Trace Formulae with Spin

Trace formulas are derived for the coefficients of a matrix differential equation representing wave propagation in a medium which supports N types of waves. In general these waves are coupled to each other but are assumed to become uncoupled as lzl -> ?o. The two coefficients of the ordinary differential equation, N x N matrices, are assumed known for z 0. A point source is located in the region z 0. The unique features of this work are a derivation of the trace formula in which the Jost functions are not needed, an alternate measurement method based on an impedance concept and a trace equation based on the impedance concept. Introduction. Motivated by the work of Trubowitz (1), trace formula methods were introduced by Deift and Trubowitz (2) to study one-dimensional inverse quantum mechanical scattering problems on the line. The trace formula expresses the potential as a functional of the reflection coefficient. Greene (3) used the same techniques to study the one-dimensional acoustical wave equation with the angular frequency playing the role of the quantum mechanical energy. Stickler and Deift (4) examined the acoustic problem of a point source in an inhomogeneous half space bounded by a pressure release surface. The sound speed was recovered from a measurement of the normal component of the acoustic velocity at the pressure release surface. The role of the quantum mechanical energy was played by a vertical wave number, which does not correspond to either a spatial or temporal derivative. Using again the analogy between vertical wave number (the wave number in the direction of the coefficients' variation) and the quantum mechanical energy, Stickler (5) showed how the trace formula methods could be used to recover the sound speed when it approaches different values at plus and minus infinity. In a quantum mechanical context this means that the potential approaches zero, say at minus infinity, but some positive constant, not necessarily known, at plus infinity. (In the Deift-Trubowitz analysis the potential approached zero in both limits.) Stickler (6) also discussed this same problem in an electromagnetic context. Tomei (7) has used the trace methods to recover the coefficients of a linear, third order equation associated with the Boussinesq equation. In the present paper the trace method is extended to a one-dimensional vector Helmholtz equation which supports N types of waves and for which the waves are coupled to each other except at infinity. The trace method yields a functional relation- ship between the coefficients of the Helmholtz equation, a reflection coefficient described below and assumed known and a specific solution of the vector Helmholtz equation with known and prescribed initial conditions. These relationships are called the trace formulas and are derived here. The coefficients of the vector Helmholtz differential equation can be replaced by these trace formulas and this results in a system of coupled, nonlinear ordinary differential equations with known initial condi- tions. Once this solution is determined, the coefficients can be recovered. The existence of a solution of this coupled system is not given.

We derive semiclassical trace formulae including Gutzwiller's trace formula using coherent states. This formulation has several advantages over the usual coordinate‐space formulation. Using a coherent‐state basis makes it immediately obvious that classical periodic orbits make separate contributions to the trace of the quantum‐mechanical time evolution operator. In addition, our approach is manifestly canonically invariant at all stages, and leads to the simplest possible derivation of Gutzwiller's formula.

Using quantum maps, we study the accuracy of semiclassical trace formulas. The role of chaos in improving the semiclassical accuracy in some systems is demonstrated quantitatively. However, our study of the standard map cautions that this may not be most general. While studying a sawtooth map we demonstrate the rather remarkable fact that at the level of the time one trace even in the presence of fixed points on singularities the trace formula may be exact, and in any case has no logarithmic divergences observed for the quantum bakers map. As a byproduct we introduce fantastic periodic curves akin to curlicues.

Path integration over closed loops and Gutzwiller's trace formula

Perturbation theory has been one of the most frequently used and most powerful tools of quantum mechanics. The very foundations of relativistic quantum theory—quantum electrodynamics—are perturbative in nature. Many-body perturbation theory has been used for electron correlation treatments since the early days of quantum chemistry, and in more recent times multireference perturbation theories have been developed to provide quantitative or semiquantitative information in very complex systems. In the beginnings of relativistic quantum mechanics, perturbation methods based on an expansion in powers of the fine structure constant, α = 1/c, were used extensively to obtain operators that would provide a connection with nonrelativistic quantum mechanics and permit some evaluation of relativistic corrections, in days well before the advent of the computer. This seems a reasonable approach, considering the small size of the fine structure constant—and for light elements it has been found to work remarkably well. Relativity is a small perturbation for a good portion of the periodic table. Perturbation expansions have their limitations, however, and as well as successes, there have been failures due to the highly singular or unbounded nature of the operators in the perturbation expansions. Therefore, in recent times other perturbation approaches have been developed to provide alternatives to the standard Breit–Pauli approach. This chapter is devoted to the development of perturbation expansions in powers of 1/c from the Dirac equation. In the previous chapter, the Pauli Hamiltonian was developed using the Foldy–Wouthuysen transformation. While this is an elegant method, it is probably simpler to make the derivation from the elimination of the small component with expansion of the denominator, and it is this approach that we use here. Another convenient approach is to make use of the modified Dirac equation in the limit of equality of the large and pseudo-large components. This approach enables us to draw on results from the modified Dirac approach in developing the two-electron terms of the Breit–Pauli Hamiltonian. We then demonstrate how the use of perturbation theory for relativistic corrections requires that multiple perturbation theory be employed for correlation effects and for properties.

The principal families of periodic orbits that emerge from the stationary points of the six-dimensional potential energy surface of the C2H2 molecular system, as well as periodic orbits from saddle-node bifurcations, have been located and propagated for an energy range up to 36 500 cm−1 above the absolute minimum of the potential. The bifurcation diagrams of these periodic orbits reveal the regions of phase space where the dynamics are regular or chaotic (with soft or hard chaos) for acetylene, vinylidene, and the region over these two isomers. An association of the structure of phase space with spectroscopic findings is made by calculating Gutzwiller’s semiclassical trace formula and classical survival probability functions.

info:eu-repo/semantics/published

Quantum and Classical Ergodicity of Spinning Particles

The sets of -component irreducible and Clifford algebraic Hermitian and unitary matrices through the two-component Pauli matrices are suggested, where s = 1/2, 3/2, 5/2, … . Using these matrix sets, the eigenvalues of which are , the -component generalized Dirac equation for a description of arbitrary half-integral spin particles is constructed. In accordance with the correspondence principle, the generalized Dirac equation suggested arises from the condition of relativistic invariance. This equation is reduced to the sets of two-component matrix equations the number of which is equal to . The new relativistic invariant equation of motion leads to an equation of continuity with a positive-definite probability density and also to the Klein–Gordon equation. This relativistic equation is causal in the presence of an external electromagnetic field interaction. It is shown that, in the case of nonrelativistic limit, the relativistic equation presented is reduced to the Pauli equation describing the motion of half-integral spin particle in the electromagnetic field.

Quantum theory of half-integer spin free particles from the perspective of the Majorana equation

The first examples of supersymmetric quantum systems we have mentioned in Sect. 2.1.1 are the Pauli and Dirac Hamiltonian. Here we will discuss some of the implications of supersymmetry for systems which are characterized by these Hamiltonians. In particular, we will consider the Pauli paramagnetism of a two-and three- dimensional non-interacting electron gas. The (IDOS regularized) Witten index is shown to be related to the zero-temperature magnetization of such systems. It is also shown that breaking of SUSY gives rise to counterexamples of the so-called paramagnetic conjecture.

The phase shift due to gravitational field and all gauge fields in the interference of two coherent beams is obtained. In the case of gravitation, it is shown, for a particle with arbitrary spin, that there exists a phase shift due to the coupling of spin to space-time curvature. This and the corresponding phase shift for gauge fields are analogous to the Aharonov-Bohm effect. The classical limit for particles moving in gravitational and gauge fields is obtained from the phase shift. For gravitation, in the absence of torsion, this is the Mathisson-Papapetrou equation, which is thereby shown to be the classical limit of the Dirac and Bargmann-Wigner wave equations, generalized to curved space-time. In the presence of torsion, a modification of this equation, given by Hehl, is obtained. It is pointed out that gravity is not a pure gauge field and that it must be placed in the more general category of an «interference field» which contains both gravity and gauge fields as special cases. The field equations for gauge fields and gravity are obtained from the heuristic assumption that a particle acts on a field in a manner which depends on how it responds to the field via the phase shift. For gauge fields, they contain the Yang-Mills equations as a special case. For gravity, a modification of Einstein’s field equations is obtained, which corresponds to the Lagrangian (1/16πK) ·, · (2Λ +R) + (1/32πf)RμνϱσRμνσϱ, where the Riemann tensor contains torsion andK, f, Λ are constants (Λ may be zero). The relevance of the phase shift, due to rotation, to the quantization of vortices in superfluid helium is pointed out. This suggests that the curl of the superfluid velocity may obey a system of equations analogous to Maxwell’s equations and the analogue of the magnetic monopole for superfluid helium is also introduced.

The authors show that any relativistic wave equations for a particle with mass m>0 and arbitrary spin s is invariant under the Lie algebra of the group GL(2s+1, C). The explicit form of basis elements of this algebra is given for any s. The complete sets of the symmetry operators of the Dirac and Maxwell equations are obtained, which belong to the classes of the first and second-order differential operators with matrix coefficients. Corresponding new conservation laws and constants of motion are found.

In this chapter we refine the global description of the endoscopic lift obtained in Corollary 4.2. The main result obtained in this chapter is Theorem 5.2, which is a special case of a global multiplicity formula conjectured by Arthur.We consider the symplectic group GSp(4) over an arbitrary totally real number field F. In the special case F = Q and for irreducible automorphic representations π with π∞ in the discrete series the proof of this theorem is given in Sects. 5.2 and 5.3. In Sect. 5.1 we explain how the local endoscopic character lift is constructed for arbitrary local base fields of characteristic zero. In Sects. 5.4 and 5.5 we explain how the Arthur–Selberg trace formula can be used to extend the results obtained in Sects. 5.2 and 5.3 to the case of arbitrary totally real number fields and arbitrary irreducible automorphic not necessarily cohomological representations ?. The proof of Theorem 5.2 is based on two ingredients: the principle of exchange and the key formula (stated in Sect. 5.3). The latter can be directly deduced from weak versions of the trace formula (see Chap. 4, Corollary 4.2, or Sect. 5.5). It deals with simultaneous changes of global representations at two places. The principle of exchange, on the other hand, deals with an exchange of the representation at one place exclusively, and its proof boils down to a special case of the Hasse–Brauer–Noether theorem (Lemma 5.4). This is based on an explicit theta lift. At this point we use [56], which unfortunately forces us to make the restrictive assumption that F is a totally real number field. Since we apply results obtained in [56] in a rather specific case, it is very likely that this restriction on F can be removed.KeywordsTrace FormulaDiscrete SeriesAutomorphic RepresentationCuspidal RepresentationQuadratic CharacterThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Trace formulas (Lagrange, Jacobi–Kronecker, Bergman–Weil) play a key role in division problems in analytic or algebraic geometry (including arithmetic aspects, see, for example, Berenstein and Yger in Am. J. Math. 121(4):723–796, 1999). Unfortunately, they usually hold within the restricted frame of complete intersections. Besides the fact that it allows one to carry local or semi-global analytic problems to a global geometric setting (think about Crofton’s formula), averaging the Cauchy kernel (from ℂn∖{z 1…z n =0}⊂ℙn(ℂ)), in order to get the Bochner–Martinelli kernel (in ℂn+1∖{0}⊂ℙn+1(ℂ)=ℂn+1∪ℙn(ℂ)), leads to the construction of explicit candidates for the realization of Grothendieck’s duality, namely BM residue currents (Passare et al. in Publ. Mat. 44:85–117, 2000; Andersson in Bull. Sci. Math. 128(6):481–512, 2004; Andersson and Wulcan in Ann. Sci. École Norm. Super. 40:985–1007, 2007), extending thus the cohomological incarnation of duality which appears in the complete intersection or Cohen–Macaulay cases. We will recall here such constructions and, in parallel, suggest how far one could take advantage of the multiplicative inductive construction introduced by Coleff and Herrera (Lecture Notes in Mathematics, vol. 633, Springer, Berlin, 1978), by relating it to the Stückrad–Vogel algorithm developed in Stückrad and Vogel (Queen Pap. Pure Appl. Math. 61:1–32, 1982), Tworzewski (Ann. Pol. Math. 62:177–191, 1995), Andersson et al. (arXiv:1009.2458, 2010) toward improper intersection theory. Results presented here were initiated all along my long-term collaboration with Carlos Berenstein. To both of us, the mathematical work of Leon Ehrenpreis certainly remained a constant and very stimulating source of inspiration. This presentation relies also deeply on my collaboration over the past years with M. Andersson, H. Samuelsson, and E. Wulcan in Göteborg.KeywordsComplete IntersectionGlobal SectionHolomorphic SectionCartier DivisorIntegration CurrentThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.