Mapping QM to CSM

Beginning in the 1970s, physicists began to realize that apparently unrelated but equally intractable difficulties in quantum field theory and in CSM could be solved together using a surprising mathematical trick known as the “Wick rotation” (Fraser 2020). To understand the resulting unification, statistical field theory, we need the idea of a very basic quantity in QM, the propagator. In QM, the square of the propagator ^vii gives the probability that a particle, starting at some point, should arrive at another point R away after time t. With the propagator in hand one can calculate predictions for any observable quantity modelled. Feynman found an appealing way to obtain the quantum propagator by performing a sum over all possible classical paths between these points, each path mathematically weighted by a quantity (technically, a “phase factor”) that involves an exponential function of t / ℏ $t/\hbar $ , where ℏ is Planck's constant divided by 2π. Mathematically, the quantum propagator of a particle starting at some arbitrary origin and reaching position R at time τ looks like this

where the first symbol to the right of the equals sign is a stylized “S” denoting summing (or integrating) over all paths. The important feature for us is the combination i t / ℏ $it/\hbar $ in the rightmost bracket, where we find t / ℏ $t/\hbar $ multiplied by the imaginary number i = − 1 $i\; = \sqrt { - 1} \;$ , that is to say, a number whose square is −1. This operation of turning the “real time” into an “imaginary time” ^viii is called a Wick rotation because in a geometric representation of complex numbers (numbers of the form x + y i $x + yi$ where x and y are real numbers) as arrows on a plane, multiplication by i appears as a counter clockwise rotation by 90°.

In CSM, the basic mathematical object is the “partition function,” which involves a sum over all possible states of a system held at some fixed temperature T. In the partition function, the temperature occurs in the combination k T $kT$ , where k is Boltzmann's constant. The analogy between “paths” in QM and “states” in CSM involves substituting 1 / k T $1/kT$ for i t / ℏ $it/\hbar $ in the propagator, giving the equation for the partition function in CSM. Symbolically, we have

For classical polymers (McLeish 2022), the abstract content of the Wick rotation becomes visualizable. The different states of a polymer, a long‐chain molecule, are different configurations of a chain, which, in turn, are the very classical paths summed over in the quantum propagator. They range from the unique configuration of the chain being completely straight, to a multitude of coiled configurations. The partition function sums over all these different “paths.” The analogy with QM is completed when positions along the path map to corresponding imaginary times in the quantum propagator. The set of paths is representatively “explored” by a polymer molecule dynamically through random thermal motions—a physical process analogous to the entirely mathematical quantum sum, which has no physical interpretation (the often‐used description “the particle goes along every path at once” does not bear up to scrutiny). When a system is in thermal equilibrium ( = macroscopically homogeneous and unchanging with time), the weighting of these paths, worked out by Ludwig Boltzmann and Josiah Willard Gibbs (Baierlein 1971), is controlled by the temperature through an exponential function of 1 / k T $1/kT$ . A century later, Sam Edwards in the UK and Pierre‐Gilles de Gennes in France first exploited this “trick” and opened up a new era for polymer physics, and earned de Gennes a Nobel Prize in 1991.

Many more (less transparent and visually striking) examples of such QM‐CSM correspondence exist. The polymer case illustrates an important dissimilarity in the physics of the correspondence, which is that time‐dependent processes in the QM realm map into thermal equilibrium (and so time‐independent) systems in the CSM realm. Philosophical responses to this disjuncture differ widely. Fraser (2020) deduces that the Wick rotation is purely formal, implying nothing for the way the universe is. At the other end of this hermeneutical spectrum, Polyakov (1987), the leading field‐theorist, conjectured that the meaning of the QM particle‐CSM polymer correspondence arises “from the deep structure of space‐time.”

The theological implications of this QM‐CSM correspondence have been little discussed to date. To our minds, it minimally implies that one needs to be very careful in claiming any kind of unique quantum holism. Technically, using Wick rotation, a path‐summing quantum system used to demonstrate entanglement/holism can be turned into a statistical mechanical state‐summing system that, at least formally, possesses the same degree of entanglement/holism. More generally, there is a large literature (Gibb, Hendry and Lancaster 2019) discussing emergence and top‐down causation that points to a statistical mechanical interconnectedness without appealing to any quantum weirdness. The same is true of any other candidate quantum result for theology. Even extremely strange consequences of quantum systems, such as the fractionally charged “anyons” that emerge in very cold many‐electron systems confined to surfaces and threaded by magnetic fields, have at least a formal correspondence in the Wick‐rotated and corresponding polymer system (McLeish, Lancaster and Pexton 2019). Recent work has pointed out that Long‐Range (greater than molecular scale) Topological Order (LRTO, or “entanglement”) is a property of both many‐particle QM and polymer CSM systems. This presence of irreducible long‐range action, also often thought to be a unique feature of QM, furnishes a purely classical example of top‐down causation (McLeish, Lancaster and Pexton 2019). This nonreductionist (since nonreducible) structure of some systems has naturally attracted theological comment (McLeish 2019). QM is therefore not unique in several salient ways: holism, irreducibility and top‐down causation.

A Role for the Observer?

Theological and philosophical discussions often claim that QM offers a unique role for the observer. With Einstein's relativity, we are introduced to observer‐dependent points of view; however, these can be reconciled by properly “transforming between frames of reference,” and all observers ultimately agree over all events. In QM, however, it is claimed that the “observer” can determine the outcome of experiments. We have argued that whatever the detailed philosophical meaning of this claim is, it only offers a “fast track to idealism” if one resorts to a linguistic sleight of hand.

On the other hand, it turns out that the description CSM offers of the natural world is indeed dependent on sentient observers. This arises from the core CSM concept of entropy, first introduced by Rudolf Clausius and others to describe the efficiency of turning heat flow into work (as in the steam engine), and then formulated statistically by Boltzmann (Baierlein 1971). Most university physics courses (and textbooks) today give the impression that the entropy is an unequivocal property of the physical world. However, Josiah Willard Gibbs, one of the inventors of CSM, has understood from the beginning that the entropy of a system is actually a description of the degree of ignorance that an observer entertains vis‐à‐vis the variables (“degrees of freedom”) describing the system (Jaynes 1957; Jaynes 1965; Baierlein 1971). Put another way, the entropy assigned by physical models of thermodynamic systems depends on how coarse‐ or fine‐grained those models are, and is calculated by summing over all possible states of variables whose values are not known, or chosen to be ignored, by the observer constructing the model. It is therefore not surprising that the 1963 Nobel physics laureate Eugene P. Wigner, otherwise famous for his work in QM, once remarked, “Entropy is an anthropomorphic concept” (Jaynes 1965). ^ix

As an illustrative example, consider two observers of a quantity of gas. A “fine‐grained observer” knows the trajectories and momenta of all the molecules. There are no coarse‐grained variables, all values of classical quantities are known at each time; this observer assigns zero entropy to the gas. A second observer has access only to local densities of the gas (defined over some coarse‐graining element of very many molecules) and their fluctuations, but not to the position of every particle. This observer does assign a finite entropy to the gas (it takes the form of a weighted sum of the logarithms of the local densities). Crucially, which observer is “correct” depends on what experiments one wants to model. Using the fine‐grained entropy will predict the wrong outcomes in an ordinary laboratory experiment in which only the classical thermodynamic variables of pressure, volume, and temperature are accessed. The kind of muddle one could get into when entropy choice and experimental reality “cross wires” is nicely illustrated by colloidal crystallization, a beautiful phenomenon responsible for opalescence. A theory that assigns entropy to a fine‐grained observer with access to the world of individual colloids rather than to the coarse‐grained world of everyday experiments predicts (wrongly) that colloids do not crystallize (Cates and Vinothan 2015). In this sense, the way CSM describes everything from superconductivity to neutron stars is irreducibly observer dependent.

It might be objected that all that has been claimed is that the values of parameters and quantities differ for models of reality simply because of different choices made by the “observers” (defined carefully as those who make those models of the world, presumably after making measurements of it), and that this is a much weaker claim than the observer dependence in QM engendered by “wave function collapse” or whichever interpretation of QM one prefers. However, a moment's reflection will clarify that, in the QM case as well, the “world” or “reality,” however realist or idealist a philosophy one holds, is only ever accessed via model‐interpreted observation. The nonlocal, top‐down, “observer‐dependent” nature of QM is no less a property of the model than in CSM. That is what “observer‐dependent” means. Yet, this is still contested in QM, but not at all the case in CSM. We know of no other case in physics where the role of the model‐choosing sentient physicist is as clearcut and has so far‐reaching consequences.

Decoherence and Hiddenness

The everyday world that sentient observers inhabit is not quantum mechanically entangled. In this world, one has to look hard to see evidence of quantum effects. Why? The answer involves “quantum decoherence.”

Consider a less emotive version of Schrödinger's cat: a particle that is in a quantum state that can, loosely, be described as “both here ( = cat alive) and there ( = cat dead).” Formally, we prepare the particle in a quantum state, a mathematical object (a “Hilbert space vector”) | Ψ ⟩ $| \Psi \rangle $ , in which we cannot say for sure whether it is in the vicinity of position (say) − x 0 / 2 $ - {x_0}/2$ or + x 0 / 2 $ + {x_0}/2$ , but only give some probability of it being in these two states. This is a wholly quantum mechanical scenario that will not “become classical” until there is a measurement of the particle's position, at which time we can say with certainty whether it is at + x 0 / 2 $ + {x_0}/2$ or − x 0 / 2 $ - {x_0}/2$ (“the cat is dead” or “the cat is alive”). Until then, | Ψ ⟩ $| \Psi \rangle $ evolves deterministically according to the time‐dependent Schrödinger equation. After an elapse of some time interval Δ t ${{\Delta}}t$ since the state | Ψ ⟩ $|\Psi \rangle $ was prepared, we can calculate the probability of the particle being at + x 0 / 2 $ + {x_0}/2$ or − x 0 / 2 $ - {x_0}/2$ exactly. Therein lies the conundrum of Schrödinger's cat: we can only give the probability of it being alive or dead, until an observation occurs.

States such as | Ψ ⟩ $| \Psi \rangle $ are known as “coherent states.” Two subatomic particles are entangled if they are in a coherent state. Nonrelativistic coherent states evolve according to the time‐dependent Schrödinger equation, whose solution allows us to calculate the probabilities ^x of obtaining various answers if we were to carry out a measurement at any time, provided that the state | Ψ ⟩ $|{{\Psi}}\rangle $ remains completely isolated. Any interaction with the environment will lead, ultimately, to the loss of the state's coherent nature, a process known as decoherence. The typical time for this to occur is the “decoherence time,” τ d ${\tau _d}$ .

To estimate τ d ${\tau _d}$ , theorists have considered a simple model in which our ‘Schrödinger's particle’ interacts with a viscous environment at temperature T. A particle of radius a and mass m moving in such an environment will come to rest after a time of the order τ r ≈ m / η a ${\tau _r} \approx m/\eta a$ . This so‐called “relaxation time” offers a measure of the degree of interaction between the particle and its (viscous) environment. Wojciech Zurek (2002) has shown that the decoherence time of our Schrödinger's particle immersed in a viscous medium is τ d ≈ τ r × ( λ th / x 0 ) 2 ${\tau _d} \approx {\tau _r} \times {( {{\lambda _{{\rm{th}}}}/{x_0}} )^2}$ , that is, the relaxation time multiplied by some number. The number is the square of a ratio of two lengths, the so‐called de Broglie thermal wavelength divided by the separation between the two possible positions of the particle in its coherent state.

Quantum effects become important if we confine a particle to a box of the size of the de Broglie wavelength or smaller. For an electron at room temperature, this is about 4 nm. A hydrogen atom is roughly 0.1 nm in diameter, so that we need QM to explain its workings. For an oxygen molecule or a nitrogen molecule at room temperature, λ_th is less than 0.02 nm. In the air we breathe, these molecules are separated on average by just over 300 nm, so that they do not confine each other enough to be quantum mechanical: air is a classical gas.

Returning to our model coherent state, for a “Schrödinger particle” of mass m = 1 $m\hspace*{0.28em}=\hspace*{0.28em}1\, $ gram at room temperature in a state | Ψ ⟩ $| {{\Psi}} \rangle $ with x 0 = 1 ${x_0} = \;1$ centimeter, the numerical ratio τ d / τ r ≈ 10 − 40 ${\tau _d}/{\tau _r} \approx {10^{ - 40}}$ . So, even if the interaction of this particle with its environment is so minimal that its relaxation time is of order of the lifetime of the present universe, or τ r ≈ 10 17 ${\tau _r} \approx {10^{17}}$  s, its decoherence time comes out to be of order τ d ≈ 10 − 23 ${\tau _d} \approx {10^{ - 23}}$ s, or about one‐tenth the lifetime of the Higgs boson. The lesson is that in the macroscopic, centimeter‐sized, world, decoherence is essentially instantaneous. For all practical purposes (and considerably beyond), the world we perceive is classical. Cats are never indeterminately alive or dead, whether observed or unobserved.

To see what it would take to make a “Mr Tompkins world” ^xi in which quantum weirdness is an everyday occurrence, consider the ratio ( λ th / x 0 ) 2 ${( {{\lambda _{{\rm{th}}}}/{x_0}} )^2}$ . If we want to consider particles separated by a macroscopic distance, say, x ₀ of the order of centimeters, then we need the thermal de Broglie wavelength to be also macroscopic to “within shouting distance,” say, about a micron (or ten thousandth of a centimeter), which is the length scale of objects just about visible in an optical microscope, so that the multiplier becomes one ten billionth (10⁻⁸). Then, a relaxation time τ r ${\tau _r}$ of about a year, which still represents very weak interaction with the environment, will translate into a decoherence time of τ d ≈ 0.1 ${\tau _d} \approx 0.1$  s—not that long, but long enough to win or lose an Olympic medal. To change the thermal de Broglie wavelength of a 1 gram particle (still considerably smaller than a cat!) at room temperature from its value in our world of just under 2 × 10 − 20 $2 \times {10^{ - 20}}$ cm to one micron, we need Planck's constant to increase by a factor of 10¹⁸ (a million trillion). However, Planck's constant, the speed of light and the universal gravitational constant together determine three fundamental quantities known as the Planck mass, Planck length and Planck time. The actual observed values of these quantities are “fine tuned” for the emergence of life on earth (Barrow and Tipler 1986, Chapter 5), so that increasing Planck's constant by a factor of a million trillion will rule out any Mr Tompkins to worry about an alternative world in which Schrödinger's cat is an everyday occurrence. ^xii

If we want sentient observers, therefore, the “fine tuning” argument dictates that the world they observe, the macroscopic world, must be a classical one in which quantum weirdness such as half‐live‐half‐dead cats only occur in science fiction (or certain kinds of quantum theologizing). In the everyday world directly available to our senses, quantum effects rarely “leak through” very occasionally. Magnets constitute an exception: they depend on electrons possessing the property of “spin,” which emerges out of Dirac's relativistic QM. Otherwise, quantum weirdness is necessarily hidden to macroscopic sentient observers (using this word in the everyday register).

If that is the case, then we may draw a theological analogy. While the Judeo‐Christian tradition believes in a God who acts in the (macroscopic) world of flesh and blood, with Christians claiming that the Son of God became “enfleshed” in the person of Jesus (John 1:14), it seems that, paralleling quantum hiddenness, it is necessary for such action to be largely hidden and not easily perceived by casual observers. That is why there are multiple pointers throughout the Old and New Testaments to the Deus absconditus, the God who is present through hiddenness, from Moses being only allowed to see YHWH's “back” on Mount Sinai (Exodus 33:18‐20) through Isaiah's exclamation that YHWH is a hidden redeemer (Isaiah 45:15) to Jesus’ cry of dereliction on the cross (Mark 15:34) and his ascension at which “a cloud hid him” from the sight of the watching apostles (Acts 1:9). From henceforth, the life of the Christian disciple on earth “is hidden with Christ in God” (Col. 3:3). The frustration of living as the ekklēsia of such a hidden God comes out into the open in many places throughout the Biblical tradition, for example, in the plea from a prophet to his hidden God, “Oh that thou wouldest rend the heavens … [and] come down” (Isaiah 64:1, KJV).

However, to ask God to render the god‐self entirely manifest in a way that will make all talk of the Deus absconditus redundant may well be tantamount to asking for an alternative creation that is incompatible with sentient flesh‐and‐blood creatures, just as asking for quantum‐entangled cats entails ruling out any world in which cats of any kind is possible. In contrast to QM, however, Christians claim that when God makes all things new in the eschaton, overtly manifest divinity will become compatible with sentient creatures. In the new (kainos) Jerusalem, “the city has no need of sun or moon to shine on it, for the glory of God is its light, and its lamp is the Lamb” (Rev. 21:23); but that will require a new (kainos—new in kind) creation. To pursue this train of thought about quantum and divine hiddenness in detail is, however, beyond our scope.

Visualizability and Speakability

Ever since the beginning, the founders of QM have pondered how they should speak about their creation. For example, Bohr tells us that in QM, we have “a formalism which defies unambiguous expression in words suited to describe classical physical pictures” (Bohn 1958, 40). Bohr here raises not only the issue of the speakability (“words suited to describe”) of QM, but also of its visualizability (“physical pictures”). Theological thinkers have, of course, wrestled with the same issues vis‐à‐vis God for two millennia and more.

Take first the difficulty of speaking about God. Two classic strategies in dealing this difficulty are well illustrated by the opening of a well‐known hymn: “Immortal, invisible, God only wise, in light inaccessible hid from our eyes.” There are two noteworthy features. First, God is spoken of in negative terms as being immortal and invisible. This way of god‐talking through denial, belongs to a long tradition of apophatic theological teaching that recommends the via negativa (Turner 1995). Second, there is a paradoxical turn that perhaps familiarity has dulled: “in light … hid from our eyes.” Religious poets and theologians have always known that to speak of God necessitates the use of what Bishop Ian Ramsay has called “odd language” involving “logical improprieties,” such as “in light … hid from our eyes.” Interestingly, Ramsay points out that such linguistic “oddness” also occurs in science (Ramsey 1957, 47–48).

So, it appears that proper speech about the complex realities dealt with in either QM or theology must “[hold] the via positiva [which necessarily involves Ramsey's logical improprieties] in creative tension with the via negativa.” (Paul 2006): “Immortal, invisible … in light … hid from our eyes;” “an electron is neither a particle nor a wave … and it is both!” Faced with complexities, quantum or divine, a positivistic language that claims to eschew paradox and not balanced by a negative language of denial will, at best, breed confusion, and, at worst, wreak conceptual havoc, not only in the respective domains of science and theology, but in discussions of their relationship (Poon 2022).

Turning to visualizability, recall that for Bohr, there are suitable words “to describe” classical physics because it is associated with “physical pictures.” This recalls a famous argument between Heisenberg and Schrödinger (Miller 1986; de Regt 1997). The latter was motivated to invent his 1926 wave mechanical formulation of QM partly because of his dissatisfaction with the lack of Anschaulichkeit, or “visualizability,” in the 1925 matrix mechanics of Heisenberg, Born and Jordan. Heisenberg's dissenting response is his 1927 article announcing the Uncertainty Relation entitled “On the anschaulichen (‘visualisable’) content of quantum theory.” ^xiii

Interestingly, quantum pedagogues since this debate have mostly opted for Schrödinger's wave mechanics in introducing quantum arcana to beginners, precisely because of its intuitive “visualisability.” ^xiv Indeed, Bern Thaler (2000) from the University of Graz (where Schrödinger once taught) published a text describing computer animations in QM for undergraduate teaching. Thaler (2000, v) tells us that after showing a movie to his students, he will “explain step‐by‐step what can be learned from the animation,” and that each movie should “provide some motivation for the effort to understand the theory behind it.” In other words, the visualizations are not of quantum reality per se, because “behind” these lies “the theory,” so that a hermeneutic is needed in order for the student to “learn from” the visualizations.

This discussion about quantum Anschaulichkeit bears comparison with the iconoclastic controversy in eighth/ninth‐century Byzantium ^xv in which theologians debated the propriety or otherwise of pictorial representations of divine realities (Louth 2007). Byzantium eventually affirmed the value of icons, albeit carefully hedged with a nuanced hermeneutic (Evdokimov 1989). Visual representations are dangerous in both domains because they may be confused with reality—visualization becomes idolatry. Nevertheless, the affirmation of visualizability in either case is an important antidote against tendencies to cut loose from materiality—to treat QM as a purely mathematical construct when it is in fact one of the most successful physical theories for describing physical phenomena, or to overspiritualize Christianity, which Archbishop William Temple (1949, 478) once described as “the most avowedly materialist of all the great religions” because of the incarnation (John 1:14). ^xvi For further development of these thoughts and to set them in wider cultural and theological contexts, see Fuller and Jasper (2014).