Hamiltonian Gauge Research Articles

Hamiltonian gauge theory with corners: constraint reduction and flux superselection

Hamiltonian gauge theory with corners: constraint reduction and flux superselection

Supercomputers against strong coupling in gravity with curvature and torsion

Many theories of gravity are spoiled by strongly coupled modes: the high computational cost of Hamiltonian analysis can obstruct the identification of these modes. A computer algebra implementation of the Hamiltonian constraint algorithm for curvature and torsion theories is presented. These non-Riemannian or Poincaré gauge theories suffer notoriously from strong coupling. The implementation forms a package (the ‘Hamiltonian Gauge Gravity Surveyor’ – HiGGS) for the xAct tensor manipulation suite in Mathematica. Poisson brackets can be evaluated in parallel, meaning that Hamiltonian analysis can be done on silicon, and at scale. Accordingly HiGGS is designed to survey the whole Lagrangian space with high-performance computing resources (clusters and supercomputers). To demonstrate this, the space of ‘outlawed’ Poincaré gauge theories is surveyed, in which a massive parity-even/odd vector or parity-odd tensor torsion particle accompanies the usual graviton. The survey spans possible configurations of teleparallel-style multiplier fields which might be used to kill-off the strongly coupled modes, with the results to be analysed in subsequent work. All brackets between the known primary and secondary constraints of all theories are made available for future study. Demonstrations are also given for using HiGGS – on a desktop computer – to run the Dirac–Bergmann algorithm on specific theories, such as Einstein–Cartan theory and its minimal extensions.

Gauge fixing for strongly correlated electrons coupled to quantum light

We discuss the problem of gauge fixing for strongly correlated electrons coupled to quantum light, described by projected low-energy models such as those obtained within tight-binding methods. Drawing from recent results in the field of quantum optics, we present a general approach to write down quantum light-matter Hamiltonian in either dipole or Coulomb gauge which are explicitly connected by a unitary transformation, thus ensuring gauge equivalence even after projection. The projected dipole gauge Hamiltonian features a linear light-matter coupling and an instantaneous self-interaction for the electrons, similar to the structure in the full continuum theory. On the other hand, in the Coulomb gauge the photon field enters in a highly non-linear way, through phase factors that dress the electronic degrees of freedom. We show that our approach generalises the well-known Peierls approximation, to which it reduces when only local, on-site orbital contributions to light-matter coupling are taken into account. As an application, we study a two-orbital model of interacting electrons coupled to a uniform cavity mode, recently studied in the context of excitonic superradiance and associated no-go theorems. Using both gauges we recover the absence of superradiant phase in the ground state and show that excitations on top of it, described by polariton modes, contain instead non-trivial light-matter entanglement. Our results highlight the importance of treating the non-linear light-matter interaction of the Coulomb gauge non-perturbatively, to obtain a well-defined ultrastrong coupling limit and to not spoil gauge equivalence.

Resolution of Gauge Ambiguities in Molecular Cavity Quantum Electrodynamics

This work provides the fundamental theoretical framework for molecular cavity quantum electrodynamics by resolving the gauge ambiguities between the Coulomb gauge and the dipole gauge Hamiltonians under the electronic state truncation. We conjecture that such ambiguity arises because not all operators are consistently constrained in the same truncated electronic subspace for both gauges. We resolve this ambiguity by constructing a unitary transformation operator that properly constrains all light-matter interaction terms in the same subspace. We further derive an equivalent and yet convenient expression for the Coulomb gauge Hamiltonian under the truncated subspace. We finally provide the analytical and numerical results of a model molecular system coupled to the cavity to demonstrate the validity of our theory.

Loop, string, and hadron dynamics in SU(2) Hamiltonian lattice gauge theories

The question of how to efficiently formulate Hamiltonian gauge theories is experiencing renewed interest due to advances in building quantum simulation platforms. We introduce a reformulation of an SU(2) Hamiltonian lattice gauge theory---a loop-string-hadron (LSH) formulation---that describes dynamics directly in terms of its loop, string, and hadron degrees of freedom, while alleviating several disadvantages of quantumly simulating the Kogut-Susskind formulation. This LSH formulation transcends the local loop formulation of $d+1$-dimensional lattice gauge theories by incorporating staggered quarks, furnishing the algebra of gauge-singlet operators, and being used to reconstruct dynamics between states that have Gauss's law built in to them. LSH operators are then factored into products of "normalized" ladder operators and diagonal matrices, priming them for classical or quantum information processing. Self-contained expressions of the Hamiltonian are given up to $d=3$. The LSH formalism makes little use of structures specific to SU(2) and its conceptual clarity makes it an attractive approach to apply to other non-Abelian groups like SU(3).

Electromagnetic coupling in tight-binding models for strongly correlated light and matter

We discuss the construction of low-energy tight-binding Hamiltonians for condensed matter systems with a strong coupling to the quantum electromagnetic field. Such Hamiltonians can be obtained by projecting the continuum theory on a given set of Wannier orbitals. However, different representations of the continuum theory lead to different low-energy formulations, because different representations may entangle light and matter, transforming orbitals into light-matter hybrid states before the projection. In particular, a multi-center Power-Zienau-Woolley transformation yields a dipolar Hamiltonian which incorporates the light-matter coupling via both Peierls phases and a polarization density. We compare this dipolar gauge Hamiltonian and the straightforward Coulomb gauge Hamiltonian for a one-dimensional solid, to describe sub-cycle light-driven electronic motion in the semiclassical limit, and a coupling of the solid to a quantized cavity mode which renormalizes the band-structure into electron-polariton bands. Both descriptions yield the same result when many bands are taken into account, but the dipolar Hamiltonian is more accurate when the model is restricted to few electronic bands, while the Coulomb Hamiltonian requires fewer electromagnetic modes.

Exact classical and quantum solutions for a covariant oscillator near the black hole horizon in Stueckelberg-Horwitz-Piron theory

We found exact solutions for canonical classical and quantum dynamics for general relativity in Horwitz general covariance theory. These solutions can be obtained by solving the generalized geodesic equation and Schrödinger-Stueckelberg-Horwitz-Piron (SHP) wave equation for a simple harmonic oscillator in the background of a two dimensional dilaton black hole spacetime metric. We proved the existence of an orthonormal basis of eigenfunctions for generalized wave equation. This basis functions form an orthogonal and normalized (orthonormal) basis for an appropriate Hilbert space. The energy spectrum has a mixed spectrum with one conserved momentum p according to a quantum number n. To find the ground state energy we used a variational method with appropriate boundary conditions. A set of mode decomposed wave functions and calculated for the Stueckelberg-Schrodinger equation on a general five dimensional blackhole spacetime in Hamilton gauge.

Semirelativistic Schrödinger Equation for Relativistic Laser-Matter Interactions.

A semirelativistic formulation of light-matter interaction is derived using the so called propagation gauge and the relativistic mass shift. We show that relativistic effects induced by a superintense laser field can, to a surprisingly large extent, be accounted for by the Schrödinger equation, provided that we replace the rest mass in the propagation gauge Hamiltonian by the corresponding time-dependent field-dressed mass. The validity of the semirelativistic approach is tested numerically on a hydrogen atom exposed to an intense extreme ultraviolet laser pulse strong enough to accelerate the electron towards relativistic velocities. It is found that while the results obtained from the ordinary (nonrelativistic) Schrödinger equation generally differ from those of the Dirac equation, demonstrating that relativistic effects are significant, the semirelativistic formulation provides results in quantitative agreement with a fully relativistic treatment.

Dirac–Bergmann constraints in physics: Singular Lagrangians, Hamiltonian constraints and the second Noether theorem

There is a review of the main mathematical properties of system described by singular Lagrangians and requiring Dirac–Bergmann theory of constraints at the Hamiltonian level. The following aspects are discussed:(i)the connection of the rank and eigenvalues of the Hessian matrix in the Euler–Lagrange equations with the chains of first- and second-class constraints;(ii)the connection of the Noether identities of the second Noether theorem with the Hamiltonian constraints;(iii)the Shanmugadhasan canonical transformation for the identification of the gauge variables and for the search of the Dirac observables, i.e. the quantities invariant under Hamiltonian gauge transformations.

Perspective: Quantum Hamiltonians for optical interactions.

The multipolar Hamiltonian of quantum electrodynamics is extensively employed in chemical and optical physics to treat rigorously the interaction of electromagnetic fields with matter. It is also widely used to evaluate intermolecular interactions. The multipolar version of the Hamiltonian is commonly obtained by carrying out a unitary transformation of the Coulomb gauge Hamiltonian that goes by the name of Power-Zienau-Woolley (PZW). Not only does the formulation provide excellent agreement with experiment, and versatility in its predictive ability, but also superior physical insight. Recently, the foundations and validity of the PZW Hamiltonian have been questioned, raising a concern over issues of gauge transformation and invariance, and whether observable quantities obtained from unitarily equivalent Hamiltonians are identical. Here, an in-depth analysis of theoretical foundations clarifies the issues and enables misconceptions to be identified. Claims of non-physicality are refuted: the PZW transformation and ensuing Hamiltonian are shown to rest on solid physical principles and secure theoretical ground.

Wave fluctuations in the system with some Yang-Mills condensates

Self-consistent dynamics of non-homogeneous fluctuations and homogeneous and isotropic condensate of Yang–Mills fields was investigated in zero, linear and quasilinear approximations over the wave modes in the framework of N = 4 supersymmetric model in Hamilton gauge in quasiclassical theory. The models with SU(2), SU(3) and SU(4) gauge groups were considered. Particle production effect and effect of generation of longitudinal oscillations were obtained.

Lagrangian geometrical optics of nonadiabatic vector waves and spin particles

Linear vector waves, both quantum and classical, experience polarization-driven bending of ray trajectories and polarization dynamics that can be interpreted as the precession of the “wave spin”. Both phenomena are governed by an effective gauge Hamiltonian vanishing in leading-order geometrical optics. This gauge Hamiltonian can be recognized as a generalization of the Stern–Gerlach Hamiltonian that is commonly known for spin-1/2 quantum particles. The corresponding reduced Lagrangians for continuous nondissipative waves and their geometrical-optics rays are derived from the fundamental wave Lagrangian. The resulting Euler–Lagrange equations can describe simultaneous interactions of N resonant modes, where N is arbitrary, and lead to equations for the wave spin, which happens to be an (N2−1)-dimensional spin vector. As a special case, classical equations for a Dirac particle (N=2) are deduced formally, without introducing additional postulates or interpretations, from the Dirac quantum Lagrangian with the Pauli term. The model reproduces the Bargmann–Michel–Telegdi equations with added Stern–Gerlach force.

Dynamics of wave fluctuations in the homogeneous Yang-Mills condensate

In the present work, the Yang-Mills (YM) quantum-wave excitations of the classical homogeneous YM have been studied in quasi-classical approximation. The formalism is initially formulated in the Hamilton gauge and is based upon canonical quantisation in the Heisenberg representation. This canonical framework is then extended and related to YM dynamics in arbitrary gauge and symmetry group containing at least one SU(2) subgroup. Such generic properties of the interacting YM system as excitation of longitudinal wave modes and energy balance between the evolving YM and have been established. In order to prove these findings, the canonical quasi-classical YM system + in the pure simplest SU(2) gauge theory has been thoroughly analysed numerically in the linear and next-to-linear approximations in the limit of small wave amplitudes. The effective gluon mass dynamically generated by wave self-interactions in the gluon plasma has been derived. A complete set of equations of motion for the YM condensate + waves system accounting for second- and third-order interactions between the has been obtained. In the next-to-linear approximation in we have found that due to interactions between the YM and the YM condensate, the latter looses its energy leading to the growth of amplitudes of the YM wave modes. A similar effect has been found in the maximally-supersymmetric N = 4 Yang-Mills theory as well as in two-condensate SU(4) model. Possible implications of these findings to Cosmology and gluon plasma physics have been discussed. (Less)

Quantum Gravity in Heisenberg Representation and Self-Consistent Theory of Gravitons in Macroscopic Spacetime

The first mathematically consistent exact equations of quantum gravity in the Heisenberg representation and Hamilton gauge are obtained. It is shown that the path integral over the canonical variables in the Hamilton gauge is mathematically equivalent to the operator equations of quantum theory of gravity with canonical rules of quantization of the gravitational and ghost fields. In its operator formulation, the theory can be used to calculate the graviton S-matrix as well as to describe the quantum evolution of macroscopic system of gravitons in the non-stationary Universe or in the vicinity of relativistic objects. In the S-matrix case, the standard results are obtained. For problems of the second type, the original Heisenberg equations of quantum gravity are converted to a self-consistent system of equations for the metric of the macroscopic spacetime and Heisenberg operators of quantum fields. It is shown that conditions of the compatibility and internal consistency of this system of equations are performed without restrictions on the amplitude and wavelength of gravitons and ghosts. The status of ghost fields in the various formulations of quantum theory of gravity is discussed.

Macroscopic Effect of Quantum Gravity: Graviton, Ghost and Instanton Condensation on Horizon Scale of Universe

We discuss a special class of quantum gravity phenomena that occur on the scale of the Universe as a whole at any stage of its evolution, including the contemporary Universe. These phenomena are a direct consequence of the zero rest mass of gravitons, conformal non-invariance of the graviton field, and one-loop finiteness of quantum gravity, i.e. it is a direct consequence of first principles only. The effects are due to graviton-ghost condensates arising from the interfereence of quantum coherent states. Each of coherent states is a state of gravitons and ghosts of a wavelength of the order of the horizon scale and of different occupation numbers. The state vector of the Universe is a coherent superposition of vectors of different occupation numbers. One-loop approximation of quantum gravity is believed to be applicable to the contemporary Universe because of its remoteness from the Planck epoch. To substantiate the reliability of macroscopic quantum effects, the formalism of one-loop quantum gravity is discussed in detail. The theory is constructed as follows: Faddeev-Popov path integral in Hamilton gauge → factorization of classical and quantum variables, allowing the existence of a self-consistent system of equations for gravitons, ghosts and macroscopic geometry → transition to the one-loop approximation, taking into account that contributions of ghost fields to observables cannot be eliminated in any way. The ghost sector corresponding to the Hamilton gauge automatically ensures of one-loop finiteness of the theory off the mass shell. The Bogolyubov-Born-Green-Kirckwood-Yvon (BBGKY) chain for the spectral function of gravitons renormalized by ghosts is used to build a self-consistent theory of gravitons in the isotropic Universe. It is the first use of this technique in quantum gravity calculations. We found three exact solutions of the equations, consisting of BBGKY chain and macroscopic Einstein’s equations. It was found that these solutions describe virtual graviton and ghost condensates as well as condensates of instanton fluctuations. All exact solutions, originally found by the BBGKY formalism, are reproduced at the level of exact solutions for field operators and state vectors. It was found that exact solutions correspond to various condensates with different graviton-ghost compositions. Each exact solution corresponds to a certain phase state of graviton-ghost substratum. We establish conditions under which a continuous quantum-gravity phase transitions occur between different phases of the graviton-ghost condensate.

Gluon chain formation in presence of static charges

We consider the origins of the gluon chain model. The model serves as a realization of the dynamics of the chromoelectric flux between static quark-antiquark sources. The derivation is based on the large-N_C limit of the Coulomb gauge Hamiltonian in the presence of a background field introduced to model magnetic confinement.

Coulomb gauge Yang-Mills theory at finite temperatures: Glueballs versus quasi-gluons

We consider a variational approach to the finite-temperature Yang-Mills theory in the Coulomb gauge. The partition function is computed in the ensemble of glueballs and quasigluons which emerge as eigenstates of the Coulomb gauge Hamiltonian. We compute the energy density and pressure and compare with results of lattice computations for both SU(2) and SU(3). The emergence of a phase transition is discussed.

Gauge invariances of higher derivative Maxwell-Chern-Simons field theory: A new Hamiltonian approach

A new method of abstracting the independent gauge invariances of higher derivative systems, recently introduced in [1], has been applied to higher derivative field theories. This has been discussed taking the extended Maxwell-Chern-Simons model as an example. A new Hamiltonian analysis of the model is provided. This Hamiltonian analysis has been used to construct the independent gauge generator. An exact mapping between the Hamiltonian gauge transformations and the U(1) symmetries of the action has been established.

Poincaré gauge symmetries, Hamiltonian symmetries, and trivial gauge transformations

We resolve a problem of finding the Poincare symmetries from hamiltonian gauge symmetries constructed through a canonical procedure of handling constrained systems. Through the use of Noether identities corresponding to the symmetries, we motivate a procedure of finding the map between the hamiltonian and Poincare gauge parameters. Using this map, we show that the Poincare and hamiltonian gauge symmetries are equivalent, modulo trivial gauge transformations.

Hamiltonian analysis of symmetries in a massive theory of gravity

We construct the generator of hamiltonian gauge symmetries in a 2+1 dimensional massive theory of gravity, proposed recently, through a systematic off-shell algorithm. Using a field dependant map among gauge parameters we show that the symmetries obtained from this generator are on-shell equivalent to the Poincar\'{e} gauge symmetries. We also clarify certain subtle issues concerning the implementation of this map.