arXiv:2501.03665

An Algebraic Proof of Thermal Wick's Theorem

Chen, Ruofan

Published Jan 7, 2025• 0 citations

Abstract

Every many-body state can be constructed from the vacuum state by iteratively applying the creation operators, therefore the thermal expectation can be transformed into a sum of vacuum expectation where the vacuum Wick's theorem holds. Based on this observation this article gives an algebraic proof of thermal Wick's theorem.

DOI: 10.48550/arxiv.2501.03665

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arXiv:2501.03665

An Algebraic Proof of Thermal Wick's Theorem

Chen, Ruofan

Published Jan 7, 2025• 0 citations

arXiv HTML PDF

Abstract

DOI: 10.48550/arxiv.2501.03665

Related papers

Wick's theorem

Bruno Juliá Díaz, Arnau Rios Huguet, Héctor Briongos Merino +2 more

In the context of many-body quantum mechanics, one often encounters the need to evaluate expectation values of operators or their matrix elements. For instance, given a Hamiltonian https://www.w3.org/1998/Math/MathML" display="inline"> H ^ , we typically want to evaluate its expectation value, which is the energy of the system, https://www.w3.org/1998/Math/MathML" display="inline"> E = ⟨ H ^ ⟩ . In second quantization, the matrix element of an operator can always be expressed as a vacuum expectation value. This generally yields an expression which is hard to evaluate. Fortunately, a powerful tool to simplify these calculations exists: Wick's theorem. This theorem provides a systematic way to break down expressions involving products of creation and annihilation operators into sums of simpler terms.In the context of many-body quantum mechanics, one often encounters the need to evaluate expectation values of operators or their matrix elements. For instance, given a Hamiltonian https://www.w3.org/1998/Math/MathML" display="inline"> H ^ , we typically want to evaluate its expectation value, which is the energy of the system, https://www.w3.org/1998/Math/MathML" display="inline"> E = ⟨ H ^ ⟩ . In second quantization, the matrix element of an operator can always be expressed as a vacuum expectation value. This generally yields an expression which is hard to evaluate. Fortunately, a powerful tool to simplify these calculations exists: Wick's theorem. This theorem provides a systematic way to break down expressions involving products of creation and annihilation operators into sums of simpler terms.

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A Wick Theorem for Spin-One Operators

H. J. Spencer

A generalization to all spin magnitudes of the drone-fermion representation (previously known for spin \textonehalf{}), is presented, and the resulting eigenstates investigated. Linear combinations of products of fermion operators acting on a vacuum state can always be found which correspond to spin states which are eigenvectors of ${\mathrm{S}}^{2}$ and ${S}^{z}$ . It is then shown that for the case of spin 1 (units of $\ensuremath{\hbar}$) it is possible to construct a Wick theorem which is analogous to the usual temperature form for standard fermion creation and destruction operators.A generalization to all spin magnitudes of the drone-fermion representation (previously known for spin \textonehalf{}), is presented, and the resulting eigenstates investigated. Linear combinations of products of fermion operators acting on a vacuum state can always be found which correspond to spin states which are eigenvectors of ${\mathrm{S}}^{2}$ and ${S}^{z}$. It is then shown that for the case of spin 1 (units of $\ensuremath{\hbar}$) it is possible to construct a Wick theorem which is analogous to the usual temperature form for standard fermion creation and destruction operators.

Physical ReviewMar 10, 196835 citations

Wick's theorem for the expectation value of an operator between many-particle states is generalized to the case of a homogeneous linear transformation operator

\^O

, which leaves invariant the

2N

-dimensional linear space composed of the annihilation and the creation operators for the N fermion or boson single-particle basis. Such a transformation generally belongs to the group

\mathrm{SO}(2N,C)

for the case of fermion, or the group

\mathrm{Sp}(2N,C)

for the case of boson. This theorem is applied to derive a reduction theorem for the matrix element of the type $〈\ensuremath{\alpha}|{\ensuremath{\alpha}}_{1}{\ensuremath{\alpha}}_{2}\ensuremath{\cdot}\ensuremath{\cdot}\ensuremath{\cdot}{\ensuremath{\alpha}}_{m}\^O{\ensuremath{\beta}}_{m+1}^{\ifmmode\dagger\else\textdagger\fi{}}{\ensuremath{\beta}}_{m+2}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{\cdot}\ensuremath{\cdot}\ensuremath{\cdot}{\ensuremath{\beta}}_{2n}^{\ifmmode\dagger\else\textdagger\fi{}}|\ensuremath{\beta}〉,$ whose many-quasiparticle states and vacua appearing in both sides are represented in different quasiparticle pictures as indicated by $\ensuremath{\alpha}$ and $\ensuremath{\beta}.$ The reduction formula is useful in calculating any nuclear transition matrix element for the quantum-number projection calculation associated with the constrained Hartree-Fock-Bogoliubov solutions, and also the generator coordinate method with quantum number projection.

{\mathrm{S}}^{2}

and

{S}^{z}

. It is then shown that for the case of spin 1 (units of $\ensuremath{\hbar}$) it is possible to construct a Wick theorem which is analogous to the usual temperature form for standard fermion creation and destruction operators.