The program of deducing quantum mechanics from the postulate of thermodynamic continuity is extended to embrace the principles of symmetry and quantum statistics, in particular the rule of conservation of symmetry type when a system of N particles in resonance interaction is augmented by one more particle. The special quantitative definition of quantum conjugacy contained in the commutation rule of Born and in the Schrödinger replacement of the momentum p by a differential operator may be replaced by a general qualitative physical definition: “p and q are conjugate observables in a mechanical system when the manifold of q reactions of the system uniquely determines the manifold of its p reactions.” When interpreted in terms of superposition mechanics this definition leads to the rules of Born and Schrödinger and from there to quantum dynamics.
The program of deducing quantum mechanics from the postulate of thermodynamic continuity is extended to embrace the principles of symmetry and quantum statistics, in particular the rule of conservation of symmetry type when a system of N particles in resonance interaction is augmented by one more particle. The special quantitative definition of quantum conjugacy contained in the commutation rule of Born and in the Schrödinger replacement of the momentum p by a differential operator may be replaced by a general qualitative physical definition: “p and q are conjugate observables in a mechanical system when the manifold of q reactions of the system uniquely determines the manifold of its p reactions.” When interpreted in terms of superposition mechanics this definition leads to the rules of Born and Schrödinger and from there to quantum dynamics.
The basic concepts and rules of quantum mechanics are shown to be immediate consequences, on the basis of simple reasoning, of a postulate of continuity in the domain of macroscopic thermodynamics. The continuity principle leads first to the concept of a fractional likeness between two states in general, as opposed to the classical alternative of either like or unlike. Then all possible states of a system can be arranged in sets of mutually unlike or “orthogonal” states, with members A and B of different sets interconnected by fractional likeness factors q(A, B) between zero and unity. These factors q are identical with the relative intensities in a splitting effect of a state into components, ruled by probability relations and by the basic principle of quantum statistics of not counting permutations of individual particles. The problem of relation between the various fractional likenesses, or probabilities of transition under analysis, is a purely mathematical problem and is solved by the introduction of probability amplitudes necessarily subject to a matrix law of multiplication, i.e., to the principle of superposition. Thus, starting from the continuity postulate of thermodynamics, quantum theory in its general outline is obtained by simple reasoning. All further details, such as the duality of waves and particles, the uncertainty relation, and the mechanics of special systems, need only one more bit of empirical information, namely, the symmetric and periodic form of the amplitude ψ of fractional likeness between a state of given coordinate and a state of given momentum.
The superposition rule for probability amplitudes ψ is not a separate ad hoc assumption of a physical character but is a formal consequence of the arrangement of the states of a system in sets of mutually orthogonal states linked by probabilities of transition. Nor do the quantum prescriptions of Born and Schrödinger have to be introduced as ad hoc assumptions; they can be deduced as consequences of (a) the superposition rule, and (b) the definition of conjugates q and p as leading to a constant probability density in qp-space, a quality known from classical statistical mechanics. Quantum theory can thus be deduced from a nonquantal background.
Abstract Quantum mechanics is one of the basic theories of modern physics. Here, the famous Schrödinger equation and the differential operators representing mechanical quantities in quantum mechanics are derived, just based on the principle that the translation invariance (symmetry) of a system in Hamiltonian mechanics should be preserved in quantum mechanics. Moreover, according to the form of the differential operators, the commutation relation in quantum mechanics between the generalized coordinate and the generalized momentum can be directly obtained. We believe that the results in this paper are very useful for understanding the physical origin of quantum mechanics.
We formulate a general principle that supplants a Boolean σ-algebra of intrinsic properties of a classical system by a σ-complex (a union of σ-algebras) of extrinsic properties of a quantum system that are elicited by interactions with other systems. We apply the canonical definitions of the concepts of classical physics of observables, states, combined systems, symmetries, and dynamics to derive the standard quantum characterizations, including the Schrödinger equation and the von Neumann-Lüder's Projection Rule. This manifestly consistent reconstruction of quantum mechanics is then used to discuss and resolve the dilemmas of the orthodox interpretation.
The thesis divides naturally into two parts. Part I raises, and in some cases answers, questions concerning symmetry in classical mechanics. The main result (Theorem 6.4) shows that the assumption of the existence of a realization puts an upper limit on the rank of the algebra. \n \n The heart of the thesis (covering three-quarters of the volume) is section II on the quantization of classical systems. §1 lists axioms desirable in any quantization rule for the 'functions of the q's'. The momentum observables are introduced in §2 prior to their quantization in §4. §5 essentially shows how conventional quantum mechanics fits into this scheme of things. By progressive specialization from a general manifold to a vector space, from a general quantization scheme to one which is linear on the linear momentum functions, and finally to an entirely well-behaved (admissible) quantization rule, into which conventional quantum mechanics fits nicely, we obtain in §7-§9 results which become progressively more and more powerful. The final theorem (Theorem 9.2) is perhaps the most significant of all. This result states that there exists a class of functions, which contains all functions of the q's and functions of the p's and all momentum observables and which is closed with respect to any linear canonical transformation L; a rule A assigning a unique self-adjoint operator to each such function f; a unitary operator WL corresponding to L and an equation \n \n𝛬(𝑓 ∘ 𝐿) = 𝑊[sub]𝐿⁻ 𝛬 𝑓 𝑊[sub]𝐿
Abstract The quantum measurement postulate remains a fundamental physical contradiction, enforcing a non-unitary, observer-dependent projection that interrupts the continuous dynamical evolution governed by the Schrödinger equation. In this work, we demonstrate that the orthodox instantaneous collapse postulate is both a thermodynamic impossibility and a logical redundancy. The observer-dependent actualization of macroscopic particle ensembles would require an instantaneous deposition of latent heat ( dt→0 d t → 0 ), triggering an unphysical, infinite power divergence that delivers a macroscopic thermal shock, fundamentally contradicting the thermodynamic stability of the apparatus. Moreover, because an unobserved apparatus must undergo continuous open-system thermodynamic relaxation prior to conscious inspection, any subsequent mathematical projection is theoretically obsolete. Furthermore, we mathematically invalidate leading alternative theories, proving that mean-field thermodynamic ensemble models (such as the Allahverdyan, Balian, and Nieuwenhuizen formalism) fundamentally contradict their own claims: while explicitly attempting to derive the probabilistic actualization dictated by the Born rule, they inadvertently force deterministic single-run outcomes. We emphasize that our critique and the theoretical framework proposed herein are explicitly directed at the objective actualization of individual quantum systems. Consequently, the purely statistical ensemble interpretation—which conceptually bypasses the single-run measurement problem—falls outside the scope of the present analysis. To replace these untenable paradigms, we establish the Open-System Measurement Framework (OSMF), shifting the focus entirely to the continuous, objective physical evolution of the individual wavefunction. We demonstrate that the quantum system-apparatus interaction generates a transient, thermodynamically unstable macroscopic superposition. Driven by its coupling to a reservoir, this state rapidly undergoes a continuous thermodynamic relaxation into a stable local minimum of its thermodynamic potential, actualizing a definite outcome. By applying the OSMF to canonical experiments, we prove that measurement is an objective, irreversible process. Crucially, this framework seamlessly reconciles Bell-type non-local correlations (the EPR paradox) with the strict causal constraints of special relativity by providing a local, continuous physical mechanism for outcome actualization.