For pt.I see ibid. vol.22, p.4877 (1989). Previously (I) the author considered a second-quantised Hamiltonian which can be used to model a large number of important strongly interacting many-body systems. He has shown that the dynamics of such systems can be described exactly using quantum fields. The equations of motion derived in this way are highly nonlinear partial differential equations (PDE). He now uses a standard field theoretical approach where initially the fields are treated as classical functions. He studies the solutions of these equations which remarkably, may be obtained making full use of very recent mathematical discoveries (the symmetry reduction method). He finds exact solutions of the equations of motion for the classical field for each of the four main cases separately found in the preceding paper. These cases may correspond to different physical situations and the last, the second-order case, corresponds to the most general physical situation. Subsequently he outlines possible quantisation procedures for these fields. He then briefly discusses the type of boundary conditions which may be applied to both the classical field and the quantum fluctuations which arise.
Numerous examples in condensed matter physics possess an effective Hamiltonian structure involving two-body anharmonic interactions between elementary excitations. The effective second-quantised Hamiltonian used to model these systems is essentially of the same form except for the particulars of the interaction parameters and the statistics of the particles which are interacting. The author investigates such systems in the limit of strongly anharmonic interactions which may possibly lead to symmetry breaking. He introduces quantum field operators and calculates their equations of motion. The interaction coefficients are expanded in a Taylor series about the critical mode to second order to illustrate the method, but he gives the expansion to infinite order for completeness. It is demonstrated that both in first and second order the equations of motion are reduced to a system of highly nonlinear coupled partial differential equations (PDE) for the field operator. He shows that the form of the second-order equation is exact due to renormalisation theory, in the case where the Hamiltonian coefficients represent operators in real space which are isotropic, and incorporates symmetries and topologies which would appear in an expansion to infinite order. This special situation arises in a very wide range of physical examples. These equations can be solved exactly using a particular ansatz in conjunction with recent mathematical developments in the field of non-linear analysis. In particular, the method of symmetry reduction for PDE provides a complete set of symmetry variables for the type of equations used in the study.
Several years ago Nelson succeeded in deriving the nonrelativistic Schrödinger equation within a stochastic model which included Newton’s second law as the fundamental dynamical rule. Unfortunately, the relativistic extension of Nelson’s work is not so straightforward as might at first be supposed. This paper examines the difficulties inherent in such a relativization and proposes supplemental axioms which resolve those difficulties. A stochastic derivation of the Klein–Gordon equation is then presented.
We study the dynamics of multiparticle Carroll-Schrödinger (CS) quantum systems in
1+1 dimensions, where
x acts as the evolution variable and
t as the configuration coordinate. We derive the
N-body theory on equal-
x slices as the Carrollian limit of a relativistic multi-time Klein-Gordon model, introducing temporal interactions via minimal coupling to the temporal energy operators. An
x-dependent gauge transformation maps this to an equivalent description with explicit many-body potentials, illustrated by a temporal coupled-oscillator model that exhibits synchronization. Adopting a complementary spatial viewpoint with a static potential
Utot(x), we show that the evolution is driven by the collective force
∑j∂xjUtot; for any translation-invariant interaction (such as a regularized Coulomb potential), these internal forces cancel, rendering the collective dynamics free and highlighting Carrollian ultralocality. We also construct a coordinate duality mapping separable Schrödinger Hamiltonians to CS generators via Schwarzian derivatives. Exchange symmetry is formulated in the time domain, yielding temporal bunching for bosons and antibunching for fermions via the second-order coherence function
g(2)(t,t′). In second quantization, the contact limit yields a temporal derivative cubic--quintic nonlinear Schrödinger equation with a theoretically fixed nonlinearity coefficient
β=−3/16. Finally, by coupling canonical pairs to external scalar and gauge fields, we establish an isomorphism with one-dimensional current-density functional theory, outlining a Carrollian Hohenberg-Kohn mapping and Kohn-Sham scheme.
• Discovery of the Schrodinger equation • Derivation (uniqueness) of the Schrodinger equation • High order equation of the Schrodinger type • Nonlocal symmetry of the Schrodinger equation [2, 7] • High-order evolution equations. Dependence of mass on velocity in the nonlocal Galilei-invariant theory [6, 19] • Galilei relativity principle and nonlinear Schrodinger-type equations [2, 5, 10–17] • Nonlocal symmetry of the linear Schrodinger–de Broglie–Klein–Gordon–Fock– Kudar–de Donder–Van Dunger [3, 18, 8] • Nonlocal Galilei symmetry of a relativistic equation [8, 9] • Nonlocal Galilei symmetry of the Dirac equation [7] • Galilei symmetry of a relativistic equation.