Dirac-Bergmann algorithm and canonical quantization of $k$-essence cosmology

TL;DR

The paper develops a general canonical quantization scheme for $k$-essence cosmology using the Dirac-Bergmann algorithm to identify and classify constraints, perform a Dirac-bracket reduction, and obtain a reduced Hamiltonian that is purely kinetic. This leads to a Wheeler-DeWitt equation of massless Klein-Gordon type in a two-dimensional minisuperspace, illustrating a universal structure across $k$-essence theories. The tachyon sector with a constant potential is analyzed in detail, yielding analytic classical solutions and a tractable quantum problem; the choice of boundary conditions critically affects singularity avoidance and the possibility of phantom crossing via quantum tunneling. The work highlights the robustness of the quantum description across $k$-essence models while emphasizing the pivotal role of coordinate choices and boundary conditions in determining physical outcomes, with prospects for extending to nonconstant potentials.

Abstract

We develop a general canonical quantization scheme for $k$-essence cosmology in scalar-tensor theory. Utilizing the Dirac-Bergmann algorithm, we construct the Hamiltonian associated with the cosmological field equations and identify the first- and second-class constraints. The introduction of appropriate canonically conjugate variables with respect to Dirac brackets, allows for the canonical quantization of the model. In these new variables, the Hamiltonian constraint reduces to a quadratic function with no potential term. Its quantum realization leads to a Wheeler-DeWitt equation reminiscent of the massless Klein-Gordon case. As an illustrative example, we consider the action of a tachyonic field and investigate the conditions under which a phantom crossing can occur as a quantum tunneling effect. For the simplified constant potential case, we investigate the consequences of different boundary conditions on the singularity avoidance and to the mean expansion rate.

Figures (5)

• Figure 1: The plots of the tachyon field and the scalar factor as functions of the cosmic time. The value of $\phi$ is bounded.
• Figure 2: The effective equation of state parameter as a function of the cosmic time.
• Figure 3: The probability density with respect to the $u$ variable for the states with $n=1$ and $n=2$. For the boundary condition \ref{['boundary1']} we obtain a tunneling effect as the universe crosses the phantom line. The blue line corresponds to $0\geq w\geq-1$, the red to $w<-1$.
• Figure 4: The total probability density including the $\phi$ variable for the states with $n=1$ and $n=2$. The crossing to the $w<-1$ region occurs at $u=0$.
• Figure 5: The probability density in the $u$ variable for the boundary condition $\Xi(u_{max})=0$. On the left (L) region the probability density stops at a finite value. The endpoint in $u$ there corresponds to $X\rightarrow-\infty$.