Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Mathematical Physics

The classical limit of ultralocal scalar fields

Chengjun Zhu, John R. Klauder

Published Jul 1, 1994• 3 citations
Publisher

Abstract

The classical limit of the quantum theory of ultralocal scalar fields is approached with the help of coherent state techniques. In the limit ℏ→0, it is demonstrated that the coherent state expectation value of the Heisenberg field operators coincide with the solutions of the corresponding classical canonical equations of motion.
DOI
10.1063/1.530475

Related papers

Quasiclassical approximation for ultralocal scalar fields

G Francisco

The author shows how to obtain the quasiclassical evolution of a class of field theories called ultralocal fields. Coherent states that follow the 'classical' orbit as defined by Klauder's weak correspondence principle and restricted action principles are explicitly shown to approximate to the quantum evolution as h(cross) to O.

Journal of Physics A Mathematical and General•Mar 11, 1986•2 citations

Quantum-classical correspondence via coherent state in integrable field theory

Jun Sato, Tsukasa Yumibayashi

We consider the problem of quantum-classical correspondence in integrable field theories. We propose a method to construct a field theoretical coherent state, in which the expectation value of the quantum field operator exactly coincides with the classical soliton. We also discuss the time evolution of this quantum state and the instability due to the nonlinearity.

arXiv:1811.03186•Nov 6, 2018•1 citation

Classical Limit of Yukawa theory from quantum state perspective

Qi Chen, Kaixun Tu, Qing Wang

We derive the quantum states corresponding to classical scalar fields in the representation expanded by the eigenstates of quantum field operators. This allows us to directly observe the spatial entanglement structure of quantum states and explore the differences and relationships between quantum superposition and classical superposition. We find that if two classical fields are identical in a certain spatial region, then their corresponding quantum states have the same reduced density matrix in that region. This indicates that knowing the classical field in a local region is sufficient to derive the reduced density matrix for that region. According to the correspondence between classical quantities and quantum states, we derive the equation of motion of the classical theory from the evolution of quantum states in Yukawa theory. This leads to the relativistic classical Yukawa theory, and we further obtain the relativistic corrections to the Yukawa potential.

arXiv:2111.00211•Oct 30, 2021•0 citations

Quantum and classical correspondence of Dirac equations

Zhi Zhang

According to Heisenberg correspondence principle,quantum matrix element of a Hermitian operator reduces to the coefficient of Fourier expansion of the corresponding classical quantity in the classical limit.Applying HCP to Dirac equations in relativistic realm,for the relativistic free particle or the charged particle moving in a constant magnetic field,it is shown that quantum operators in Dirac theory reduces to the classical quantities in the classical limit and the classical limit of quantum expectation value is the time average of the classical quantity.

Journal of Atomic and Molecular Physics•Jan 1, 2008•0 citations

Interacting classical and quantum ensembles

Michael J. W. Hall, Marcel Reginatto

A consistent description of interactions between classical and quantum systems is relevant to quantum measurement theory and to calculations in quantum chemistry and quantum gravity. A solution is offered here to this long-standing problem, based on a universally applicable formalism for ensembles on configuration space. This approach overcomes difficulties arising in previous attempts and in particular allows for a back reaction on the classical ensemble, conservation of probability and energy, and the correct classical equations of motion in the limit of no interaction. Applications include automatic decoherence for quantum ensembles interacting with classical measurement apparatuses, a generalization of coherent states to hybrid harmonic oscillators, and an equation for describing the interaction of quantum matter fields with classical gravity, which implies that the radius of a Robertson-Walker universe containing a quantum massive scalar field can be sharply defined only for particular ``quantized'' values.

arXiv:quant-ph/0509134•Physical Review A•Dec 9, 2005•106 citations

Coherent states

Christopher C. Gerry, P. L. Knight

At the end of the preceding chapter, we showed that the photon number states | n 〉 have a uniform phase distribution over the range 0 to 2π. Essentially, then, there is no well-defined phase for these states and, as we have already shown, the expectation value of the field operator for a number state vanishes. It is frequently suggested (see, for example, Sakurai) that the classical limit of the quantized field is the limit in which the number of photons becomes very large such that the number operator becomes a continuous variable. However, this cannot be the whole story since the mean field 〈 n | Ê x | n 〉 = 0 no matter how large the value of n . We know that at a fixed point in space a classical field oscillates sinusoidally in time. Clearly this does not happen for the expectation value of the field operator for a number state. In this chapter we present a set of states, the coherent states, which do give rise to a sensible classical limit; and, in fact, these states are the “most classical” quantum states of a harmonic oscillator, as we shall see. Eigenstates of the annihilation operator and minimum uncertainty states In order to have a non-zero expectation value of the electric field operator or, equivalently, of the annihilation and creation operators, we are required to have a superposition of number states differing only by ±1.

Cambridge University Press eBooks•Oct 28, 2004•1 citation
We derive the quantum states corresponding to classical scalar fields in the representation expanded by the eigenstates of quantum field operators. This allows us to directly observe the spatial entanglement structure of quantum states and explore the differences and relationships between quantum superposition and classical superposition. We find that if two classical fields are identical in a certain spatial region, then their corresponding quantum states have the same reduced density matrix in that region. This indicates that knowing the classical field in a local region is sufficient to derive the reduced density matrix for that region. According to the correspondence between classical quantities and quantum states, we derive the equation of motion of the classical theory from the evolution of quantum states in Yukawa theory. This leads to the relativistic classical Yukawa theory, and we further obtain the relativistic corrections to the Yukawa potential.
According to Heisenberg correspondence principle,quantum matrix element of a Hermitian operator reduces to the coefficient of Fourier expansion of the corresponding classical quantity in the classical limit.Applying HCP to Dirac equations in relativistic realm,for the relativistic free particle or the charged particle moving in a constant magnetic field,it is shown that quantum operators in Dirac theory reduces to the classical quantities in the classical limit and the classical limit of quantum expectation value is the time average of the classical quantity.
A consistent description of interactions between classical and quantum systems is relevant to quantum measurement theory and to calculations in quantum chemistry and quantum gravity. A solution is offered here to this long-standing problem, based on a universally applicable formalism for ensembles on configuration space. This approach overcomes difficulties arising in previous attempts and in particular allows for a back reaction on the classical ensemble, conservation of probability and energy, and the correct classical equations of motion in the limit of no interaction. Applications include automatic decoherence for quantum ensembles interacting with classical measurement apparatuses, a generalization of coherent states to hybrid harmonic oscillators, and an equation for describing the interaction of quantum matter fields with classical gravity, which implies that the radius of a Robertson-Walker universe containing a quantum massive scalar field can be sharply defined only for particular ``quantized'' values.
At the end of the preceding chapter, we showed that the photon number states | n 〉 have a uniform phase distribution over the range 0 to 2π. Essentially, then, there is no well-defined phase for these states and, as we have already shown, the expectation value of the field operator for a number state vanishes. It is frequently suggested (see, for example, Sakurai) that the classical limit of the quantized field is the limit in which the number of photons becomes very large such that the number operator becomes a continuous variable. However, this cannot be the whole story since the mean field 〈 n | Ê x | n 〉 = 0 no matter how large the value of n . We know that at a fixed point in space a classical field oscillates sinusoidally in time. Clearly this does not happen for the expectation value of the field operator for a number state. In this chapter we present a set of states, the coherent states, which do give rise to a sensible classical limit; and, in fact, these states are the “most classical” quantum states of a harmonic oscillator, as we shall see. Eigenstates of the annihilation operator and minimum uncertainty states In order to have a non-zero expectation value of the electric field operator or, equivalently, of the annihilation and creation operators, we are required to have a superposition of number states differing only by ±1.