Although spatial locality is explicit in the Heisenberg picture of quantum dynamics, spatial locality is not explicit in the Schrödinger picture equations of motion. The gauge picture is a modification of Schrödinger’s picture such that locality is explicit in the equations of motion. In order to achieve this explicit locality, the gauge picture utilizes (1) a distinct wavefunction associated with each patch of space, and (2) time-dependent unitary connections to relate the Hilbert spaces associated with nearby patches. In this work, we show that by adding an additional spatially-local term to the gauge picture equations of motion, we can effectively interpolate between the gauge and Schrödinger pictures, such that when this additional term has a large coefficient, all of the gauge picture wavefunctions approach the Schrödginer picture wavefunction (and the connections approach the identity).
Although spatial locality is explicit in the Heisenberg picture of quantum dynamics, spatial locality is not explicit in the Schrödinger picture equations of motion. The gauge picture is a modification of Schrödinger’s picture such that locality is explicit in the equations of motion. In order to achieve this explicit locality, the gauge picture utilizes (1) a distinct wavefunction associated with each patch of space, and (2) time-dependent unitary connections to relate the Hilbert spaces associated with nearby patches. In this work, we show that by adding an additional spatially-local term to the gauge picture equations of motion, we can effectively interpolate between the gauge and Schrödinger pictures, such that when this additional term has a large coefficient, all of the gauge picture wavefunctions approach the Schrödginer picture wavefunction (and the connections approach the identity).
Although local Hamiltonians exhibit local time dynamics, this locality is not explicit in the Schrödinger picture in the sense that the wavefunction amplitudes do not obey a local equation of motion. We show that geometric locality can be achieved explicitly in the equations of motion by "gauging" the global unitary invariance of quantum mechanics into a local gauge invariance. That is, expectation values ⟨ψ|A|ψ⟩ are invariant under a global unitary transformation acting on the wavefunction |ψ⟩→U|ψ⟩ and operators A→UAU† , and we show that it is possible to gauge this global invariance into a local gauge invariance. To do this, we replace the wavefunction with a collection of local wavefunctions |ψJ⟩ , one for each patch of space J . The collection of spatial patches is chosen to cover the space; e.g. we could choose the patches to be single qubits or nearest-neighbor sites on a lattice. Local wavefunctions associated with neighboring pairs of spatial patches I and J are related to each other by dynamical unitary transformations UIJ . The local wavefunctions are local in the sense that their dynamics are local. That is, the equations of motion for the local wavefunctions |ψJ⟩ and connections UIJ are explicitly local in space and only depend on nearby Hamiltonian terms. (The local wavefunctions are many-body wavefunctions and have the same Hilbert space dimension as the usual wavefunction.) We call this picture of quantum dynamics the gauge picture since it exhibits a local gauge invariance. The local dynamics of a single spatial patch is related to the interaction picture, where the interaction Hamiltonian consists of only nearby Hamiltonian terms. We can also generalize the explicit locality to include locality in local charge and energy densities.
This chapter explores the Schrödinger and Heisenberg pictures in quantum mechanics, emphasizing their mathematical equivalence and physical consistency. The two formulations are shown to be connected by a unitary transformation. The Heisenberg equation of motion is derived, and the time evolution of operators—particularly those not explicitly dependent on time—is expressed in terms of their initial values. It is demonstrated that scalar products, expectation values, matrix elements, and commutation relations remain invariant under the change of picture. Drawing a close analogy with classical mechanics, Dirac's insight is presented: the quantum mechanical counterpart of the classical Poisson bracket is the operator commutator divided by iℏ . The chapter concludes by deriving the Lorentz force law within the quantum framework using commutator algebra. Metadata/Keywords: Schrödinger picture, Heisenberg picture, unitary transformation, Heisenberg equation of motion, Poisson bracket, Lorentz force, time evolution, expectation values
The Schrodinger equation has been considered to be a postulate of quantum physics, but it is also perceived and derived heuristically as the quantum equivalent of the classical energy relation. We indicate that the Schrodinger equation cannot be a physical postulate, and we show that the heuristic derivations wrongly consider that the particle kinetic energy is proportional to the square of the momentum operator acting on the wavefunction. The correct term is proportional to the square of the momentum. Analyzing particle interactions, we realize that particles have multiple virtual motions and that each motion is accompanied by a wave that has constant amplitude. Accordingly, we define the wavefunction as the superposition of the virtual waves of the particle. As a result, quantum mechanics becomes a local and global theory, while in the traditional formulation it is a local theory. In several interaction settings we can tell what particle motions arise and explain the outcomes in simple and tangible terms. Most importantly, the mathematical foundation of quantum mechanics becomes clear and justified, and we derive the Schrodinger, Dirac, etc. equations as the conditions the wavefunction must satisfy at each space-time point in order to fulfill the respective total energy equation.
Explicitly time-dependent Hamiltonians are ubiquitous in applications of quantum theory. It is therefore necessary to solve the time-dependent Schrödinger equation directly. The system’s dynamics is associated with a unitary time-evolution operator (a propagator), formally given as an infinite Dyson series. Time-dependent observables are invariant under unitary time-dependent transformations, where it is sometimes useful to transform the time-evolution from the states into the corresponding operators. This is carried out in part (in full) by transforming to the interaction (Heisenberg) picture. The corresponding equations of motion for the time-dependent operators are introduced. For quadratic potential energy functions, the time evolution of quantum expectation values coincides with the corresponding classical dynamics. This is demonstrated and analyzed in detail for Gaussian wave packets and a coherent state. Finally, we derive exact and approximate expressions for time-dependent transition probabilities and transition rates between quantum states. The validity of time-dependent perturbation theory is analyzed by comparison to exact dynamics.
This chapter demonstrates how one can find the time evolution of the operators from the knowledge of the time evolution of the state vector, and vice versa. It introduces a unitary operator and discusses the concept of a unitary transformation to show how one could transfer the time dependence from the state vectors to the operators. The possibility that either the state vectors or the operators can depend explicitly on time will lead to introduce the Schrodinger, Heisenberg, and interaction pictures. The chapter discusses the fundamental differences between these pictures. It also discusses the Ehrenfest theorem, which shows under which conditions quantum mechanics predicts the same results for measured physical quantities as classical physics. It illustrates unitary transformations of an arbitrary operator Â(t) which can depend explicitly on time. Quantum mechanics produces the same result as classical mechanics for a system in which particles are represented by well-localized wave functions.
Quantum mechanics is conventionally formulated in terms of complex Hilbert spaces, linear operators, and the Schrödinger equation. Although geometric structures play an increasingly important role in modern formulations of quantum theory, the wavefunction itself is generally regarded as a fundamental object. In this work, we investigate an alternative formulation in which the fundamental object is a coherence coordinate , where Omega represents radial coherence and Theta represents phase coherence associated with geometric transport. Starting from three assumptions—a coherence coordinate, a probability density generated by radial coherence, and a geometric Hamiltonian—we reconstruct the continuity equation, the quantum Hamilton–Jacobi equation, and ultimately the Schrödinger equation. Within this framework, the wavefunction appears as a derived quantity, while Hilbert spaces emerge naturally as spaces of square-integrable functions on a coherence manifold. The resulting construction establishes a differential-geometric representation of quantum mechanics and provides a common geometric language connecting quantum states, probability flow, geometric phases, holonomies, and gauge transport.