Abstract As previously shown, quantum physics for single pairs of creation and annihilation processes may be derived from first principles. Quantum physics at all can be therefore considered as an interplay of such elementary processes. This is easily possible if the number of pairs of processes is finite. Difficulties arise only for infinite numbers. The difficulties are similar to those occurring in the derivation of the equation for an oscillating string from that for an oscillator chain. It is true that the spectra of both systems are not continuously connected. However, a weaker theorem is more important: The chain eigenvalue of each order converges to the string one of the same order for an infinitely growing number of oscillators of a certain kind. Therefore both systems are continuously connected in the sense of semiconvergency. Exhausting the space continuum with a sequence of lattices equably becomming infinitely large and fine, the infinitely dimensional Hilbertspace is steadily connected with the finitely dimensional one in the sense of semiconvergency. It will be shown that the Hilbert spaces in the sequence of lattices yield the suitable tool for quantum physics as an interplay in the mentioned sense. This kind of Hilbert space, the so-called rational one, must be preferred in physics rather than the real one introduced by Hilbert, since all theories in physics are based on a finite number of data. In particular, we formulate Dirac's equation in the rational Hilbert space. It is shown that, even in quantum physics, a theorem of classical physics remains true, according to which relativity results from certain principles formulating most obvious experiences. We obtain the Lorentz invariant Dirac equation mainly from a modification of Newtons definition II according to which p = Hυ/c2 (instead of p = m υ).
Abstract As previously shown, quantum physics for single pairs of creation and annihilation processes may be derived from first principles. Quantum physics at all can be therefore considered as an interplay of such elementary processes. This is easily possible if the number of pairs of processes is finite. Difficulties arise only for infinite numbers. The difficulties are similar to those occurring in the derivation of the equation for an oscillating string from that for an oscillator chain. It is true that the spectra of both systems are not continuously connected. However, a weaker theorem is more important: The chain eigenvalue of each order converges to the string one of the same order for an infinitely growing number of oscillators of a certain kind. Therefore both systems are continuously connected in the sense of semiconvergency. Exhausting the space continuum with a sequence of lattices equably becomming infinitely large and fine, the infinitely dimensional Hilbertspace is steadily connected with the finitely dimensional one in the sense of semiconvergency. It will be shown that the Hilbert spaces in the sequence of lattices yield the suitable tool for quantum physics as an interplay in the mentioned sense. This kind of Hilbert space, the so-called rational one, must be preferred in physics rather than the real one introduced by Hilbert, since all theories in physics are based on a finite number of data. In particular, we formulate Dirac's equation in the rational Hilbert space. It is shown that, even in quantum physics, a theorem of classical physics remains true, according to which relativity results from certain principles formulating most obvious experiences. We obtain the Lorentz invariant Dirac equation mainly from a modification of Newtons definition II according to which p = Hυ/c2 (instead of p = m υ).
The question is often asked how to interprete quantum physics. That question does not arise in classical physics, since Newton's axioms are immediately connected with basic ideas and experiences. The same is possible in quantum physics, if we remember how elementary particle physicists describe their experiments. As Helmholtz has pointed out. the basic assumption of classical physics is that of geneidentity. That means: Bodies remain the same during their motion. Obviously, that is no longer true in quantum physics. Particles can be created and annihilated. Therefore creation and annihilation must be considered as basic processes. Motion only occurs, if a particle is annihilated in a certain point, if an equal one is created in an infinitesimally neighbouring point, and if this process is continuously going on during a certain time. Motions of that kind are compatible with the existence of some manifest creation and annihilation processes. If we accept this idea, quantum physics can be derived from first principles. As in classical physics, we know therefore what happens from the very beginning. Thus questions of interpretation become dispensable. A particular mathematical method is used to exhaust continua. The theory is formulated in a finite lattice, whose point density and extension equally go to infinity. All calculations are therefore performed in a finite dimensional Hilbert space. The results are however related to an infinite dimensional one. Earlier calculations may, therefore, be essentially correct, though they must be rejected in theories which are based on manifestly infinite dimensional Hilbert spaces. Here limiting processes do not occur in the state space. They are only admissible for numerical results.
AbstractStandard quantum mechanics is formulated using complex Hilbert spaces, real-valued probabilities, continuous one-parameter dynamics, and—where particles and fields have continuous degrees of freedom —function spaces constructed over continua such as R3 . In that precise mathematical sense, the real and complex continua are deeply embedded in the standard theory. It does not follow, however, that experiments have demonstrated the physical existence of an uncountable set of exact real numbers, still less that individual quantum systems physically “store” infinitely many digits. Every preparation, control operation, and measurement has finite precision, and many quantum predictions can be approximated arbitrarily closely using countable or finite descriptions.This distinction matters because the continuum is sometimes blamed for quantum theory's conceptual strangeness. That diagnosis is mostly unsupported. Superposition, interference, uncertainty, noncommutativity, entanglement, contextuality, Bell-inequality violations, and no-cloning already occur in finite-dimensional quantum systems and do not depend in any straightforward way on continuous position space or arbitrary irrational numbers. Restricting a Hilbert space to rational, algebraic, computable, or finitely specified states therefore does not automatically produce classical physics. Indeed, dense restricted subsets can approximate ordinary quantum predictions to arbitrary finite accuracy, although turning such a subset into a complete physical theory requires much more: closed dynamics, a composition rule, a measurement theory, robustness, and new empirical predictions.A particularly developed contemporary alternative is Tim Palmer's Invariant Set Theory and its recent formulation as Rational Quantum Mechanics (RaQM). RaQM is not accurately described as simply “quantum mechanics with irrational numbers banned.” It restricts the bases in which states are physically defined through number-theoretic rationality conditions, represents admissible states using finite bit strings, invokes a fractal invariant state space equipped with a non-Euclidean, p-adic-like metric, and violates Bell measurement independence at the level of exact counterfactual settings. Palmer's 2026 proposal further predicts a finite qubit-information capacity and an eventual saturation of quantum computational advantage. These are genuine, potentially falsifiable proposals, but they remain speculative and have not displaced standard quantum theory.The strongest conclusion is consequently asymmetric. There is a serious foundational question about whether the continuum is an ontological feature of nature or an extraordinarily efficient mathematical completion. Finite precision prevents existing experiments from proving a literal continuum. But there is presently little evidence that continuum mathematics causes quantum phenomena. Any discrete replacement must still account quantitatively for interference, contextuality, Bell correlations, entanglement, spectroscopy, atomic structure, and quantum computation. Discreteness can change quantum physics only when it is accompanied by additional physical postulates—and those postulates, rather than discreteness alone, must ultimately face experiment.
Hardy's axiomatic approach to the quantum theory of discrete Hilbert Spaces reveals that just one principle distinguishes it from classical probability theory: there should be continuous (and hence infinitesimal) reversible transformations between any pair of pure states - the single word `continuous' giving rise to quantum theory. This raises the question: Can one formulate a finite theory of qubit physics (FTQP) - necessary different from quantum theory - which can replicate the tested predictions of quantum theory of qubits to experimental accuracy? Here we show that an FTQP based on complex Hilbert vectors with rational squared amplitudes and rational phase angles is possible, provided the metric of state space, gp, is based on p-adic rather than Euclidean distance. A key number theorem describing an incompatibility between rational angles and rational cosines accounts for quantum complementarity in this FTQP. Dynamical evolution is described by a deterministic mapping on the set of p-adic integers and the measurement problem is trivially solved in terms of a nonlinear clustering of states in state space. Based on gp, causal deterministic analyses of quantum interferometry, GHZ, the sequential Stern-Gerlach experiment, Leggett-Garg and the Bell Theorem are described. The close relationship between fractals and p-adic integers suggest the existence of a primal fractal-like 'invariant set' geometry IU in cosmological state space, from which space-time and the laws of physics in space-time are emergent.
A physical theory is proposed that obeys both the principles of special relativity and of quantum mechanics. Time and space are treated on exactly the same footing, namely as quantum mechanical observables on a Hilbert space. The theory is not based upon Lagrangian or Hamiltonian mechanics and cannot be formulated within the framework of unitary dynamics. As a most important aspect, the theory breaks with the concept of a continuously flowing time in favour of a discrete jump process in spacetime. All physical statements are formulated in terms of detector events rather than particle states. The physical object under consideration is a spinless particle in empty space. Yet the theory also accounts for particle-antiparticle pair creation and annihilation, and is therefore not a single-particle theory in the strict sense. The Maxwell equations are derived as a straightforward consequence of some fundamental commutation relations. In the non-relativistic limit, and in the limit of infinitely small time uncertainties, one obtains the familiar Schr\\"odinger equation of a spinless particle exposed to an external electromagnetic field.
An interpretation and re-formulation of modern physics which removes the presumption of the space-time continuum, and bases physical theory on a small number of rational and empirical principles. After briefly describing the philosophical principles underlying the theory, we rigorously construct a discrete model of quantum mechanics. Kets are names, or labels which categorise states of matter but do not directly describe them. The principle of superposition is seen as a definitional truism in the categorisation of states. This resolves the measurement problem of quantum mechanics by attributing the collapse of the wave function to information. The probability interpretation has a natural meaning in which the configuration of interacting particles plays the role of a hidden variable. The model supports a form of relativistic quantum field theory which does not depend on quantisation or second quantisation from classical mechanics. Continuous laws of wave mechanics are found in a discrete metaphysic which does not involve waves. Classical law is the expected behaviour of many elementary particles. The constraints on the theory are sufficient to establish that particles are point-like entities with specific properties found in nature. Newton's first law and conservation of momentum are established from the principle of homogeneity. Maxwell's equations are derived from the simple interaction in which a Dirac particle emits or absorbs a photon. Feynman rules are calculated for the discrete theory and differ from the standard rules by the removal of the ultraviolet divergence and the use of proper loop integrals. They give finite and unambiguous results.
It is not surprising that our language should be incapable of describing processes occurring within atoms, for, as has been remarked, it was invented to describe the experiences of daily life, … Fortunately, mathematics is not subject to this limitation, and it has been possible to invent a mathematical scheme–the quantum theory–which seems entirely adequate for the treatment of atomic processes. Werner Heisenberg, The Physical Principles of the Quantum Theory (1930) In this chapter we introduce the mathematical structure of quantum theory, for a finite number of systems and finite dimension, starting from three simple Hilbert-space postulates, and based on the intuitive notions of state, transformation, and effect, within a causal context. Such notions will be formalized in a general non-causal context in Chapter 3, which is devoted to the informational framework. We will assume the reader has a basic knowledge of matrix analysis and of operators over Hilbert spaces. Comments about the extension to infinite dimensions, continuous outcome-spaces, and infinitely many systems can be found in the notes at the end of the chapter. From the three simple Hilbert-space postulates, the present chapter extracts six principles, which will become the postulates for quantum theory in the axiomatic derivation presented in Part III of the book. In addition, we will take the opportunity to present and explore many relevant features of the theory – including its convex and causal structure, the Choi–Jamiołkowski isomorphism, and relevant notions, such as those of POVM, complete positivity of maps, quantum operation and channels, entangled states, state discrimination, tomography of states, and transformations. The purification principle will open the stage to the dilation theorems of the theory of quantum open systems, along with other consequences, such as the existence of faithful and steering states, the theorem of no information without disturbance, quantum teleportation, and more. We will review the quantum no-cloning and no-programming theorems, and finally provide a thorough derivation of the theory of quantum channel inversion, which is the backbone of the general theory of quantum error correction in quantum information. Additional interesting topics will be classical information theory as a restriction of quantum information theory, and a critical reconsideration of the von Neumann postulate and the notion of observable.