## The Planck scale

My first career was in physics, with an emphasis in general relativity, so a copy of *Physics Meets Philosophy at the Planck Scale*^{1}Craig Callender and Nick Huggett (eds.), Physics Meets Philosophy at the Planck Scale: Contemporary theories in quantum gravity (Cambridge, 2001) in a public library recently caught my eye. As a retired but still active pastor, it struck me that theological comment on a meeting at the Planck scale might interest readers of *Covalence*.

Before I explain the Planck scale, I’ll tell you where we’re going. Some physicists today are searching for a correct quantum theory of gravitation, and some hope to unify all basic physical interactions into what is sometimes called “a theory of everything.” I’ll conclude with a few modest theological comments about the problems and goals of this enterprise.

The Planck scale is named for physicist Max Planck who, in the late 1890s, was studying the sharing of energy between radiation and matter. Physicists then thought that all physical processes were continuous — “Nature doesn’t make jumps.” But Planck got nowhere with that assumption. Finally he guessed that radiation with frequency f did share energy in jumps, or “quanta,” of size hf, where h was a number that would be called Planck’s constant. He then derived an equation that matched the observations of his experimental colleagues.

That began the quantum revolution. In his 1900 paper, Planck noted one result of introducing h. There already were two recognized constants of nature — c, the speed of light, and G, which gives the strengths of gravitational forces. Like h, those constants depend on the units of length, mass, and time in which we express them.

But we could adopt “Planck units” in which c, G, and h all have the value one. The Planck length is 2 x 10^{—35} meters, a decimal point followed by 35 zeroes before 2. The Planck time is that length divided by the speed of light, 5 x 10^{—44} seconds. Such ultra-tiny lengths and times “at the Planck scale” are far smaller than anything that physicists confront on earth. But what we can surmise about the start of the Big Bang 14 billion years ago indicates that conditions there would have been at that scale.

Planck’s introduction of “his” constant led to modern quantum theory. Before then, Albert Einstein developed his special and general theories of relativity, the latter a view of gravitation that went beyond Newton. Quantum theory and relativity have both been very successful, and today’s physicists and philosophers are thinking about what they may reveal when considered together at the Planck scale. Maybe the long sought “theory of everything” will result. Over twenty years have passed since the publication of the essays to which I referred, and no genuine “Aha!” moment has emerged.

In the next two sections I’ll sketch the development of our understanding of the four basic physical interactions — electromagnetism, gravitation, and the strong and weak interactions. Readers familiar with those topics may skim or skip them. Then we’ll look at issues arising at the Planck scale. My theological comments will follow.

## Unification in the macroworld

Correct rules for how things happen in the world are enough for some people, but theorists ask why those rules, and how one rule relates to another. Copernicus’ proposal that the earth was one of the bodies moving around the sun meant that the heavens and the earth were no longer separate realms. The old phrase, “As above, so below” got a new meaning when it was realized that regularities in one realm held true in the other. Maybe humans could gain a unified picture of the physical world.

In the seventeenth century, Isaac Newton’s absolute space and time and his laws of motion and universal gravitation were a step toward such a picture. With them, astronomical phenomena and much of what happens on earth could be explained. But Newton himself knew that his law of gravitation didn’t explain how or why two bodies attract one another. About that, he wrote, “I frame no hypotheses.”^{2}Sir Isaac Newton, Principia, Vol. II, The System of the World (University of California, 1934), p.547.

Early in the nineteenth century, moving magnets were found to excite electric currents in nearby wires, and electric currents acted as magnets. Michael Faraday introduced the idea of a continuous field to describe such phenomena, and James Clerk Maxwell developed equations for electric and magnetic fields. Their solutions representing waves traveling at the speed of light suggested that light was an electromagnetic phenomenon. Maxwell had united electrical and magnetic forces into a single entity, the electromagnetic field — the first unified field theory.

Since material oscillations were then the only waves known, light was thought to move through a “luminiferous aether.” Light’s speed should then have been affected by an observer’s motion through the aether, but it wasn’t. In 1905 Albert Einstein solved that problem by replacing Newton’s absolute space and time with relative spaces and times of observers moving with uniform velocity with respect to one another. Because of that limitation, this theory is called special or restricted relativity.

Einstein’s theory made it clear that the electromagnetic field itself was “waving” in a light wave. An aether was superfluous. Equivalence of mass and energy, E = mc^{2}, was an important result of this theory. Hermann Minkowski contributed the concept of four dimensional “space-time” as the natural setting for relativity theory.^{3}Articles by Einstein, Lorentz, Weyl, and Minkowski relevant to the development of relativity theory are in The Principle of Relativity (Dover 1923).

Einstein then posed two further questions; can relativity be extended to accelerated reference frames, and can gravity be described by a field theory? He saw that those questions were related because of something known before Newton’s time but whose significance hadn’t been appreciated; all bodies in a given location fall at the same rate under the influence of gravitation if air resistance is removed.

You can get rid of gravity in your immediate vicinity by going into free fall. Anything near you falls with you, so you can’t tell that you’re falling without viewing distant objects. On the other hand, if your car is braking, objects inside behave as though there’s some artificial gravity toward the front. Gravitation can be altered and even eliminated by choosing your coordinate system appropriately.

Einstein came to see that gravitation couldn’t be dealt with adequately if space-time was described by Euclidean geometry. A friend brought him up to speed on non-euclidean geometries, and in 1915 Einstein had satisfactory equations for gravitation. As Maxwell’s equations relate electromagnetic fields to electric charges and currents, Einstein’s relate space-time geometry to energy and momentum. In 1919, he became the first rockstar scientist when his theory gave correct values for the gravitational deflection of starlight passing near the sun during a solar eclipse.

The successes of those two field theories led to attempts to unify them. But the attention of physicists in the early 1920s focused on possibilities opened up by Planck’s quantum concept for understanding the atomic and sub-atomic realms.

## Unification in the microworld

New phenomena for which Planck’s quantum concept seemed relevant were discovered between 1895 and 1925. Light spectra from different elements, the electron as a constituent of atoms, radioactive decays of some atoms, and the stability of Ernst Rutherford’s “solar system” model of an atom with negative electrons orbiting a positive nucleus, were just a few things begging for explanation.

Phenomena having contradictory features posed a great challenge. For a century, light had been understood as a wave because it displayed interference phenomena. But in the photoelectric effect, light shone on a metal ejected electrons from it. In 1905 Einstein explained that as a result of collisions with electrons by light particles (“photons”^{4}The word “photon,” from the Greek for “light,” was coined by the American chemist Gilbert Lewis in 1926.) whose energies were given by Planck’s frequency — energy relation. So was light a wave or a particle?

Twenty years later Louis de Broglie matched that proposal by suggesting that things like electrons that were thought to be particles could also behave as waves. The theories of Einstein and of de Broglie were confirmed by experiments, and both physicists won Nobel Prizes. “Wave-particle duality” was a universal property of matter that traditional physics couldn’t handle.

In 1925 and 1926 Werner Heisenberg and Erwin Schrödinger published two versions of a new quantum mechanics. Heisenberg’s approach, later worked out with Max Born and Pascual Jordan, used matrix algebra developed in the previous century. Schrödinger created an equation for the behavior of the “matter waves” that de Broglie had introduced. The two theories were proved to be mathematically equivalent, and both could be used to explain atomic and molecular structures and behaviors.

Heisenberg called attention to an “uncertainty principle” as an important result of quantum theory. By analyzing ways to measure a particle’s position or momentum, he showed that either one of those quantities, but not both, could be determined precisely. The product of uncertainties in momentum and position can’t be much smaller than Planck’s constant.^{5}The exact minimum value is h/4π. There’s a similar relationship between measurements of a system’s energy and the time interval in which the measurement takes place.

In Newton’s physics, prediction of a physical system’s development in time requires that we know its initial state. The uncertainty principle says we can’t know that state exactly. We’ll see the importance of this principle at the Planck scale soon.

Paul Dirac then developed an extension of Schrödinger’s wave equation that was consistent with special relativity. Quite unexpectedly, it predicted a positively charged counterpart of the electron. Such positrons were soon found in cosmic rays, the first example of antimatter. Dirac also applied quantum mechanical principles to electromagnetic fields interacting with electrons to produce a quantum field theory.

In 1932, James Chadwick discovered the neutron, an electrically neutral particle slightly heavier than a proton. Neutrons and protons together make up atomic nuclei. In 1935, Hideki Yukawa published a theory in which attractive forces between nucleons were transmitted by exchanging particles with a mass between those of electrons and of nucleons. This gave rise to a force, the strong interaction, which could overcome the electrostatic repulsion between nuclear protons and hold nuclei together. Ten years later, the particles Yukawa predicted, pions, which were found in cosmic ray debris.

Finally, there’s the weak interaction. A neutron can undergo radioactive beta decay and apparently be converted into a proton and an electron. But energy didn’t seem to be conserved in that process. In Enrico Fermi’s theory of beta decay, the neutron emits an electron and a neutrino which carries off some energy and becomes a proton. Over twenty years later, the weakly interacting neutrinos were detected.

In the years following World War II, many new particles were found in cosmic rays and experiments with particle accelerators. Eventually it became clear that there were four basic physical interactions. They are, in decreasing order of strength, the strong, electromagnetic, weak, and gravitational interactions.

In quantum electrodynamics, electromagnetic interactions between particles are transmitted by exchanges of photons. Similar, though more complicated theories were later developed for the weak and strong interactions. For brevity, we can call the fields for those three interactions “matter fields.” In them the forces between particles (themselves quanta of fields) are transmitted by other particles.

In the 1970s, the weak and electromagnetic forces were united into a theory of electroweak interactions. There are hopes for a “grand unified theory” (GUT) in which the strong and electroweak interactions are combined in a natural way. But we can’t yet achieve particle energies needed in order to test and perhaps refine such a theory.

With all of that, what about a quantum theory of gravitational interactions? And is hope for theoretical unification of gravitation with the other three interactions at all realistic? We’ll know chances of achieving one or both of those possibilities better when we come to understand the Planck scale more fully.

## Problems at the Planck Scale

Gravitation is the weakest of the four basic interactions, but enough mass packed together can overcome all the other forces. This leads to gravitational collapse of massive stars and formation of black holes from which light can’t escape. When we approach the Planck scale, gravitation can be expected to dominate.

But we don’t yet have a good quantum theory of gravitation. When attempts are made to solve the equations of field theories, the answers are sometimes “infinity.” There’s a well-defined procedure, “renormalization,” for removing the infinite baggage. But gravitation can’t be renormalized. The lack of a theory of quantum gravity makes it hard to speak of conditions at the Planck scale where gravitation is a critical factor.^{6}https://www.britannica.com/science/renormalization

Years ago I noted that general relativity and quantum theory together limit time measurements at the Planck scale.^{7}George L. Murphy, “The Fundamental Length of Quantized Gravitation,” American Journal of Physics 42, 958, November 1974. Einstein’s prediction of gravity’s effect on the rate of a clock is now confirmed. If we consider a simple model of a clock and take into account the gravitational effect of its total energy, we find that the uncertainty relation between energy and time rules out measurement of time and space intervals smaller than the Planck values. Theories that refer to such values are therefore suspect.

Most of the essays in Reference #1 pose scientific and philosophical questions about approaches to uniting general relativity with quantum theory. Significantly, none of the essays argue for a grand theory uniting gravitation with the other interactions. This isn’t surprising if gravity has its origins in the geometry of space-time and the other three don’t originate in that way. Three essays deal with string theory, which today isn’t regarded as highly as it was 25 years ago.^{8}E.g., Lee Smolin, Trouble with Physics: The Rise of String Theory, the Fall of a Science, and What Comes Next (Houghton Mifflin, 2006). The last two deal with relations between gravity and quantum mechanics, a subject that deserves comment here.

Quantized general relativity might have startling features. Quantum states change with time, so the state of an electron whose spin initially points UP can evolve into a state with 50% probabilities of spin UP and spin DOWN. If we measure the spin when it’s in the latter state, we’ll find it either all UP or all DOWN. A traditional view is that measurement “collapses” the state to one or the other value, but that’s debated.

Schrödinger had doubts about that traditional view and parodied it with his “cat paradox,”^{9}https://www.aps.org/publications/apsnews/200203/history.cfm now a popularized technicality. A cat in a sealed room has a 50% chance of being killed within the next hour by an “infernal machine” triggered by decay of a radioactive atom. The cat’s state is initially ALIVE, but develops until after an hour it’s in a 50% ALIVE plus 50% DEAD state. On observation, the cat will be either dead or alive. How that is supposed to come about has been hotly debated.

Oxford University researcher Joy Christian points out the implications of applying Schrödinger’s argument not to a feline’s fate but to ideas of quantum gravity in which observing the world would make it one kind of world rather than another. This helps to explain the article’s title, “Why the quantum must yield to gravity.”

Unlike the Planck length and time, we can visualize a Planck mass, 2 x 10^{—8 }kg. A speck of carbon of that size would be “seeable” to a person with good eyes. Christian (p.306) quotes Richard Feynman to the effect that quantum theory may fail for masses larger than this. It could then be neglected for cats or the entire universe.

That’s a plausible argument but not a compelling one. There are, after all, macroscopic phenomena like superfluids that show distinctively quantum features.

## Theological reflections

Georges Lemaître, a physicist and Catholic priest, found the first solutions of Einstein’s equations representing what would later be called a big bang cosmology. He denied that he did this to support religious belief in creation, saying his that his view “was consonant with the wording of Isaiah’s [45:15] speaking of the ‘Hidden God’ hidden even in the beginning of the universe.”^{10}Helge Kragh, Cosmology and Controversy (Princeton University, 1996), p.60. But it’s appropriate for those who believe that “in the beginning God created the heavens and earth” (Genesis 1:1) to reflect on the theological significance of what looks like the universe’s beginning.

If the four basic interactions could be unified, would we have a literal “theory of everything” that explained everything? The answer is a clear “No.” Kurt Gödel’s incompleteness theorems published in 1931 say that in any mathematical system at least as complex as elementary arithmetic, there will be propositions that can neither be proved nor disproved within the system.^{11}Kurt Gödel, On Formally Undecidable Propositions of Principia Mathematica and Related Systems (Dover, 1992). It’s worth noting that Gödel wasn’t only a logician. In 1950 he contributed an article to the *Festschrift* for Einstein’s 70th birthday [Reviews of Modern Physics 21.3, July 1949, pp.447-450.] with a solution to Einstein’s equations representing a rotating universe in which some observer could get into their own pasts! Interestingly, the *Festschrift *also included an article by Georges Lemaître on relativistic cosmology. That rules out a literal “theory of everything.”

A theory unifying all four basic interactions seems remote in any case. I was enamored of that idea in my early work in physics, but Einstein’s successful theory of a dynamic geometry and the successful theories of the other three forces now seem like different beasts. They inhabit the same universe, but we could think of them as beasts that evolved on different planets and have different chemistries. The Christian tradition has sometimes focused only on God’s interest in our own species. But scripture celebrates the variety of living things on the earth, as in Psalm 104 and Job 39-40.

Christians believe that there is one God who created this one universe. (I’ll save speculations about other universes for later.) The one God is not, however, a monolithic unity but the Holy Trinity, Father, Son and Holy Spirit. The variety in the one God’s creation should not really be surprising.

George Murphy received his Ph.D. in physics from Johns Hopkins for work on general relativity in 1972. He taught at Westminster College (PA), The University of Western Australia and Luther College and did research for eleven years before entering Wartburg seminary. Ordained in 1983, he has served as a pastor in Lutheran and Episcopal congregations. His first article on theology and science was published in 1977 and he has since published six books and numerous articles and continues to speak and lead workshops in this area. His most recent book, Models of Atonement: Speaking about Salvation in a Scientific World (Lutheran University Press, 2013), discusses ways of understanding the saving work of Christ in an evolving world.