## Cosmological constant as “gauge-fixing” parameter

This note represents scratch work on an idea that will probably become a research letter, leaving aside the issue of publishing. It might be edited until it is substantive and complete enough to become that letter. The aim here is explain the ideas in as accessible a way as possible to the most general possible audience. Expect a “hip shot” to lead and supporting citations to follow.

The cosmological constant is a term that may be added to solutions to the Einstein field equations governing general relativity to produce new families of solutions. The cosmological constant gives “empty” space a constant energy density in these new solutions. For a few decades, physicists have suspected that our actual universe has an effectively nonzero cosmological constant. This leads us to question, “Where is that energy coming from?” since it is natural to assume that “empty” space actually carries no energy, momentum, or stress.

There is a “textbook” example of quantum mechanical behavior called the “simple quantum harmonic oscillator.” Decoding the jargon, all the name really means is that quantum particles sometimes act as though they were pendulums or attached to idealized springs. This example is “textbook” because it is both simple and found all throughout nature. One example of it is in quantum fields. We can think of space as being filled with something like a medium that represents every possible state of every particle we know, and this is basically a quantum field. Generally, we can think of these fields as being made of many “simple quantum harmonic oscillators,” potential “pendulums” that fill all of space that real particles can oscillate with. Part of the “textbook” treatment of these oscillators is that, if the point at which the “pendulum” doesn’t “swing” at all is taken to be the point of zero energy, then the quantum treatment requires a small positive energy for any “pendulum,” because a quantum mechanical pendulum can never quite totally stop its motion. We can never take this energy away from the field. (That is, unless we remove potential states, as is seen in the “Casimir effect,” but we will not need to talk about this effect here.)

So the natural conceptual leap is to identify this tiny minimum required “motion” of the field as exactly the constant energy density that the cosmological constant resembles, where we assume that the point of zero energy would be exactly where all the “motion” of the “pendulums” stops. This seems theoretically reasonable, but from experimental observation we know that it is *fantastically* wrong, or at least an incomplete picture. The estimate we arrive at this way is about 120 orders of magnitude too large to be the cosmological constant we measure. (120 orders of magnitude would be a multiplicative factor of a number written with a one followed by 120 zeroes.) It has literally been called the worst theoretical prediction in physics ever.

Attempts have been made to rectify this huge difference, such as the supposition of “supersymmetry.” Among its other features, supersymmetry could lead to cancellation of the minimum required energies of different fields against each other. However, with every round of experiments at the newest and most powerful particle colliders, we push the boundaries for finding evidence of supersymmetry further out, with no great experimental success to date.

I have led you through the natural assumptions, here. We assume that space without fields is truly “empty,” carrying no energy, momentum, or stress. We assume that the “true” point of zero energy for quantum field oscillators is the (physically unattainable) point where all oscillation stops. I propose that we throw either or both of these assumptions out the window, and I’m about to explain how abandoning either one or both leads to basically the same consequences.

First, I want to state that “textbook” quantum mechanics is basically agnostic to the nature of gravity, and “textbook” general relativity is basically agnostic to the nature of quantum mechanics. One can be applied to the other, presumably, but the two theories were not developed with clear rules for doing so, or even the necessity for one to be applied to the other. There is ambiguity, here. In quantum mechanical consideration of energy, it is basically only *differences* in energy which are significant. In general relativity, we have to know the *absolute* quantity of the energy, among other things, to determine how things gravitate, and quantum mechanics doesn’t necessarily tell us what this absolute quantity is.

In classical as well as quantum mechanics, rather than assuming that the point of exactly zero oscillation is the point of zero *absolute* energy, this point could just as well be a billion units of energy or a negative billion units of energy. We can add this billion or negative billion units to the ground state energy consistently, and it makes absolutely no mechanical difference whatsoever. It only changes the zero-point of our accounting, while the motions of particles remain exactly the same whatever this zero-point might be. We call this “gauge invariance,” because it reminds us of something like the zero-point on a car tire pressure gauge. We can set this zero at an idealized absolute zero pressure or set the zero at atmospheric pressure, but all we really measure are *differences* between the pressure of tire and our reference zero-point. For the purposes of gravity, then, it’s actually ambiguous what the absolute energy of the zero-point is coming from quantum mechanics, because quantum mechanics doesn’t depend on absolute measure of the energy. However, it’s arguably natural to assume that the gravitational absolute zero-point is the (physically unattainable) point where all field oscillation hypothetically stops.

In general relativity, we can add a cosmological constant to a known acceptable solution to the theory’s governing equations and produce another acceptable solution, with no consideration of quantum mechanics whatsoever. We can change this constant energy density in a way that is basically arbitrary. General relativity does not tell us or require of us that “empty” space is truly empty except for the zero-point of quantum fields. Conversely, our freedom in choosing the cosmological constant while still keeping an acceptable physical solution could suggest that “empty” space is not required to carry zero energy at all. However, unlike in the case of gauge invariance, changing the cosmological constant intrinsically changes the mechanics.

On an aside, the cosmological constant has always resembled a gauge parameter to me, coming from the gauge invariance described above. That is, it looks like an arbitrary zero point for the energy. It is not arbitrary, though. It can’t be, because the strength of the gravitational attraction depends on an absolute zero-point for the energy, momentum, and stress. Rather, I think it might be a parameter that fixes a preferred gauge, a preferred absolute zero-point, for quantum fields which otherwise don’t depend on an absolute zero-point of energy for their mechanics.

Ontologically, we can think of the cosmological constant as either being the minimum energy of the quantum fields when their gauge is fixed by the true gravitational zero-point, or we can think of the constant as an intrinsic energy density for “empty” space before we add quantum fields. In either case, so long as the “dark energy” density we measure in the universe remains fixed, there is no mechanical difference between these two ontologies.

Quantum mechanics does not tell us the zero-point of energy of matter fields for gravitational accounting purposes, and general relativity does not tell us this either. There might be no 120 order of magnitude difference between the true absolute zero-point of the fields and the observed dark energy density then, at all. Neither quantum mechanics nor general relativity actually give us enough information to say what the zero-point of the quantum fields are in the first place, or what the energy density of space is without any quantum fields at all.