Real-time Feynman path integral with Picard–Lefschetz theory and its applications to quantum tunneling

Tanizaki, Yuya; Koike, Takayuki

doi:10.1016/j.aop.2014.09.003

Cited by 101 publications

(110 citation statements)

References 34 publications

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“…Both the complex saddle and the HTA are crucial for the argument. This example also demonstrates that in using Lefschetz thimbles, for example, either in Euclidean semi-classics or real time semi-classics (with sign problems) [110,111] or in lattice simulations [101][102][103][104], all thimbles whose Stokes multipliers are non-zero must be summed over. Numerical evidence for the correctness of this perspective is also given in [105][106][107][108][109].…”

Section: Hidden Topological Angles and Complex Saddlesmentioning

confidence: 99%

New Nonperturbative Methods in Quantum Field Theory: From Large-NOrbifold Equivalence to Bions and Resurgence

Dunne

Ünsal

2016

Annu. Rev. Nucl. Part. Sci.

112

170

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We present a broad conceptual introduction to some new ideas in non-perturbative QFT. The large-N orbifold-orientifold equivalence connects a natural large-N limit of QCD to QCD with adjoint fermions. QCD(adj) with periodic boundary conditions and double-trace deformation of Yang-Mills theory satisfy large-N volume independence, a type of orbifold equivalence. Certain QFTs that satisfy volume independence at N = ∞ exhibit adiabatic continuity at finite-N , and also become semi-classically calculable on small R 3 × S 1 . We discuss the role of monopole-instantons, and magnetic and neutral bion saddles in connection to mass gap, and center and chiral symmetry realizations. Neutral bions also provide a weak coupling semiclassical realization of infrared-renormalons. These considerations help motivate the necessity of complexification of path integrals (Picard-Lefschetz theory) in semi-classical analysis, and highlights the importance of hidden topological angles. Finally, we briefly review the resurgence program, which potentially provides a novel non-perturbative continuum definition of QFT. All these ideas are continuously connected.Keywords: large-N orbifold and orientifold equivalence, volume independence, double-trace deformations, adiabatic continuity, semi-classical calculability, magnetic and neutral bions, Picard-Lefschetz theory of path integration, and resurgence arXiv:1601.03414v1 [hep-th]

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Section: Hidden Topological Angles and Complex Saddlesmentioning

confidence: 99%

New Nonperturbative Methods in Quantum Field Theory: From Large-NOrbifold Equivalence to Bions and Resurgence

Dunne

Ünsal

2016

Annu. Rev. Nucl. Part. Sci.

112

170

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“…The main approaches which have been pursued are complex Langevin dynamics and integration along Lefschetz thimbles [26][27][28][29][30][31][32][33][34][35]. In the latter, the original path of integration is deformed in order to pass through the fixed (or critical) points of the complex action, which typically reside in the complexified space.…”

Section: Introductionmentioning

confidence: 99%

Some remarks on Lefschetz thimbles and complex Langevin dynamics

Aarts

Bongiovanni

Seiler

et al. 2014

J. High Energ. Phys.

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Lefschetz thimbles and complex Langevin dynamics both provide a means to tackle the numerical sign problem prevalent in theories with a complex weight in the partition function, e.g. due to nonzero chemical potential. Here we collect some findings for the quartic model, and for U(1) and SU(2) models in the presence of a determinant, which have some features not discussed before, due to a singular drift. We find evidence for a relation between classical runaways and stable thimbles, and give an example of a degenerate fixed point. We typically find that the distributions sampled in complex Langevin dynamics are related to the thimble(s), but with some important caveats, for instance due to the presence of unstable fixed points in the Langevin dynamics.

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“…Other examples are discussed in [18][19][20]. We will perform a similar analytic continuation in n f , the number of fermion flavors in the theory.…”

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confidence: 99%

Hidden Topological Angles in Path Integrals

et al. 2015

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We demonstrate the existence of hidden topological angles (HTAs) in a large class of quantum field theories and quantum mechanical systems. HTAs are distinct from theta-parameters in the lagrangian. They arise as invariant angle associated with saddle points of the complexified path integral and their descent manifolds (Lefschetz thimbles). Physical effects of HTAs become most transparent upon analytic continuation in n f to non-integer number of flavors, reducing in the integer n f limit to a Z 2 valued phase difference between dominant saddles. In N = 1 super Yang-Mills theory we demonstrate the microscopic mechanism for the vanishing of the gluon condensate. The same effect leads to an anomalously small condensate in a QCD-like SU(N) gauge theory with fermions in the two-index representation. The basic phenomenon is that, contrary to folklore, the gluon condensate can receive both positive and negative contributions in a semi-classical expansion. In quantum mechanics, a HTA leads to a difference in semi-classical expansion of integer and half-integer spin particles.Introduction. Providing a non-perturbative continuum definition of the path integral in quantum field theory is a challenging but important problem [1]. There is growing evidence that, if an ordinary integral or a path integral admits a Lefschetz-thimble decomposition [2,3] or resurgent transseries expansion [4][5][6][7][8][9][10][11][12] then either of these methods gives this long-sought non-perturbative definition. If this is indeed the case, then we expect that these new methods will provide new and deep insight into quantum field theory and quantum mechanics formulated in terms of path integrals. In this article we introduce a new phenomenon of this kind, the appearance of hidden topological angles (HTAs).The main prescription associated with the Lefschetzthimble decomposition or the resurgent expansion is the following: Even if an ordinary integral or a path integral is formulated over real fields, the natural space that the critical points (saddles) ρ σ live in is the complexification of the original space of fields. However, the dimension of the crit-

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Real-time Feynman path integral with Picard–Lefschetz theory and its applications to quantum tunneling

Cited by 101 publications

References 34 publications

New Nonperturbative Methods in Quantum Field Theory: From Large-NOrbifold Equivalence to Bions and Resurgence

New Nonperturbative Methods in Quantum Field Theory: From Large-NOrbifold Equivalence to Bions and Resurgence

Some remarks on Lefschetz thimbles and complex Langevin dynamics

Hidden Topological Angles in Path Integrals

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