We show that the semi-classical analysis of generic Euclidean path integrals necessarily requires complexification of the action and measure, and consideration of complex saddle solutions. We demonstrate that complex saddle points have a natural interpretation in terms of the Picard-Lefschetz theory. Motivated in part by the semi-classical expansion of QCD with adjoint matter on @math , we study quantum-mechanical systems with bosonic and fermionic (Grassmann) degrees of freedom with harmonic degenerate minima, as well as (related) purely bosonic systems with harmonic non-degenerate minima. We find exact finite action non-BPS bounce and bion solutions to the holomorphic Newton equations. We find not only real solutions, but also complex solution with non-trivial monodromy, and finally complex multi-valued and singular solutions. Complex bions are necessary for obtaining the correct non-perturbative structure of these models. In the supersymmetric limit the complex solutions govern the ground state properties, and their contribution to the semiclassical expansion is necessary to obtain consistency with the supersymmetry algebra. The multi-valuedness of the action is either related to the hidden topological angle or to the resurgent cancellation of ambiguities. We also show that in the approximate multi-instanton description the integration over the complex quasi-zero mode thimble produces the most salient features of the exact solutions. While exact complex saddles are more difficult to construct in quantum field theory, the relation to the approximate thimble construction suggests that such solutions may be underlying some remarkable features of approximate bion saddles in quantum field theories.
There are earlier important works in quantum mechanics in which complexification plays some role. The main distinction between our present study and these works is the following. In our work, we have shown that the ground state properties of a generic quantum mechanical system are governed by complex (sometimes even multi-valued) saddles. In the quantum mechanical papers mentioned below, the ground state properties are always governed by real saddles, such as instantons, but complex classical solutions become important when considering the entire spectrum, for example the spectral resolvent. Gutzwiller @cite_48 @cite_2 pioneered the idea of summing over all classical solutions in semiclassical expansions of Green's functions and spectral problems. This was further developed by many authors, from a general formulation in @cite_11 @cite_55 , to specific analyses of the symmetric double-well @cite_49 @cite_69 @cite_102 and periodic potentials @cite_128 . A complex version of WKB analysis of the pure quartic oscillator also requires inclusion of complex semiclassical configurations in addition to the familiar real Bohr-Sommerfeld ones @cite_84 @cite_41 @cite_23 .
@cite_69 @cite_128 @cite_41 @cite_48 @cite_55 @cite_102 @cite_84 @cite_49 @cite_2 @cite_23 @cite_11
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Abstract Complex saddle points in the double well anharmonic oscillator are derived. The boundary conditions τ ( T )=− ϕ (− T )=1 at finite time T required for the computation of tunneling amplitudes lead to a countable set of saddle points. In the limit T → ∞, these saddle points behave as a superposition of instantons and anti-instantons and their action tends to the action associated with the quasi-solutions used in the standard procedure. Saddle points with periodic boundary conditions are also investigated. Abstract The semiclassical limit of the Green function for a particle in the one-dimensional sine-Gordon potential is obtained by summing over complex classical paths. The results are the same as those obtained in the less physically intuitive WKB approach. In addition to being of practical utility for solving quantum mechanical problems involving tunnelling, the classical path method may show how to deal with dense configurations of instantons. In Euclidean path integrals, quantum mechanical tunneling amplitudes are associated with instanton configurations. We explain how tunneling amplitudes are encoded in real-time Feynman path integrals. The essential steps are borrowed from Picard-Lefschetz theory and resurgence theory. The relation between the solutions of the time‐independent Schrodinger equation and the periodic orbits of the corresponding classical system is examined in the case where neither can be found by the separation of variables. If the quasiclassical approximation for the Green's function is integrated over the coordinates, a response function for the system is obtained which depends only on the energy and whose singularities give the approximate eigenvalues of the energy. This response function is written as a sum over all periodic orbits where each term has a phase factor containing the action integral and the number of conjugate points, as well as an amplitude factor containing the period and the stability exponent of the orbit. In terms of the approximate density of states per unit interval of energy, each stable periodic orbit is shown to yield a series of δ functions whose locations are given by a simple quantum condition: The action integral differs from an integer multiple of h by half the stability a... Abstract In 1967 M.C. Gutzwiller succeeded to derive the semiclassical expression of the quantum energy density of systems exhibiting a chaotic Hamiltonian dynamics in the classical limit. The result is known as the Gutzwiller trace formula. The scope of this review is to present in a self-contained way recent developments in functional determinant theory allowing to revisit the Gutzwiller trace formula in the spirit of field theory. The field theoretic setup permits to work explicitly at every step of the derivation of the trace formula with invariant quantities of classical periodic orbits. R. Forman's theory of functional determinants of linear, nonsingular elliptic operators yields the expression of quantum quadratic fluctuations around classical periodic orbits directly in terms of the monodromy matrix of the periodic orbits. The phase factor associated to quadratic fluctuations, the Maslov phase, is shown to be specified by the Morse index for closed extremals, also known as Conley and Zehnder index. Process for the preparation of 5-formyl-2-aminothiazole hydrobromide and 2-amino-5-formylthiazole which comprises reacting bromomalonaldehyde and thiourea in the substantial absence of acid or base to form the 5-formyl-2-aminothiazole hydrobromide, and then treating the same with base to form the aminothiazole.