Feynman Path Integrals, Trajectories and Copenhagen Interpretation. Note Quote.

As the trajectory exists by precept in the trajectory representation, there is no need for Copenhagen’s collapse of the wave function. The trajectory representation can describe an individual particle. On the other hand, Copenhagen describes an ensemble of particles while only rendering probabilities for individual particles.

The trajectory representation renders microstates of the Schrödinger’s wave function for the bound state problem. Each microstate by the equation

ψ = (2m)^1/4cos(W/h ̄)/(W′)^1/2[a − c2/(4b)]^1/2

(aφ² + bθ² + cφθ)^1/2/[a − c²/(4b)]^1/2 cos[arctan(b(θ/φ) + c/2)/(ab − c²/4)^1/2 = φ

is sufficient by itself to determine the Schrödinger’s wave function. Thus, the existence of microstates is a counter example refuting the Copenhagen assertion that the Schrödinger’s wave function be an exhaustive description of non-relativistic quantum phenomenon. The trajectory representation is deterministic. We can now identify a trajectory and its corresponding Schrödinger wave function with sub-barrier energy that tunnels through the barrier with certainty. Hence, tunneling with certainty is a counter example refuting Born’s postulate of the Copenhagen interpretation that attributes a probability amplitude to the Schrödinger’s wave function. As the trajectory representation is deterministic and does not need ψ, much less to assign a probability amplitude to it, the trajectory representation does not need a wave packet to describe or localize a particle. The equation of motion,

t − τ = ∂W/∂E, where t is the trajectory time, relative to its constant coordinate τ, and given as a function of x;

for a particle (monochromatic wave) has been shown to be consistent with the group velocity of the wave packet. Normalization, as previously noted herein, is determined by the nonlinearity of the generalized Hamilton-Jacobi equation for the trajectory representation and for the Copenhagen interpretation by the probability of finding the particle in space being unity. Though probability is not needed for tunneling through a barrier, the trajectory interpretation for tunneling is still consistent with the Schrödinger representation without the Copenhagen interpretation. The incident wave with compound spatial modulation of amplitude and phase for the trajectory representation,

has only two spectral components which are the incident and reflected unmodulated waves of the Schrödinger representation.

Trajectories differ with Feynman’s path integrals in three ways. First, trajectories employ a quantum Hamilton’s characteristic function while a path integral is based upon a classical Hamilton’s characteristic function. Second, the quantum Hamilton’s characteristic function is determined uniquely by the initial values of the quantum stationary Hamilton-Jacobi equation, while path integrals are democratic summing over all possible classical paths to determine Feynman’s amplitude. While path integrals need an infinite number of constants of the motion even for a single particle in one dimension, motion in the trajectory representation for a finite number of particles in finite dimensions is always determined by only a finite number of constants of the motion. Third, trajectories are well defined in classically forbidden regions where path integrals are not defined by precept.

Heisenberg’s uncertainty principle shall remain premature as long as Copenhagen uses an insufficient subset of initial conditions (x, p) to describe quantum phenomena. Bohr’s complementarity postulates that the wave-particle duality be resolved consistent with the measuring instrument’s specific properties.

Heisenberg’s uncertainty principle shall remain premature as long as Copenhagen uses an insufficient subset of initial conditions (x, p) to describe quantum phenomena. Bohr’s complementarity postulates that the wave-particle duality be resolved consistent with the measuring instrument’s specific properties. Anonymous referees of the Copenhagen school have had reservations concerning the representation of the incident modulated wave as represented by the equation

before the barrier. They have reported that compoundly modulated wave represented by the above equation is only a clever superposition of the incident and reflected unmodulated plane waves. They have concluded that synthesizing a running wave with compound spatial modulation from its spectral components is nonphysical because it would spontaneously split. By the superposition principle of linear differential equations, the spectral components may be used to synthesize a new pair of independent solutions with compound modulations running in opposite directions. Likewise, an unmodulated plane wave running in one direction can be synthesized from two waves with compound modulation running in the opposite directions for mappings under the superposition principle are reversible.

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Gauge Geometry and Philosophical Dynamism

Weyl was dissatisfied with his own theory of the predicative construction of the arithmetically definable subset of the classical real continuum by the time he had finished his Das Kontinuum, when he compared it with Husserl’s continuum of time, which possessed a “non-atomistic” character in contradistinction with his own theory. No determined point of time could be exhibited, only approximate fixing is possible, just as in the case of “continuum of spatial intuition”. He himself accepted the necessity that the mathematical concept of continuum, the continuous manifold, should not be characterized in terms of modern set theory enriched by topological axioms, because this would contradict the essence of continuum. Weyl says,

It seems that set theory violates against the essence of continuum, which, by its very nature, cannot at all be battered into a single set of elements. not the relationship of an element to a set, but a part of the whole ought to be taken as a basis for the analysis of the continuum.

For Weyl, single points of continuum were empty abstractions, and made him enter a difficult terrain, as no mathematical conceptual frame was in sight, which could satisfy his methodological postulate in a sufficiently elaborative manner. For some years, he sympathized with Brouwer’s idea to characterize points in the intuitionistic one-dimensional continuum by “free choice sequences” of nested intervals, and even tried to extend the idea to higher dimensions and explored the possibility of a purely combinatorial approach to the concept of manifold, in which point-like localizations were given only by infinite sequences of nested star neighborhoods in barycentric subdivisions of a combinatorially defined “manifold”. There arose, however, the problem of how to characterize the local manifold property in purely combinatorial terms.

Weyl was much more successful on another level to rebuild differential geometry in manifolds from a “purely infinitesimal” point of view. He generalized Riemann’s proposal for a differential geometric metric

ds²(x) = ∑ⁿ _{i, j = 1} g_ij(x) dxⁱ dx^j

From his purely infinitesimal point of view, it seemed a strange effect that the length of two vectors ξ(x) and η(x’) given at different points x and x’ can be immediately and objectively compared in this framework after the calculation of

|ξ(x)|² = ∑ⁿ _{i, j = 1} g_ij(x) ξⁱ ξ^j,

|η(x’)|² = ∑ⁿ _{i, j = 1} g_ij(x’) ηⁱ η^j

In this context, it was, comparatively easy for Weyl, to give a perfectly infinitesimal characterization of metrical concepts. He started from a well-known structure of conformal metric, i.e. an equivalence class [g] of semi-Riemannian metrics g = g_ij(x) and g’ = g’_ij(x), which are equal up to a point of dependent positive factor λ(x) > 0, g’ = λg. Then, comparison of length made immediate sense only for vectors attached to the same point x, independently of the gauge of the metric, i.e. the choice of the representative in the conformal class. To achieve comparability of lengths of vectors inside each infinitesimal neighborhood, Weyl introduced the conception of length connection formed in analogy to the affine connection, Γ, just distilled from the classical Christoffel Symbols Γ^k_ij of Riemannian geometry by Levi Civita. The localization inside such an infinitesimal neighborhood was given, as would have been done already by the mathematicians of the past, by coordinate parameters x and x’ = x + dx for some infinitesimal displacement dx. Weyl’s length connection consisted, then, in an equivalence class of differential I-forms [Ψ], Ψ ≡ ∑ⁿ_{i = 1} Ψ_idxⁱ, where an equivalent representation of the form is given by Ψ’ ≡ Ψ – d log λ, corresponding to a change of gauge of the conformal metric by the factor λ. Weyl called this transformation, which he recognized as necessary for the consistency of his extended symbol system, the gauge transformation of the length connection.

Weyl established a purely infinitesimal gauge geometry, where lengths of vectors (or derived metrical concepts in tensor fields) were immediately comparable only in the infinitesimal neighborhood of one point, and for points of finite distance only after an integration procedure. this integration turned out to be, in general, path dependent. Independence of the choice of path between two points x and x’ holds if and only if the length curvature vanishes. the concept of curvature was built in direct analogy to the curvature of the affine connection and turned out to be, in this case, just the exterior derivative of the length connection f ≡ dΨ. This led Weyl to a coherent and conceptually pleasing realization of a metrical differential geometry built upon purely infinitesimal principles. moreover, Weyl was convinced of important consequences of his new gauge geometry for physics. The infinitesimal neighborhoods understood as spheres of activity as Fichte might have said, suggested looking for interpretations of the length connection as a field representing physically active quantities. In fact, building on the mathematically obvious observation df ≡ 0, which was formally identical with the second system of the generally covariant Maxwell equations, Weyl immediately drew the conclusion that the length curvature f ought to be identified with the electromagnetic field.

He, however, gave up the belief in the ontological correctness of the purely field-theoretic approach to matter, where the Mie-Hilbert theory of a combined Lagrange function L(g,Ψ) for the action of the gravitational field (g) and electromagnetism (Ψ) was further geometrized and technically enriched by the principle of gauge invariance (L), substituting in its place a philosophically motivated a priori argumentation for the conceptual superiority of his gauge geometry. The goal of a unified description of gravitation and electromagnetism, and the derivation of matter structures from it, was nothing specific to Weyl. In his theory, the purely infinitesimal approach to manifolds and the ensuing possibility to geometrically unify the two-known interaction fields gravitation and electromagnetism took on a dense and conceptually sophisticated form.