A correspondence of classical to quantum physics studied by Schrö\-dinger and Ehrenfest applies without the necessity of technical conjecture that classical observables are associated with Hermitian Hilbert space operators. This correspondence provides appropriate nonrelativistic classical interpretations to realizations of relativistic quantum physics that are incompatible with the canonical formalism. Using this correspondence, Newtonian mechanics for a $1/r$ potential provides approximations for the dynamics of nonrelativistic classical particle states within unconstrained quantum field theory (UQFT). Using this correspondence, Newtonian mechanics for a 1 / r 1 𝑟 1/r potential provides approximations for the dynamics of nonrelativistic classical particle states within unconstrained quantum field theory (UQFT). Keywords: Relativistic quantum physics, foundation of quantum mechanics, generalized functions. 1 Introduction Early in the development of quantum mechanics, Schrödinger and Ehrenfest [ 1 , 2 , 3 ] studied classical limits as approximations for the trajectories exhibited by the expected values of locations and momenta. Significantly, this correspondence of features does not require that Hilbert space operators satisfy the technical conjecture of the canonical formalism [ 4 ]. The canonical formalism conjectures a further correspondence: that classical fields correspond to Hermitian Hilbert space field operators. The canonical formalism generalizes the Dirac-von Neumann formulation of nonrelativistic quantum mechanics. In the Dirac-von Neumann formulation, classical dynamic quantities correspond to Hermitian operators in a Hilbert space realization of quantum mechanics [ 5 ]. The correspondence of features in the classical limit of quantum mechanics studied by Schrödinger and Ehrenfest does not require that multiplication by real fields defines a self-adjoint Hilbert space operator, nor is the canonical formalism’s extrapolation of classical Hamiltonians to high energies and short distances imposed. These assertions for the technical properties of operations in Hilbert space realizations of relativistic quantum mechanics are problematic [ 12 , 13 , 14 ]. A classical limit may be more limited than the general correspondence of classical dynamic quantities with Hermitian Hilbert space operators. The canonical formalism remains an applicable procedure when the additional technical properties are satisfied, notably in nonrelativistic quantum mechanics and in free field theory. In this note, the methods of Schrödinger and Ehrenfest are applied to classical correspondences for relativistic quantum physics. Classical correspondences are established by approximation of the evolution of regions of the dominant support of states as classical trajectories. This correspondence applies in nonrelativistic, particle-like instances. For this note, particular selections for classical particle-like states and a particular UQFT are developed. The selected states are generated from Gaussian minimum packet product states, selected because they achieve the Heisenberg bound for simultaneous knowledge of location and momentum, and because time translations of minimum packet functions are in family for the nonrelativistic limit of the UQFT Hamiltonian. A UQFT with a single Lorentz scalar field is selected for convenience in the analysis. Results suggest that the temporal evolution of UQFT states are approximated in nonrelativistic classical particle instances by Newtonian mechanics with a − g / r 𝑔 𝑟 -g/r potential. At the level of approximation achieved here, Newtonian mechanics suffices for the classical correspondences. In Lagrangian QFT and nonrelativistic quantum mechanics, an interaction term in the Hamiltonian is specified and determines the dynamics, while in UQFT, the interaction results from the form of the Hilbert space scalar product and not the generator of time translation. Rather than the derivation of quantum dynamics from a classical interaction, UQFT dynamics are constrained only to achieve the characteristics of relativistic physics. A UQFT does not necessarily model a single classical force, nor any classical force. Associations with classical dynamics are a test of the physical relevance of UQFT. Newtonian mechanics is imposed as an interpretation of nonrelativistic classical particle approximations to the quantum dynamics. These classical associations apply in likelihood and only for a limited family of states that associate with nonrelativistic classical particles. Ehrenfest’s theorem is applied to associate an interaction Hamiltonian and nonrelativistic quantum mechanics to UQFT using the common associations with classical trajectories. Other than this association, UQFT lacks an interaction Hamiltonian. First, a digression to establish notation. Particle-like cases with initially large L 0 ​ ( 0 ) subscript 𝐿 0 0 L_{0}(0) transitioning to large L 0 ​ ( λ ) subscript 𝐿 0 𝜆 L_{0}(\lambda) result in negligible interaction. To satisfy the nonrelativistic classical particle bounds, the trajectories must remain many times L 0 subscript 𝐿 0 L_{0} apart and then the plane wave limit places the trajectories at too great a separation to interact significantly, and the quantum corrections of item 2 obscure the associations of states with classical trajectories at great ranges. Plane wave scattering The nonrelativistic limit of the corresponding differential cross section d ​ σ / d ​ Ω 𝑑 𝜎 𝑑 Ω d\sigma/d\Omega is not the Mott cross section for particles interacting with a 1 / r 1 𝑟 1/r potential despite the association of the nonrelativistic classical particle trajectories with 1 / r 1 𝑟 1/r potentials. There are scalar field UQFT with elastic cross sections that have nearly (regularized) 1 / r 1 𝑟 1/r equivalent plane wave scattering potentials in first Born approximation [ 11 ], but evaluation of the scalar products ( 14 ) in these cases is beyond the scope of this study.

Quantum Mechanics of Klein-Gordon Fields II: Relativistic Coherent States 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. Published as: AnnalsPhys.321:2210-2241,2006 DOI: 10.1016/j.aop.2006.02.008 arXiv categories: quant-ph gr-qc hep-th math-ph math.MP 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. 1 Introduction The study of the relationship between classical and quantum mechanics (QM) has been among the most important issues of modern theoretical physics. The organization of the article is as follows. In Section 2 we outline a general construction for coherent states of a charged relativistic particle. In Section 3, we focus our attention on the coherent states of a free relativistic particle and examine their physical properties and classical and nonrelativistic limits. In Section 4, we study the coherent states of a neutral scalar particle. In Section 5, we consider the consequences of coupling a complex scalar field to a constant homogeneous magnetic field. Finally, in Section 6 we present our concluding remarks. Throughout this paper we will occasionally refer to Ref. [ 12 ] as paper I and use the label (I-n) to denote Eq. (n) of paper I. We also derive the associated minimum uncertainty relations and study their time-evolution and their nonrelativistic limit. It is well-known that the physical quantities such as transition amplitudes and expectation values of observables are independent of the choice of representation of the quantum system [ 30 , 26 ] . 1 λ ​ 1 + 1 4 ​ λ 4 ​ τ 2 , Δ ​ x ​ ( τ ) ​ Δ ​ p ​ ( τ ) = 1 2 ​ 1 + 1 4 ​ λ 4 ​ τ 2 . 1 𝜆 1 1 4 superscript 𝜆 4 superscript 𝜏 2 Δ x 𝜏 Δ p 𝜏 1 2 1 1 4 superscript 𝜆 4 superscript 𝜏 2 \displaystyle\frac{1}{\lambda}\sqrt{1+\frac{1}{4}\lambda^{4}\tau^{2}}\,,~{}~{}~{}~{}~{}~{}\Delta{\rm x}(\tau)\Delta{\rm p}(\tau)=\frac{1}{2}\sqrt{1+\frac{1}{4}\lambda^{4}\tau^{2}}\,. (60) Now, we are in a position to plot these quantities and perform a relativistic-to-nonrelativistic and quantum-to-classical comparisons. Fig. For B = 0 𝐵 0 B=0 (or Λ = 0 Λ 0 \Lambda=0 ) Eq. ( 109 ) tends to the well-known result for the free particle (compare with ( 63 )). These nonrelativistic expectation values do not depend on the widths λ ⟂ subscript 𝜆 perpendicular-to \lambda_{\perp} and λ 3 subscript 𝜆 3 \lambda_{3} and coincide with the corresponding classical quantities. Using our numerical method, we have compared the relativistic and For all values of these parameters ⟨ x ˙ 3 ⟩ nr / x ˙ cl . 3 = 1 subscript delimited-⟨⟩ superscript ˙ x 3 nr subscript superscript ˙ 𝑥 3 cl 1 \langle\dot{\rm x}^{3}\rangle_{\rm nr}/\dot{x}^{3}_{\rm cl.}=1 . Hence the data confirms that our relativistic calculations have the correct nonrelativistic limit. 6 Conclusion In [ 12 ] we give a formulation of the quantum mechanics of first quantized scalar fields which is based on the construction of a genuine Hilbert space. This is determined by a one-parameter family of inner products ( ⋅ , ⋅ ) a subscript ⋅ ⋅ 𝑎 (\cdot,\cdot)_{a} where a ∈ ( − 1 , 1 ) 𝑎 1 1 a\in(-1,1) . Our strategy is to construct coherent states in the two-component Foldy representation and pull them back using the appropriate unitary transformation to obtain coherent KG fields. In contrast to the earlier approaches to this problem, ours is free from the problems associated with the charge-superselection rule. The general behavior of our coherent states are similar to that of a classical particle in both free and interacting cases. Moreover, in the nonrelativistic limit our results coincide with those of nonrelativistic quantum mechanics. References [1] E. Schrödinger, Naturwissenschaften 14 , 664 (1926); See also the interesting historical paper: F. Steiner, Physica B 151 , 323 (1988). Figure 4: Graphs of the energy expectation value ⟨ E ⟩ delimited-⟨⟩ 𝐸 \langle E\rangle (left) and its dispersion ( Δ ​ E ) Δ 𝐸 (\Delta E) (right) as functions of the momentum expectation value ⟨ p ⟩ delimited-⟨⟩ p \langle{\rm p}\rangle for the coherent states of a free particle with different λ 𝜆 \lambda : The graphs of the corresponding classical (nonquantum) and nonrelativistic (quantum) curves are also given. For small values of λ 𝜆 \lambda the coherent state displays completely classical behavior. 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 → ∞ → 𝑐 c\rightarrow\infty ), the curves do not depend on either of the widths λ ⟂ subscript 𝜆 perpendicular-to \lambda_{\perp} and λ 3 subscript 𝜆 3 \lambda_{3} or the magnetic field parameter Λ Λ \Lambda . They tend to the corresponding nonrelativistic curves obtained from Eqs. ( 107 ) and ( 108 ) which agree with the predictions of the classical theory. ◄ Feeling lucky? Conversion report Report an issue View original on arXiv ►