## Who's On First? Relativity, Time, and Quantum Theory

Nov

9

Einstein’s special theory of relativity calls for radical renovation of common-sense ideas about time. Different observers, moving at constant velocity relative to one another, require different notions of time, since their clocks run differently. Yet each such observer can use his “time” to describe what he sees, and every description will give valid results, using the same laws of physics. In short: According to special relativity, there are many quite different but equally valid ways of assigning times to events.

Einstein himself understood the importance of breaking free from the idea that there is an objective, universal “now.” Yet, paradoxically, today’s standard formulation of quantum mechanics makes heavy use of that discredited “now.” Playing with paradoxes is part of a theoretical physicist’s vocation, as well as high-class recreation. Let’s play with this one.

First, some background. Despite special relativity’s freedom in assigning times, for each choice there is a definite ordering of events into earlier and later. In a classic metaphor, time flows like a river through all space, and the flow never reverses.1 Figures 1, 2, and 3 tell the central story.

Figure 1

Figure 2

Figure 3

To organize our thoughts, let us make a definite choice of time; in the jargon, let us fix a frame of reference. Then we can frame the history of the world as shown in Figure 1. Here time runs vertically, while space runs horizontally. Since we’re going to be considering several versions of time, we’ll name this one t1. For convenience in drawing, we are restricting attention to a one-dimensional slice of space—in other words, a line. One-dimensional “spaces” of events sharing the same value of time t1 would appear as horizontal lines (which I haven’t drawn). The meaning of the colored regions and their labels will be elucidated presently.

Observers moving at constant velocity with respect to our frame of reference will need to use their own physically appropriate, different versions of “time,” corresponding to how their clocks run. Figures 2 and 3 display the lines for which two different versions of time, t2 and t3, are constant. t2 is the appropriate measure of time for observers moving at a certain constant velocity toward the right, while t3 is the appropriate measure of time for observers moving at a certain velocity toward the left—that is, in our figures, in the horizontal, “spatial” direction—relative to our reference frame. For observers with higher speeds, the tilt of these lines will be steeper. But the tilt never exceeds 45 degrees, because 45 degrees corresponds to the limiting speed, namely the speed of light.

With this background, we are ready to appreciate the distinctions shown in Figure 1. In the center of the diagram is a blue point b representing a specific event. Some events—those that lie in the green future region of space-time—occur at a later time than b, whether we use t1, t2, t3, or any other allowed observer’s measure of time. We say that these events are in b’s causal future (or, if there is no danger of confusion, simply b’s future). What happens at b can affect events in b’s causal future, without upsetting any observer’s sense that a cause—b—must occur before its effect. Closely connected is the fact that signals from b can reach events in b’s future without ever exceeding the speed of light. We call such physically allowed signals “subluminal” signals.

Similarly, we can define b’s causal past, depicted in red. It consists of all events that can affect b. There is a nice symmetry here: If we draw cones emanating from an event a in b’s causal past, we will find b in the upper colored region. An event a is in b’s causal past, if and only if b is in a’s causal future.

But many events fall into neither of those regions; they are neither in b’s causal future, nor in b’s causal past. We say that such events are “space-like” with respect to b. The event a, which appears in Figures 2 and 3, is of that kind. According to t2, a occurs after b; but according to t3, a occurs before b. Neither a nor b can send subluminal signals to the other.

In a similar way, we can consider the regions that are future, past, or space-like with respect to a. This leads us to a more elaborate division of space-time, illustrated in Figure 4. The orange region contains events in the common (causal) past of both a and b, the purple region their common future, and so forth. This colorful diagram hints at a potentially rich subject, the geometry of causation, that could be developed much further. (Specifically, it could add some spice to high-school geometry and analytical geometry courses, and provide material for independent projects.)

Figure 4

As we’ve seen, if a and b are space-like separated, then either can come before the other, according to different moving observers. So it is natural to ask: If a third event, c, is space-like separated with respect to both a and b, can all possible time-orderings, or “chronologies,” of a, b, c be achieved? The answer, perhaps surprisingly, is No. We can see why in Figures 5 and 6. Right-moving observers, who use up-sloping lines of constant time, similar to the lines of constant t2 in Figure 2, will see b come before both a and c (Figure 5). But c may come either after or before a, depending on how steep the slope is. Similarly, according to left-moving observers (Figure 6), a will always come before b and c, but the order of b and c varies. The bottom line: c never comes first, but other than that all time-orderings are possible.

Figure 5

Figure 6

These exercises in special relativity are entertaining in themselves, but there are also serious issues in play. They arise when we combine special relativity with quantum mechanics.

Two distinct kinds of difficulties arise as we attempt to combine those two great theories. They are the difficulties of construction and the difficulties of interpretation.

The difficulties of construction dominated 20th century physics. (One measure of this: By my conservative count six separate Nobel Prizes, shared by 12 individuals, were awarded primarily for advances on this problem.) The tough issues that arose here, in the construction of relativistic quantum theories, are in some sense technical. Combining special relativity and quantum mechanics leads to quantum field theory, and the equations of quantum ﬁeld theory are dicey to solve. If you try to solve those equations in a straightforward way, you find nonsensical results—for example, inﬁnitely strong forces. In fact it emerged, after many adventures, that most quantum ﬁeld theories really don’t make sense! They are mathematically inconsistent. Those that do make sense can only be defined using tricky mathematical procedures. Passing in silence over that epic, we reach the bottom line: After heroic struggles, the difficulties of construction were eventually (mostly) overcome, and today quantum ﬁeld theory forms the foundation of our immensely successful Standard Model.

The difficulties of interpretation have a different flavor. Closely related to our issues with time-orderings, they arise because labeling events by time plays an absolutely central role in the conventional formulation of quantum mechanics.

The quantum state of the world is represented by its wave function, which is a mathematical object defined on surfaces of constant time. Furthermore, measurements “collapse” the wave function, introducing a drastic, discontinuous change. Suppose, for example, that we decide to use t1 as our time. Then a measurement at t1 = 0 changes the wave function everywhere at all times subsequent to t1 = 0.

But what if we had chosen t2 or t3? The occurrence of that sort of collapse implies that there is a drastic difference between the formal descriptions of quantum mechanics based on our choice of reference frame. If we work with t2, then measurements at b will collapse the wave function seen at a, since b comes before a. For the same reason, measurements at b do not collapse the wave function at a. But if we work with t3, since the time-ordering between a and b is reversed, the situation is just the opposite!

Yet special relativity demands that either t2 or t3 can be used in a valid description of nature. Have we discovered a contradiction?

Not necessarily.

The point is that quantum-mechanical wave functions are tools for describing nature, rather than nature herself. Mathematically, quantum-mechanical wave functions contain a lot of excess (unobservable) baggage and redundancy, so that wave functions that look drastically different can nevertheless give the same results for most, or possibly all feasible physical observations.

While it falls short of outright contradiction, there remains, it seems fair to say, considerable tension at the interface between quantum mechanics and special relativity. During the long struggle to construct quantum ﬁeld theories, several physicists speculated that the inﬁnitely strong forces they calculated were surface symptoms of a fundamentally rotten core, whose rottenness was indicated more directly by the difficulties with interpretation. It didn’t work out that way. We have been able to construct theories that are not only consistent but also immensely successful, despite their near-contradictions and excess baggage.

As new technologies for probing the nano-world render possible what were once purely thought experiments, we have wonderful new opportunity to ask creative questions, confronting the paradoxes of quantum mechanics head on. Maybe we’ll ﬁnd some surprising answers—that’s what makes paradoxes fun.

1 There are more speculative possibilities: that time exhibits cycles, or branches, or even has several dimensions of it own. In general relativity we let time bend together with space, and in describing the Big Bang and black holes we encounter singularities, where time begins or ends. This is fascinating stuff! But “flat, unidirectional” time is the basis for almost all practical physics, and it already provides rich food for thought, so that’s what I’ll be considering here.

Go Deeper