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We will read, through the Emmy Noether paper and the two concepts of `proper and `improper conservation laws, the problem, posed by Hilbert, of the nature of the law of conservation of energy in the theory of General Relativity. Epistemological issues involved with the two kind of conservation laws will be enucleate.
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In this communication, we show that both infinite-dimension
When discussing consequences of symmetries of dynamical systems based on Noethers first theorem, most standard textbooks on classical or quantum mechanics present a conclusion stating that a global continuous Lie symmetry implies the existence of a time independent conserved Noether charge which is the generator of the action on phase space of that symmetry, and which necessarily must as well commute with the Hamiltonian. However this need not be so, nor does that statement do justice to the complete scope and reach of Noethers first theorem. Rather a much less restrictive statement applies, namely that the corresponding Noether charge as an observable over phase space may in fact possess an explicit time dependency, and yet define a constant of the motion by having a commutator with the Hamiltonian which is nonvanishing, thus indeed defining a dynamical conserved quantity. Furthermore, and this certainly within the Hamiltonian formulation, the converse statement is valid as well, namely that any dynamical constant of motion is necessarily the Noether charge of some symmetry leaving the systems action invariant up to some total time derivative contribution. The present contribution revisits these different points and their consequences, straightaway within the Hamiltonian formulation which is the most appropriate for such issues. Explicit illustrations are also provided through three general but simple enough classes of systems.
The classical quantization of a family of a quadratic Li{e}nard-type equation (Li{e}nard II equation) is achieved by a quantization scheme (M.~C. Nucci. {em Theor. Math. Phys.}, 168:994--1001, 2011) that preserves the Noether point symmetries of the underlying Lagrangian in order to construct the Schrodinger equation. This method straightforwardly yields the Schrodinger equation as given in (A.~Ghose~Choudhury and Partha Guha. {em J. Phys. A: Math. Theor.}, 46:165202, 2013).
Classical mechanics, relativity, electrodynamics and quantum mechanics are often depicted as separate realms of physics, each with its own formalism and notion. This remains unsatisfactory with respect to the unity of nature and to the necessary number of postulates. We uncover the intrinsic connection of these areas of physics and describe them using a common symplectic Hamiltonian formalism. Our approach is based on a proper distinction between variables and constants, i.e. on a basic but rigorous ontology of time. We link these concept with the obvious conditions for the possibility of measurements. The derived consequences put the measurement problem of quantum mechanics and the Copenhagen interpretation of the quantum mechanical wavefunction into perspective. According to our (onto-) logic we find that spacetime can not be fundamental. We argue that a geometric interpretation of symplectic dynamics emerges from the isomorphism between the corresponding Lie algebra and the representation of a Clifford algebra. Within this conceptional framework we derive the dimensionality of spacetime, the form of Lorentz transformations and of the Lorentz force and fundamental laws of physics as the Planck-Einstein relation, the Maxwell equations and finally the Dirac equation.
The hard problem in consciousness is the problem of understanding how physical processes in the brain could give rise to subjective conscious experience. In this paper, I suggest that in order to understand the relationship between consciousness and the physical world, we need to probe deeply into the nature of physical reality. This leads us to quantum physics and to a second explanatory gap: that between quantum and classical reality. I will seek a philosophical framework that can address these two gaps simultaneously. Our analysis of quantum mechanics will naturally lead us to the notion of a hidden reality and to the postulate that consciousness is an integral component of this reality. The framework proposed in the paper provides the philosophical underpinnings for a theory of consciousness while satisfactorily resolving the interpretation problem in quantum mechanics without the need to alter its mathematical structure. I also discuss some implications for a scientific theory of consciousness.
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