9.9 Imaginary Explanations
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. 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. All directions are relative.
This "relativity of angular reference frames" in quantum mechanics closely parallels the relativity of translational reference frames in special relativity. In fact, we can construct a "Bell's inequality" for velocities in the context of special relativity. First, let's briefly review the quantum mechanical case. Suppose we perform spin measurements on a series of particle pairs. Sometimes we measure along the vertical direction A, sometimes at a 45 degree angle along the B direction, and sometimes at a 90 degree angle along the C direction as shown below:
We then determine the correlation in the results for various combinations of measurement angles at the two ends of the experiment. Assuming each particle has a spin axis with an absolute direction, it follows that the correlation between two measurements can only be a linear function of the relative angle between the measurements. This implies that the change in correlation from A to C should equal the sum of the change in correlation from A to B plus the change from B to C. However, the measured results do not satisfy this additive property, i.e., they violate Bell's inequality.
Now consider three objects in different translational frames of reference as shown below:
The object B is stationary, while objects A and C move away from B in opposite directions at high speed. Intuitively we would expect the relative velocity between A and C to equal the sum of the relative velocities between A and B and between B and C. This is equivalent to Bell's Theorem for translational frames of reference. However, when we measure the velocity between A and C we find that it does not satisfy this additive property, i.e., it violates "Bell's inequality" for special relativity.
The parallel with 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(q ) where q 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 f = arctanh(v) where arctanh is the inverse hyperbolic tangent (see Part 6 of the Appendix). By analogy, we might expect the "correlation" between measurements performed with respect to two basis systems with this relative angle would be
which of course is Lorentz-Fitzgerald factor that scales the transformation of space and time intervals from one system of inertial coordinates to another. 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 formally identified 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 "EPRB 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 it squares to a negative value in the line element, and yields an invariant null-cone structure for light propagation, yielding invariant light speed.
On the other hand, 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. But how can it be both? 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.
Interestingly, 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, though in different ways. 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 sqrt(-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.