What is the energy of vacuum, plus or minus? I am learning QFT. If we used scalar field, the energy of the vacuum turns out to be $+\infty$. However, if we use the spinor field, it turns out to be $-\infty$. Both of the infinite energy are caused by the integral to infinity.
Now I am really confused. No matter what field we choose, we are describing the vacuum. Why we get different sign for this? It is really contradictory theoretically, regardless of the fact that the zero-point energy is real or can be observed.
Can anyone help me with this?
 A: The vacuum of the Universe is unique: it is whatever it is. But our description of the vacuum is not unique: every physical model has its own characteristics, and they offer different descriptions of each aspect of reality. And adding/removing fields changes the model, and therefore it changes what the model predicts.
For example, the muon, as a concept, is unique: it is whatever it is. But the QED description of the muon is not the same as the full Standard Model description of the muon. In the latter there are weak and strong decay modes that are absent in the former.
The very same thing happens to the vacuum: if you change the theory, the vacuum changes. Every theory has its own description of the vacuum, and some theories are better than others. There is nothing contradictory about this.
The "true" vacuum of the Universe has a certain energy, the cosmological constant. You may be under the impression that this energy can be predicted using QFT; for example, according to you, a scalar field theory predicts $\Lambda=+\infty$, and a spinor theory predicts $\Lambda=-\infty$. Both predictions are "obviously" wrong; but, most importantly, there is nothing theoretically inconsistent about the fact that these predictions disagree: different models can and usually do predict different things. In any case, and as a matter of fact, the cosmological constant is not really something that you can predict using QFT (cf. here), so your premise is actually false anyway.
