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Journal of Mathematical Physics

Method for Construction of Unitary Operators in Quantum Field Theory

Paul H. Moravek, David W. Joseph

Published Nov 1, 1963• 6 citations
Publisher

Abstract

A method for the construction of explicit representations of unitary Hillbert-space operators in the particle-number representation is presented and illustrated by application to several examples.
DOI
10.1063/1.1703913

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The unitary matrix that brings a Hermitian H into block-diagonal form can be uniquely determined under very simple and transparent conditions. Int his work the block-diagonalization problem is investigated in the framework of the second quantization formalism. Starting with an operator Ḩ which in any n-particle Fock space has a well-defined matrix representation an attemt was made to answer the question whether the transformation matrices T which can be separately given in the various n-particle spaces can be considered matrix representations of the same operator. T̂. Interestingly, the very important result was reached that the block-diagonalization operator T̂ exists and is unique. As a particular example, attention was concentrated on the case of an operator Ĥ given by a one-particle operator. In this case the block-diagonalization operator can be constructed and given in explicit from. This approach is applied to the theory of Green’s functions where the block-diagonalization of the Hamiltonian has interesting consequences that are illustrated in some details.

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ON PHASE SPACE REPRESENTATIONS IN QUANTUM FIELD THEORY

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Submitted to•Jan 1, 1982•0 citations
The unitary matrix that brings a Hermitian H into block-diagonal form can be uniquely determined under very simple and transparent conditions. Int his work the block-diagonalization problem is investigated in the framework of the second quantization formalism. Starting with an operator Ḩ which in any n-particle Fock space has a well-defined matrix representation an attemt was made to answer the question whether the transformation matrices T which can be separately given in the various n-particle spaces can be considered matrix representations of the same operator. T̂. Interestingly, the very important result was reached that the block-diagonalization operator T̂ exists and is unique. As a particular example, attention was concentrated on the case of an operator Ĥ given by a one-particle operator. In this case the block-diagonalization operator can be constructed and given in explicit from. This approach is applied to the theory of Green’s functions where the block-diagonalization of the Hamiltonian has interesting consequences that are illustrated in some details.
Non-locality and quantum appear together entitling scientific projects and papers quite usually.These are two hardly detachable concepts and modern quantum theories are built up taking into account this aspect explicitly.However, that characteristic turns challenging from a fundamental point of view and can be unhelpful for practical purposes.Here, we construct quantum field theory with deliberately local aspects for a 1+1 dimensional Dirac field in flat spacetime generalizing a recently developed formalism for scalar fields.That purpose is achieved exploiting the non-uniqueness of quantization process.This local construction leads to a natural notion of local quanta and provides a local algebra of operators.As an application, it is shown that the standard vacuum state is a boiling soup of local particles and this fact is connected with a still Gedankenexperiment consisting in slamming down and removing a mirror in an empty cavity. I. MOTIVATION AND BACKGROUNDNon-locality is a feature as intuitively challenging as genuine in the quantum world.The so-successful ordinary representation of quantum field theory (QFT) is based on the information carried by global operators (creation-annihilation operators) related to field excitations with well defined momentum, and therefore, completely delocalized in space.This picture accounts for an innumerable amount of phenomena with an astonishing accuracy, as particle physics or solid state physics shows.However, this conventional formalism can hardly deal with tasks related to local properties of quantum field states.Simple questions as "where is that particle?"or "how are these particles distributed in space?" are unsuccessfully answered in this background.What is more, it is difficult to reconcile the common understanding of a particle as a minute entity in space with the particle notion arising from the ordinary Fock construction, which rather describes excitations spreading in whole space.Furthermore, no localized states can be constructed in such QFT where occupation number representation of a quantum field can be understood as a "momentum" representation in the sense that the number of particles informs about the momentum carried by the field.The problem at this point is that it does not exist a reciprocal "position" representation, we have no local (in space) creation-annihilation operators.This result can be understood as a consequence of Hegerfeldt's theorem.