Quantum Leaps

There is, thoughtful students agree, no entirely satisfactory interpretation of quantum mechanics. Freeman Dyson argued that the theory applies only to predictions about the future. The past is not probabilistic. What’s done is done. Quantum mechanics does not apply; the past must be described classically. I do not think many other physicists are willing to go this far.

I spent the winter and spring of 1989 at the European Center for Nuclear Research (CERN). Among other things, I was hoping to conduct a series of interviews with John Bell. I had known him for twenty years, but I knew little about his personal history. He told me that he came from a very modest Protestant family in Ireland. Under the normal evolution of things, he would have been expected to drop out of school at fourteen and begin working to support his family. There was no free high school education in Ireland. Realizing that there was something special about Bell, his mother scraped together enough money to allow him to continue his education. He worked his way through college, but soon after he got his undergraduate degree, he went to work for the British nuclear energy program.

Bell’s discontent with the standard formulations of quantum mechanics dated from his undergraduate education. During my interviews, Bell often complained about existing textbooks. None of them discussed the foundations of quantum mechanics coherently. Why didn’t he write his own? Bell said that he was stuck by the incompatibility of quantum mechanics and special relativity. I regret that I did not pursue this with him, although in essays such as “Against ‘Measurement,’” published in 1990, he spells out his dilemma very clearly.¹

If it is difficult to reconcile quantum mechanics and special relativity, it is far more difficult to reconcile quantum mechanics and general relativity in a quantum theory of gravity. Léon Rosenfeld took the position that the project was impossible, and so unnecessary. Gravity is macroscopic; quantum events, microscopic. Most physicists argue today that quantum theory must include gravity, simply because quantum mechanics is universal.

Quantum mechanics was created in the mid-1920s and soon after, attempts were made to quantize the electromagnetic field. Quantum electrodynamics contained the mathematics for describing this. In calculating the self-energy of the electron, the theory demanded a parameter for its mass. What was used initially was its bare mass—the mass the electron would have if all its electromagnetic interactions were turned off. Were this to happen, the electron could no longer respond to electric and magnetic fields, and so it would remain opaque to any experiment designed to measure its mass. But its calculation inevitably seemed to involve logarithmically divergent integrals. Things remained this way until just before the Second World War, when Hendrik Kramers introduced a new idea. He suggested that the bare mass of an electron should, in calculations, be replaced by its observable mass. Infinities were swept away. This procedure came to be known as renormalization. After the war, physicists were able to calculate finite perturbations expansions in quantum electrodynamics by renormalizing both the mass and the charge.

In 1952, Dyson argued that while the terms in the series expansion

$\alpha =e^2 / ħc ~ 1 / 137$

can be rendered finite in every order, the series itself could not converge. His argument was a masterpiece of simplicity. If the series did converge, it would be an analytic function at $\alpha =0$. It could then be extended to $-\alpha$ by analytic continuation. In that domain, electrons attract electrons and positrons attract positrons. Because of the Coulomb potential, these states have a larger and larger negative energy as the number of particles, N, increases. Pairs of particles are added to the negative energy as the square of N. The kinetic energy, on the other hand, is positive and increases linearly with N itself. Negative energy outpaces positive energy. In the end, the system collapses.

Divergent series notwithstanding, quantum electrodynamics yielded results of remarkable accuracy. Consider the magnetic moment of the electron. This calculation, which has been calculated up to the fifth order in $\alpha$, agrees with experiment to ten parts in a billion. If one continued the calculation to higher and higher orders, at some point the series would begin to break down. There is no sign of that as yet. Why not carry out a similar program for gravitation? One can readily write down the Feynman graphs that represent the terms in the expansion. Yet there remains an irremediable difficulty. Every order reveals new types of infinities, and no finite number of renormalizations renders all the terms in the series finite.

The theory is not renormalizable.

Bell and Bohr

Niels Bohr never published a full version of what came to be known as the Copenhagen interpretation of quantum mechanics; disciples spread the word. At the heart of the Copenhagen interpretation is the separation of the world into a measuring apparatus and the system that it measures. The apparatus must be classical. Werner Heisenberg’s uncertainty principle does not apply. The classical world cannot be derived as a limit of quantum mechanics. The distinction accepted, a problem remains. How does a classical apparatus perform measurements on a quantum mechanical system? This process cannot be described by quantum mechanics itself. This infuriated Bell because Bohr was never precise about the distinction between classical and quantum systems.

Erwin Schrödinger originally thought that solutions to his wave equation were real waves oscillating in time and space. Physicists soon realized that this interpretation was hopelessly flawed. While the radial solution representing the electron’s ground state in a hydrogen atom falls off exponentially as the distance from the proton increases, it never vanishes. This suggests that Schrödinger’s equation may well guide the electron to infinity over the course of time. This does not happen. Max Born suggested that the wave function represented probabilities, and not waves in space and time. The Born interpretation is now canonical. This resolved another issue. The wave function of a single particle is defined on a four-dimensional space. When more than one particle is under analysis, the dimensions of space and time begin to expand. This is a problem for real waves, but one that disappears if Schrödinger’s equation governs waves of probability.

In addition to the wave function, a quantum system is characterized by its observables, represented, in turn, by operators. Observables take a set of allowed values. The object of measurement is to discover which of these values characterizes the system. These values do not exist prior to measurement. Suppose that they did. To each of the allowed values, a, in a quantum system, $\mathrm{\Psi }$, there is an associated wave function ${\phi }_a$. Expanding the wave function yields

$\mathrm{\Psi }={\mathrm{\Sigma }}_a{c}_a{\phi }_a$,

where c ranges over the complex numbers. The absolute value of their amplitudes yields the probability that a measurement will reveal a particular value of the observable. What happens to the wave function as the result of measurement? The complete wave function disappears, leaving only a component that reflects the result of measurement. Such is the collapse of the wave function. This interpretive procedure is not derived from the theory; it is tacked on. This is no very serious objection, because it is common to all physical theories. Isaac Newton’s laws do not interpret themselves.

It is considerably more worrisome that quantum mechanical measurements cannot be described by quantum mechanics. Measurement is irreversible. Given a projected wave function, it is impossible to reconstruct the wave function prior to measurement. This is in conflict with the standard interpretation of Schrödinger’s equation. The wave function is reversible in time, and preserves its norm going backwards or forwards. Bohr seems to have understood this.

John von Neumann produced a model for the collapse of the wave function, one requiring a modification of quantum mechanical rules. This had no effect on experiment and still less of an effect on theoreticians. An exception was David Bohm, who dealt with the issue by denying the distinction between measurements and other sorts of interactions. On Bohm’s theory, the wave function plays an entirely different role. There is no collapse. I tried to read Bohm’s papers when they appeared in the early 1950s, and they seemed too complicated. I remember saying this to Julian Schwinger. He said that they seemed too simple.

Hugh Everett III came to Princeton in 1953 as a graduate student in mathematics, but switched to theoretical physics. He was fortunate to find a thesis advisor in John Wheeler, who had a remarkable tolerance for unusual ideas. Together with Bryce DeWitt, Everett was responsible for the many-worlds interpretation of quantum mechanics. In 1959, Wheeler arranged for Everett to visit Copenhagen, where he presented his ideas to Bohr and Rosenfeld. Their meeting was a disaster. Bohr had no interest in his ideas and Rosenfeld called him an idiot. After this, Everett rarely spoke about quantum mechanics in public. He died suddenly at the age of fifty-one.

Bell had, of course, studied Everett’s ideas and he had a sympathetic view of them. Still, he found the many-worlds interpretation flawed. In the first place, it was solipsistic. It assigned to an experimenter the power to bring new and unobservable universes into existence. Bell also noticed a conflict between the many-worlds interpretation and special relativity. On the other hand, Everett did inspire a new generation of physicists to think about the foundations of quantum mechanics. Bell died on October 10, 1990, at the age of sixty-two. He did not live to see the work of Murray Gell-Mann and James Hartle, which extended Everett’s ideas,² but his reservations about many-worlds would, I suspect, have remained unchanged.

Many-Worlds from the Beginning

In 1933, Paul Dirac published a paper entitled “The Lagrangian in Quantum Mechanics” in an obscure Russian journal.³ It was pretty much ignored until Richard Feynman revisited it in his PhD thesis, prepared under Wheeler’s supervision. Dirac had been impressed by the way in which the formalism of quantum mechanics seemed a generalization of classical mechanics. His first paper on the subject, published in 1925, showed how the Poisson brackets of classical mechanics became the canonical commutation relations of quantum mechanics.⁴ Until Dirac’s 1933 paper, quantum mechanics had been formulated as a generalization of Hamiltonian mechanics. Dirac came to the conclusion that the Lagrangian was more fundamental. No matter its formalism, quantum mechanics must determine the evolution of quantum states. It is the transition amplitude that is of the essence, and this amplitude is given by the scalar product of two state vectors. Dirac produced an expression for this transition amplitude:

$⟨q_t|q_T⟩$ corresponds to exp$\left[i{\int }_t^{T}Ldt\right]$.

Here q denotes some set of observables, L is a Lagrangian, and the exponent is the quantum version of the classical action. Dirac then replaced this integral with a product of factors representing possible intermediate paths, which were computed separately. Dirac never wrote down an equation and never worked out an example.

Feynman converted Dirac’s correspondence into an equation, one prefaced by a constant that he derived from examples. The integral emerged as a sum over all paths consistent with constraints on the system. In simple cases, such as the harmonic oscillator, this path integral can be calculated. At the limit of a dynamically decreasing Planck’s constant, the only remaining path is classical.

In this way, the classical world is rediscovered. Suppose a system is in some initial state. Quantum mechanics yields the ensuing state transition probabilities, the system’s complete history. Probabilities are summed over them all. Like waves, histories can interfere or become entangled with one another. Interactions can also cause these histories to decohere, and under the right circumstances decoherent histories behave classically. Probabilities cannot be assigned to entangled histories. This formalism was expanded by Gell-Mann and Hartle. It was Gell-Mann, in particular, who introduced the idea of strong decoherence, a state or condition of a system in which quantum entanglements become negligible. A measurement picks out one of the histories, leaving the rest to evolve separately.

Some years ago I heard Eugene Wigner lecture about the measurement problem. It was a surprise to hear someone of his generation admit that there was a problem. Wigner suggested a small addition to the Schrödinger wave equation, one evoking its collapse. The addition would be non-Hermitian, so circumventing the issue of reversibility, but would have no other effect. A related but different version of this idea was proposed by Giancarlo Ghirardi, Alberto Rimini, and Tullio Weber (GRW). Like Wigner, GRW added something to Schrödinger’s equation. But in GRW, the requisite parameter is stochastic. Bell did not have anything against this idea, although it was not his favorite. It was to the de Broglie–Bohm pilot wave that he gave his allegiance.

We owe to Louis de Broglie the wave theory of matter. When Bohm began publishing, de Broglie at once claimed priority. Schrödinger’s wave equation reappears in Bohmian mechanics, together with a wave function that it determines. The observables are classical particles with real Newtonian trajectories. These trajectories are governed by a twofold force. The first represents those Newtonian forces that would act on a particle in the absence of quantum effects. The second represents a quantum potential determined by the wave function. The scheme is equivalent to ordinary quantum mechanics. There are no special operations for measurements because the wave function never collapses. It is the statistical nature of the system’s initial conditions that gives rise to the uncertainty principle.

It is instructive to take a specific example. If $\mathrm{\Psi }$ is the wave function that satisfies the Schrödinger equation, then the quantum potential is given by

$-\frac{{ħ}^2}{2m}\left(\frac{{\nabla }^{2}R}{R}\right)$,

where R is the amplitude of the wave function,

$\mathrm{\Psi }=R \mathrm{e}\mathrm{x}\mathrm{p}\left(iS / ħ\right)$.

The equation for a free particle reduces to Newton’s law,⁵ and, in a more general sense, Bohm’s theory reproduces nonrelativistic quantum mechanics. This is all to the good and it is also all to the bad. Bell showed that in the case of interacting particles, the results are hopelessly nonlocal. The influence of various particles on each other involves superluminal signals. Bell could not figure out how to make this theory relativistic. This might suggest that there is no satisfactory interpretation of quantum theory. We are waiting for experimental guidance. It is clear that the instantaneous collapse of the wave function is not relativistically covariant. This matters if the wave function represents something more than a notepad for recording probabilities.

In the 1930’s, von Neumann argued that special relativity and quantum mechanics were incompatible. In quantum mechanics, position, but not time, is designated by an operator. Much effort was expended in trying to find a temporal operator. Most physicists now believe this to have been a waste of time. Quantum mechanics cannot ever be made compatible with special relativity. When collision energies become relativistic, pairs of particles are produced from the vacuum. Quantum field theory is needed to accommodate this.

Bell’s Theorem

When I took Schwinger’s course in quantum mechanics, there was no assigned textbook; indeed, there were very few texts on quantum mechanics at all. I looked at what was available and chose Bohm’s Quantum Theory, which had been published in 1951.⁶ This was standard quantum theory with a Copenhagen interpretation—no hint of Bohmian mechanics. What I liked about this text was Bohm’s discussion of the meaning of the theory. The penultimate chapter was entitled “Quantum Theory of the Measurement Process.” His analysis of the 1922 Stern–Gerlach experiment has become standard. The experiment was designed to measure the angular momentum of a silver atom. Neither Otto Stern nor Walter Gerlach realized that the ground state has an angular momentum of ½. They began with a particle in the state

${\mathrm{\Psi }}_0=f_0\left(z\right)\left(c_{+}{v}_{+}+c_{-}{v}_{-}\right)$,

where c designates numbers, and v, spin functions. The particle now impinges on a strong magnetic field that is inhomogeneous in the z direction. The particles are coupled to the field by their spin magnetic moments,

$H_I=\mu \left(\sigma \cdot H\right)$,

where H is the magnetic field, and

$\mu =-\frac{e}{ħ 2mc}$.

If this were classical physics, each particle would follow a trajectory determined by the sign of its spin. Since there are only two spins, there are only two possible trajectories. A suitably placed detector could determine whether the particle’s spin was up or down. Knowing nothing about quantum mechanics, this is just what Stern and Gerlach did.

Quantum mechanics shows where this is bound to lead. Consider the interaction Hamiltonian

$H_I=\mu \left(H_0+z{H}_0^{\text{'}}\right)$,

where H represents the magnetic fields. The time dependence for each spin state is exp$\left(iz{H}_0^{\text{'}}t\right)$, with a different sign in the exponential for each spin state. Measurement requires averaging, and these averages produce cross terms. This is quantum entanglement.

In the case at hand, the exponential oscillates very rapidly and the cross terms can be neglected. This is what decoherence means. Stern and Gerlach were able to proceed without quantum mechanics because they had recovered a classical situation with two noninterfering wave packets.⁷

With the Stern–Gerlach experiment in hand, I would like to use it to explain Bell’s theorem. Imagine first performing spin measurements along the z-axis, when particles are moving in the y direction, and rotate a magnet through an angle θ in the xz plane. The original Pauli matrix is

$\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$.

In the new system this becomes

$\left[\begin{array}{cc}\cos \left(\mathrm{\theta }\right)& \sin \left(\mathrm{\theta }\right)\\ \sin \left(\mathrm{\theta }\right)& -\cos \left(\mathrm{\theta }\right)\end{array}\right]$,

with eigenvectors

$\left[\begin{array}{c}\cos \left(\mathrm{\theta }/2\right)\\ \sin \left(\mathrm{\theta }/2\right)\end{array}\right]$ and $\left[\begin{array}{c}-\sin \left(\mathrm{\theta }/2\right)\\ \cos \left(\mathrm{\theta }/2\right)\end{array}\right]$.

Expanding the first column vector,

$\left[\begin{array}{c}1\\ 0\end{array}\right]=a_{+}\left[\begin{array}{c}\cos \left(\mathrm{\theta }/2\right)\\ \sin \left(\mathrm{\theta }/2\right)\end{array}\right]+a_{-}\left[\begin{array}{c}-\sin \left(\mathrm{\theta }/2\right)\\ \cos \left(\mathrm{\theta }/2\right)\end{array}\right]$.

This implies that a₊ = cos(θ/2) and a_– = –sin(θ/2). The probability of finding the spin up in a rotated magnet is cos(θ/2)², while the probability of finding spin down is sin(θ/2)². In entangled singlet particles, any spin down measurement in one magnet implies that the probability of measuring the same result in a rotated magnet is sin(θ/2)². The probability of measuring the opposite spin is cos(θ/2)². The quantum mechanical correlation is sin(θ/2)² – cos(θ/2)²= –cos(θ). The two spins are aligned a sin(θ/2)² fraction of the time. This is in agreement with quantum mechanics. If both magnets are rotated in opposite directions by the same angle, they will be aligned 2sin(θ/2)² of the time. Quantum mechanics predicts that this agreement occurs sin(θ)² percent of the time.

We can summarize Bell’s result as follows. Let a and b be two unit vectors that characterize, say, the directions of the magnetic fields in the Stern Gerlach experiment. Then we have shown that the correlation predicted by quantum mechanics is a⋅b. If λ is a hidden variable that determines the result of the experiment, Bell’s theorem says that there exists no functions A(a, λ), B(b, λ), which reproduce the quantum correlations.

This is Bell’s theorem.⁸