arXiv subjects

Alexander D. Popov

Publications and source records attributed to Alexander D. Popov.

At least 19 recordsLinked to original sources

arXiv2026

The Geometry Underlying the Quantum Harmonic Oscillator

We consider two-dimensional harmonic oscillator in the complex Bargmann-Fock-Segal representation with $T^*{\mathbb R}^{2}={\mathbb C}^2$ as classical phase space. We show that the eigenfunctions $ψ_n$ of the quantum Hamiltonian correspond to complex radial coordinates in the reduced phase space ${\mathbb C}^2/{\mathbb Z}_n\subset{\mathbb C}^2$. They describe ${\mathbb Z}_n$-invariant motion of particle along a circle $S^1$ in lens space $S^3/{\mathbb Z}_n\subset{\mathbb C}^2/{\mathbb Z}_n$, where ${\mathbb Z}_n$ is the cyclic group of rotation by an angle $2π/n$ on the circle $S^1$, $n=1,2,...\,$. Thus the general solution of the Schrödinger equation carries information about an infinite number of admissible classical states $ψ_n$ that can be mapped to other states after lifting into the quantum bundle. We show that in the Kepler/hydrogen atom problem there is a similar correspondence between classical and quantum states.

math-ph↗

arXiv2026

On Dirac equations on phase spaces

We consider Dirac equations on relativistic phase spaces $T^*{\mathbb R}^{p-1,1}$, where ${\mathbb R}^{p-1,1}$ is Minkowski space with $p=2,4$. We use the geometric quantization approach in which the wave functions are polarized sections of a complex line bundle $L_{\sf{v}}$ over $T^*{\mathbb R}^{p-1,1}$. The covariant derivatives with connection $A_{\sf{vac}}$ in this bundle define canonical commutation relations. Fermions are charged with respect to the field $A_{\sf{vac}}$, so lifting the Dirac equations from space-time ${\mathbb R}^{p-1,1}$ to phase space $T^*{\mathbb R}^{p-1,1}$ results in their solutions being localized in the space ${\mathbb R}^{p-1}$ or in space-time ${\mathbb R}^{p-1,1}$. We describe the explicit form of these solutions.

hep-th↗

arXiv2026

Discrete symmetries in classical and quantum oscillators

We consider the nature of the wave function using the example of a harmonic oscillator. We show that the eigenfunctions $ψ_n{=}z^n$ of the quantum Hamiltonian in the complex Bargmann-Fock-Segal representation with $z\in\mathbb C$ are the coordinates of a classical oscillator with energy $E_n=\hbarωn$, $n=0,1,2,...\,$. They are defined on conical spaces ${\mathbb C}/{\mathbb Z}_n$ with cone angles $2π/n$, which are embedded as subspaces in the phase space $\mathbb C$ of the classical oscillator. Here ${\mathbb Z}_n$ is the finite cyclic group of rotations of the space $\mathbb C$ by an angle $2π/n$. The superposition $ψ=\sum_n c_nψ_n$ of the eigenfunctions $ψ_n$ arises only with incomplete knowledge of the initial data for solving the Schrödinger equation, when the conditions of invariance with respect to the discrete groups ${\mathbb Z}_n$ are not imposed and the general solution takes into account all possible initial data parametrized by the numbers $n\in\mathbb N$.

quant-ph↗

arXiv2025

Dirac particles, spin and photons

We describe relativistic particles with spin as points moving in phase space $X=T^* R^{1,3}\times C^2_L\times C^2_R$, where $T^* R^{1,3}=R^{1,3}\times R^{1,3}$ is the space of coordinates and momenta, and $C^2_L$ and $C^2_R$ are the spaces of representation of the Lorentz group of type $(\frac12 , 0)$ and $(0, \frac12)$. Passing from relativistic mechanics with a Lorentz-invariant Hamiltonian function $H$ on the phase space $X$ to quantum mechanics with a Hamiltonian operator $\hat H$, we introduce two complex conjugate line bundles $L_C^+$ and $L_C^-$ over $X$. Quantum particles are introduced as sections $Ψ_+$ of the bundle $L_C^+$ holomorphic along the space $C^2_L\times C^2_R$, and antiparticles are sections $Ψ_-^{}$ of the bundle $L_C^-$ antiholomorphic along the internal spin space $C^2_L\times C^2_R$. The wave functions $Ψ_\pm$ are characterized by conserved charges $q_{\sf{v}}=\pm 1$ associated with the structure group U(1)$_{\sf{v}}$ of the bundles $L_C^\pm$. Wave functions $Ψ_\pm$ are governed by relativistic analogue of the Schrödinger equation. We show how fields with spin $s=0$ (Klein-Gordon), spin $s=\frac12$ (Dirac) and spin $s=1$ (Proca fields) arise from these equations in the zeroth, first, and second order expansions of the functions $Ψ_\pm^{}$ in the coordinates of the spin space $C^2_L\times C^2_R$. The Klein-Gordon, Dirac and Proca equations for these fields follow from the Schrödinger equation on the extended phase space $T^* R^{1,3}\times C^2_L\times C^2_R$. Using these results, we also introduce equations describing first quantized photons. We show that taking into account the charges $q_{\sf{v}}=\pm 1$ of the fields $Ψ_\pm$ changes the definitions of the inner products and currents, which eliminates negative energies and negative probabilities from relativistic quantum mechanics.

hep-th↗

arXiv2025

Antiparticles in non-relativistic quantum mechanics

Non-relativistic quantum mechanics was originally formulated to describe particles. Using ideas from the geometric quantization approach, we show how the concept of antiparticles can and should be introduced in the non-relativistic case without appealing to quantum field theory. We discuss this in detail using the example of the one-dimensional harmonic oscillator.

math-ph↗

arXiv2024

Supersymmetric Klein-Gordon and Dirac oscillators

We have recently shown that the space of initial data (covariant phase space) of the relativistic oscillator in Minkowski space $\mathbb{R}^{3,1}$ is a homogeneous Kähler-Einstein manifold $Z_6$=AdS$_7$/U(1)=U(3,1)/U(3)$\times$U(1). It was also shown that the energy eigenstates of the quantum relativistic oscillator form a direct sum of two weighted Bergman spaces of holomorphic (particles) and antiholomorphic (antiparticles) square-integrable functions on the covariant phase space $Z_6$ of the classical oscillator. Here we show that the covariant phase space of the supersymmetric version of the relativistic oscillator (oscillating spinning particle) is the odd tangent bundle of the space $Z_6$. Quantizing this model yields a Dirac oscillator equation on the phase space whose solution space is a direct sum of two spinor spaces parametrized by holomorphic and antiholomorphic functions on the odd tangent bundle of $Z_6$. After expanding the general solution in Grassmann variables, we obtain components of the spinor field that are holomorphic and antiholomorphic functions from Bergman spaces on $Z_6$ with different weight functions. Thus, the supersymmetric model under consideration is exactly solvable, Lorentz covariant and unitary.

hep-th↗

arXiv2024

Klein-Gordon oscillators and Bergman spaces

We consider classical and quantum dynamics of relativistic oscillator in Minkowski space $\mathbb{R}^{3,1}$. It is shown that for a non-zero frequency parameter $ω$ the covariant phase space of the classical Klein-Gordon oscillator is a homogeneous Kähler-Einstein manifold $Z_6=\mathrm{Ad}S_7/\mathrm{U}(1)=\mathrm{U}(3,1)/\mathrm{U}(3)\times \mathrm{U}(1)$. In the limit $ω\to 0$, this manifold is deformed into the covariant phase space $T^*H^3$ of a free relativistic particle, where $H^3=H^3_+\cup H_-^3$ is a two-sheeted hyperboloid in momentum space. Quantization of this model with $ω\ne 0$ leads to the Klein-Gordon oscillator equation which we consider in the Segal-Bargmann representation. It is shown that the general solution of this model is given by functions from the weighted Bergman space of square-integrable holomorphic (for particles) and antiholomorphic (for antiparticles) functions on the Kähler-Einstein manifold $Z_6$. This relativistic model is Lorentz covariant, unitary and does not contain non-physical states.

hep-th↗

arXiv2024

Quantum connection, charges and virtual particles

Geometrically, quantum mechanics is defined by a complex line bundle $L_\hbar$ over the classical particle phase space $T^*{R}^3\cong{R}^6$ with coordinates $x^a$ and momenta $p_a$, $a,...=1,2,3$. This quantum bundle $L_\hbar$ is endowed with a connection $A_\hbar$, and its sections are standard wave functions $ψ$ obeying the Schrödinger equation. The components of covariant derivatives $\nabla_{A_\hbar}^{}$ in $L_\hbar$ are equivalent to operators ${\hat x}^a$ and ${\hat p}_a$. The bundle $L_\hbar=: L_{C}^+$ is associated with symmetry group U(1)$_\hbar$ and describes particles with quantum charge $q=1$ which is eigenvalue of the generator of the group U(1)$_\hbar$. The complex conjugate bundle $L^-_{C}:={\overline{L_{C}^+}}$ describes antiparticles with quantum charge $q=-1$. We will lift the bundles $L_{C}^\pm$ and connection $A_\hbar$ on them to the relativistic phase space $T^*{R}^{3,1}$ and couple them to the Dirac spinor bundle describing both particles and antiparticles. Free relativistic quarks and leptons are described by the Dirac equation on Minkowski space ${R}^{3,1}$. This equation does not contain interaction with the quantum connection $A_\hbar$ on bundles $L^\pm_{C}\to T^*{R}^{3,1}$ because $A_\hbar$ has non-vanishing components only along $p_a$-directions in $T^*{R}^{3,1}$. To enable the interaction of elementary fermions $Ψ$ with quantum connection $A_\hbar$ on $L_{C}^\pm$, we will extend the Dirac equation to the phase space while maintaining the condition that $Ψ$ depends only on $t$ and $x^a$. The extended equation has an infinite number of oscillator-type solutions with discrete energy values as well as wave packets of coherent states. We argue that all these normalized solutions describe virtual particles and antiparticles living outside the mass shell hyperboloid. The transition to free particles is possible through squeezed coherent states.

hep-th↗

arXiv2022

Geometric confinement in gauge theories

In 1978, Friedberg and Lee introduced the phenomenological soliton bag model of hadrons, generalizing the MIT bag model developed in 1974 shortly after the formulation of QCD. In this model, quarks and gluons are confined due to coupling with a real scalar field $ρ$ which tends to zero outside some compact region $S\subset{\mathbb R}^3$ determined dynamically from the equations of motion. The gauge coupling in the soliton bag model is running as the inverse power of $ρ$ already at the semiclassical level. We show that this model arises naturally as a consequence of introducing the warped product metric ${\mathrm{d}}s^2_M + ρ^2{\mathrm{d}}s^2_G$ on the principal $G$-bundle $P(M,G)\cong M\times G$ with a non-Abelian group $G$ over Minkowski space $M={\mathbb R}^{3,1}$. Confinement of quarks and gluons in a compact domain $S\subset{\mathbb R}^3$ is a consequence of the collapse of the bundle manifold $M\times G$ to $M$ outside $S$ due to shrinking of the group manifold $G$ to a point. We describe the formation of such regions $S$ as a dynamical process controlled by the order parameter field $ρ$.

hep-th↗

arXiv2022

Stueckelberg and Higgs Mechanisms: Frames and Scales

We consider Yang-Mills theory with a compact gauge group $G$ on Minkowski space ${\mathbb R}^{3,1}$ and compare the introduction of masses of gauge bosons using the Stueckelberg and Higgs mechanisms. The Stueckelberg field $ϕ$ is identified with a $G$-frame on the gauge vector bundle $E$ and the kinetic term for $ϕ$ leads to the mass of the gauge bosons. The Stueckelberg mechanism is extended to the Higgs mechanism by adding to the game a scalar field describing rescaling of metric on fibres of $E$. Thus, we associate Higgs fields as well as running coupling parameters with conformal geometry on fibres of gauge bundles. In particular, a running coupling tending to zero or to infinity is equivalent to an unbounded expansion of $G$-fibres or its contraction to a point. We also discuss scale connection, space-time dependent Higgs vacua and compactly supported gauge and quark fields as an attribute of confinement.

hep-th↗

arXiv2022

On Exact Solvability of $\mathcal N$=4 super Yang-Mills

We consider the ambitwistor description of $\mathcal N$=4 supersymmetric extension of U($N$) Yang-Mills theory on Minkowski space $\mathbb R^{3,1}$. It is shown that solutions of super-Yang-Mills equations are encoded in real analytic U($N$)-valued functions on a domain in superambitwistor space ${\mathcal L}_{\mathbb R}^{5|6}$ of real dimension $(5|6)$. This leads to a procedure for generating solutions of super-Yang-Mills equations on $\mathbb R^{3,1}$ via solving a Riemann-Hilbert-type factorization problem on two-spheres in $\mathcal L_{\mathbb R}^{5|6}$.

hep-th↗

arXiv2021

Yang-Mills-Stueckelberg Theories, Framing and Local Breaking of Symmetries

We consider Yang-Mills theory with a compact structure group $G$ on a Lorentzian 4-manifold $M={\mathbb R}\timesΣ$ such that gauge transformations become identity on a submanifold $S$ of $Σ$ (framing over $S\subsetΣ$). The space $S$ is not necessarily a boundary of $Σ$ and can have dimension $k\le 3$. Framing of gauge bundles over $S\subsetΣ$ demands introduction of a $G$-valued function $ϕ_S$ with support on $S$ and modification of Yang-Mills equations along ${\mathbb R}\times S\subset M$. The fields $ϕ_S$ parametrize nonequivalent flat connections mapped into each other by a dynamical group ${\mathcal G}_S$ changing gauge frames over $S$. It is shown that the charged condensate $ϕ_S$ is the Stueckelberg field generating an effective mass of gluons in the domain $S$ of space $Σ$ and keeping them massless outside $S$. We argue that the local Stueckelberg field $ϕ_S$ can be responsible for color confinement. We also briefly discuss local breaking of symmetries in gravity. It is shown that framing of the tangent bundle over a subspace of space-time makes gravitons massive in this subspace.

hep-th↗

arXiv2021

A low-energy limit of Yang-Mills theory on de Sitter space

We consider Yang--Mills theory with a compact structure group $G$ on four-dimensional de Sitter space dS$_4$. Using conformal invariance, we transform the theory from dS$_4$ to the finite cylinder ${\cal I}\times S^3$, where ${\cal I}=(-π/2, π/2)$ and $S^3$ is the round three-sphere. By considering only bundles $P\to{\cal I}\times S^3$ which are framed over the temporal boundary $\partial{\cal I}\times S^3$, we introduce additional degrees of freedom which restrict gauge transformations to be identity on $\partial{\cal I}\times S^3$. We study the consequences of the framing on the variation of the action, and on the Yang--Mills equations. This allows for an infinite-dimensional moduli space of Yang--Mills vacua on dS$_4$. We show that, in the low-energy limit, when momentum along ${\cal I}$ is much smaller than along $S^3$, the Yang--Mills dynamics in dS$_4$ is approximated by geodesic motion in the infinite-dimensional space ${\cal M}_{\rm vac}$ of gauge-inequivalent Yang--Mills vacua on $S^3$. Since ${\cal M}_{\rm vac}\cong C^\infty (S^3, G)/G$ is a group manifold, the dynamics is expected to be integrable.

hep-th↗

arXiv2021

A Twistor Space Action for Yang-Mills Theory

We consider the twistor space ${\cal P}^6\cong{\mathbb R}^4{\times}{\mathbb C}P^1$ of ${\mathbb R}^4$ with a non-integrable almost complex structure ${\cal J}$ such that the canonical bundle of the almost complex manifold $({\cal P}^6, {\cal J})$ is trivial. It is shown that ${\cal J}$-holomorphic Chern-Simons theory on a real $(6|2)$-dimensional graded extension ${\cal P}^{6|2}$ of the twistor space ${\cal P}^6$ is equivalent to self-dual Yang-Mills theory on Euclidean space ${\mathbb R}^4$ with Lorentz invariant action. It is also shown that adding a local term to a Chern-Simons-type action on ${\cal P}^{6|2}$, one can extend it to a twistor action describing full Yang-Mills theory.

hep-th↗

arXiv2019

Skyrme and Faddeev models in the low-energy limit of 4d Yang-Mills-Higgs theories

Firstly, we consider Yang-Mills theory on ${\mathbb R}^{3,1}$ with an adjoint Higgs field spontaneously breaking a compact gauge group $G$ to a subgroup $H$, so that the Higgs vacuum manifold forms the coset $G/H$. It is shown that in the low-energy limit, when the Higgs vacuum value is large, the 4d Yang-Mills-Higgs theory reduces to the Faddeev sigma model on ${\mathbb R}^{3,1}$ with $G/H$ as target. Its action contains the standard two-derivative sigma-model term as well as the four-derivative Skyrme-type term, which stabilizes solutions against scaling. Secondly, we put the Higgs field in the bi-fundamental representation of $G=\textrm{U}_+(N)\times\textrm{U}_-(N)$, realizing the simplest $A_2$-type quiver gauge theory. Breaking $G$ to $H{=}\,\textrm{diag}(G)$, the vacuum manifold $G/H\cong\textrm{U}(N)$ is a group. In this case, when the Higgs vacuum value is large, the 4d $A_2$-quiver gauge theory reduces to the Skyrme sigma model on ${\mathbb R}^{3,1}$ with U$(N)$ as target. Thus, both the Skyrme and the Faddeev model arise as effective field theories in the infrared of Yang-Mills-Higgs models.

hep-th↗

arXiv2019

Sasakian quiver gauge theories and instantons on cones over round and squashed seven-spheres

We study quiver gauge theories on the round and squashed seven-spheres, and orbifolds thereof. They arise by imposing $G$-equivariance on the homogeneous space $G/H=\mathrm{SU}(4)/\mathrm{SU}(3)$ endowed with its Sasaki-Einstein structure, and $G/H=\mathrm{Sp}(2)/\mathrm{Sp}(1)$ as a 3-Sasakian manifold. In both cases we describe the equivariance conditions and the resulting quivers. We further study the moduli spaces of instantons on the metric cones over these spaces by using the known description for Hermitian Yang-Mills instantons on Calabi-Yau cones. It is shown that the moduli space of instantons on the hyper-Kahler cone can be described as the intersection of three Hermitian Yang-Mills moduli spaces. We also study moduli spaces of translationally invariant instantons on the metric cone $\mathbb{R}^8/\mathbb{Z}_k$ over $S^7/\mathbb{Z}_k$.

hep-th↗

arXiv2018

Dual infrared limits of 6d $\cal N$=(2,0) theory

Compactifying type $A_{N-1}$ 6d ${\cal N}{=}(2,0)$ supersymmetric CFT on a product manifold $M^4\timesΣ^2=M^3\times\tilde{S}^1\times S^1\times{\cal I}$ either over $S^1$ or over $\tilde{S}^1$ leads to maximally supersymmetric 5d gauge theories on $M^4\times{\cal I}$ or on $M^3\timesΣ^2$, respectively. Choosing the radii of $S^1$ and $\tilde{S}^1$ inversely proportional to each other, these 5d gauge theories are dual to one another since their coupling constants $e^2$ and $\tilde{e}^2$ are proportional to those radii respectively. We consider their non-Abelian but non-supersymmetric extensions, i.e. SU($N$) Yang-Mills theories on $M^4\times{\cal I}$ and on $M^3\timesΣ^2$, where $M^4\supset M^3=\mathbb R_t\times T_p^2$ with time $t$ and a punctured 2-torus, and ${\cal I}\subsetΣ^2$ is an interval. In the first case, shrinking ${\cal I}$ to a point reduces to Yang-Mills theory or to the Skyrme model on $M^4$, depending on the method chosen for the low-energy reduction. In the second case, scaling down the metric on $M^3$ and employing the adiabatic method, we derive in the infrared limit a non-linear SU($N$) sigma model with a baby-Skyrme-type term on $Σ^2$, which can be reduced further to $A_{N-1}$ Toda theory.

hep-th↗

arXiv2018

Skyrme-Faddeev model from 5d super-Yang-Mills

We consider 5d Yang-Mills-Higgs theory with a compact ADE-type gauge group $G$ and one adjoint scalar field on $\mathbb{R}^{3,1}\times\mathbb{R}_+$, where $\mathbb{R}_+=[0,\infty)$ is the half-line. The maximally supersymmetric extension of this model, with five adjoint scalars, appears after a reduction of 6d ${\cal N}{=}\,(2,0)$ superconformal field theory on $\mathbb{R}^{3,1}\times\mathbb{R}_+\times S^1$ along the circle $S^1$. We show that in the low-energy limit, when momenta along $\mathbb{R}^{3,1}$ are much smaller than along $\mathbb{R}_+$, the 5d Yang-Mills-Higgs theory reduces to a nonlinear sigma model on $\mathbb{R}^{3,1}$ with a coset $G/H$ as its target space. Here $H$ is a closed subgroup of $G$ determined by the Higgs-field asymptotics at infinity. The 4d sigma model describes an infinite tower of interacting fields, and in the infrared it is dominated by the standard two-derivative kinetic term and the four-derivative Skyrme-Faddeev term.

hep-th↗