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Energy Quantisation and Time Parameterisation
Full text linkWe show that if space is compact, then trajectories
cannot be defined in the framework of the quantum
Hamilton–Jacobi (HJ) equation. The starting point is the
simple observation that when the energy is quantised it is
not possible to make variations with respect to the energy,
and the time parameterisation t − t_0 = ∂S_0/∂E, implied by
Jacobi’s theorem, which leads to the group velocity, is ill
defined. It should be stressed that this follows directly from
the quantum HJ equation without any axiomatic assumption
concerning the standard formulation of quantum mechanics.
This provides a stringent connection between the quantum HJ
equation and the Copenhagen interpretation. Together with
tunnelling and the energy quantisation theorem for confining
potentials, formulated in the framework of quantum HJ equation,
it leads to the main features of the axioms of quantum
mechanics from a unique geometrical principle. Similar to
the case of the classical HJ equation, this fixes its quantum
analog by requiring that there exist point transformations,
rather than canonical ones, leading to the trivial hamiltonian.
This is equivalent to a basic cocycle condition on the states.
Such a cocycle condition can be implemented on compact
spaces, so that continuous energy spectra are allowed only
as a limiting case. Remarkably, a compact space would also
imply that the Dirac and von Neumann formulations of quantum
mechanics essentially coincide.We suggest that there is
a definition of time parameterisation leading to trajectories
in the context of the quantum HJ equation having the probabilistic
interpretation of the Copenhagen School
The Equivalence Postulate of Quantum Mechanics, Dark Energy, and the Intrinsic Curvature of Elementary Particles
Full text linkThe equivalence postulate of quantum mechanics offers an axiomatic approach
to quantum field theories and quantum gravity. The equivalence hypothesis
can be viewed as adaptation of the classical Hamilton-Jacobi formalism
to quantum mechanics. The construction reveals two key identities that
underlie the formalism in Euclidean or Minkowski spaces. The first is a cocycle
condition, which is invariant under
D
-dimensional Möbius transformations
with Euclidean or Minkowski metrics. The second is a quadratic identity which
is a representation of the
D
-dimensional quantum Hamilton-Jacobi equation.
In this approach, the solutions of the associated Schrödinger equation are used
to solve the nonlinear quantum Hamilton-Jacobi equation. A basic property
of the construction is that the two solutions of the corresponding Schrödinger
equation must be retained. The quantum potential, which arises in the formalism,
can be interpreted as a curvature term. The author proposes that the quantum
potential, which is always nontrivial and is an intrinsic energy term characterising
a particle, can be interpreted as dark energy. Numerical estimates of
its magnitude show that it is extremely suppressed. In the multiparticle case
the quantum potential, as well as the mass, is cumulative
Toward the classification of the realistic free fermionic models
Full text linkThe realistic free fermionic models have had remarkable success in providing plausible explanations for various properties of the Standard Model which include the natural appearance of three generations, the explanation of the heavy top quark mass and the qualitative structure of the fermion mass spectrum in general, the stability of the proton and more. These intriguing achievements makes evident the need to understand the general space of these models. While the number of possibilities is large, general patterns can be extracted. In this paper the author presents a detailed discussion on the construction of the realistic free fermionic models with the aim of providing some insight into the basic structures and building blocks that enter the construction. The role of free phases in the determination of the phenomenology of the models is discussed in detail. The author discusses the connection between the free phases and mirror symmetry in (2,2) models and the corresponding symmetries in the case of (2,0) models. The importance of the free phases in determining the effective low energy phenomenology is illustrated in several examples. The classification of the models in terms of boundary condition selection rules, real world-sheet fermion pairings, exotic matter states and the hidden sector is discussed
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