The i Is in the Equation

Contents

1. The emblem is not Newton’s
2. What “classical” actually means
3. The obvious objection
4. The repair, and it is sharper
5. What the i is actually doing
6. Two things the popular story gets wrong
7. So what is the difference?
8. For the philosophers
9. For the theologians
10. On “math is why”
11. Exercises

01 The emblem is not Newton’s

Write F = ma on the board and the room relaxes. Everyone has met it. It is the most famous sentence physics has ever written, and it is not a sentence Newton wrote.

The Principia‘s second law reads: Mutationem motus proportionalem esse vi motrici impressæ, & fieri secundum lineam rectam qua vis illa imprimitur — a change in motion is proportional to the motive force impressed, and takes place along the straight line in which that force is impressed. Notice what that is. It is a proportion, stated geometrically, and its measure of “change of motion” is a directed quantity that Newton deliberately left neutral between a single impulse and a force acting continuously. There is no equals sign, no m, no a.

The algebraic form arrives over the following sixty years — anticipated by Varignon in 1700 and by Hermann’s Phoronomia in 1716, and given its general, coordinate-by-coordinate, works-for-any-body form by Euler in a paper read to the Berlin Academy in 1750 and printed in 1752. Newton had seen the algebraic version in print and did not adopt it.

I start here because it makes the right point about what we are studying. F = ma is not a fact someone discovered. It is a proposal about what kind of thing a world is, and it took three generations to get into the form we now treat as obvious. The proposal is this: a world contains things with definite positions and definite momenta; forces push them; and given the pushes and the starting conditions, everything that follows is settled.

Laplace drew the consequence in 1814, in a passage philosophers know well and usually quote incompletely. An intelligence that knew all the forces and all the positions at one instant, he wrote, could embrace past and future in a single formula: “nothing would be uncertain and the future, as the past, would be present to its eyes.” He called it une intelligence, not a demon — the label came later. And the sentence that almost always gets dropped is the one immediately after, where he says the human mind will always remain infinitely removed from any such thing.

F = ma is not an observation. It is a claim about what a world is made of, and it is written entirely in real numbers.

02 What “classical” actually means

Before the comparison, kill a misunderstanding that will otherwise wreck it. “Classical” does not mean old. It is not a period label.

In physics, classical means: the theory in which Planck’s constant plays no role and observable quantities all commute — the limit ħ → 0. That is a claim about a theory’s structure, not its date. Which has a consequence students find genuinely startling: general relativity is a classical theory. Einstein’s field equations contain no ħ and no i. Curved spacetime, black holes, gravitational waves — all classical physics, all twentieth century.

Figure 01

The audit: where the i is, and where it is not

Classical — real coefficients throughout

• F = maNewton, in Euler’s algebra
• ∂L/∂q − d/dt(∂L/∂q̇) = 0Euler–Lagrange
• ∇×E = −∂B/∂tMaxwell
• ρ(∂v/∂t + v·∇v) = −∇p + μ∇²vNavier–Stokes
• G_μν = 8πG T_μνEinstein field equations — 1915, and classical
• dS ≥ 0the second law

Quantum — the i is structural

• iħ ∂ψ/∂t = ĤψSchrödinger
• [x̂, p̂] = iħthe canonical commutator
• p̂ = −iħ ∂/∂xmomentum — see part four
• (iħγ^μ∂_μ − mc)ψ = 0Dirac
• K = ∫𝒟x e^iS/ħthe path integral weight

03 The obvious objection

Any engineer in the room will object, and they will be right. Classical physics uses complex numbers constantly. Alternating-current circuits are analysed entirely with complex impedances. Fourier analysis runs on e^iωt. Optics uses a complex refractive index whose imaginary part is absorption. Every linear differential equation in the classical canon is solved by guessing a complex exponential.

So “quantum has i and classical does not” is simply false as stated, and a philosopher should not accept it. Worse, the tempting repair — classical takes the real part at the end and quantum does not — is also false, because quantum mechanics plainly does extract a real number at the end. It is called |⟨φ|ψ⟩|², and every theory extracts real numbers at the end. That is what measurement is.

The real distinction is one level down, and it is worth the extra step.

04 The repair, and it is sharper

Ask not what you do at the end. Ask what the equation is made of.

A classical equation has real coefficients. That single fact does all the work. If you feed it a complex solution Ψ, then because the coefficients are real, the real part and the imaginary part of Ψ each satisfy the equation separately. Your complex solution was never one object; it was two real solutions travelling together for convenience. You may take the real part at the end, or at the beginning, or never — and nothing changes.

Now try it on the Schrödinger equation.

Figure 02

The test that actually separates them

Classical — the i is scaffolding

m Ψ″ = −k Ψ   →   m(Re Ψ)″ = −k(Re Ψ)  ✓

Real coefficients, so the real part is itself a solution. Kick the scaffolding away and the building stands.

↓  same test, one equation later  ↓

Quantum — the i is load-bearing

iħ ψ̇ = Ĥψ   →   iħ(Re ψ)̇ = Ĥ(Re ψ)  ✗

The i sits inside the equation, so real and imaginary parts are coupled. Re ψ is not a solution. There is no basis-independent “real part” of a quantum state to take.

Schrödinger himself hated this. Writing to Lorentz in June 1926, months after publishing the equation that bears his name: “What is unpleasant here, and indeed directly to be objected to, is the use of complex numbers. Ψ is surely fundamentally a real function.” He was wrong, and it took the field about two years to be sure he was wrong. That is a useful thing for students to know about how science actually goes.

05 What the i is actually doing

Fine — but why? What breaks if you insist on real numbers?

Here is the cleanest answer, and it is one philosophers can hold onto because it is about the relationship between change and knowledge.

Whatever drives time evolution has to preserve total probability — the state cannot leak away. In a real vector space the operator that generates such a motion must be antisymmetric. And an antisymmetric operator is precisely the sort of thing that has no real eigenvalues. It generates rotations perfectly well; it just cannot be an observable. There is nothing to measure.

Multiply by i — the quarter turn of part six — and the antisymmetric generator becomes a self-adjoint one. Now it has a real spectrum. Now it is a quantity you can go and measure. We call it the energy.

That is worth saying twice. In a real formulation you can have either a well-behaved generator of change or a measurable energy, but not the same object doing both jobs. The i is the hinge that identifies them. The Hamiltonian is simultaneously what pushes the world forward and what you read off an instrument, and it is the complex structure that makes those the same thing.

06 Two things the popular story gets wrong

Now two corrections, because the version of this argument circulating in popular science is overstated in both directions and your students will meet it.

“You need complex numbers to get interference” — false

Signed real numbers interfere perfectly well: (a₁+a₂)² has a cross term that goes negative whenever the signs differ. Stronger still, a quantum computer built from only real gates — Hadamard and Toffoli — is universal. Cancellation needs something that can be negative. It does not need something that can be complex.

“Experiment has proved nature requires complex numbers” — not settled

In 2021 Renou and colleagues showed that a precisely specified real quantum theory makes different predictions from the standard one in multi-source networks, and two experiments in 2022 came down against the real version. Genuine and beautiful. But the theorem assumes that composite systems combine by the tensor product — and in 2026 a paper in Physical Review Letters replaced exactly that assumption with a different composition rule and recovered all the multipartite predictions over the reals. Renou himself co-authored the accompanying commentary. So the honest position today is: complex numbers are the natural and standard choice, and the case that they are forced is live research rather than closed.

And classical mechanics can be dressed in complex clothes too

In the tradition running from Koopman and von Neumann in 1931–32, classical mechanics is written on a complex Hilbert space with a Schrödinger-shaped equation. It produces no interference — but only because the classical observables all commute and one imposes a rule about what counts as measurable. The classicality is put in by hand as a restriction, not derived. Which is a very good thing for a philosophy class to sit with.

So the honest formulation is: the i is the fingerprint, not the crime. It is the most reliable way to tell at a glance which kind of theory you are reading. It is not the thing that makes quantum mechanics strange.

07 So what is the difference?

Three answers, in increasing order of depth, none of which mention the letter i.

Amplitudes add, and then get squared. Classically, if a particle can arrive by route 1 or route 2, the probabilities add: P = P₁ + P₂, and since probabilities are never negative, nothing can ever cancel. Quantum mechanically the amplitudes add and the squaring happens afterwards:

Figure 03

What the cross term looks like on a screen

• classical: the two routes simply add
• quantum: the same envelope, cut by the cross term

Observables do not commute. Classically, position and momentum are both just numbers attached to a system; you can specify both, exactly, at once. Quantum mechanically x̂p̂ ≠ p̂x̂, and the difference is the iħ of Figure 01. Order of asking changes the answer.

And the deepest one, which needs no interpretation at all. A classical state space is a simplex: every mixed state decomposes into pure states in exactly one way. If I hand you a classical system in a 50/50 mixture, there is a fact about which two things it is a mixture of. The quantum state space is not a simplex. A quantum mixed state has infinitely many decompositions, all equally valid, and nothing distinguishes them. That is a difference in the shape of the space of possibilities, and it survives every argument about what the theory means.

08 For the philosophers

Now the part you came for — and three corrections to what you have probably been told.

First: the Schrödinger equation is deterministic. Give me ψ now and a Hamiltonian, and ψ at every future and past moment is fixed, uniquely, reversibly — every bit as rigidly as anything in Newton. Quantum mechanics is not the theory that broke determinism. The indeterminism lives entirely in the other postulate, the one about what happens when a measurement occurs. Laplace’s intelligence, handed the wavefunction of the universe, would have no trouble at all until someone looked at something.

Which is exactly why the measurement problem is a problem, and the sharpest statement of it belongs to a philosopher rather than a physicist.

Figure 04

Maudlin’s trilemma: three claims, and you may keep two

1. Deny the first The wavefunction is a complete description of a system. → Bohmian mechanics
   add particles
2. Deny the second The wavefunction always evolves by the linear equation. → GRW / collapse
   add a real random jump
3. Deny the third Measurements have single determinate outcomes. → Everett
   keep every branch

Second: the uncertainty principle is not about disturbance. This is the single most common misunderstanding among non-physicists, and Heisenberg is partly to blame for it. His 1927 paper argued from a thought experiment about a microscope kicking an electron — a claim about measurement disturbing a system. The rigorous inequality, proved by Kennard later that same year and generalised by Robertson in 1929, is about something else entirely: it concerns the standard deviations of results over an ensemble of identically prepared systems. It is a statement about what can be prepared, not about how clumsy your instrument is. Any single position measurement can be as precise as you like. Heisenberg conflated the two himself, and the confusion has been downstream of him ever since.

Third: quantum mechanics does not settle determinism. Bohmian mechanics is fully deterministic. Everett is deterministic at the level of the universal wavefunction. GRW is genuinely stochastic. These make the same experimental predictions. So there is no single “quantum mechanics” from which to read a verdict about the fundamental furniture of the world. And note carefully what Bell’s theorem did and did not do: it ruled out local hidden variables. It did not rule out hidden variables, and it did not rule out determinism — Bohm’s theory survives it by being non-local, and Bell’s own conclusion was non-locality, not indeterminism.

What did change is subtler and, I think, more interesting for your purposes. Heisenberg reached for Aristotle to say it. The quantum state, he wrote in Physics and Philosophy, “contains statements about possibilities or better tendencies (‘potentia’ in Aristotelian philosophy), and these statements are completely objective, they do not depend on any observer.”

That is a substantial claim and worth pressing on. Potentia is the Latin for Aristotle’s dynamis, and Heisenberg means it in the Metaphysics Θ.6 sense — potentiality, the way an acorn is potentially an oak — rather than the Θ.1 sense of a power to act on something else. The two get run together constantly, and the analogy only works with the first. What he is proposing is a third category between the possible and the actual, objectively there in the world, and this is exactly what a probability amplitude behaves like: not a fact about outcomes, not merely our ignorance, but a real disposition that the world carries around and that interferes with itself. Mere ignorance does not interfere.

09 For the theologians

And now the temptation, which I want to name precisely rather than either endorse or sneer at, because there is real scholarship here and there is also a great deal of nonsense.

Here is what I would want a student to be able to say about it.

The scientific objections are real and they are not hostile. It is interpretation-dependent: on Bohm or Everett there are no collapse events to act at, so the proposal has no place to stand. There is an amplification problem: quantum indeterminacies must be scaled up to matter at the level of history, and real physical systems mostly average them into background noise rather than magnifying them. Decoherence does not select outcomes, so “a quantum event” is not as well defined as the argument needs. And the structure is uncomfortably close to a god of the gaps — a gap that a future theory might close.

The theological objections are, to my eye, weightier. Ignacio Silva’s is the sharpest: locating divine action inside quantum indeterminacy quietly demotes God to a secondary cause, one agent among others operating in a gap, which is precisely what the Thomistic doctrine of primary and secondary causation was built to avoid. On that view no gap is needed, because God is not the sort of cause that competes for room. William Stoeger, a Jesuit and an astrophysicist who worked on the project itself, resisted the quantum route for a related reason: it removes divine action from the level at which persons live. And there is a rather deflating structural point — if God determines every collapse everywhere at every moment, in what sense is that less interventionist than the alternative it was designed to avoid?

The same discipline applies to free will, where the temptation is stronger and the argument weaker. Quantum indeterminacy does not deliver agency, because randomness is not freedom. Hold the agent’s character, history and reasons fixed, replay the moment a thousand times, and if the outcome varies, what varies is luck — and luck is not something you are responsible for. Robert Kane has the most serious libertarian answer, locating self-forming actions in genuinely torn deliberation, and most philosophers judge it unsuccessful. It is worth telling students that Peter van Inwagen, who formulated the sharpest version of this objection, was himself a libertarian, and concluded that free will is a mystery rather than that it is an illusion. And it is worth pointing at Timothy O’Connor’s agent causation as the contrast case — a libertarian account that does not lean on physics at all, and so cannot be knocked over by it.

The mathematics licenses less than people want from it. That is not a loss. A theology that needed a gap in physics was always renting.

10 On “math is why”

Which brings the series back to its premise. On Max Tegmark’s reading, the mathematical structure is not a description laid over the world — it is the world, and physical properties are positions inside it.

Take that seriously and the question “why is there an i in the quantum equations and not the classical ones” stops being a question about notation. The number field is not a convention chosen by physicists for convenience; it is a claim about what the world is made of. Classical physics says: real numbers suffice, everything that exists has a definite value, and change is a push. Quantum physics says: the state space carries a complex structure, and the thing that drives change is the same object as the thing you measure.

And then the honest coda, which I like better than a tidy ending. We do not know that the world had to be complex. There is a theorem — Solèr’s — saying that a quantum logic of the right kind must be built over the reals, the complex numbers, or the quaternions. Three candidates. Nature picked the middle one, the case that it was forced to is still being argued in the literature this year, and nobody has a satisfying account of why. That is not a gap to be filled with anything. It is a genuinely open question about the structure of what there is, and the right response to it is work.

11 Exercises

1. Run the test yourself Take the classical oscillator mΨ″ = −kΨ and verify that if Ψ solves it so does Re Ψ. Then attempt the same with iħψ̇ = Ĥψ and identify the exact line at which it fails.
2. Find the second assumption Write out the classical two-route argument in full and identify both assumptions. Then construct a purely classical scenario in which the second one fails and the probabilities do not add. What does that tell you about how much of quantum strangeness is really about quantum?
3. Separate the two uncertainties State Heisenberg’s 1927 microscope argument and Kennard’s 1927 inequality as two distinct propositions. Which one does an ensemble of prepared systems test? Which one would a single very careful measurement test? Why did conflating them matter?
4. Take the trilemma seriously For each leg of Figure 04, write the strongest one-paragraph case for giving up that claim. Then say honestly which you found hardest to argue, and whether that is evidence about the world or about you.
5. Argue the other side Take the position that quantum indeterminacy really does open a legitimate space for divine action, and answer the four scientific objections in section 09 on their own terms. Then say what the proposal costs you theologically if it succeeds.