9.8 Quaedam Tertia Natura Abscondita

The square root of 9 may be either +3 or -3, because a plus times a plus or a minus times a minus yields a plus. Therefore the square root of -9 is neither +3 nor -3, but is a thing of some obscure third nature.

Girolamo Cardano, 1545

In a certain sense the peculiar aspects of quantum spin measurements in EPR-type experiments can be regarded as a natural extension of the principle of special relativity. Classically a particle has an intrinsic spin about some axis with an absolute direction, and the results of measurements depend on the difference between this absolute spin axis and the absolute measurement axis. In contrast, quantum theory says there are no absolute spin angles, only relative spin angles. In other words, the only angles that matter are the differences between two measurements, whose absolute values have no physical significance. Furthermore, the relations between measurements vary in a non-linear way, so it's not possible to refer them to any absolute direction.

This "relativity of angular reference frames" in quantum mechanics closely parallels the relativity of translational reference frames in special relativity. This shouldn’t be too surprising, considering that velocity “boosts” are actually rotations through imaginary angles. Recall from Section 2.4 that the relationship between the frequencies of a given signal as measured by the emitter and absorber depends on the two individual speeds v e and v a relative to the medium through which the signal propagates at the speed c s , but as this speed approaches c (the speed of light in a vacuum), the frequency shift becomes dependent only on a single variable, namely, the mutual speed between the emitter and absorber relative to each other. This degeneration of dependency from two independent “absolute” variables down to a single “relative” variable is so familiar today that we take it for granted, and yet it is impossible to explain in classical Newtonian terms. Schematically we can illustrate this in terms of three objects in different translational frames of reference as shown below:

The object B is stationary (corresponding to the presumptive medium of signal propagation), while objects A and C move relative to B in opposite directions at high speed. Intuitively we would expect the velocity of A in terms of the rest frame of C (and vice versa) to equal the sum of the velocities of A and C in terms of the rest frame of B. If we allowed the directions of motion to be oblique, we would still have the “triangle inequality” placing limits on how the mutual speeds are related to each other. This could be regarded as something like a “Bell inequality” for translational frames of reference. When we measure the velocity of A in terms of the rest frame of C we find that it does not satisfy this additive property, i.e., it violates "Bell's inequality" for special relativity.

Compare the above with the actual Bell's inequality for entangled spin measurements in quantum mechanics. Two measurements of the separate components of an entangle pair may be taken at different orientations, say at the angles A and C, relative to the presumptive common spin axis of the pair, as shown below:

We then determine the correlations between the results for various combinations of measurement angles at the two ends of the experiment. Just as in the case of frequency measurements taken at two different boost angles, the classical expectation is that the correlation between the results will depend on the two measurement angles relative to some reference direction established by the mechanism. But again we find that the correlations actually depend only on the single difference between angles A and C, not on their two individual values relative to some underlying reference.

More fundamentally, we could regard the triangle inequality in ordinary Euclidean geometry to be analogous to Bell’s inequality in quantum mechanics. The pairwise distances AB, BC, and CA between three points A,B,C in Euclidean space must satisfy the inequality AB + BC ≥ CA, but of course this inequality is violated for intervals in Minkowski spacetime. In this context, rather than interpreting the violation as proof of the impossibility of “local realism”, we simply acknowledge that locality in a pseudo-metric manifold has a different structure than that of a Euclidean manifold.

The close parallel between the “boost inequalities” in special relativity and the Bell inequalities for spin measurements in quantum mechanics is more than just superficial. In both cases we find that the assumption of an absolute frame (angular or translational) leads us to expect a linear relation between observable qualities, and in both cases it turns out that in fact only the relations between one realized event and another, rather than between a realized event and some absolute reference, govern the outcomes. Recall from Section 9.5 that the correlation between the spin measurements (of entangled spin-1/2 particles) is simply −cos(θ) where θ is the relative spatial angle between the two measurements. The usual presumption is that the measurement devices are at rest with respect to each other, but if they have some non-zero relative velocity v, we can represent the "boost" as a complex rotation through an angle ϕ = arctanh(v) where arctanh is the inverse hyperbolic tangent. By analogy, we might expect the "correlation" between measurements performed with respect to two basis systems with this relative angle would be

which is just the Lorentz-Fitzgerald factor that scales the transformation of space and time intervals from one system of inertial coordinates to another, leading to the relativistic Doppler effect, and so on. In other words, this factor represents the projection of intervals in one frame onto the basis axes of another frame, just as the correlation between the particle spin measurements is the projection of the spin vector onto the respective measurement bases. Thus the "mysterious" and "spooky" correlations of quantum mechanics can be placed in close analogy with the time dilation and length contraction effects of special relativity, which once seemed equally counterintuitive. The spinor representation, which uses complex numbers to naturally combine spatial rotations and "boosts" into a single elegant formalism, was discussed in Section 2.6. In this context we can formulate a generalized "EPR experiment" allowing the two measurement bases to differ not only in spatial orientation but also by a boost factor, i.e., by a state of relative motion. The resulting unified picture shows that the peculiar aspects of quantum mechanics can, to a surprising extent, be regarded as aspects of special relativity.

In a sense, relativity and quantum theory could be summarized as two different strategies for accommodating the peculiar wave-particle duality of physical phenomena. One of the problems this duality presented to classical physics was that apparently light could either be treated as an inertial particle emitted at a fixed speed relative to the source, ala Newton and Ritz, or it could be treated as a wave with a speed of propagation fixed relative to the medium and independent of the source, ala Maxwell. But how can it be both? Relativity essentially answered this question by proposing a unified spacetime structure with an indefinite metric (viz, a pseudo-Riemannian metric). This is sometimes described by saying time is imaginary, so its square contributes negatively to the line element, and yields an invariant null-cone structure for light propagation, yielding invariant light speed.

But waves and particles also differ with regard to interference effects, i.e., light can be treated as a stream of inertial particles with no interference (though perhaps "fits and starts) ala Newton, or as a wave with fully wavelike interference effects, ala Huygens. Again the question was how to account for the fact that light exhibits both of these characteristics. Quantum mechanics essentially answered this question by proposing that observables are actually expressible in terms of probability amplitudes, and these amplitudes contain an imaginary component which, upon taking the norm, can contribute negatively to the probabilities, yielding interference effects.

Thus we see that both of these strategies can be expressed in terms of the introduction of imaginary (in the mathematical sense) components in the descriptions of physical phenomena, yielding the possibility of cancellations in, respectively, the spacetime interval and superposition probabilities (i.e., interference). They both attempt to reconcile aspects of the wave-particle duality of physical entities. The intimate correspondence between relativity and quantum theory was not lost on Niels Bohr, who remarked in his Warsaw lecture in 1938

Even the formalisms, which in both theories within their scope offer adequate means of comprehending all conceivable experience, exhibit deep-going analogies. In fact, the astounding simplicity of the generalisation of classical physical theories, which are obtained by the use of multidimensional [non-positive-definite] geometry and non-commutative algebra, respectively, rests in both cases essentially on the introduction of the conventional symbol √-1. The abstract character of the formalisms concerned is indeed, on closer examination, as typical of relativity theory as it is of quantum mechanics, and it is in this respect purely a matter of tradition if the former theory is considered as a completion of classical physics rather than as a first fundamental step in the thorough-going revision of our conceptual means of comparing observations, which the modern development of physics has forced upon us.

Of course, Bernhard Riemann, who founded the mathematical theory of differential geometry that became general relativity, also contributed profound insights to the theory of complex functions, the Riemann sphere (Section 2.6), Riemann surfaces, and so on. (Here too, as in the case of differential geometry, Riemann built on and extended the ideas of Gauss, who was among the first to conceive of the complex number plane.) More recently, Roger Penrose has argued that some “complex number magic” seems to be at work in many of the most fundamental physical processes, and his twistor formalism is an attempt to find a framework for physics that exploits the special properties of complex functions at a fundamental level.

Modern scientists are so used to complex numbers that, in some sense, the mystery is now reversed. Instead of being surprised at the physical manifestations of imaginary and complex numbers, we should perhaps wonder at the preponderance of realness in the world. The fact is that, although the components of the state vector in quantum mechanics are generally complex, the measurement operators are all required – by fiat – to be Hermitian, meaning that they have strictly real eigenvalues. In other words, while the state of a physical system is allowed to be complex, the result of any measurement is always necessarily real. So, we can’t claim that nature is indifferent to the distinction between real and imaginary numbers. This is suggestive of a connection between the “measurement problem” in quantum mechanics and the ontological status of imaginary numbers.