Alexander D. Popov

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 $n\in\mathbb N$.

We have recently shown that the space of initial data (covariant phase space) of the relativistic oscillator in Minkowski space $$\mathbb {R}^{3,1}$$ R 3 , 1 is a homogeneous Kähler–Einstein manifold $$Z_6=\textrm{AdS}_7/\textrm{U}(1) =\textrm{U}(3,1)/\textrm{U}(3)\times \textrm{U}(1)$$ Z 6 = AdS 7 / U ( 1 ) = U ( 3 , 1 ) / U ( 3 ) × 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 $$Z_6$$ 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 $$Z_6$$ 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 $$Z_6$$ 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 $$Z_6$$ Z 6 with different weight functions. Thus, the supersymmetric model under consideration is exactly solvable, Lorentz covariant and unitary.

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.

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.

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 $\rho$, 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 runs as the inverse power of $\rho$, already at the semiclassical level. We show that this model arises naturally as a consequence of introducing the warped product metric $\mathrm{d}{s}_M^2+{\rho }^2\mathrm{d}{s}_G^2$ 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 $\rho$.

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 $\rho$ 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 $\rho$ already at the semiclassical level. We show that this model arises naturally as a consequence of introducing the warped product metric ${\mathrm{d}}s^2_M + \rho^2{\mathrm{d}}s^2_G$ 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 $\rho$.