Planck energy is the unit of energy in the system of Planck units, calculated from the three main fundamental constants in physics: the speed of light, the gravitational constant, and Planck’s constant, with a value of around 1.2×10^19 GeV.
It is in fact the only combination of these constants that yields the dimensions of energy, which can be proven by dimensional analysis.
Its value is enormous for high-energy physics. For comparison, the collision energies at the LHC are just 13600 GeV. Still, it shouldn’t be thought of as a maximum energy limit, because other Planck units such as the Planck charge is just about 12 times the elementary charge of an electron. It is commonly thought to be a particular energy scale, at which quantum effects of gravity become significant. An example of this is that theoretically, a photon with Planck energy has its wavelength shorter than its own Schwarzschild radius.
Since we cannot currently test this energy scale, there is the question of whether it is a fundamental energy scale in our theories, or if it is just a curiosity of dimensional analysis and has no physical relevance in the laws of nature as we known them. We will now explore where Planck energy is used and how it fails to reproduce observable quantities and its implications.
The Planck energy is used as a cutoff energy in the calculation of the vacuum energy density in quantum electrodynamics. The ground state (the state with the lowest energy) of the quantum harmonic oscillator has a nonvanishing zero-point energy which is half the oscillation frequency of the corresponding classical harmonic oscillator times Planck’s constant. The energy density in quantum electrodynamics can be easily derived as a summed zero-point energy for each oscillator mode. The energy density equation is divergent and becomes infinite unless an ultraviolet frequency cut-off is imposed, signifying up to which frequency range one believes the quantum field theory framework is effectively valid. If one sets this limit to the scale of the Planck energy, the result is 120 orders of magnitude higher than the observation cosmological constraint of dark energy as the energy of the vacuum, leading to the vacuum catastrophe problem in theoretical physics (the equation can also be reached through dimensional analysis). And this is just accounting for the vacuum contribution of quantum electrodynamics. In theory, it is expected that the total vacuum energy of the complete quantum field theory model is roughly the sum of the vacuum energy contributions of the individual fields.
The Planck energy is also used in the theoretical calculation of the Higgs mass. The observable Higgs mass of around 125 GeV is the sum of the Higgs bare mass and its quantum corrections, which are sensible to the ultraviolet cutoff in the same way as the calculation of the energy of the vacuum in quantum electrodynamics.
The ultraviolet cutoff is used to regularize the integral equation for the correction of the Higgs mass by the Planck energy and without this cutoff, the Higgs mass correction and the observable value would be infinite. An extreme fine tuning with the quantum correction term must take place in order for the observable mass not to be at the energy scale of Planck mass and energy. This constitutes a so-called hierarchy problem.
Thus, it is clear that the use of the Planck energy in physics leads to problems with both the measured cosmological constant and the measured Higg’s mass. But why is this the case? The speed of light and Planck’s constant, are used together in relativistic quantum mechanics and the gravitational constant and the speed of light are used together in General Relativity, and both combinations provide accurate predictions.
But the Planck energy is the only case where the three are used together, excluding the Hawking radiation equation and Bekenstein’s entropy equation of black holes, for which there is no experimental evidence.
Problems arise when combining the gravitational constant and Planck’s constant. The gravitational constant is known with far less precision than the other two fundamental constants, and gravity has only been precisely tested for separations ranging from the scale of the solar system down to a few millimeters. In contrast, Coulomb’s Law and its electroweak generalization has been tested for separations down to 10^-18m. Plus, there is evidence that gravity might behave differently in the larger scales, such as dark matter and dark energy. In particular, modified gravity suggests that gravity is stronger in small accelerations or equivalently, in small gravitational field intensities. This is similar to what happens for the masses of fundamental particles in quantum field theory, in which even though they are regarded as fundamental constants, their values vary with the energy scale through renormalization.
Is the gravitational constant truly constant? Does it vary with time or energy scale? Is it fundamental or is it a combination of more different constants or a constant and a varying parameter?
It is worth noting that Planck’s constant, the gravitational constant and the speed of light, are also present in the Tolman-Oppenheimer-Volkoff limit as an upper bound of the mass of cold and non-rotating neutron stars before their collapse. But for this case, the combination of constants is not used as a cutoff as for the two cases described before. Oppenheimer and Tolman supposed that the mass limit was equivalent to a black hole with the density of nuclear matter, including in this way the Schwarzschild radius with the gravitational constant and nuclear density calculations of Yukawa, that is, Planck’s constant.
References:
S.E. Rugh and H. Zinkernagel, The Quantum Vacuum and the Cosmological Constant Problem, 2002