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Työväentutkimus Vuosikirja

Sources in quantum mechanics and quantum field theory

Eric Toombs

Published Jul 31, 2026• 0 citations
OpenAlex

Abstract

The Schrödinger equation is modified. A source term, or \emph{source}, is added, making it non-homogeneous. A time-$L^2$ constraint $\int_{-\infty}^\infty t |\psi(t)|^2 \in \R$ and an unphysical decay are also added. The result is called the \emph{blue equation} because ``non-homogeneous Schrödinger equation with unphysical decay'' is too long. The source term is a way to set the initial condition of a system, thus a source term can be used to specify a time-dependent quantum state. The ramifications of representing quantum states with sources are explored throughout. Measurement, in the form of probability amplitudes between sources, is used to normalise the sources. Sources are used to simplify scattering theory. The ``in'' and ``out'' states of the Lippmann-Schwinger equation are normalised. The Neumann series is used to do time-dependent perturbation theory. A new kind of $S$-matrix, the red operator, is defined, which works the same way, only on sources. This red operator is used to interpret sources as physical quantities of their own. Applications of sources to quantum field theory are considered. It is shown that a source is dual to a field operator, thus sources can be contracted directly with the field operators in a correlation function. The result is the probability amplitude between sources. In this way, one can avoid the use of the LSZ reduction formula. A path integral is defined and used to demonstrate that the duality between source and field operators exists in the context of path integrals as well. The source reveals a causal nature which was hidden or absent before. There is a causal connection from a source to the generated quantum state. A naïve definition of causality based off of this connection is considered. This definition cannot be used, however, due to special relativity. Epstein and Glaser's definition of causality, as used in causal perturbation theory, is used instead. It is proven that a \emph{time-local} system is causal, according to this definition.

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The Schrödinger equation is modified. A source term, or \emph{source}, is added, making it non-homogeneous. A time-L2L^2L2 constraint ∫−∞∞t∣ψ(t)∣2∈R\int_{-\infty}^\infty t |\psi(t)|^2 \in \R∫−∞∞​t∣ψ(t)∣2∈R and an unphysical decay are also added. The result is called the \emph{blue equation} because non-homogeneous Schrödinger equation with unphysical decay'' is too long. The source term is a way to set the initial condition of a system, thus a source term can be used to specify a time-dependent quantum state. The ramifications of representing quantum states with sources are explored throughout. Measurement, in the form of probability amplitudes between sources, is used to normalise the sources. Sources are used to simplify scattering theory. The in'' and ``out'' states of the Lippmann-Schwinger equation are normalised. The Neumann series is used to do time-dependent perturbation theory. A new kind of SSS-matrix, the red operator, is defined, which works the same way, only on sources. This red operator is used to interpret sources as physical quantities of their own. Applications of sources to quantum field theory are considered. It is shown that a source is dual to a field operator, thus sources can be contracted directly with the field operators in a correlation function. The result is the probability amplitude between sources. In this way, one can avoid the use of the LSZ reduction formula. A path integral is defined and used to demonstrate that the duality between source and field operators exists in the context of path integrals as well. The source reveals a causal nature which was hidden or absent before. There is a causal connection from a source to the generated quantum state. A naïve definition of causality based off of this connection is considered. This definition cannot be used, however, due to special relativity. Epstein and Glaser's definition of causality, as used in causal perturbation theory, is used instead. It is proven that a \emph{time-local} system is causal, according to this definition.
In the standard formulation of quantum mechanics, observables are represented by operators in a Hilbert space and states by vectors in it. Quantum field theory too is formulated in terms of field operators. States in field theory include the vacuum state, one particle states, two particle states and so on. One does not measure operators or state vectors, however; measurements involve eigenvalues of operators or transition probabilities between states. It turns out that one can formulate quantum mechanics and field theory without using operators. Indeed, the probability amplitude for a transition from a position q1 at time t1 to a position q2 at time t2 can be expressed as a path integral ∫ dt L(q(t),q˙(t)), (5.1) where q(t) runs over all possible functions satisfying the conditions q(t1) = q1, q(t2) = q2 (5.2) and the integral over q(t) is an integral over all such paths of the exponential evaluated for each path. The measure of integration is not well defined a priori, but it is possible to give prescriptions that make sense. In classical mechanics, positions at two different times determine a path from the second order differential equation of motion, which yields a definite solution if two conditions are imposed. In contrast, quantum mechanics does not allow us to talk about trajectories. The path integral formulation manifestly refers to the myriad paths that may be taken by a particle in going from one point to another. It goes so far as to assign a probability amplitude to each path, and it is related to the classical action corresponding to that path. However, it is complex and it also involves a superposition, characteristic of quantum mechanics.
Some aspects of the interpretation of quantum theory are discussed.It is emphasized that quantum theory is formulated in a Cartesian coordinate system; in other coordinates the result obtained with the help of the Hamiltonian formalism and commutator relations between 'canonically conjugated' coordinate and momentum operators leads to a wrong version of quantum mechanics.In this connection the Feynman integral formalism is also discussed.In this formalism the measure is not well-defined and there is no idea how to distinguish between the true version of quantum mechanics and an incorrect one; it is rather a mnemonic rule to generate perturbation series from an undefined zero order term.The origin of time is analyzed in detail by the example of atomic collisions.It is shown that the time-dependent Schrödinger equation for the closed three-body (two nuclei + electron) system has no physical meaning since in the high impact energy limit it transforms into an equation with two independent time-like variables; the time appears in the stationary Schrödinger equation as a result of extraction of a classical subsystem (two nuclei) from a closed three-body system.Following the Einstein-Rosen-Podolsky experiment and Bell's inequality the wave function is interpreted as an actual field of information in the elementary form.The relation between physics and mathematics is also discussed.
Abstract The new quantum mechanics consists of a scheme of equations which are very closely analogous to the equations of classical mechanics, with the funda­mental difference that the dynamical variables do not obey the commutative law of multiplication, but satisfy instead the well-known quantum conditions. It follows that one cannot suppose the dynamical variables to be ordinary numbers (c-numbers), but may call them numbers of a special type (q-numbers). The theory shows that these q-numbers can in general be represented by matrices whose elements are c-numbers (functions of a time parameter). When one has performed the calculations with the q-numbers and obtained all the matrices one wants, the question arises how one is to get physical results from the theory, i. e., how can one obtain c-numbers from the theory that one can compare with experimental values ? Hitherto this has been done with the help of a number of special assumptions. In Heisenberg’s original matrix mechanics it was assumed that the elements of the diagonal matrix that repre­sents the energy are the energy levels of the system, and the elements of the matrix that represents the total polarisation, which are periodic functions of the time, determine the frequencies and intensities of the spectral lines in analogy to the classical theory. Schrodinger’s wave representation of the quantum mechanics has provided new ways of obtaining physical results from the theory, based on the assumption that the square of the amplitude of the wave function can in certain cases be interpreted as a probability. From this assumption one can, for instance, work out the probability of a transition being produced in a system (or the number of transitions produced in an assembly of like systems) by an arbitrary external perturbing force, and can thus, by supposing the perturbation to consist of incident radiation, obtain directly Einstein’s B coefficients. Again in Born’s treatment of collision problems it is assumed that the square of the amplitude of the wave function scattered in any direction determines the probability of the colliding electron (or other body) being scattered in that direction.
The Schrödinger equation and the Born rule constitute two of the most successful structures in quantum mechanics. The former governs the continuous evolution of the wave function, while the latter connects its squared amplitude to the statistical distribution of experimental outcomes. Yet these two structures operate at different descriptive levels: the Schrödinger state is continuous, spatially extended, and phase-sensitive, whereas individual experiments terminate in localized physical events. This paper develops a theoretical framework for reconstructing the physical relation between these two levels. Rather than proceeding only in the conventional direction from Schrödinger dynamics to Born probabilities, we introduce an inverse explanatory strategy: from Born back to Schrödinger. Born statistics are treated as an empirical constraint from which the physical role of the wave function is reconsidered. We propose that, for the single-particle regime examined here, the wave function may be interpreted functionally as a dynamical propagation structure rather than as the localized particle itself. Its amplitude, phase, conserved current, gauge-covariant propagation momentum, and associated velocity field jointly define a structured family of possible motions. This interpretation leads to three separations: propagation from localization, propagation from physical population, and arrival from detector registration. A physical preparation generates a source measure (\mu_0^{\rm phys}); Schrödinger dynamics generates a propagation map (\Phi_\Psi); and microscopic detector dynamics is represented by a response map (\mathcal R_D). The Born problem can then be formulated as the closure condition RD ⁣[(ΦΨ)∗μ0phys]=?μBorn,D. \boxed{\mathcal R_D\!\left[(\Phi_\Psi)_*\mu_0^{\rm phys}\right] \overset{?}{=} \mu_{\rm Born,D}.} RD​[(ΦΨ​)∗​μ0phys​]=?μBorn,D​.​ This relation is not claimed here as a derived law, but as the central dynamical problem that a physical reconstruction of quantum probability must solve without inserting the Born measure at either boundary. The paper synthesizes six preceding studies on wave-packet localization, Born's scattering interpretation, the emergence of probability, the physical meaning of the wave function, source-to-detector transport, and unified propagation dynamics. It places the resulting framework in dialogue with Madelung and Hamilton–Jacobi formulations, Bohmian mechanics, decoherence, Quantum Darwinism, structural and symmetry-based derivations of the Born rule, Everettian approaches, objective-collapse theories, and detector models. The resulting proposal is best understood as a propagation-first theoretical reconstruction program: Schrödinger dynamics organizes possible propagation, while Born statistics constrain how that propagation becomes physically realized. Keywords: Schrödinger equation; Born rule; wave function; quantum foundations; propagation structure; quantum probability; quantum trajectories; source-to-detector transport; detector response; wave–particle duality; Bohmian mechanics; quantum measurement
The Schrodinger equation is not covariant. Nevertheless, quantum field theory is often formulated using the Schrodinger equation to describe the time evolution of the system, which is equivalent to using Feynman path integrals. It is well known that scattering theory gives covariant results provided that the interaction vanishes in the asymptotic limit of t→±∞t \rightarrow \pm \inftyt→±∞, but there are other situations of interest that do not satisfy those conditions. As an example, the Aharonov-Bohm effect is derived here using second-quantized field theory to describe all the electrons as well as the electromagnetic field, which allows the effects of retarded vector potentials to be calculated in a straightforward way using the Feynman propagator. The results are in agreement with the usual expression for the Aharonov-Bohm effect when the retardation of the electromagnetic field is negligible, but they predict a fractional phase shift, a lack of covariance, and violations of causality when retardation effects are significant. These results suggest that quantum field theory based on the Schrodinger equation or Feynman path integrals is incomplete in the sense that it cannot give a correct description of all observable phenomena.
Due to the space and time dependence of the wavefunction in the time-dependent Schrödinger equation, different boundary conditions are possible. The equation is usually solved as an `initial-value problem', by fixing the value of the wavefunction in all space at a given instant. We compare this standard approach to `source boundary conditions' that fix the wave at all times in a given region, in particular at a point in one dimension. In contrast to the well known physical interpretation of the initial-value-problem approach, the interpretation of the source approach has remained unclear, since it introduces negative energy components, even for `free motion', and a time-dependent norm. This work provides physical meaning to the source method by finding the link with equivalent initial-value problems.