**Spontaneous creation of the universe from nothing:**

An interesting idea is that the universe could be spontaneously created from nothing, but no rigorous proof has been given. In this paper, we present such a proof based on the analytic solutions of the Wheeler-DeWitt equation (WDWE). Explicit solutions of the WDWE for the special operator ordering factor p = −2 (or 4) show that, once a small true vacuum bubble is created by quantum fluctuations of the metastable false vacuum, it can expand exponentially no matter whether the bubble is closed, flat or open. The exponential expansion will end when the bubble becomes large and thus the early universe appears. With the de Broglie-Bohm quantum trajectory theory, we show explicitly that it is the quantum potential that plays the role of the cosmological constant and provides the power for the exponential expansion of the true vacuum bubble. So it is clear that the birth of the early universe completely depends on the quantum nature of the theory.

**I. 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.