Lorentz Invariance and the Zero-Point Stress-Energy Tensor

Abstract

Some 67 years ago (1951), Wolfgang Pauli noted that the net zero-point energy density could be set to zero by a carefully fine-tuned cancellation between bosons and fermions. In the current article, I will argue in a slightly different direction: the zero-point energy density is only one component of the zero-point stress energy tensor, and it is this tensor quantity that is in many ways the more fundamental object of interest. I shall demonstrate that Lorentz invariance of the zero-point stress energy tensor implies finiteness of the zero-point stress energy tensor, and vice versa. Under certain circumstances (in particular, but not limited to, the finite quantum field theories [QFTs]), Pauli's cancellation mechanism will survive the introduction of particle interactions. I shall then relate the discussion to beyond standard model [BSM] physics, to the cosmological constant, and to Sakharov-style induced gravity.

The Theory of Local Mass Dimension One Fermions of Spin One Half

“…The new zero point energy comes with a minus sign, and is infinite. For an insightful discussion on this point we refer the reader to Matt Visser's recent e-print on the general subject [46]). Above the zero point energy, all the four degrees of freedom contribute the same energy E(p) to the field f(x) for a given momentum p.…”

The Pauli sum rules imply BSM physics

“…This is merely a book-keeping device, it is not per se an appeal to supersymmetry. (See particularly reference [16] and the more recent extensive discussion in reference [3]. Any of the options of softly but explicitly broken supersymmetry, the known nonsupersymmetric UV-finite QFTs [17][18][19][20][21][22], or mis-aligned supersymmetry [23][24][25][26] can be used to motivate introducing this book-keeping device.…”

“…From However we do know that the cosmological constant can be estimated by [3] ρ cc = ρ zpe = −p zpe = − At least the energy scale for the cosmological constant is not off by the extremely naive factor 10 123 ; it is now more like 10 55 . Roughly speaking, the ρ cc ∼ O(M 4 Planck ) guesstimate has been replaced by a ρ cc ∼ O(m 4 Higgs ) estimate.…”

“…When viewed with hindsight these three Pauli conditions are necessary and sufficient for ensuring both the Lorentz invariance and finiteness of the zero-point stress-energy tensor [3]. Specifically, (see for instance reference [3] and many references therein), once the three sum rules above are imposed one deduces that the net zero-point energy density and net zero-point pressure are given by the finite expression:…”

“…Finally one sums over all particle species indexed by n. It is the physical relevance of this sum over the entire particle physics spectrum, leading to significant bose-fermi cancellations, that is Pauli's key physical insight. (In related discussion in reference [2] the bose-fermi sign factor (−1) 2S has been absorbed into the degeneracy factor g.) Note that the bose-fermi cancellations coming from the sum over particle species can be thought of as a physical, unitarity preserving, form of Pauli-Villars regularization -see discussion in reference [3].…”