present treatment. A hint from loop quantum gravity (LQG) theory is that when one wants to remove the ambiguities from LQG, the ordering factor should take the value p = −2 [18]. VIII. DISCUSSION AND CONCLUSION In summary, we have presented a mathematical proof that the universe can be created spontaneously from nothing. When a small true vacuum bubble is created by quantum fluctuations of the metastable false vacuum, it can expand exponentially if the ordering factor takes the value p = −2 (or 4). In this way, the early universe appears irreversibly. We have shown that it is the quantum potential that provides the power for the exponential expansion of the bubble. Thus, we can conclude that the birth of the early universe is completely determined by quantum mechanism. One may ask the question when and how space, time and matter appear in the early universe from nothing. With the exponential expansion of the bubble, it is doubtless that space and time will emerge. Due to Heisenberg’s uncertainty principle, there should be virtual particle pairs created by quantum fluctuations. Generally speaking, a virtual particle pair will annihilate soon after its birth. But, two virtual particles from a pair can be separated immediately before annihilation due to the exponential expansion of the bubble. Therefore, there would be a large amount of real particles created as vacuum bubble expands exponentially. A rigorous mathematical calculation for the rate of particle creation with the exponential expansion of the bubble will be studied in our future work. IX. ACKNOWLEDGMENT We thank the referees for their helpful comments and suggestions that significantly polish this work. Financial support from NSFC under Grant No. 11074283, and NBRPC under Grant Nos. 2013CB922003 is appreciated. [1] B. A. Bassett, Rev. Mod. Phys. 78, 537(2006). [2] A. A. Starobinsky, JETP Lett. 30, 682 (1979) [Pisma Zh. Eksp. Teor. Fiz. 30,719 (1979)]. [3] A. A. Starobinsky, Phys. Lett. B 91, 99 (1980). [4] A. H. Guth, Phys. Rev. D 23, 347 (1981). [5] B. S. DeWitt, Phys. Rev. 160, 1113 (1967). [6] A. Vilenkin, Phys. Rev. D 50, 2581 (1994). [7] N. Pinto-Neto and J. C. Fabris, Classical Quantum Gravity 30, 143001 (2013). [8] N. Pinto-Neto, F. T. Falciano, R. Pereira, and E. S. Santini, Phys. Rev. D 86, 063504 (2012). [9] S. P. Kim, Phys. Lett. A 236, 11 (1997). [10] S. W. Hawking, Nucl. Phys. B 239, 257 (1984). [11] D. Bohm, Phys. Rev. 85, 166 (1952). [12] P. R. Holland, The quantum Theory of Motion. Cambridge University Press, Cambridge (1993). [13] A. Vilenkin, Phys. Rev. D 37, 888 (1988). [14] L. P. Grishchuk, Classical Quantum Gravity 10, 2449 (1993). [15] J. B. Hartle, S. W. Hawking, and T. Hertog, J. Cosmol. Astropart. Phys. 01 (2014) 015. [16] D. H. Coule, Classical Quantum Gravity 22, R125 (2005). [17] J. B. Hartle and S. W. Hawking, Phys. Rev. D 28, 2960 (1983). [18] W. Nelson and M. Sakellariadou, Phys. Rev. D 78, 024006 (2008). [19] When x ? |ν 2 − 1/4|, Bessel functions take the asymptotic forms, Iν(x) ∼ e x / √ 2πx, Kν(x) ∼ e −x / √ 2πx, Jν (x ? |ν 2 − 1/4|) ∼ p 2/πx cos(x − νπ/2 − π/4), and Yν (x ? |ν 2−1/4|) ∼ p 2/πx sin(x−νπ/2−π/4). We can use these asymptotic forms to calculate S(a) and a(t) for large bubbles