For time-independent Hamiltonians, the dynamics of quantum expectation values of observables in coherent states | $\bar A_T$ | can be easily represented as an integral formula involving forward and backward propagators | $K_\pm$ |⁠. In the semiclassical regime, an approximate formula | $\bar{A}_T^{\rm sc}$ | can be constructed via the replacement of | $K_\pm$ | by their semiclassical versions, followed by a consistent integration procedure. Alternatively, one can keep the original propagators and rewrite the integral formula for | $\bar A_T$ | as a truncated series expansion, thus introducing a new approximate formula | $\bar A_T^{\rm se}$ |⁠. Yet a third approximation | $\bar A_T^{\rm cl}$ | can be derived by use of a classical statistical approach based on the Liouville equation and Gaussian probability distributions. In the present paper, we develop these three approximate formulas for expectation values, apply them to simple systems, and evaluate their accuracy.For time-independent Hamiltonians, the dynamics of quantum expectation values of observables in coherent states |$\bar A_T$| can be easily represented as an integral formula involving forward and backward propagators |$K_\pm$|⁠. In the semiclassical regime, an approximate formula |$\bar{A}_T^{\rm sc}$| can be constructed via the replacement of |$K_\pm$| by their semiclassical versions, followed by a consistent integration procedure. Alternatively, one can keep the original propagators and rewrite the integral formula for |$\bar A_T$| as a truncated series expansion, thus introducing a new approximate formula |$\bar A_T^{\rm se}$|⁠. Yet a third approximation |$\bar A_T^{\rm cl}$| can be derived by use of a classical statistical approach based on the Liouville equation and Gaussian probability distributions. In the present paper, we develop these three approximate formulas for expectation values, apply them to simple systems, and evaluate their accuracy.

Progress of Theoretical and Experimental PhysicsAug 29, 20173 citations

Effective Equations for Expectation Values

Andrei Olegovich Barvinsky, G. A. Vilkovisky

Jan 1, 19880 citations

On the finite volume expectation values of local operators in the sine-Gordon model

Árpád Hegedűs

In this paper we present sets of linear integral equations which make possible to compute the finite volume expectation values of the trace of the stress energy tensor (Θ) and the U ( 1 ) current ( J μ ) in any eigenstate of the Hamiltonian of the sine-Gordon model. The solution of these equations in the large volume limit allows one to get exact analytical formulas for the expectation values in the Bethe-Yang limit. These analytical formulas are used to test an earlier conjecture for the Bethe-Yang limit of expectation values in non-diagonally scattering theories. The analytical tests have been carried out upto three particle states and gave agreement with the conjectured formula, provided the definition of polarized symmetric diagonal form-factors is modified appropriately. Nevertheless, we point out that our results provide only a partial confirmation of the conjecture, and further investigations are necessary to confirm all of its subtleties. The most important missing piece in the confirmation is the mathematical proof of the finiteness of the symmetric diagonal limit of form-factors in a non-diagonally scattering theory.In this paper we present sets of linear integral equations which make possible to compute the finite volume expectation values of the trace of the stress energy tensor (Θ) and the U ( 1 ) current ( J μ ) in any eigenstate of the Hamiltonian of the sine-Gordon model. The solution of these equations in the large volume limit allows one to get exact analytical formulas for the expectation values in the Bethe-Yang limit. These analytical formulas are used to test an earlier conjecture for the Bethe-Yang limit of expectation values in non-diagonally scattering theories. The analytical tests have been carried out upto three particle states and gave agreement with the conjectured formula, provided the definition of polarized symmetric diagonal form-factors is modified appropriately. Nevertheless, we point out that our results provide only a partial confirmation of the conjecture, and further investigations are necessary to confirm all of its subtleties. The most important missing piece in the confirmation is the mathematical proof of the finiteness of the symmetric diagonal limit of form-factors in a non-diagonally scattering theory.

arXiv:1901.01806Nuclear Physics BAug 30, 20195 citations

Discussion on Upper and Lower Bounds for Expectation Values

A. Weinstein

Reviews of Modern PhysicsJul 1, 19631 citation

OPERATORS AND EXPECTATION VALUES

WORLD SCIENTIFIC eBooksFeb 1, 20244 citations

For time-independent Hamiltonians, the dynamics of quantum expectation values of observables in coherent states |

\bar A_T

| can be easily represented as an integral formula involving forward and backward propagators |

K_\pm

|⁠. In the semiclassical regime, an approximate formula |

\bar{A}_T^{\rm sc}

| can be constructed via the replacement of |

K_\pm

| by their semiclassical versions, followed by a consistent integration procedure. Alternatively, one can keep the original propagators and rewrite the integral formula for |

\bar A_T

| as a truncated series expansion, thus introducing a new approximate formula |

\bar A_T^{\rm se}

|⁠. Yet a third approximation |

\bar A_T^{\rm cl}

| can be derived by use of a classical statistical approach based on the Liouville equation and Gaussian probability distributions. In the present paper, we develop these three approximate formulas for expectation values, apply them to simple systems, and evaluate their accuracy.