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Quantum Mechanics of Klein-Gordon Fields II: Relativistic Coherent States

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A. Mostafazadeh & F. Zamani

Quantum Physics · Submitted 20 Feb 2006 · 42 references

Pith's reading: The authors build relativistic coherent states for Klein-Gordon fields by pulling ordinary coherent states through the Foldy representation, and analyze their properties for free, neutral, and magnetically coupled cases.

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arXiv IDquant-ph/0602161 (version 1)
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CitationA. Mostafazadeh & F. Zamani. "Quantum Mechanics of Klein-Gordon Fields II: Relativistic Coherent States." arXiv:quant-ph/0602161 (2006). https://pith.science/paper/R6V3AODG
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Claim paper
The paper

The paper claims

The authors build relativistic coherent states for Klein-Gordon fields by pulling ordinary coherent states through the Foldy representation, and analyze their properties for free, neutral, and magnetically coupled cases.

The reading

Coherent states are special quantum states that keep their shape and follow classical paths; they are the closest quantum analogue to classical particles. For relativistic particles described by the Klein-Gordon equation, building such states has been complicated because the usual position operators can mix positive and negative energy (charge) sectors. This paper uses a formulation from the authors' earlier work where the solution space is a genuine Hilbert space with an inner product depending on a parameter a, and maps it to the two-component Foldy representation.

In the Foldy representation, position and momentum are simple, so the authors define an annihilation operator and its coherent states. Pulling these back to the Klein-Gordon Hilbert space gives coherent fields with definite charge. For a free particle, these states have the standard minimum uncertainty product, spread in time, and reproduce classical relativistic motion when the width parameter is small. For neutral fields, they combine positive and negative charge states appropriately. For a charged particle in a constant magnetic field, they compute expectation values of position, momentum, energy, radius, and angular momentum using a basis of Landau-type eigenfunctions.

The results are mostly consistent with classical helix motion, with smaller wave packets and weaker magnetic fields giving closer classical correspondence. The authors report that the initial probability density of their coherent field does not depend on the magnetic field strength, and use this to propose their states as the appropriate nonrelativistic coherent states for a charged particle in a magnetic field.

Extended reading notes

Core claim

The paper claims that the states defined by A_a psi = zeta psi and C psi = epsilon psi (Eq. (10)), equivalently |zeta_epsilon, epsilon> = |zeta_epsilon> tensor e_epsilon in the Foldy representation (Eq. (13)), provide a physically viable set of relativistic coherent states for first-quantized Klein-Gordon fields, free from charge-superselection problems, with correct classical and nonrelativistic limits. The central assertion appears in the abstract: 'we offer an explicit construction of coherent states for both charged and neutral free Klein-Gordon fields as well as for charged fields interacting with a constant magnetic field.'

Load-bearing premise

The construction rests on the assumption, taken from paper I and [26], that the operator X_a = U_a^{-1}(x tensor sigma0)U_a (Eq. (6)) is the physical position operator for a first-quantized Klein-Gordon particle. This position operator is charge-superselection-respecting, but its physical validity is not independently established in this paper; if the physical position operator is not the one used to build the annihilation operator A_a (Eq. (7)), then the states defined as eigenstates of A_a are not the physically relevant relativistic coherent states. This premise enters in Section 2, Eqs. (5)-(7), and underlies the entire construction.

Original abstract

We use the formulation of the quantum mechanics of first quantized Klein-Gordon fields given in the first of this series of papers to study relativistic coherent states. In particular, we offer an explicit construction of coherent states for both charged and neutral (real) free Klein-Gordon fields as well as for charged fields interacting with a constant magnetic field. Our construction is free from the problems associated with charge-superselection rule that complicated the previous studies. We compute various physical quantities associated with our coherent states and present a detailed investigation of their classical (nonquantum) and nonrelativistic limits.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,607 input tokens.

What the abstract itself states, as probabilities: claims first 0.08 · states limitation 0.10 · human participants 0.01 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.85 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We use the formulation of the quantum mechanics of first quantized Klein-Gordon fields given in the first of this series of papers to study relativistic coherent states. In particular, we offer an explicit construction of coherent states for both charged and neutral (real) free Klein-Gordon fields as well as for charged fields interacting with a constant magnetic field. Our construction is free from the problems associated with charge-superselection rule that complicated the previous studies. We compute various physical quantities associated with our coherent states and present a detailed investigation of their classical (nonquantum) and nonrelativistic limits.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We use the formulation of the quantum mechanics of first quantized Klein-Gordon fields given in the first of this series of papers to study relativistic coherent states. contribution 0.50 0.88 0.06 asserted 0.98
s1 In particular, we offer an explicit construction of coherent states for both charged and neutral (real) free Klein-Gordon fields as well as for charged fields interacting with a constant magnetic field. contribution 0.98 0.95 0.25 asserted 1.00
s2 Our construction is free from the problems associated with charge-superselection rule that complicated the previous studies. contribution 0.55 0.85 0.31 asserted 1.00
s3 We compute various physical quantities associated with our coherent states and present a detailed investigation of their classical (nonquantum) and nonrelativistic limits. contribution 0.88 0.96 0.16 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.87 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read
  • arXiv listing abstract v1
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. Graphs of the velocity expectation value
Figure 2. Plots of the dispersion in position ∆x as a function
Figure 3. Plots of (∆x)(∆p) in terms of time for various
Figure 4. Graphs of the energy expectation value hEi (left) and its dispersion (∆E) (right) as functions of the momentum expectation value hpi for the coherent states of a free particle with different λ: The graphs of the corresponding classical (nonquantum) and nonrelativistic (quantum) curves are also given. For small values of λ the coherent state displays completely classical behavior
Figure 5. Plots of |fcoh(ǫ; x 0 0 , x)| 2 and |ψ (ǫ,ζ~ǫ) a (x 0 0 , x)| 2 for various λ and momentum expectation values hpi: |fcoh(ǫ; x 0 0 , x)| 2 represents the nonrelativistic probability density. For small values of the momentum expectation value |ψ (ǫ,~ζǫ) a (x 0 0 , x)| 2 tends to |fcoh (ǫ; x 0 0 , x)| 2 . Here we have set κ(1 + ǫa) = 1. 25
Figure 6. Graphs showing the time-evolution of the probabil
Figure 7. Graphs showing the time-evolution of the probabil
Figure 8. Typical trajectories traced by the expectation va
Figure 9. Plots of hx 1 (τ )i (solid curve) and hx 2 (τ )i+ 1 (dashed curve) for various widths λ⊥ and λ3 with Λ = 0.01, and hΠi = 2.0. hx 1 (τ )i and hx 2 (τ )i are scaled with the classical radius Rcl. and oscillate, with a decreasing amplitude, respectively like damped sine and cosine functions of the scaled time τ /τcl. . τcl. is classical period of precession. λ⊥ ≃ √ Λ means |λ⊥ − √ Λ| ≤ 10−4 . For λ⊥ → √ Λ and small value of λ3 the curves approach to the corresponding classical curves. The effect of λ3 dominates that of λ⊥. Also changing the initial transverse momentum does not affect…
Figure 10. Plots of hx 1 (τ )i (solid curve) and hx 2 (τ )i + 1 (dashed curve) for two values of the magnetic field parameter Λ = 0.1, 0.001 with the same initial momentum hΠi = 3,hp3i = 2.4, and widths λ3 = 10−3 , λ⊥ ≃ √ Λ. For λ⊥ → √ Λ and small value of λ3 the curves traced by the expectation value of the position operator approach the classical curves. 28
Figure 11. Plots of hx 1 (τ )i (solid curve) and hx 2 (τ )i + 1 (dashed curve) for Λ = 0.01, hΠi = 1 and 4, λ3 = λ⊥ = 0.25 and hp3i = 0.8hΠi. hx 1 (τ )i and hx 2 (τ )i are scaled with the classical radius and time is scaled with the classical period of precession. The higher the initial momentum becomes the closer the curves get to the corresponding classical curves
Figure 12. Plots of hx 1 (τ )i (solid curve) and hx 2 (τ )i + 1 (dashed curve) for Λ = 0.1 and 10−4 , hΠi = 2,hp3i = 1.6, and λ3 = λ⊥ = 0.25. hx 1 (τ )i and hx 2 (τ )i are scaled with the classical radius and time is scaled with the classical period of precession. As the magnetic field decreases the curves approach the corresponding classical curves. 29
Figure 13. (a) and (b) are plots of hp1(τ )i − 1 (solid curve), hp2(τ )i (dashed curve), and (c) and (d) are plots of the transverse kinetic momentums hΠ1(τ )i (solid curve) and hΠ2(τ )i (dashed curve), for Λ = 0.1 and 10−4 , hΠi = 2,hp3i = 1.6, and λ3 = λ⊥ = 0.25. The expectation value of the momentum operators are scaled with the corresponding classical transverse kinetic momentum Π⊥ = hp1i and the time is scaled with the classical period of precession. The behavior of hp1(τ )i − 1 and hΠ1(τ )i (resp. hp2(τ )i, and hΠ2(τ )i) reminisce the damped cosine (resp. sine) function. According to…
Figure 14. Plots of (∆x3 )(∆p3) as a function of time for different initial momentum expectation values, magnetic field parameter Λ, and the widths λ3 and λ⊥. (∆x3 (τ ))(∆p3(τ )) is an increasing function of the magnetic field. The presence of the magnetic field enhances the spreading of the wave packet. Also faster moving wave packets have a slower spreading rate. 31
Figure 15. Plots of |ψ (ζ~) a (x 0 0 ; x i )| 2 as a function of x i for Λ = 0.01, where |ψ (ζ~) a (x 0 0 ; x i )| 2 means |ψ ( ~ζ) a (x 0 0 ; ~x)| 2 with x j = 0 if j 6= i. We has set κ(1 + ǫa) = 1. Since we consider the same widths along all three dimensions, (graphs of) |ψ ( ~ζ) a (x 0 0 ; x i )| 2 are identical for all i = 1, 2, 3. |fcoh(x 0 0 ; x i )| 2 , which is shown by the solid line, gives the nonrelativistic probability density. For smaller values of the momentum expectation value, i.e., in the nonrelativistic limit, |ψ ( ~ζ) a (x 0 0 ; x i )| 2 tends to |fcoh (x 0 0 ; x i )| 2
Figure 16. Plots of |ψ (ζ~) a (x 0 0 ; x i )| 2 (dashed curve) with hΠi = 1 and λ⊥ = λ3 = 0.25, for three values of the magnetic field parameter: Λ = 0.1 (a), Λ = 10−3 (b) and Λ = 10−6 (c). The solid curve shows the nonrelativistic counterpart |fcoh (x 0 0 ; x i )| 2 . |ψ (ζ~) a (x 0 0 ; x i )| 2 does not depend on the magnetic field parameter. 32
Figure 17. (a) – (c) are plots of hx 1 (τ )i (solid curve) and hx 2 (τ )i + 1 (dashed curve); (d) is the plots of hp1(τ )i−1 (solid curve) and hp2(τ )i (dashed curve), for hΠi = 10−3 , different values of the magnetic field parameter Λ, and widths λ3, λ⊥. The expectation value of position and momentum operators are scaled respectively with the classical radius and classical transverse kinetic momentum. Time is scaled with the classical period of precession. In the nonrelativistic limit (c → ∞), the curves do not depend on either of the widths λ⊥ and λ3 or the magnetic field parameter Λ. They t

Figures reproduced from arXiv:quant-ph/0602161 by the authors.

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Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

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