Symmetries and motions in manifolds

In these lectures the relations between symmetries, Lie algebras, Killing vectors and Noether's theorem are reviewed. A generalisation of the basic ideas to include velocity-dependent co-ordinate transformations naturally leads to the concept of Killing tensors. Via their Poisson brackets these tensors generate an a priori infinite-dimensional Lie algebra. The nature of such infinite algebras is clarified using the example of flat space-time. Next the formalism is extended to spinning space, which in addition to the standard real co-ordinates is parametrised also by Grassmann-valued vector variables. The equations for extremal trajectories (“geodesics”) of these spaces describe the pseudo-classical mechanics of a Dirac fermion. We apply the formalism to solve for the motion of a pseudo-classical electron in Schwarzschild space-time.     

Absolutely anticommuting (anti-)BRST symmetry transformations for topologically massive Abelian gauge theory

We demonstrate the existence of the nilpotent and absolutely anticommuting Becchi–Rouet–Stora–Tyutin (BRST) and anti-BRST symmetry transformations for the four (3+1)-dimensional (4D) topologically massive Abelian U(1) gauge theory that is described by the coupled Lagrangian densities (which incorporate the celebrated (B∧F) term). The absolute anticommutativity of the (anti-) BRST symmetry transformations is ensured by the existence of a Curci–Ferrari type restriction that emerges from the superfield formalism as well as from the equations of motion which are derived from the above coupled Lagrangian densities. We show the invariance of the action from the point of view of the symmetry considerations as well as superfield formulation. We discuss, furthermore, the topological term within the framework of superfield formalism and provide the geometrical meaning of its invariance under the (anti-)BRST symmetry transformations.     

Pseudointegrable systems in classical and quantum mechanics

We study particles moving in planar polygonal enclosures with rational angles, and show by several methods that trajectories in the classical phase space explore two-dimensional invariant surfaces which are generically not tori as in integrable systems but instead have the topology of multiply-handled spheres. The quantum mechanics of one such ‘pseudointegrable system’ is studied in detail by computing energy levels using an exact formalism. This system consists of motion on a unit coordinate torus containing a square reflecting obstacle with side L. We find that neighbouring levels avoid degeneracies as L varies, and that the probability distribution for the spacing S of adjacent levels vanishes linearly as S→0 (‘level repulsion’). The Weyl area rule plus edge and corner corrections gives a very accurate approximation for the mean level density. Oscillatory corrections to the mean level density are given as a sum over closed classical paths; for pseudointegrable systems these closed paths form families covering part of the phase-space invariant surfaces.     

Role of surface integrals in the Hamiltonian formulation of general relativity

It is shown that if the phase space of general relativity is defined so as to contain the trajectories representing solutions of the equations of motion then, for asymptotically flat spaces, the Hamiltonian does not vanish but its value is given rather by a nonzero surface integral. If the deformations of the surface on which the state is defined are restricted so that the surface moves asymptotically parallel to itself in the time direction, then the surface integral gives directly the energy of the system, prior to fixing the coordinates or solving the constraints. Under more general conditions (when asymptotic Poincaré transformations are allowed) the surface integrals giving the total momentum and angular momentum also contribute to the Hamiltonian. These quantities are also identified without reference to a particular fixation of the coordinates. When coordinate conditions are imposed the associated reduced Hamiltonian is unambiguously obtained by introducing the solutions of the constraints into the surface integral giving the numerical value of the unreduced Hamiltonian. In the present treatment there are therefore no divergences that cease to be divergences after coordinate conditions are imposed. The procedure of reduction of the Hamiltonian is explicity carried out for two cases: (a) Maximal slicing, (b) ADM coordinate conditions.A Hamiltonian formalism which is manifestly covariant under Poincaré transformations at infinity is presented. In such a formalism the ten independent variables describing the asymptotic location of the surface are introduced, together with corresponding conjugate momenta, as new canonical variables in the same footing with the g_ij, π^ij. In this context one may fix the coordinates in the “interior” but still leave open the possibility of making asymptotic Poincaré transformations. In that case all ten generators of the Poincaré group are obtained by inserting the solution of the constraints into corresponding surface integrals.     

The Quantum Harmonic Oscillator in the ESR Model

The ESR model proposes a new theoretical perspective which incorporates the mathematical formalism of standard (Hilbert space) quantum mechanics (QM) in a noncontextual framework, reinterpreting quantum probabilities as conditional on detection instead of absolute. We have provided in some previous papers mathematical representations of the physical entities introduced by the ESR model, namely observables, properties, pure states, proper and improper mixtures, together with rules for calculating conditional and overall probabilities, and for describing transformations of states induced by measurements. We study in this paper the relevant physical case of the quantum harmonic oscillator in our mathematical formalism. We reinterpret the standard quantum rules for probabilities, provide new expressions for absolute probabilities, and show how the standard state transformations must be modified according to the ESR model.     

Electromagnetic and Force Field Equations

In classical physics the electromagnetic equations are described by Maxwell's equations. Maxwell's equations proved to be invariant under gauge, or Lorentz transformations. Also, Einstein's equations of the special theory of relativity are invariant under Lorentz transformations. On the other hand classical mechanics and quantum mechanics laws are invariant under Galilean transformations. This means that, there are two different dynamical structures describing our universe. Einstein's unified field theory failled in putting our universe in one dynamical structure. New electromagnetic and force field equations are going to be derived. They have the same shape like Maxwell's equations, but with different dynamical structure. Those equations are invariant under Galilean transformations and in the density matrix formalism of quantum mechanics.     

A Hamilton-Jacobi formalism for thermodynamics

We show that classical thermodynamics has a formulation in terms of Hamilton-Jacobi theory, analogous to mechanics. Even though the thermodynamic variables come in conjugate pairs such as pressure/volume or temperature/entropy, the phase space is odd-dimensional. For a system with n thermodynamic degrees of freedom it is 2n+1-dimensional. The equations of state of a substance pick out an n-dimensional submanifold. A family of substances whose equations of state depend on n parameters define a hypersurface of co-dimension one. This can be described by the vanishing of a function which plays the role of a Hamiltonian. The ordinary differential equations (characteristic equations) defined by this function describe a dynamical system on the hypersurface. Its orbits can be used to reconstruct the equations of state. The ‘time’ variable associated to this dynamics is related to, but is not identical to, entropy. After developing this formalism on well-grounded systems such as the van der Waals gases and the Curie-Weiss magnets, we derive a Hamilton-Jacobi equation for black hole thermodynamics in General Relativity. The cosmological constant appears as a constant of integration in this picture.     

Nonlocal conserved quantities,balance laws,and equations of motion

A formalism is developed whereby balance laws are directly obtained from nonlocal (integrodifferential) linear second-order equations of motion for systems described by several dependent variables. These laws augment the equations of motion as further useful information about the physical system and, under certain conditions, are shown to reduce to conservation laws. The formalism can be applied to physical systems whose equations of motion may be relativistic and either classical or quantum. It is shown to facilitate obtaining global conservation laws for quantities which include energy and momentum. Applications of the formalism are given for a nonlocal Schrödinger equation and for a system of local relativistic equations of motion describing particles of arbitrary integral spin.     

A probabilistic analysis of the difficulties of unifying quantum mechanics with the theory of relativity

A procedure is given for the transformation of quantum mechanical operator equations into stochastic equations. The stochastic equations reveal a simple correlation between quantum mechanics and classical mechanics: Quantum mechanics operates with “optimal estimations,” classical mechanics is the limit of “complete information.” In this connection, Schrödinger's substitution relationsp _x → -i? ?/?x, etc, reveal themselves as exact mathematical transformation formulas. The stochastic version of quantum mechanical equations provides an explanation for the difficulties in correlating quantum mechanics and the theory of relativity: In physics “time” is always thought of as a numerical parameter; but in the present formalism of physics “time” is described by two formally totally different quantities. One of these two “times” is a numerical parameter and the other a random variable. This last concept of time shows all the properties required by the theory of relativity and is therefore to be considered as the relativistic time.     

Canonical formulation of the spherically symmetric einstein-Yang-Mills-Higgs system for a general gauge group

The dynamics of the spherically symmetric system of gravitation interacting with scalar and Yang-Mills fields is presented in the context of the canonical formalism. The gauge group considered is a general (compact and semisimple) N parameter group. The scalar (Higgs) field transforms according to an unspecified M-dimensional orthogonal representation of the gauge group. The canonical formalism is based on Dirac's techniques for dealing with constrained hamiltonian systems. First the condition that the scalar and Yang-Mills fields and their conjugate momenta be spherically symmetric up to a gauge is formulated and solved for global gauge transformations, finding, in a general gauge, the explicit angular dependence of the fields and conjugate momenta. It is shown that if the gauge group does not admit a subgroup (locally) isomorphic to the rotation group, then the dynamical variables can only be manifestly spherically symmetric. If the opposite is the case, then the number of allowed degrees of freedom is connected to the angular momentum content of the adjoint representation of the gauge group. Once the suitable variables with explicit angular dependence have been obtained, a reduced action is derived by integrating away the angular coordinates. The canonical formulation of the problem is now based on dynamical variables depending only on an arbitrary radial coordinate r and an arbitrary time coordinate t. Besides the gravitational variables, the formalism now contains two pairs of N-vector variables (R, π_r), (Θ, π_Θ), corresponding to the allowed Yang-Mills degrees of freedom and one pair of M-vector variables, (h, π_h), associated with the original scalar field. The reduced Hamiltonian is invariant under a group of r-dependent gauge transformations such that R plays the role of the gauge field (transforming in the typically inhomogeneous way) and in terms of which the gauge covariant derivatives of Θ and h naturally appear. No derivatives of R appear in the Hamiltonian and the gauge freedom allows us to define a gauge in which R is zero. Also the r and t coordinates are fixed in a way consistent with the equations of motion. Some nontrivial static solutions are found. One of these solutions is given in closed form; it is singular and corresponds to a generalization of the singular solution found in the literature with different degrees of generality and the geometry is described by the Reissner-Nordström metric. The other solution is defined through its asymptotic behavior. It generalizes to curved space the finite energy solution discyssed by Julia and Zee in flat space.     

An extension of the coupled cluster formalism to excited states (I)

The expS method (coupled cluster formalism) is extended to excited states of finite and infinite systems. We obtain equations which are formally similar to the known ground-state equations of the expS theory. The method is applicable to Fermi as well as Bose systems.     

S Q E D 4 and Q E D 4 on the Null-Plane

We study the scalar electrodynamics (S Q E D ₄) and the spinor electrodynamics (Q E D ₄) in the null-plane formalism. We follow Dirac’s technique for constrained systems to analyze the constraint structure in both theories in detail. We impose the appropriate boundary conditions on the fields to fix the hidden subset first class constraints that generate improper gauge transformations and obtain a unique inverse of the second-class constraint matrix. Finally, choosing the null-plane gauge condition, we determine the generalized Dirac brackets of the independent dynamical variables, which via the correspondence principle give the (anti)-commutators for posterior quantization.     

The entropy function for the extremal Kerr-(anti-)de Sitter black holes

Based on Sen's entropy function formalism, we consider the Bekenstein-Hawking entropy of the extremal Kerr-(anti-)de Sitter black holes in 4-dimensions. Unlike the extremal Kerr black hole case with flat asymptotic geometry, where the Bekenstein-Hawking entropy S is proportional to the angular momentum J, we get a quartic algebraic relation between S and J by using the known solution to the Einstein equation. We recover the same relation in the entropy function formalism. Instead of full geometry, we write down an ansatz for the near horizon geometry only. The exact form of the unknown functions and parameters in the ansatz are obtained by solving the differential equations which extremize the entropy function. The results agree with the nontrivial relation between S and J.We also study the Gauss-Bonnet correction to the entropy exploiting the entropy function formalism. We show that the term, though being topological thus does not affect the solution, contributes a constant addition to the entropy because the term shifts the Hamiltonian by that amount.     

Statistical mechanics of Coulomb gases as quantum theory on Riemann surfaces

Statistical mechanics of a 1D multivalent Coulomb gas can be mapped onto non-Hermitian quantum mechanics. We use this example to develop the instanton calculus on Riemann surfaces. Borrowing from the formalism developed in the context of the Seiberg-Witten duality, we treat momentum and coordinate as complex variables. Constant-energy manifolds are given by Riemann surfaces of genus g ≥ 1. The actions along principal cycles on these surfaces obey the ordinary differential equation in the moduli space of the Riemann surface known as the Picard-Fuchs equation. We derive and solve the Picard-Fuchs equations for Coulomb gases of various charge content. Analysis of monodromies of these solutions around their singular points yields semiclassical spectra as well as instanton effects such as the Bloch bandwidth. Both are shown to be in perfect agreement with numerical simulations.     

On electromagnetic fields in the Hamiltonian description of continua

The group theoretical structure of an infinite dimensional Hamiltonian formulation of continuum mechanics is studied using as an example the Maxwell-Vlasov system. In contrast to earlier works, electromagnetism and charged matter are coupled via Poisson brackets without using the vector potential. The charged matter is described on the group of canonical transformations on R⁶ and we show that its evolution arises from a symplectic structure, modified by the magnetic field. The configurations of the electromagnetic field must be constrained by the physical requirement of the Gauss law. With the energy-functional taken as a Hamiltonian this leads - even for relativistic particles - to the well-known equations of motion.     

Objectivity in Perspective: Relationism in the Interpretation of Quantum Mechanics

Pekka Lahti is a prominent exponent of the renaissance of foundational studies in quantum mechanics that has taken place during the last few decades. Among other things, he and coworkers have drawn renewed attention to, and have analyzed with fresh mathematical rigor, the threat of inconsistency at the basis of quantum theory: ordinary measurement interactions, described within the mathematical formalism by Schrödinger-type equations of motion, seem to be unable to lead to the occurrence of definite measurement outcomes, whereas the same formalism is interpreted in terms of probabilities of precisely such definite outcomes. Of course, it is essential here to be explicit about how definite measurement results (or definite properties in general) should be represented in the formalism. To this end Lahti et al. have introduced their objectification requirement that says that a system can be taken to possess a definite property if it is certain (in the sense of probability 1) that this property will be found upon measurement. As they have gone on to demonstrate, this requirement entails that in general definite outcomes cannot arise in unitary measuring processes.In this paper we investigate whether it is possible to escape from this deadlock. As we shall argue, there is a way out in which the objectification requirement is fully maintained. The key idea is to adapt the notion of objectivity itself, by introducing relational or perspectival properties. It seems that such a “relational perspective” offers prospects of overcoming some of the long-standing problems in the interpretation of quantum mechanics.     

广义Raitzin正则方程的Hojman守恒定理

利用时间不变的无限小变换下的Lie对称性，研究广义经典力学中Raitzin正则方程的Hojman 守恒定理。建立广义Raitzin正则方程。给出无限小变换下Lie对称性的确定方程。建立系统的Hojman守恒定理，并举例说明结果的应用。