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Vacuum states of different quantum fields corresponds to different energy levels. Some sources say they are mostly zero, some other says that it is a limit imposed by heisenberg uncertainty principle, while some other also say that it is an instrinsic energy level for some fields which is not related to heisenberg uncertainty principle at all. In a sense of absolute energy level which of these claims is correct and where did they come from?

never took courses but why
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1 Answers1

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Vacuum states of different quantum fields corresponds to different energy levels.

It is not entirely clear what this statement is supposed to mean in the actual formalism, but it is almost certainly incorrect. The vacuum energy - or zero-point energy - of a quantum field theory is not observable in non-gravitational theories (in which it corresponds instead to the cosmological constant), and is in fact a renormalization parameter in such theories that can be set to an arbitrary value.

It has no physical content, and, loosely spoken, all other energy levels are ordinarily measured relative to the energy of the vacuum. You also cannot meaningfully compare the energy levels of different quantum field theories because their states live in different Hilbert space with different Hamiltonian, i.e. the very definition of what energy means differs between theories.

(In fact, the above is not strictly true. When your theory has more than one perturbative vacuum, such as in the case of instantons, it may become meaningful to talk about their energy or at least their energy differences. But such phenomena are specific to certain types of quantum field theories and there is in my opinion little use in trying to make any sort of generic statement about these cases.)

Anyways, the uncertainty principle is entirely unrelated to the vacuum energy. And actually, it is often misapplied, especially in popular presentation, when it comes to energies. See this excellent answer by joshphysics for the proper meaning of an uncertainty principle that involves energy.

ACuriousMind
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  • Is the zero-point energy meaningfully different from the vacuum expectation value (vev) of a particular field? Because, for example, the Higgs mass is dependent on the Higgs field vev, which I would argue means the vev has physical content. – probably_someone Apr 24 '19 at 17:49
  • @ACuriousMind "is in fact a renormalization parameter in such theories that can be set to an arbitrary value." <- Could you point me to a reference for that statement? – ungerade Apr 24 '19 at 18:29
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    @ungerade See e.g chapter 2.8.1 and 8.3 in Weigand's QFT notes. The idea is that the vacuum energy is the same as the resummation of all diagrams with no external legs, and in the absence of gravity/a cosmological constant there is no physical parameter you could measure to fix the renormalization parameter for the divergence of this resummation (unlike e.g. for the resummation of diagrams with two external legs, i.e. the propagator, where the renormalization parameter is the physical mass). – ACuriousMind Apr 25 '19 at 17:05
  • @probably_someone Yes, the zero-point energy and VEVs are different concepts. – ACuriousMind Apr 25 '19 at 17:05
  • @ACuriousMind would they take on definite values if we consider general relativity? – never took courses but why Apr 26 '19 at 15:00
  • @nevertookcoursesbutwhy In principle yes, but this is an area fraught with perils since there is no accepted theory of quantum gravity. – ACuriousMind Apr 26 '19 at 15:48