The Seminars 2023: Term I

Sandra Klevansky,  (Universität Heidelberg, Germany) 26/01/2023,15:00 London time (=15:00 GMT)

Underdetermined Dyson-Schwinger equations

This talk examines the effectiveness of the Dyson-Schwinger (DS) equations as a calculational tool in quantum field theory. The DS equations are an infinite sequence of coupled equations that are satisfied exactly by the connected Green’s functions G_n of the field theory. These equations link lower to higher Green’s functions and, if they are truncated, the resulting finite system of equations is underdetermined. The simplest way to solve the underdetermined system is to set all higher Green’s function(s) to zero and then to solve the resulting determined system for the first few Green’s functions. The G₁ or G₂ so obtained can be compared with exact results in solvable models to see if the accuracy improves for high-order truncations. Five D=0 models are studied: Hermitian φ⁴ and φ⁶ and non-Hermitian iφ³, -φ⁴, and iφ⁵ theories. The truncated DS equations give a sequence of approximants that converge slowly to a limiting value but this limiting value always differs from the exact value by a few percent. More sophisticated truncation schemes based on mean-field-like approximations do not fix this formidable calculational problem.

Based on joint work with Carl M. Bender and Christos Karapoulitidis

Wen-Yuan Ai (King’s College London, UK), 09/02/2023, 15:00 London time (=15:00 GMT)

A conjectural relation for the PT-symmetric –g f⁴ theory

Abstract: While many PT-symmetric quantum-mechanical theories are well studied now, their generalizations to quantum field theory remain to be understood. Conventional methods based on the Schrödinger equation or the construction of the C operator can hardly be applied to field theories. On the other hand, the path-integral formulation can be extremely powerful in field theory. In this talk, I will discuss a conjectural relation between the Euclidean partition function of the PT-symmetric –g f⁴ theory and that of its ordinary Hermitian cousin (with a positive quartic potential). The relation is built on analytic continuation in the quartic coupling in the ordinary Hermitian theory and ensures a real energy spectrum for the non-Hermitian PT-symmetric theory. I will show how this relation is motivated and how it can be partly proved.

This talk is based on my recent joint work with Carl Bender and Sarben Sarkar: Phys. Rev. D 106 (2022) 12 , arXiv:2209.07897 hep-th

Philip Mannheim (University of Connecticut, US), 16/02/2023, 15:00 London time (=15:00 GMT)

Determining the normalization of the quantum field theory vacuum, with implications for quantum gravity

Abstract: In a standard quantum field theory the norm ⟨Ω|Ω⟩ of the vacuum state is taken, apparently without proof, to be finite. In this paper we provide a procedure, based on constructing an equivalent wave mechanics, for determining whether or not the vacuum norm is finite. We provide an example based on a second-order plus fourth-order scalar field theory, a prototype for quantum gravity, in which it is not. In this example the Minkowski path integral with a real measure diverges though the Euclidean path integral does not. Even though contributions from the Wick rotation contour are, again apparently without proof, ordinarily ignored, in this case they cannot be. Since ⟨Ω|Ω⟩ is not finite, use of the standard Feynman rules is not valid. And while these rules not only lead to states with negative norm, they in fact lead to states with infinite negative norm. However, if the fields in that theory are continued into the complex plane, we show that then there is a domain in the complex plane known as a Stokes wedge in which one can define an appropriate time-independent, positive and finite inner product, viz. the ⟨L|R⟩ overlap of left-eigenstates and right-eigenstates of the Hamiltonian; with the vacuum state then being normalizable, and with there being no states with negative or infinite ⟨L|R⟩ norm. In this Stokes wedge it is the Euclidean path integral that diverges while the Minkowski path integral does not. The concerns that we raise in this paper only apply to bosons since the matrices associated with their creation and annihilation operators are infinite dimensional. Since the ones associated with fermions are finite dimensional, the fermion theory vacuum is automatically normalizable. We discuss some general implications of our results for quantum gravity studies.

Based on arXiv: 2301.13029

Takanobu Taira (The University of Tokyo, Japan), 02/03/2023, 15:00 London time (=15:00 GMT)

Moduli spaces for PT-regularized solitons

Abstract: One of the characteristics of soliton (including solitary wave) is its “particle-like” dynamics. However, it is non-trivial why such behaviour is observed when the degree of freedom of soliton is expected to be infinite. This behaviour is due to the finite degree of freedom of the moduli or collective coordinate of the soliton. Recently, the moduli have been used to predict the soliton collision in the phi4 theory and successfully captured the “wobbling” effect [1,2].
Unfortunately, the moduli analysis can not be applied to every soliton as the analysis fails when the moduli space has a singularity. In this talk, we will show that extending the soliton to a PT-symmetric complex soliton, one can avoid the singularity and perform the moduli analysis. Furthermore, we observed that the moduli analysis could predict the non-trivial triple bouncing behaviour of the Bullough-Dodd soliton. 

This talk is based on our recent work [3].

[1] I. Takyi, and H. Weigel. “Collective coordinates in one-dimensional soliton models revisited.” Physical Review D 94.8 (2016): 085008.

[2] N.S. Manton, et al. “Kink moduli spaces: Collective coordinates reconsidered.” Physical Review D 103.2 (2021): 025024.

[3] F. Correa, A. Fring, A. and T. Taira “Moduli spaces for PT-regularized solitons.” Journal of High Energy Physics 2022.10 (2022): 1-20.

Junggi Yoon (Asia Pacific Center for Theoretical Physics, South Korea), 09/03/2023, 15:00 London time (=15:00 GMT)

Higher-derivative Fermionic Theories

Abstract: Higher derivative bosonic theories often suffer from Ostrogradski instability. In this talk, I begin with a simple quantum mechanical toy model with higher derivative fermion which is motivated by TTbar deformation of two-dimensional free fermion. Although this model seemingly has negative norm state, I will show that a new inner product can cure the negative norm problem. Then I will discuss the Hermiticity of Hamiltonian and the unitarity of the model.

Based on this toy model, I will present two simple examples where the higher derivative fermion appear. The first example is a field redefinition of (0+1) dimensional free fermion. I will explain the role of path integral measure to eliminate the spurious degrees of freedom. The second example is the two-dimensional free symplectic fermion. I will explain how this model has non-negative norm in spite of negative central charge.

References:

Kyungsun Lee, Piljin Yi and Junggi Yoon, “TT-deformed fermionic theories revisited” JHEP07 (2021) 217 arXiv:2104.09529.

Shinsei Ryu and Junggi Yoon, “Unitarity of Symplectic Fermion in alpha-vacua with Negative Central Charge,” arXiv:2208.12169.

Özlem Yeşiltaş (Gazi University, Turkey), 06/04/2023, 15:00 London time (=15:00 GMT)

Revisited Pseudosupersymmetric Approach To The Dirac Fields In Different Geometries

Abstract: In the previous work, pseudo-supersymmetry of the Dirac Hamiltonian in three -dimensional curved space-time was studied with a metric for an expanding de Sitter space-time which is two spheres [1]. This study includes pseudo-supersymmetric approach to the fermion motion in different geometries which are represented by static conformally flat, Schwarzchild and Friedmann–Robertson–Walker metrics.

[1] Ö. Yeşiltaş, Non-Hermitian Dirac Hamiltonian in Three-Dimensional Gravity and Pseudosupersymmetry, Advances in High Energy Physics 2015 Article ID 484151 2015.

Latévi Lawson (African Institute for Mathematical Sciences, Ghana), 20/04/2023, 15:00 London time (=15:00 GMT)

Path integral in position-deformed Heisenberg algebra with maximal length uncertainty

Abstract: The deformed Heisenberg algebra with uncertainty principle is one of the candidate approaches to probe quantum gravity at the Planck scale. It consists of deforming the ordinary Heisenberg algebra in momentum or in position operators. In their elegant paper [1], Fring and his colleagues introduced a position-deformed Heisenberg algebra in 2D with minimal length uncertainty. It had been shown that any fundamental objects introduced in this deformed space are string-like. Recently, we extended this seminal work to the case that there is also a maximal length uncertainty from this position-deformed Heisenberg [2]. We have shown that the emergence of this maximal length strongly deforms the quantum levels allowing particles to jump from state to another with weak external excitation and with high energy probability densities [3,4,5].

In the present talk, we confirm the previous result by studying the effects of this deformation on the trajectories of a classical system moving from one point to another. As result, we show that this system can travel very fast in this deformed space with very low energies.

[1] A. Fring L. Gouba and F. Scholtz, Strings from position-dependent noncommutativity, J. Phys. A: Math. Theor. 43, 345401 (2010)

[2] L. Lawson, Minimal and maximal lengths from position-dependent noncommutativity, J. Phys. A: Math. Theor. 53, 115303 (2020)

[3] L. Lawson, Minimal and maximal lengths of quantum gravity from non-Hermitian position-dependent non commutativity, Scientific Report 12, 20650 (2022)

[4] L. Lawson, Position-dependent mass in strong quantum gravitational background fields, Phys. A Math. Theor 55, 105303 (2022)

[5] L. Lawson and P. Osei, Gazeau-Klauder coherent states in position-deformed Heisenberg algebra, J. Phys. Commun. 6, 085016 (2022)