Exact canonical collective-coordinate theory of density-wave motion: the unresolved completion of Tomonaga's 1955 elementary theory of quantum-mechanical collective motion

Tomonaga's two-part 1955 paper 'Elementary Theory of Quantum-Mechanical Collective Motion of Particles' introduced the microscopic collective-coordinate method: collective coordinates defined as normalized density Fourier sums over particle positions, approximately conjugate momenta, and a separation of the many-body Hamiltonian into collective and internal parts valid to leading order in a long-wavelength, large-N expansion. Part II applied the method to longitudinal density oscillations with Coulomb or non-Coulomb two-body forces, recovering the Bohm-Pines plasma frequency and Bloch's one-dimensional sound-wave frequency, and gave an auxiliary-field representation equivalent to Bohm's field Hamiltonian. What the 1955 method leaves open is the following: uncontrolled 1/sqrt(N) errors in the canonical commutation relations and in the coordinate-independence of the separated coefficients; validity only for interactions that are not too short-ranged; unresolved linear coupling between collective and internal motion in three dimensions (stability conditions and mode broadening); unexplored nonlinear collective terms generated by many-body forces; and an undeveloped dynamics for the nonlocal auxiliary-field formulation. The seventy-year literature lineage comprises exact Bose-system treatments (Sunakawa 1962), hydrodynamic derivations (Jepsen and Ter Haar 1962), the nuclear self-consistent collective-coordinate program, and the exact-canonical-momenta program of Nishiyama and da Providencia (2013-2021), which has so far completed the construction only in restricted sectors. The surviving open problem is the exact canonical collective-coordinate theory of density-wave motion in two and three dimensions - exact conjugate momenta with mutually consistent subsidiary conditions, the decoupled collective subspace, the quantitative collective-internal coupling, the nonlinear collective Hamiltonian, and the nonlocal-field dynamics - for arbitrary interaction range and statistics, reducing to the 1955 leading-order results in the appropriate limits.

Background

A recurring task of quantum many-body theory is to separate collective motion - coherent oscillations involving many particles at once, such as plasma oscillations of an electron gas, zero sound in a degenerate Fermi gas (a gas obeying the Pauli principle, with all single-particle states filled up to the Fermi momentum), or surface vibrations of an atomic nucleus - from the internal (intrinsic) motion of the individual particles. Tomonaga's two-part paper 'Elementary Theory of Quantum-Mechanical Collective Motion of Particles' (Progress of Theoretical Physics 13, 467 and 482, 1955) gave a microscopic construction of such variables for a general system of N particles with two-body forces, extending his earlier one-dimensional treatment of sound waves in a many-fermion system (1950), which realized an idea due to Bloch (1933) that a Fermi gas can, in an approximate sense, be described by a quantized field of sound waves obeying Bose statistics; that earlier treatment already noted that the method fails for forces of too short a range.

In Part II the method is applied to longitudinal (density-wave) oscillations. For a plane-wave mode of wave number k, the collective coordinate is the normalized density Fourier sum $\xi^{+}=(1/N)\sum_{n}e^{ikx_{n}}$ (with conjugate $\xi^{-}=(\xi^{+})^{\dagger}$), and a conjugate momentum $\pi^{+}$ is built as a symmetrized sum of single-particle momentum operators. In the large-N limit with the center of mass at rest, these obey canonical commutation relations $[\pi^{\pm},\xi^{\pm}]=-i\hbar$ up to fluctuations of order $1/\sqrt{N}$; this approximation underlies the whole theory. Expanding the two-body interaction, whose Fourier kernel is denoted $f(K)$, in a long-wavelength (small k) and large-N expansion, the Hamiltonian separates into an internal part and a collective part $H_{c}$ whose potential term is $N^{2}f(k)\xi^{+}\xi^{-}$; the internal particles then interact through a screened potential $V_{0}$ from which the Fourier components at the collective wave numbers have been removed. The collective inertia is $1/2I=k^{2}/Nm$. For the Coulomb kernel $f(K)=4\pi e^{2}/K^{2}$ the theory reproduces the Bohm-Pines plasma frequency, and for a one-dimensional almost-degenerate non-interacting Fermi gas it reproduces the sound-wave frequency $\nu=N\hbar k/2m$ obtained earlier by Bloch; both comparisons are carried out in the source paper. Finally, embedding the particles in an extended canonical system with auxiliary field variables constrained by subsidiary conditions (operator constraints selecting the physical subspace) reproduces, for the Coulomb kernel, Bohm's Hamiltonian formulation of plasma oscillations in which the electric field is an independent canonical variable and the subsidiary condition is Gauss's law; for a general kernel the auxiliary-field energy is built from the nonlocal operator $f(\sqrt{-\Delta})$.

What these derivations leave open is the following: (i) the coefficients of the separated Hamiltonian, such as the screened potential $V_{0}$, are only approximately independent of the collective coordinates, and no error bounds for the omitted terms - or for their effect on the collective frequencies - were obtained; (ii) the mean-value replacements used in the expansion require the kernel $f(K)$ to be small at large K and the wave-number transfers to be small compared with inverse correlation lengths, so genuinely short-range interactions lie outside the established validity; (iii) in three spatial dimensions the kinetic energy contains terms linear in the collective coordinates that couple collective and internal motion, and neither the conditions on the interaction strength under which a sharp collective mode is stable nor the quantitative broadening or destruction of the mode at strong coupling were determined; (iv) many-body forces generate powers of the collective coordinates beyond quadratic, that is, nonlinear collective dynamics, which was not worked out; (v) the dynamics, quantization, and hydrodynamic limits of the auxiliary nonlocal-field formulation were left undeveloped.

Successor frameworks grew around these questions. The random phase approximation (RPA - the linearized time-dependent mean-field theory coupling density fluctuations to particle-hole excitations, the direct descendant of the Bohm-Pines plasma theory) became the standard tool for linear collective modes in electronic and nuclear systems. The generator-coordinate method represents collective motion by superposing mean-field wavefunctions parametrized by collective variables. In nuclear physics, the self-consistent collective-coordinate (SCC) method determines the collective coordinates dynamically instead of choosing them a priori, and underlies the modern microscopic derivation of the Bohr-Mottelson collective Hamiltonian (the standard phenomenological Hamiltonian for slow quadrupole surface vibrations and rotations of an atomic nucleus). A direct continuation line - Sunakawa's exact collective description of Bose systems (1962), in which the collective momenta are fixed as solutions of discrete integral equations instead of being assumed canonical up to $1/\sqrt{N}$ errors, and the exact-canonical-momenta program of Nishiyama and da Providencia (2013-2021) - replaces the approximate canonical variables by exact ones at the price of subsidiary conditions defining the collective subspace, and has been completed only in restricted sectors.

Problem Statement

Complete, on exact footing, Tomonaga's 1955 collective-coordinate description of longitudinal density oscillations, within the collective-coordinate framework that the source paper established. Concretely, construct the exact theory whose long-wavelength, large-N leading terms Tomonaga computed:

(a) collective momenta exactly canonically conjugate to the density-wave collective coordinates (exact $[\pi,\xi]=-i\hbar$, with mutually commuting momenta), with the collective-subspace subsidiary conditions - the constraints that define the collective subspace, to be distinguished from the auxiliary-field constraints that select the physical subspace of Background - made explicit and shown mutually consistent, for two- and three-dimensional systems, for Fermi as well as Bose statistics, and for two-body interactions whose Fourier kernel $f(K)$ is not assumed small at large K (genuinely short-range interactions), together with the extension to many-body forces; the exact theory must also determine the corrections to the 1955 leading-order results, and thereby the quantitative effect of the non-leading terms discarded in the 1955 separation of the Hamiltonian;

(b) the quantitative account of the coupling between collective and internal motion in three dimensions: the conditions on the interaction strength (the magnitude of $N^{2}f(k)$) under which a sharp collective oscillation is stable against the linear coupling terms, and the quantitative broadening or fragmentation of the collective mode when the coupling is strong;

(c) the nonlinear collective Hamiltonian - the higher powers of the collective coordinates generated by many-body forces and by large amplitudes - and its dynamics (the large-amplitude collective-motion problem);

(d) the auxiliary-field representation for general kernels: the dynamical equations, quantization, and hydrodynamic limits of the nonlocal field theory with energy built from $f(\sqrt{-\Delta})$, and its consistency with the auxiliary-field subsidiary conditions for non-Coulomb interactions.

The exact theory reduces, in the appropriate limits, to the 1955 leading-order results: the plasma frequency for the Coulomb kernel, Bloch's one-dimensional frequency $\nu=N\hbar k/2m$ for the non-interacting almost-degenerate Fermi gas, and the screened internal interaction with the collective Fourier components removed. Several constructions that bear on this theory are already established in the literature: the exact Bose-system treatment (Sunakawa), the quadrupole-type construction of exact conjugate momenta together with its two-dimensional collective-subspace analysis (Nishiyama and da Providencia 2014, 2015), the one-dimensional neutron-proton construction, and the three-dimensional isoscalar irrotational sector. The exact theory is to be worked out sector by sector over the remaining cases (statistics, dimensionality, interaction range, inclusion of many-body forces): for each sector, either the corresponding exact structure is constructed, or the exact construction is proved obstructed for a precisely stated class of collective variables and systems (as occurs in the isovector sector of the three-dimensional neutron-proton system, where the commutator algebra does not close), together with the delimitation of which sectors remain constructible. The remaining sectors are the combinations of the four axes - statistics, dimensionality, interaction range, inclusion of many-body forces - that those constructions do not cover.

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