# Does zero point energy really contribute to the cosmological constant?

The zero point energy is usually supposed to contribute to the cosmological constant. And the mismatch between the small cosmological constant compared with the huge zero point energy is deemed as one of the most serious problems in physics.

However, the zero point energy and the cosmological constant are totally different animals.

The energy-momentum tensor $$T^{\mu\nu}_\Lambda$$ of the cosmological constant is of the form: $$T^{00}_\Lambda = \rho_\Lambda,$$ and $$T^{11}_\Lambda = T^{22}_\Lambda = T^{33}_\Lambda= p_\Lambda$$ with $$p_\Lambda = -\rho_\Lambda.$$

And what does the zero point energy look like? Let's take a massless fermion for example, the vacuum energy-momentum tensor $$T^{\mu\nu}_F$$ can be calculated as (see details on page 55 of Philip Mannheim's paper here): $$T^{00}_F = \rho_F = -\frac{2\hbar}{(2\pi)^3}\int k d^3k,$$ and $$T^{11}_F = T^{22}_F = T^{33}_F= p_F = -\frac{2\hbar}{3(2\pi)^3}\int k d^3k$$ Therefore, with a proper regularization/cutoff, one has $$p_F = \frac{\rho_F}{3}$$ which is categorically different from the case of cosmological constant $$p_\Lambda = -\rho_\Lambda$$.

Hence there is no similarity between the zero point energy and the cosmological constant at all!

In cosmological nomenclature $$p = w\rho$$ where $$w$$ is called equation of state parameter, which is $$-1$$ for the cosmological constant and $$1/3$$ (radiation-like) for the above massless fermion example.

Changing to massive fermion/boson will not help the case either. You would get a $$w$$ somewhere between mater-like ($$w=0$$) and radiation-like ($$w=1/3$$) depending on the energy-momentum level, but you would not get a cosmological-constant-like ($$w=-1$$) equation of state parameter. Interested readers are encouraged to verify independently.

Some readers may challenge the above calculation, specifically about the Lorentz invariance of the regularization. Well I would like to challenge anyone to provide a specifically Lorentz invariant regularization and prove that equation of state parameter $$w$$ is the same between cosmological constant and zero point energy of massless mater.

To corroborate the notion let's quote Jerome Martin's paper (page 12) Everything You Always Wanted To Know About The Cosmological Constant Problem (But Were Afraid To Ask);

It is clear from the previous expressions that $$p/\rho \neq -1$$ which indicates that the stress energy tensor is not of the form ∝ $$-\rho g_{\mu\nu}$$. In the limit m → 0, as can be easily shown from Eqs. (75) and (78), the equation of state is in fact $$p/\rho = 1/3$$. This would mean that the zero point fluctuations do not behave like a cosmological constant but rather like radiation.

The paper goes on to discuss how to fix this with dimensional regularization. But the dimensional regularization usually kills off non-logarithmic divergences and the divergent integral in hand is quartical, so I am not particularly convinced.

Note that the Higgs potential : $$V_{H} \sim (-m_H^2 |\phi|^2 + \lambda |\phi|^4)$$ may indeed contribute to cosmological constant since it has a equation of state parameter $$w=-1$$, should the Higgs field $$\phi$$ develop a non-zero VEV upon spontaneous symmetry breaking. Mind you that the Higgs potential contribution is a separate story from the zero point energy ppl usually talk about.

• Not only does the stress-energy from your/Mannheim's calculation break Lorentz invariance, but the preferred frame that it picks out is whatever frame you happened to do the calculation in. Showing that the QFT ground state breaks Lorentz invariance would be one thing (a Nobel-prize-winning thing), but this calculation isn't even internally consistent. I think that if there's a real question here, it's "how can I calculate the correct Lorentz-invariant vacuum stress-energy other than by dimensional regularization?" Jan 30, 2021 at 2:34
• Yes, I am seriously trying to achieve "a Nobel-prize-winning thing" by posting a question anonymously on a social media site. Thanks for reminding me that I should set aside enough money in case I need to buy a ticket to Stockholm. Jan 30, 2021 at 16:16
• The giddy prospect aside, let's go back to some trivial and boring stuff. @benrg do you consider the density terms on the right side of the Friedmann equation (other than the cosmological constant term) preserving Lorentz invariance? Jan 30, 2021 at 16:17
• The real cosmological constant problem isn't the $O(\Lambda^4)$ zero point energy where $\Lambda$ is the cutoff scale, it's the fact that any physics with mass scale $m$ shifts the cosmological constant by $O(m^4)$, which is far too much even if you just count the electron. You can get this conclusion with either dimensional regularization or the leading $O(m^4)$ contribution in cutoff regularization. Jan 30, 2021 at 21:56
• @benrg : Then what's the argument, and what shows that its stress-energy tensor has exactly the same form as the one for $\Lambda$, so that $\Lambda$ can be equated without mathematical inconsistency to it and the conundrum posed? Remember, you have to derive it from RQFT only, and show the two coincide. (Note that I don't expect this to fit in comments, so citation of a standard reference work would be acceptable, provided it mentions the final formula.) Feb 1, 2021 at 2:50

I think you and your sources are right, in the sense calculations of $$T_{\mu\nu}$$ for zero point fluctuations of a field have trouble to recover anything resembling the cosmological term.

Introducing frequency cutoff means we have a preferred frame, so energy density isn't Lorentz invariant. Such cut fluctuations can't have the form of the cosmological term.

If we allow the integration to run to infinity, then we get something that formally looks Lorentz invariant. For example, zero point fluctuations of EM field have spectral function $$k\omega^3$$ and its value at any frequency is Lorentz invariant.

For such infinite fluctuations all diagonal entries of $$T$$ have the same magnitude... Perhaps this gives some motivation to the idea that cosmological term is due to vacuum fluctuations.

But they are infinite! Clearly, explaining the cosmological term in terms of vacuum fluctuations is not very functional mathematically, and is completely inconsistent with observations (which imply zero or small cosmological term).

I think it better to lay this idea that zero point fluctuations have real big positive energy to rest. There is no known phenomenon which requires existence of big positive energy of zero point fluctuations. Even the Casimir effect turned out to be better explained in terms of interaction forces between material bodies (retarded EM forces, sometimes called London forces). Besides the classical Jaffe paper, I like how Grundler explains this. [1],[2] and regarding the stress-energy tensor not having the form of the cosmological term, see [3].

[1] Grundler G., The Casimir-Effect: No Manifestation of Zero-Point Energy, https://arxiv.org/abs/1303.3790

[2] Grundler G., What the Casimir-Effect really is telling about Zero-Point Energy, https://www.astrophys-neunhof.de/mtlg/se93327.pdf

[3] Grundler G., Canonical quantization of elementary fields, https://arxiv.org/abs/1506.08647

One possibility which I sometimes wonder about is that maybe zero point fluctuations of EM field do exist, like postulated in Stochastic Electrodynamics, in the sense $$\langle E^2\rangle$$ is non-zero even at $$T=0K$$ and has the above mentioned spectrum. But perhaps Poynting formula does not correctly give EM energy on the microscale, so the field does not need to have substantial positive energy to fluctuate and there is no UV catastrophe. If that was so, we could have zero-point fluctuations with the Lorentz invariant spectrum, keep the SED results and there would be no associated energy/cosmological term problem.

• Thanks for providing the links to Grundler's papers. The regular text books failed to mention these strong arguments that you pointed out, and give the readers a false impression that zero point energy and cosmological constant are indisputably connected. The connection is actually still being contested among physicists. Feb 1, 2021 at 2:45
• You're welcome. I've added third link where Grundler addresses the cosmological term directly. Feb 1, 2021 at 7:48