamic group representation is strictly a transformation theory from the fermion representation to continuous variable or boson representations of the dynamic group. Examples are given for showing the essence and the prospects of applications of the generator coordinate method approach to the dynamic group representation.The generator coordinate method approach to the dynamic group representation is discussed in general. In various cases, either in group space or in coset space, representations of the dynamic group can readily be obtained with the generator coordinate method. Boson representation is just one form of the generator coordinate method approach to the dynamic group representation. Various representations of the dynamic group are described by the generator coordinate method approach to the dynamic group representation in a unified way. Not only is the algebraic structure of generators preserved, but also conditions imposed by (a) the Pauli principle, (b) symmetry properties, (c) dynamic properties, and (d) other concrete properties of nuclear systems are well incorporated in these representations, so that the original fermion representation is faithfully realized. The generator coordinate method approach to the dynamic group representation is strictly a transformation theory from the fermion representation to continuous variable or boson representations of the dynamic group. Examples are given for showing the essence and the prospects of applications of the generator coordinate method approach to the dynamic group representation.

Physical Review CNov 1, 198711 citations

Representation theory of Gaussian unitary transformations for bosonic and fermionic systems

Tommaso Guaita, Lucas Hackl, Thomas Quella

Gaussian unitary transformations are generated by quadratic Hamiltonians, i.e., Hamiltonians containing quadratic terms in creations and annihilation operators, and are heavily used in many areas of quantum physics, ranging from quantum optics and condensed matter theory to quantum information and quantum field theory in curved spacetime. They are known to form a representation of the metaplectic and spin group for bosons and fermions, respectively. These groups are the double covers of the symplectic and special orthogonal group, respectively, and our goal is to analyze the behavior of the sign ambiguity that one needs to deal with when moving between these groups and their double cover. We relate this sign ambiguity to expectation values of the form $\langle 0|\exp{(-i\hat{H})}|0\rangle$ , where $|0\rangle$ is a Gaussian state and $\hat{H}$ an arbitrary quadratic Hamiltonian. We provide closed formulas for $\langle 0|\exp{(-i\hat{H})}|0\rangle$ and show how we can efficiently describe group multiplications in the double cover without the need of going to a faithful representation on an exponentially large or even infinite-dimensional space. Our construction relies on an explicit parametrization of these two groups (metaplectic, spin) in terms of symplectic and orthogonal group elements together with a twisted U(1) group.Gaussian unitary transformations are generated by quadratic Hamiltonians, i.e., Hamiltonians containing quadratic terms in creations and annihilation operators, and are heavily used in many areas of quantum physics, ranging from quantum optics and condensed matter theory to quantum information and quantum field theory in curved spacetime. They are known to form a representation of the metaplectic and spin group for bosons and fermions, respectively. These groups are the double covers of the symplectic and special orthogonal group, respectively, and our goal is to analyze the behavior of the sign ambiguity that one needs to deal with when moving between these groups and their double cover. We relate this sign ambiguity to expectation values of the form $\langle 0|\exp{(-i\hat{H})}|0\rangle$, where $|0\rangle$ is a Gaussian state and $\hat{H}$ an arbitrary quadratic Hamiltonian. We provide closed formulas for $\langle 0|\exp{(-i\hat{H})}|0\rangle$ and show how we can efficiently describe group multiplications in the double cover without the need of going to a faithful representation on an exponentially large or even infinite-dimensional space. Our construction relies on an explicit parametrization of these two groups (metaplectic, spin) in terms of symplectic and orthogonal group elements together with a twisted U(1) group.

arXiv:2409.11628Annals of PhysicsOct 17, 20263 citations

Group representation for quantum theory

正人林

Springer eBooksJan 1, 20176 citations

Properties of Quantum Statistical Expectation Values

D. A. Uhlenbrock

The semigroup of statistical operators, and the unitary group of time translation operators generated by the same Hamiltonian of a nonrelativistic fermion field, are naturally imbedded in a holomorphic half-plane semigroup. The statistical expectation values of products of time-dependent operators are then boundary values of holomorphic functions which allow a Bochner integral representation with a Cauchy kernel. The general time-temperature-dependent Green's functions permit a concise spectral representation. It is suggested that a thermodynamic perturbation theory should treat the Heisenberg and Bloch equations simultaneously in terms of a perturbation theory for holomorphic half-plane semigroups generated by semibounded self-adjoint Hamiltonians.The semigroup of statistical operators, and the unitary group of time translation operators generated by the same Hamiltonian of a nonrelativistic fermion field, are naturally imbedded in a holomorphic half-plane semigroup. The statistical expectation values of products of time-dependent operators are then boundary values of holomorphic functions which allow a Bochner integral representation with a Cauchy kernel. The general time-temperature-dependent Green's functions permit a concise spectral representation. It is suggested that a thermodynamic perturbation theory should treat the Heisenberg and Bloch equations simultaneously in terms of a perturbation theory for holomorphic half-plane semigroups generated by semibounded self-adjoint Hamiltonians.

Journal of Mathematical PhysicsMay 1, 19663 citations

Group Representation for Quantum Theory

Masahito Hayashi

Springer eBooksNov 18, 201635 citations

\langle 0|\exp{(-i\hat{H})}|0\rangle

, where

|0\rangle

is a Gaussian state and

\hat{H}

an arbitrary quadratic Hamiltonian. We provide closed formulas for

\langle 0|\exp{(-i\hat{H})}|0\rangle

and show how we can efficiently describe group multiplications in the double cover without the need of going to a faithful representation on an exponentially large or even infinite-dimensional space. Our construction relies on an explicit parametrization of these two groups (metaplectic, spin) in terms of symplectic and orthogonal group elements together with a twisted U(1) group.