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Multi-Hamiltonian systems are investigated by using the Hamilton–Jacobi method. Integration of a set of total differential equations which includes the equations of motion and the action integral function is discussed. It is shown that this set is integrable if and only if the total variations of the Hamiltonians vanish. Two examples are studied.

We consider a homogeneous and isotropic Universe, described by the minisuperspace Lagrangian with the scale factor as a generalized coordinate. We show that the energy of a closed Universe is zero. We apply the uncertainty principle to this Lagrangian and propose that the quantum uncertainty of the scale factor causes the primordial fluctuations of the matter density. We use the dynamics of the early Universe in the Einstein–Cartan theory of gravity with spin and torsion, which eliminates the big-bang singularity and replaces it with a nonsingular bounce. Quantum particle production in highly curved spacetime generates a finite period of cosmic inflation that is consistent with the Planck satellite data. From the inflated primordial fluctuations we determine the magnitude of the temperature fluctuations in the cosmic microwave background, as a function of the numbers of the thermal degrees of freedom of elementary particles and the particle production coefficient which is the only unknown parameter.

In this paper, we investigate the behavior of the energy eigenvalues of the Schrödinger equation by using the canonical quantization method. We obtain the Hamiltonian of the Schrödinger equation by the Lagrangian in terms of the new coordinates. Then we calculate the partition function by the eigenvalues and the thermodynamic properties of the system in the superstatistics formalism for the modified Dirac delta and the Gamma distributions. All results in the limiting cases satisfy that of the harmonic oscillator. Furthermore, the effects of the all parameters in the problem of energy eigenvalues and thermodynamic properties are calculated and shown graphically.

In the one-dimensional stationary case, we construct a mechanical Lagrangian describing the quantum motion of a nonrelativistic spinless system. This Lagrangian is written as a difference between a function T, which represents the quantum generalization of the kinetic energy and which depends on the coordinate x and the temporal derivatives of x up the third order, and the classical potential V(x). The Hamiltonian is then constructed and the corresponding canonical equations are deduced. The function T is first assumed to be arbitrary. The development of T in a power series together with the dimensional analysis allow us to fix univocally the series coefficients by requiring that the well-known quantum stationary Hamilton–Jacobi equation be reproduced. As a consequence of this approach, we formulate the law of the quantum motion representing a new version of the quantum Newton law. We also analytically establish the famous Bohm relation outside the framework of the hydrodynamical approach and show that the well-known quantum potential, although it is a part of the kinetic term, plays really the role of an additional potential as assumed by Bohm.

Peierls brackets are part of the space-time approach to quantum field theory, and provide a Poisson bracket which, being defined for pairs of observables which are group invariant, is group invariant by construction. It is therefore well suited for combining the use of Poisson brackets and the full diffeomorphism group in general relativity. The present paper provides an introduction to the topic, with applications to gauge field theory.

After the last missing piece, the Higgs particle, has probably been identified, the Standard Model of the subatomic particles appears to be a quite robust structure, that can survive on its own for a long time to come. Most researchers expect considerable modifications and improvements to come in the near future, but it could also be that the Model will stay essentially as it is. This, however, would also require a change in our thinking, and the question remains whether and how it can be reconciled with our desire for our theories to be “natural”.

Several applications of Lie symmetries and its generalisation are presented: from turning butterflies into tornados, to its applications in epidemics, population dynamics, and ultimately converting classical problems into the quantum realm. Applications of nonclassical symmetries are also illustrated.