scholarly journals PT -symmetric quantum electrodynamics and unitarity

2013 ◽  
Vol 371 (1989) ◽  
pp. 20120057 ◽  
Author(s):  
Kimball A Milton ◽  
E. K. Abalo ◽  
Prachi Parashar ◽  
Nima Pourtolami ◽  
J. Wagner
Keyword(s):  
Field Theory ◽  
Quantum Mechanics ◽  
Quantum Electrodynamics ◽  
Discrete Symmetry ◽  
New Approach ◽  
Higher Dimensional ◽  
The Status ◽  
Symmetric Quantum ◽  
Quantum Mechanical Systems ◽  
Invariant Quantum

More than 15 years ago, a new approach to quantum mechanics was suggested, in which Hermiticity of the Hamiltonian was to be replaced by invariance under a discrete symmetry, the product of parity and time-reversal symmetry, . It was shown that, if is unbroken, energies were, in fact, positive, and unitarity was satisfied. Since quantum mechanics is quantum field theory in one dimension—time—it was natural to extend this idea to higher-dimensional field theory, and in fact an apparently viable version of -invariant quantum electrodynamics (QED) was proposed. However, it has proved difficult to establish that the unitarity of the scattering matrix, for example, the Källén spectral representation for the photon propagator, can be maintained in this theory. This has led to questions of whether, in fact, even quantum mechanical systems are consistent with probability conservation when Green’s functions are examined, since the latter have to possess physical requirements of analyticity. The status of QED will be reviewed in this paper, as well as the general issue of unitarity.

NON-HERMITIAN QUANTUM FIELD THEORY

2005 ◽  
Vol 20 (19) ◽  
pp. 4646-4652 ◽  
Author(s):  
CARL. M. BENDER
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Quantum Mechanics ◽  
Degrees Of Freedom ◽  
Real Spectrum ◽  
Quantum Field ◽  
Unitary Theory ◽  
Symmetric Quantum ◽  
Quantum Mechanical Systems ◽  
Bounded Below

In my talk at the Seventh QCD Workshop held in Villefranche in January 2003 I showed that a non-Hermitian Hamiltonian H possessing an unbroken [Formula: see text] symmetry (i) has a real spectrum that is bounded below, and (ii) defines a unitary theory of quantum mechanics with positive norm. The proof of unitarity requires a linear operator [Formula: see text], which was originally defined as a sum over the eigenfunctions of H. However, using this definition to calculate [Formula: see text] is cumbersome in quantum mechanics and impossible in quantum field theory. I describe here an alternative method for calculating [Formula: see text] directly in terms of the operator dynamical variables of the quantum theory. This new method is general and applies to a variety of quantum mechanical systems having several degrees of freedom. More importantly, this method gives the [Formula: see text] operator in quantum field theory. The [Formula: see text] operator is a new time-independent observable in [Formula: see text]-symmetric quantum field theory.

scholarly journals Generation of families of spectra in PT -symmetric quantum mechanics and scalar bosonic field theory

2013 ◽  
Vol 371 (1989) ◽  
pp. 20120049 ◽  
Author(s):  
Steffen Schmidt ◽  
S. P. Klevansky
Keyword(s):  
Field Theory ◽  
Quantum Mechanics ◽  
Quantum Theory ◽  
Complex Analysis ◽  
One Dimension ◽  
Quantum Mechanical ◽  
Bosonic Field ◽  
Symmetric Quantum ◽  
The Difference ◽  
Do So

This paper explains the systematics of the generation of families of spectra for the -symmetric quantum-mechanical Hamiltonians H = p 2 + x 2 (i x ) ϵ , H = p 2 +( x 2 ) δ and H = p 2 −( x 2 ) μ . In addition, it contrasts the results obtained with those found for a bosonic scalar field theory, in particular in one dimension, highlighting the similarities to and differences from the quantum-mechanical case. It is shown that the number of families of spectra can be deduced from the number of non-contiguous pairs of Stokes wedges that display symmetry. To do so, simple arguments that use the Wentzel–Kramers–Brillouin approximation are used, and these imply that the eigenvalues are real. However, definitive results are in most cases presently only obtainable numerically, and not all eigenvalues in each family may be real. Within the approximations used, it is illustrated that the difference between the quantum-mechanical and the field-theoretical cases lies in the number of accessible regions in which the eigenfunctions decay exponentially. This paper reviews and implements well-known techniques in complex analysis and -symmetric quantum theory.

scholarly journals The principle of stationary variance in quantum field theory

2014 ◽  
Vol 29 (05) ◽  
pp. 1450026 ◽  
Author(s):  
Fabio Siringo
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Quantum Mechanics ◽  
Quantum System ◽  
Quantum Electrodynamics ◽  
Best Approximation ◽  
Variational Approach ◽  
Heisenberg Model ◽  
Gauge Theories ◽  
Quantum Field

The principle of stationary variance is advocated as a viable variational approach to quantum field theory (QFT). The method is based on the principle that the variance of energy should be at its minimum when the state of a quantum system reaches its best approximation for an eigenstate. While not too much popular in quantum mechanics (QM), the method is shown to be valuable in QFT and three special examples are given in very different areas ranging from Heisenberg model of antiferromagnetism (AF) to quantum electrodynamics (QED) and gauge theories.

Quantum mechanics

2018 ◽  
Author(s):  
Michael Kachelriess
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Quantum Mechanics ◽  
Relativistic Quantum ◽  
Transition Amplitude ◽  
External Source ◽  
Quantum Field ◽  
Damping Term ◽  
Relativistic Quantum Theory ◽  
Generating Functional

After a brief review of the operator approach to quantum mechanics, Feynmans path integral, which expresses a transition amplitude as a sum over all paths, is derived. Adding a linear coupling to an external source J and a damping term to the Lagrangian, the ground-state persistence amplitude is obtained. This quantity serves as the generating functional Z[J] for n-point Green functions which are the main target when studying quantum field theory. Then the harmonic oscillator as an example for a one-dimensional quantum field theory is discussed and the reason why a relativistic quantum theory should be based on quantum fields is explained.

QLB for Quantum Many-Body and Quantum Field Theory

2018 ◽  
Author(s):  
Sauro Succi
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Quantum Mechanics ◽  
Quantum Computing ◽  
Single Particle ◽  
Body Problem ◽  
Three Dimensions ◽  
Quantum Field ◽  
Many Body ◽  
Quantum Particles

Chapter 32 expounded the basic theory of quantum LB for the case of relativistic and non-relativistic wavefunctions, namely single-particle quantum mechanics. This chapter goes on to cover extensions of the quantum LB formalism to the overly challenging arena of quantum many-body problems and quantum field theory, along with an appraisal of prospective quantum computing implementations. Solving the single particle Schrodinger, or Dirac, equation in three dimensions is a computationally demanding task. This task, however, pales in front of the ordeal of solving the Schrodinger equation for the quantum many-body problem, namely a collection of many quantum particles, typically nuclei and electrons in a given atom or molecule.

scholarly journals On the Modular Operator of Mutli-component Regions in Chiral CFT

2021 ◽  
Author(s):  
Stefan Hollands
Keyword(s):  
Field Theory ◽  
Integral Equation ◽  
Hilbert Problem ◽  
Matrix Elements ◽  
New Approach ◽  
Conformal Field ◽  
The Matrix ◽  
Kms Condition ◽  
Modular Operator ◽  
Riemann Hilbert Problem

AbstractWe introduce a new approach to find the Tomita–Takesaki modular flow for multi-component regions in general chiral conformal field theory. Our method is based on locality and analyticity of primary fields as well as the so-called Kubo–Martin–Schwinger (KMS) condition. These features can be used to transform the problem to a Riemann–Hilbert problem on a covering of the complex plane cut along the regions, which is equivalent to an integral equation for the matrix elements of the modular Hamiltonian. Examples are considered.

scholarly journals Gauge invariant canonical symplectic algorithms for real-time lattice strong-field quantum electrodynamics

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Qiang Chen ◽  
Jianyuan Xiao ◽  
Peifeng Fan
Keyword(s):  
Field Theory ◽  
Real Time ◽  
Quantum Electrodynamics ◽  
Symplectic Form ◽  
Strong Field ◽  
Relativistic Quantum ◽  
High Order ◽  
High Quality ◽  
Geometric Structures ◽  
Gauge Invariant

Abstract A class of high-order canonical symplectic structure-preserving geometric algorithms are developed for high-quality simulations of the quantized Dirac-Maxwell theory based strong-field quantum electrodynamics (SFQED) and relativistic quantum plasmas (RQP) phenomena. With minimal coupling, the Lagrangian density of an interacting bispinor-gauge fields theory is constructed in a conjugate real fields form. The canonical symplectic form and canonical equations of this field theory are obtained by the general Hamilton’s principle on cotangent bundle. Based on discrete exterior calculus, the gauge field components are discreted to form a cochain complex, and the bispinor components are naturally discreted on a staggered dual lattice as combinations of differential forms. With pull-back and push-forward gauge covariant derivatives, the discrete action is gauge invariant. A well-defined discrete canonical Poisson bracket generates a semi-discrete lattice canonical field theory (LCFT), which admits the canonical symplectic form, unitary property, gauge symmetry and discrete Poincaré subgroup, which are good approximations of the original continuous geometric structures. The Hamiltonian splitting method, Cayley transformation and symmetric composition technique are introduced to construct a class of high-order numerical schemes for the semi-discrete LCFT. These schemes involve two degenerate fermion flavors and are locally unconditional stable, which also preserve the geometric structures. Admitting Nielsen-Ninomiya theorem, the continuous chiral symmetry is partially broken on the lattice. As an extension, a pair of discrete chiral operators are introduced to reconstruct the lattice chirality. Equipped with statistically quantization-equivalent ensemble models of the Dirac vacuum and non-trivial plasma backgrounds, the schemes are expected to have excellent performance in secular simulations of relativistic quantum effects, where the numerical errors of conserved quantities are well bounded by very small values without coherent accumulation. The algorithms are verified in detail by numerical energy spectra. Real-time LCFT simulations are successfully implemented for the nonlinear Schwinger mechanism induced e-e+ pairs creation and vacuum Kerr effect, where the nonlinear and non-perturbative features captured by the solutions provide a complete strong-field physical picture in a very wide range, which open a new door toward high-quality simulations in SFQED and RQP fields.

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