Classically general covariance is found from the idea that a vector is a physical quantity which exists independently of choice of coordinate system and is unchanged by a change of coordinate system. It is often assumed that there exists some form of absolute mathematical space or space-time, and that in a flat space approximation vectors can be imagined between defined points in this space-time, much as we can imagine an arrowed line drawn on a piece of paper. However, while classical vector quantities can be represented on paper, in the quantum domain physical quantities do not in general exist with precise values except in measurement; a change of apparatus, for example by rotating it, may affect the outcome of the measurement, so the condition for general covariance does not apply. The purpose of this paper is to re-examine covariance within the context of an orthodox, Dirac-Von Neumann interpretation of quantum mechanics, to replace it with a new condition, here called quantum covariance, and to show that quantum covariance is the required condition for the unification of general relativity with quantum mechanics for non-interacting particles.
Classically general covariance is found from the idea that a vector is a physical quantity which exists independently of choice of coordinate system and is unchanged by a change of coordinate system. It is often assumed that there exists some form of absolute mathematical space or space-time, and that in a flat space approximation vectors can be imagined between defined points in this space-time, much as we can imagine an arrowed line drawn on a piece of paper. However, while classical vector quantities can be represented on paper, in the quantum domain physical quantities do not in general exist with precise values except in measurement; a change of apparatus, for example by rotating it, may affect the outcome of the measurement, so the condition for general covariance does not apply. The purpose of this paper is to re-examine covariance within the context of an orthodox, Dirac-Von Neumann interpretation of quantum mechanics, to replace it with a new condition, here called quantum covariance, and to show that quantum covariance is the required condition for the unification of general relativity with quantum mechanics for non-interacting particles.
We have seen in the previous chapter how useful wave mechanics can be in solving physical problems. But wave mechanics has several limitations. It describes physical states by means of wave functions, which are functions of the positions of the particles of the system, but why should we single out position as the fundamental physical observable? For instance, we might want to describe states in terms of probability amplitudes for particles to have certain values of the momentum or energy rather than the position. A more fundamental limitation is that there are attributes of physical systems that cannot be described at all in terms of the positions and momenta of a set of particles. One of these attributes is spin , which will be a chief subject of Chapter 4. Another is the value of the electric or magnetic field at some point in space, treated in Chapter 11. This chapter will describe the principles of quantum mechanics in a formalism which is essentially the “transformation theory” of Dirac, mentioned briefly in Section 1.4. This formalism generalizes both the wave mechanics of Schrödinger and the matrix mechanics of Heisenberg, and is sufficiently comprehensive to apply to any sort of physical system. States The first postulate of quantum mechanics is that physical states can be represented as vectors in a sort of abstract space known as Hilbert space . Before getting into Hilbert space, I need to say a bit about vectors in general. In kindergarten we learn that vectors are quantities with both magnitude and direction. Later, when we study analytic geometry, we learn instead to describe a vector in d dimensions as a string of d numbers, the components of the vector. The latter approach lends itself well to calculation, but in some respects the kindergarten version is better, because it allows us to describe relations among vectors without specifying a coordinate system. For instance, a statement that one vector is parallel to a second vector, or perpendicular to a third, has nothing to do with how we choose our coordinate system. Here we will formulate what we mean by vector spaces in general, and Hilbert space in particular, in a way that is independent of the coordinates we use to describe directions in these spaces.
We summarize below some of the postulates and definitions basic to our formalism, and present some important results based on these postulates. The formalism is purely mathematical in nature with very little physics input, but it provides the structure within which the physical concepts that will be discussed in the later chapters will be framed. State vectors It is important to realize that the Quantum Theory is a linear theory in which the physical state of a system is described by a vector in a complex, linear vector space. This vector may represent a free particle or a particle bound in an atom or a particle interacting with other particles or with external fields. It is much like a vector in ordinary three-dimensional space, following many of the same rules, except that it describes a very complicated physical system. We will be elaborating further on this in the following. The mathematical structure of a quantum mechanical system will be presented in terms of the notations developed by Dirac. A physical state in this notation is described by a “ket” vector, |〉, designated variously as |α〉 or |ψ〉 or a ket with other appropriate symbols depending on the specific problem at hand. The kets can be complex. Their complex conjugates, |〉 * , are designated by 〈| which are called “bra” vectors. Thus, corresponding to every ket vector there is a bra vector. These vectors are abstract quantities whose physical interpretation is derived through their so-called “representatives” in the coordinate or momentum space or in a space appropriate to the problem under consideration.
We have seen in the previous chapter how useful wave mechanics can be in solving physical problems. But wave mechanics has several limitations. It describes physical states by means of wave functions, which are functions of the positions of the particles of the system, but why should we single out position as the fundamental physical observable? For instance, we might want to describe states in terms of probability amplitudes for particles to have certain values of the momentum or energy rather than the position. A more fundamental limitation: There are attributes of physical systems that cannot be described at all in terms of the positions and momenta of a set of particles. One of these attributes is spin , which will be the subject of Chapter 4. Another is the value of the electric or magnetic field at some point in space, treated in Chapter 11. This chapter will describe the principles of quantum mechanics in a formalism which is essentially the “transformation theory” of Dirac, mentioned briefly in Section 1.4. This formalism generalizes both the wave mechanics of Schrödinger and the matrix mechanics of Heisenberg, and is sufficiently comprehensive to apply to any sort of physical system. States The first postulate of quantum mechanics is that physical states can be represented as vectors in a sort of abstract space known as Hilbert space . Before getting into Hilbert space, I need to say a bit about vectors in general. In kindergarten we learn that vectors are quantities with both magnitude and direction. Later, when we study analytic geometry, we learn instead to describe a vector in d dimensions as a string of d numbers, the components of the vector. This latter approach lends itself well to calculation, but in some respects the kindergarten version is better, because it allows us to describe relations among vectors without specifying a coordinate system. For instance, a statement that one vector is parallel to a second vector, or perpendicular to a third, has nothing to do with how we choose our coordinate system.
Non-commuting quantities and hidden parameters – Wave-corpuscular dualism and hidden parameters – Local or nonlocal hidden parameters – Phase space in quantum mechanics – Weyl, Wigner, and Moyal – Von Neumann’s theorem about the absence of hidden parameters in quantum mechanics and Hermann – Bell’s objection – Quantum-mechanical and mathematical incommeasurability – Kochen – Specker’s idea about their equivalence – The notion of partial algebra – Embeddability of a qubit into a bit – Quantum computer is not Turing machine – Is continuality universal? – Diffeomorphism and velocity – Einstein’s general principle of relativity – „Mach’s principle“ – The Skolemian relativity of the discrete and the continuous – The counterexample in § 6 of their paper – About the classical tautology which is untrue being replaced by the statements about commeasurable quantum-mechanical quantities – Logical hidden parameters – The undecidability of the hypothesis about hidden parameters – Wigner’s work and и Weyl’s previous one – Lie groups, representations, and psi-function – From a qualitative to a quantitative expression of relativity − psi-function, or the discrete by the random – Bartlett’s approach − psi-function as the characteristic function of random quantity – Discrete and/ or continual description – Quantity and its “digitalized projection“ – The idea of „velocity−probability“ – The notion of probability and the light speed postulate – Generalized probability and its physical interpretation – A quantum description of macro-world – The period of the as-sociated de Broglie wave and the length of now – Causality equivalently replaced by chance – The philosophy of quantum information and religion – Einstein’s thesis about “the consubstantiality of inertia ant weight“ – Again about the interpretation of complex velocity – The speed of time – Newton’s law of inertia and Lagrange’s formulation of mechanics – Force and effect – The theory of tachyons and general relativity – Riesz’s representation theorem – The notion of covariant world line – Encoding a world line by psi-function – Spacetime and qubit − psi-function by qubits – About the physical interpretation of both the complex axes of a qubit – The interpretation of the self-adjoint operators components – The world line of an arbitrary quantity – The invariance of the physical laws towards quantum object and apparatus – Hilbert space and that of Minkowski – The relationship between the coefficients of -function and the qubits – World line = psi-function + self-adjoint operator – Reality and description – Does a „curved“ Hilbert space exist? – The axiom of choice, or when is possible a flattening of Hilbert space? – But why not to flatten also pseudo-Riemannian space? – The commutator of conjugate quantities – Relative mass – The strokes of self-movement and its philosophical interpretation – The self-perfection of the universe – The generalization of quantity in quantum physics – An analogy of the Feynman formalism – Feynman and many-world interpretation – The psi-function of various objects – Countable and uncountable basis – Generalized continuum and arithmetization – Field and entanglement – Function as coding – The idea of „curved“ Descartes product – The environment of a function – Another view to the notion of velocity-probability – Reality and description – Hilbert space as a model both of object and description – The notion of holistic logic – Physical quantity as the information about it – Cross-temporal correlations – The forecasting of future – Description in separable and inseparable Hilbert space – „Forces“ or „miracles“ – Velocity or time – The notion of non-finite set – Dasein or Dazeit – The trajectory of the whole – Ontological and onto-theological difference – An analogy of the Feynman and many-world interpretation − psi-function as physical quantity – Things in the world and instances in time – The generation of the physi-cal by mathematical – The generalized notion of observer – Subjective or objective probability – Energy as the change of probability per the unite of time – The generalized principle of least action from a new view-point – The exception of two dimensions and Fermat’s last theorem
A type of mechanics will be presented that possesses some distinctive properties. On the one hand, its physical description & rules of operation are readily comprehensible & intuitively clear. On the other, it fully satisfies all observable predictions of non-relativistic quantum mechanics. Within it, particles exist at points in space, follow continuous, piecewise differentiable paths, and their linear momentum is equal to their mass times their velocity along their path. Yet the probabilities for position and momentum, conditioned on the state of the particle's environment, follow the rules of quantum theory. Indeed, all observable consequences of quantum theory are satisfied; particles can be entangled, have intrinsic spin, this spin is not local to the particle, particle identity can effect probabilities, and so forth. All the rules of quantum mechanics are obeyed, and all arise in a straightforward fashion. After this is established, connections will be drawn out between this type of mechanics and other types of quantum worlds; those that obey Bohmian mechanics, stochastic mechanics, the many worlds interpretation, and physical collapse. In the final section, a relativistic version of the mechanics will be presented.