Deviation from the standard uncertainty principle and the dark energy problem

Abstract

Quantum fluctuations of a real massless scalar field are studied in the context of the generalized uncertainty principle (GUP). The dynamical finite vacuum energy is found in spatially flat Friedmann–Robertson–Walker spacetime which can be identified as dark energy to explain late time cosmic speed-up. The results show that a tiny deviation from the standard uncertainty principle is necessary on cosmological ground. By using the observational data we have constraint the GUP parameter even more stronger than ever.

Appendix A: The deformed density of states and the Liouville theorem

In this Appendix, we consider time evolution of the deformed density of states (6) and we show that Liouville theorem is satisfied in the GUP framework. The classical limit of the deformed commutation relations (3), (4) and (5) can be obtained by replacing the operators with their classical counterparts and Dirac commutators with Poisson brackets as $\frac{1}{i}[,\,]\rightarrow \{,\,\}$. In $D$-dimensions, the deformed Poisson algebra is given by [9]

$$\begin{aligned} \{\,x_i,\,p_j\}&= \big (1-\alpha \,p+{2\alpha ^2} \,p^2\big )\delta _{ij},\nonumber \\ {\{\,x_i,\,x_j\,\}}&= \alpha \left( 4\alpha -\frac{1}{p}\right) \big (p_i x_j-p_j x_i\big ),\nonumber \\ {\{\,p_i,\,p_j\,\}}&= \,0, \end{aligned}$$

(40)

where we have defined $\alpha =\frac{\alpha _0}{M_{_{Pl}}}$. The deformation to the phase space due to the deformed commutation relations (40) can be obtained through a general transformation in the corresponding phase space which deforms the phase space volume as [29, 30]

$$\begin{aligned} \frac{d^{D}x\,d^{D}p}{J}, \end{aligned}$$

(41)

where $J(x,p)$ is the Jacobian of the transformation which can be expressed in terms of the Poisson brackets as [31]

$$\begin{aligned} J=\frac{1}{2^{^D}D!}\sum _{i_1...i_{2D}=1}^{2D} \varepsilon _{i_1...i_{2D}}\{J_{i_1},J_{i_2}\} ...\{J_{i_{2D-1}},\,J_{i_{2D}}\}, \end{aligned}$$

(42)

where $\varepsilon$ is the Levi-Civita symbol and $J_i$ denotes the new phase space variables so that for odd $i$ it is a coordinate and for even $i$ it is a conjugate momentum. In the Jacobian (42), the coordinate–coordinate Poisson brackets are always multiplied by the momentum–momentum Poisson brackets. So, the non-zero coordinate–coordinate Poisson brackets have no contribution in the Jacobian because the momenta commutes through relation (40). Consequently, the Jacobian (42) simplifies to [29, 30]

$$\begin{aligned} J=\prod _{i=1}^{D}\{x_i,\,p_i\}=\Big (1-\alpha \, p+{2\alpha ^2}\,p^2\Big )^{D}. \end{aligned}$$

(43)

Using the above Jacobian in relation (41) gives the deformed phase space volume in the GUP framework

$$\begin{aligned} \frac{d^{D}x\,d^{D}p}{\Big (1-\alpha \,p+\,2 \alpha ^2\,p^2\Big )^D}. \end{aligned}$$

(44)

In the next step we consider the time evolution of the deformed phase space volume (44).

The classical equations of motion can be represented with Poisson brackets in the Hamiltonian formalism as

$$\begin{aligned} \dot{x}_i&= -\{x_i,p_j\}\,\frac{\partial {\mathcal {H}}}{\partial p_j}+\{x_i,x_j\}\,\frac{\partial {\mathcal {H}}}{\partial x_j},\nonumber \\ {\dot{p}_i}&= -\{x_j,p_i\}\, \frac{\partial {\mathcal {H}}}{\partial x_j}, \end{aligned}$$

(45)

where ${\mathcal {H}}(x,p)$ is the Hamiltonian of the system. Consider an infinitesimal change for the phase space variables under time evolution

$$\begin{aligned} x_i'&= x_i + \delta x_i,\nonumber \\ p_i'&= p_i + \delta p_i. \end{aligned}$$

(46)

The dynamics of $\delta x_i$ and $\delta p_i$ is given by relations (45)

$$\begin{aligned} \delta {x_i}&= \{x_i,p_j\}\,\frac{\partial {\mathcal {H}}}{\partial p_j}\delta {t}+\{x_i,x_j\}\,\frac{\partial {\mathcal {H}}}{\partial x_j}\delta {t},\nonumber \\ {{\delta }}p_i&= -\{x_j,p_i\}\,\frac{\partial {\mathcal {H}}}{\partial x_j}\delta {t}. \end{aligned}$$

(47)

An infinitesimal phase space volume evolves through relations (46) as

$$\begin{aligned} d^D{x'}\,d^D{p'}=\bigg |\frac{\partial (x'_i,p'_i)}{\partial (x_i,p_i)}\bigg |\,d^D{x}\,d^D{p}. \end{aligned}$$

(48)

The Jacobian can be obtained by using the relations (46) and (47). Up to the first order of $\delta {t}$, we have [14]

$$\begin{aligned} \bigg |\frac{\partial (x'_i,p'_i)}{\partial (x_i,p_i)}\bigg |&= 1+\bigg (\frac{\partial }{\partial x_i}\{x_i,x_j\} -\frac{\partial }{\partial p_i}\{x_j,p_i\}\bigg )\frac{ \partial {\mathcal {H}}}{\partial x_j}\delta {t}\nonumber \\&= 1-{\alpha }D\,\left( 4\alpha -\frac{1}{p}\right) \,p_j\frac{\partial {\mathcal {H}}}{\partial x_j}\delta {t} \end{aligned}$$

(49)

where we have used the relations (40). Substituting the above relation in the relation (48) gives

$$\begin{aligned} d^D{x'}\,d^D{p'}=\bigg (1-{\alpha }D\,\left( 4\alpha -\frac{1}{p}\right) \,p_j\frac{\partial {\mathcal {H}}}{\partial x_j} \delta {t}\bigg )\,d^{D}x\,d^{D}p. \end{aligned}$$

(50)

On the other hand, we should consider the time evolution of the term on the denominator of the relation (44). Using relations (45) and (47), to first order of $\delta {t}$, we have

$$\begin{aligned} {p'}^2=\sum _{i}p'_i\,p'_i=p^2-2(1-\alpha {p}+2{\alpha ^2} {p}^2)\,p_j\frac{\partial {\mathcal {H}}}{\partial x_j} \,\delta {t}, \end{aligned}$$

(51)

and

$$\begin{aligned} p'=\sqrt{{p'}^2}=p-(1-\alpha {p}+2{\alpha ^2}{p}^2)\,\frac{p_j}{p}\frac{\partial {\mathcal {H}}}{\partial x_j}\delta {t}, \end{aligned}$$

(52)

where we have used the fact that $\delta {t}$ is small. Using the relations (51) and (52) we have

$$\begin{aligned} 1-\alpha {p'}+2{\alpha ^2}{p'}^2=(1-\alpha {p}+2{\alpha ^2}{p^2}) {\bigg (1-\alpha \left( 4\alpha -\frac{1}{p}\right) \,p_j \frac{\partial {\mathcal {H}}}{\partial x_j}\delta {t}\bigg )}, \end{aligned}$$

(53)

which gives the result

$$\begin{aligned} \big (1-\alpha {p'}+2{\alpha ^2}{p'}^2\big )^D=\big (1-\alpha {p} +2{\alpha ^2}{p^2}\big )^D\,\bigg (1-\alpha {D}\,\left( 4\alpha -\frac{ 1}{p}\right) \,p_j\frac{\partial H}{\partial x_j}\delta {t}\bigg ).\nonumber \\ \end{aligned}$$

(54)

Using the relations (50) and (54) we find

$$\begin{aligned} \frac{d^{D}x'\,d^{D}p'}{\Big (1-\alpha \,p'+\,2 \alpha ^2\,p'^2\Big )^D}=\frac{d^{D}x\,d^{D}p}{\Big (1-\alpha \,p+\,2 \alpha ^2\,p^2\Big )^D}\, \end{aligned}$$

(55)

which ensures that the phase space volume (44) is invariant under time evolution and consequently the Liouville theorem is satisfied in this setup. Also, it is important to note that the deformed algebra (40) induces the maximal momentum (UV cutoff) as $p_{_{max}}=1/{2\alpha }=\,M_{_{Pl}}/2{\alpha _0}$. So the range of integrals in the deformed phase space with phase space volume (44) should be changed as has been indicated in Ref. [9]. Taking this results into account and using the invariant phase space volume (44), one can obtain the deformed density of states (6).