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Journal of Physics A Mathematical and General

Einstein-Podolsky-Rosen pair states and the charge-amplitude representation for complex scalar fields

Hong-Yi Fan, Zeng‐Bing Chen

Published Mar 6, 2000• 1 citation
Publisher

Abstract

For complex scalar fields ( x ) and ( x ), we introduce the entangled states which are their common eigenvectors. The states possessing complete and orthonormal properties are a field-theoretical generalization of the Einstein-Podolsky-Rosen pair states in quantum mechanics. Based on the new representation, we obtain the complete and orthonormal states { q }, r , the common eigenvectors of the charge operator and ( x ) ( x ) , with which the charge lowering and raising operators can be naturally defined. The applications of our formulation are briefly discussed.

DOI
10.1088/0305-4470/33/10/313

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For complex scalar fields ( x ) and ( x ), we introduce the entangled states which are their common eigenvectors. The states possessing complete and orthonormal properties are a field-theoretical generalization of the Einstein-Podolsky-Rosen pair states in quantum mechanics. Based on the new representation, we obtain the complete and orthonormal states { q }, r , the common eigenvectors of the charge operator and ( x ) ( x ) , with which the charge lowering and raising operators can be naturally defined. The applications of our formulation are briefly discussed.
We demonstrate the Einstein-Podolsky-Rosen (EPR) correlations in the Gaussian trial wavefunctional widely used in the Schrödinger representation of the self-interacting complex scalar fields (CSF). This demonstration is based on the explicit eigenstates ||ξ⟩, which possess the precise EPR entanglement and constitute a complete and orthonormal representation, of CSF ϕ and ϕ † in the Fock space. To show the entanglement of the Gaussian trial wavefunctional, we evaluate in the ⟨ξ||-representation its `entangled' Wigner functional resembling the Wigner function of the two-mode squeezed vacuum state, whose quantum entanglement is well known.
We give the explicit form of the common eigenvectors of the relative position Q1{\mathit{Q}}_{1}Q1​-Q2{\mathit{Q}}_{2}Q2​ and the total momentum P1{\mathit{P}}_{1}P1​+P2{\mathit{P}}_{2}P2​, of two particles which were considered by Einstein, Podolsky, and Rosen [Phys. Rev. 47, 777 (1935)] in their argument that the quantum-mechanical state vector is not complete. Orthonormality and completeness of such eigenvectors, as well as their use in constructing the unitary operator for simultaneously squeezing Q1{\mathit{Q}}_{1}Q1​-Q2{\mathit{Q}}_{2}Q2​ and P1{\mathit{P}}_{1}P1​+P2{\mathit{P}}_{2}P2​, are derived by using the technique of integration within an ordered product of operators.
Our primary purpose of this work is to explicitly construct the general multipartite Einstein–Podolsky–Rosen (EPR) entangled state in multi-mode Fock space for a system with different masses of particles, which makes up a new quantum mechanical representation owing to completeness relation and orthogonal property. Its entanglement can be seen more clearly by analyzing its standard Schmidt decomposition. In addition, some applications of the multipartite entanglement are proposed including deriving the generalized Wigner operator and squeezing operator.