Quantum connection, charges and virtual particles
- URL: http://arxiv.org/abs/2310.06507v4
- Date: Mon, 12 Feb 2024 15:16:39 GMT
- Title: Quantum connection, charges and virtual particles
- Authors: Alexander D. Popov
- Abstract summary: A quantum bundle $L_hbar$ is endowed with a connection $A_hbar$ and its sections are standard wave functions $psi$ obeying the Schr"odinger equation.
We will lift the bundles $L_Cpm$ and connection $A_hbar$ on them to the relativistic phase space $T*R3,1$ and couple them to the Dirac spinor bundle describing both particles and antiparticles.
- Score: 65.268245109828
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: 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 $\psi$ obeying the Schr\"odinger 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 $\Psi$ with quantum connection $A_\hbar$ on $L_{C}^\pm$, we
will extend the Dirac equation to the phase space while maintaining the
condition that $\Psi$ 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.
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