## Tuesday, October 13, 2009 ... //

### Causality, fate, and the arrow of time

A top science journalist in the New York Times has written a bizarre article about a couple of even stranger preprints by famous authors - including an early co-father of string theory - that have argued that a mysterious fate guarantees that any attempt to build the Superconducting Supercollider has to fail, and any other collider similar to the Large Hadron Collider has to break as well in order for us not to find the Higgs boson because the Higgs boson is the God particle and God wants to protect Her own face. Or something along these lines.

They apparently believe that there must exist a cosmic conspiracy that has guaranteed that Ronald Reagan's Superconducting Supercollider had to be killed by an inevitably emerging lack of interest from former vice-president Al Gore (who prefers junk science over big science) who teamed up with the usual G.O.P. suspects (who think that big science is the same thing as a big government).

I won't include links to these texts because they don't satisfy the intellectual criteria to be promoted at the TRF main page. But feel free to post the links in the comments. Instead, I want to explain some general facts about causality and the fate.

Recent topics: thermodynamics

In the recent months, many texts have been written about the thermodynamic arrow of time. Arguments analogous to Boltzmann's proof of the H-theorem were employed to prove that the thermodynamic arrow of time (the arrow that defines the future as the moment when the entropy is higher than in the past) has to coincide with the logical arrow of time.

It feels logical to say a few things about the logical arrow of time, the relationships between the causes and their consequences, and the modifications of all these facts by quantum mechanics, special relativity, general relativity, quantum field theory, and string theory. Can our Universe be driven by a predetermined destiny? Of course, the short answer is "No" but a longer answer is the goal of the following part of my text.

History

The philosophers have said many vague things about the time in the past. As people were getting more familiar with the logic of the world that surrounds us, the philosophers were slowly getting upgraded to the physicists. Aristotle's rants that were composed of a uniform mixture of good observations and nonsense were being gradually superseded by scientific texts that carefully studied what propositions were well-defined, which of them were proved, which of them were disproved, and what were the exact relationships between all these statements.

Of course, not all philosophers have been upgraded so even in 2009, we still find people who can't become physicists because they're deeply confused about certain elementary things. Fairly enough, they still call themselves philosophers. We also find people who call themselves physicists even though they haven't been upgraded to the logical reasoning, either.

Basics of the logical arrow of time

The well-defined events in our world can be arranged along a time axis; we will talk about the evolving geometry later. Although the differential equations may be T-symmetric or CPT-symmetric (and the basic equations of relativistic quantum theories have to be CPT-symmetric in general - and they are approximately T-symmetric, too), the interpretation of the past and the future is very different.

Let me enumerate a few logical statements in which the past and the future differ, for example:

1. We may remember the past but we can't remember the future
2. We may change the future but we know that we can no longer change the past
3. We learn about events in their future, i.e. after these events take place, not in their past, i.e. before they take place
4. We look forward to the future or we are afraid of the future
5. We want the future to be better than the past
We could continue for a while. You may see that these observations are not quite independent from each other. But they are independent of the differential equations that determine the microscopic dynamical physical laws. Indeed, those laws don't have an arrow of time.

But aside from these equations - which are typically T or CPT-symmetric - one must employ a framework defining how the fundamental equations may be used to learn something about the reality: a system to interpret the mathematical concepts in practice. And this logical framework adds an additional "structure" to the spacetime. This logical structure includes a logical arrow of time - something that can distinguish the past from the future at every region of spacetime. This arrow of time doesn't contradict the CPT-symmetric fundamental laws at all. And it is, in fact, necessary for the world to operate.

The five conditions above may sound too "psychological" to you. Can we replace them by a set of "mechanistic" principles? Well, strictly speaking, we can't. The difference between the past and the future "is" psychological, in a sense. But if you want me to get close to such a mechanistic reinterpretation, I would write that
the future always evolves from the past. In other words, the laws of physics or the "cosmic design" may only restrict the "initial conditions" in the past, and not the "final conditions" in the future (our destiny).
So the reason why we're here is that a hospitable planet was created and intelligent life gradually evolved on it in the past. The reason is not that God or Nature have a big plan to do something with us in the future and our partially advanced contemporary civilization is needed for such a future outcome.

In other words, scientific explanations of phenomena may include non-trivial information about the (possibly T-symmetric) laws how the Universe evolves. And they may include special information about the initial state (in the past). But they can never include special information about the future (except for restrictions that should be valid at each moment, not only in the future).

Note that special initial conditions can't contradict the differential equations for the evolution: they are independent pieces of information. It is necessary for most applications to say something about the initial state. On the other hand, it would be a contradiction to constrain, aside from the differential equations, both the initial and the final states. A chosen initial state simply won't evolve into a chosen final state by the given differential equations.

Only one of the "t=const" boundary conditions may be specified, and we call the slice where the conditions are specified "the past". The future evolves out of this past according to the laws of Nature. If someone uses the word "future" for the slice where the state is predetermined and if he pretends that he is calculating "the past" out of it, he is doing something else than Nature. Nature, by the very definition of the past and the future (according to the logical arrow of time), is always doing the usual thing: it is calculating the future from the past. The only thing one gains by describing any situation in the opposite way is terminological havoc.

While you might consider the previous paragraphs to be a description of a "mechanism" how Nature operates - it generates the future out of the past - there is a corresponding rule how a rational person thinks whenever he needs to use subjective probabilities and inference. He always calculates the "posterior probabilities" from the "prior probabilities", using the Bayesian formula or its equivalents.

Once again, a rational person never assumes "posteriors" - e.g. the information about the future - to figure out what's happening in the past. Only "priors" appear as parameters in the calculations of probabilities. And "priors" are geometrically connected to the past, not to the future. Science can also reconstruct or retrodict the past, but as we have repeatedly explained on this blog, the logical rules of retrodiction of probabilities are different from the rules of prediction.

The predictions of the future are "straightforward". Given a known initial state in the past and the differential equations or other laws that evolve the states, we can calculate the future state - or its probabilities (from the squared complex amplitudes, in the quantum case) and the resulting numbers (or probabilities) are completely objective. Every scientist must end up with the same answers.

It's not the case when you try to retrodict the past. Retrodicting the past is equivalent to finding the probabilities of different hypotheses - because the initial state in the past is a part of a hypothesis of ours. Such a retrodiction requires a logical inference, and the probabilities calculated in this way will always depend on the priors which are always at least partially subjective (although some choices of priors may be more insane than others - especially when a person "dogmatically eliminates" the right answer by choosing an effectively vanishing prior from the very scratch).

To reiterate this point, the world operates according to a "cosmic design" which includes a choice of the evolution laws but also some information about the "initial conditions". The "final state" of the system is never a part of a rational hypothesis: it is always - exactly or probabilistically - calculated out of the evolution laws and the initial conditions.

Causes precede their effects

In other words, there is no destiny. The future will be whatever it will be. It will depend on the way how the world will evolve from the present - according to its "differential equations" or other evolution laws. And we feel that we may affect the future by our "free will", too. Of course, our will is not quite free because it is determined by our hormones and brain processes - which can be predicted by neurobiology or psychology (specific small parts of physics), at least probabilistically and at least in principle.

But the point is that the Universe can't be constrained by the requirement that the "final state" in the future looks one way or another. Because you may be irritated by this statement, let me weaken it a little bit. Even if such a "destiny" were a part of the cosmic design, we couldn't find it by the evidence-based rational reasoning because the evidence-based rational reasoning always boils down to the evidence which can be summarized as observations in the past - and to various "patterns" that we can find in these observations of the past.

No-prophet theorem

Even though some people such as Moses or James Hansen have described themselves as prophets, there is no direct observation of the future. The only rationally justified way to say something about the future is to observe the distant past and the recent past, induce the likely laws of physics (the verb "induce" differs from the verb "deduce" by allowing for an uncertainty of the conclusions: "induction" is a kind of generalization combined with logical inference), assume that these laws will probably continue to hold for a while, and use them to predict the future from the present.

These principles dictate vastly different methods to say something about the past or something about the future, especially when irreversible processes are important. I claim that these principles are completely essential components of every scientific or rational reasoning - both in pure science as well as in applied science or everyday life - and I claim that whoever rejects these principles (at least in the context of ordinary everyday events) is mentally ill, at least until he gives a justification for the greatest breakthrough in science that such a modification would probably mean.

(The authors of the paper about the black hole final state get an exemption because such a final state would have a very small effect, except for a close vicinity of the final singularity. But the paper is almost certainly wrong, anyway.)

Whoever doesn't realize that the methods to learn something about the past and about the future are and must be vastly different couldn't have possibly used his or her brain in the everyday life rationally, or at least he or she hasn't realized how such an achievement was done. That includes Sean Carroll and many others.

What can you affect: changing geometry

From now on, let me assume that the reader understands why everything we seem to know about the past or about the future always boils down to the observations of the past (this discrimination against the future is necessary because we don't know anything about it directly), and the laws of evolution that were induced from the observations of the past as well. For example, the reader should already know why the article by Holger Nielsen et al. and by Dennis Overbye are pure hogwash. ;-)

I want to say a few well-known things about the "geometry" of causality.

In classical physics, the world was believed to evolve deterministically. Given the known initial state, one could calculate the state at any later moment exactly. Because only one type of "boundary conditions in time" may be adopted and we have called them "initial conditions in the past", we see that an event can only influence the events in the future.

Because classical physics was deterministic, one could have calculated the future with a complete certainty. At least in principle, there was a one-to-one map between the states at one moment "t1" and the states at another moment "t2". It followed that in principle, one could have known everything about the system and the probabilities of all statements were either 0% or 100%. That's the only case when Bayesian inference may become independent of the priors: the inference would include a "complete learning of the truth". Once it becomes independent of the priors, you can also retrodict the past in the same way as you predict the future and the arrow of time would become invisible (assuming a complete knowledge of the system).

However, both statistical physics and quantum mechanics invalidate the reasoning. In the case of statistical physics, we want to derive "macroscopic" conclusions about a physical system that require the initial state (and therefore also the final state) to be known incompletely, and a sensible choice of the priors is important (for example, one is not allowed to assume that the initial state is conspired to lead to a special or low-energy final state). In the case of quantum mechanics, the evolution laws are statistical in character: all theories compatible with the postulates of quantum mechanics can only predict the probabilities of future events. They can't say accurately what the events will be.

In Newtonian physics, one could say that an event at time "t1" can only influence events at time "t2" if "t2" is greater than "t1". Special relativity confirmed this statement but it has actually strengthened it: the refined proposition says that an event at point "P" can only influence another event at point "Q" in the spacetime if "Q" belongs to the future light cone of "P".

In other words, superluminal (faster-than-light) action is as impossible as the action that influences (rewrites) the past. It's not hard to understand why: according to special relativity, the ordering of two space-like separated points "P", "Q" in spacetime (i.e. two points that need a superluminal signal to be connected) depends on the chosen reference frame, and is therefore not "absolute" or agreed by all observers. In one frame, "P" comes before "Q"; in another frame, it's the other way around.

So if you forbid "influences upon your past" in one reference frame (which is necessary to avoid logical paradoxes where you can kill your grandmother while she was still a virgin), you must also forbid "influences upon all space-like separated points" i.e. all "superluminal influences", too. That's required by the principle of relativity because it says that all the laws of Nature must be independent of the chosen inertial frame. This principle or "meta-law" must apply to the causality rules, too. And because all space-like vectors look like "vectors directed to the past" in many different reference frames, the causal action along space-like separations must be forbidden as well.

So the principle of relativity, together with causality ("causes precede their effects": philosophers could call this requirement "antecedence" rather than "causality"), imply a kind of "locality" which says that one can't directly and immediately influence distant, space-like separated regions. In Newtonian physics, causality only cared about the ordering of events in time ("causes precede effects") and locality only cared about the ordering of events in space ("something had to happen somewhere in between the places of the cause and the effect in space").

However, in relativity, locality and causality are closely linked to one another. They're not independent anymore. That's just another example (or consequence) of the rule that relativity says that (almost) whatever holds for space, must also hold for time (and vice versa). It should also be obvious how the rule of causality is generalized in general relativity: you may still deduce what is the future light cone of a point, even if you deal with a curved spacetime.

Quantum field theory

But it's important to realize that even in quantum field theory - and at least morally in all theories that extend them (string theory is the only known example, and probably the only mathematically possible example, of a consistent framework that generalizes QFT, while respecting its paramount consistency constraints, but is not strictly a QFT) - causality and locality strictly hold.

That's easy to see in Heisenberg's picture. Take an interacting relativistic quantum field theory such as QED or QCD. Write the equations for the field operators, i.e. the Heisenberg equations. They will resemble the ordinary, Maxwell-like classical equations. But there will be hats.

In classical Maxwell's theory, a change of an electric field at point "P" (or a small region around it) will only influence the values of the fields at the future light cone of "P". That's trivial to see in the case of the free (wave) equations. But the interactions don't change anything about it because at very short time separations, they still respect the light cones.

You can check that the same thing will hold in the quantized theory. Because all the predictions may be formulated as predicted probabilities - expectation values of projection operators - and all these projection operators may be written as functionals of the field operators, it follows that all probabilities of all observations around the point "Q" will be unaffected by changes near the point "P" unless "Q" belongs to the future light cone of "P".

People like to say a lot of incorrect things about "nonlocality" of quantum mechanics. The reality is that the theories that describe "almost" all phenomena we have ever observed - quantum field theories - are completely local and causal. And at least morally, this conclusion applies even to the "theory of everything", string theory. A separate long article is needed to clarify what aspects of locality and causality are preserved in string theory i.e. quantum gravity and what parts are not: I will say a few words about this topic at the very end.

In fact, experts often use the term "local quantum field theory" to describe the familiar QCD-like quantum field theories. There's nothing "nonlocal" about the quantum entanglement etc. No "nonlocal" propagation of signals ever occurs. Such a "nonlocal propagation" would be needed if you wanted to emulate the world by a mechanistic theory which assumes that there exists a well-defined state of the world at each moment - a state that evolves deterministically (or quasi-deterministically, allowing some random generators to be used in the equations).

But such a model is incorrect. The world is described by probabilistic laws only. And once you understand that probabilities are the only things you can predict and all the mechanistic, morally classical models are deeply misguided, you should be able to see that no physical prediction for a region "Q" can ever be affected by decisions made in a space-like separated region "P": superluminal propagation of information is strictly impossible.

Also, there are all kinds of experiments that show "something moving superluminally". But even when it comes to sophisticated optics in exotic materials, all these experiments are just about an "illusion". In properly chosen variables, the equations of the Standard Model may be shown to strictly preserve the relativistic causality and locality. Of course, the locality will be violated if you make a nonlocal field redefinition. That's essentially what's going on in all these experiments.

In some materials, one can prepare an initial state with long-range correlations. The information about the initial state is stored in a big region "P" while the experimenter pretends that the excitation of the initial state is only assigned to a point "p" inside "P" (e.g. the center of "P"). By the proper rules of causality, the changes made in "P" may affect the union of all the future light cones of all the points in "P": some of these points "q" may lie outside the future light cone of the point "p".

But you will never be able to create a "layer of material" that could speed up the propagation of genuine signals through it above the speed of light. Whatever materials you will use in the fibers, you will never be able to transmit genuine packets from London to Boston in less than 17 milliseconds. In this sense, the "superluminality" is always the same kind of an illusion as a "trace of light on the wall" coming from a quickly rotating laser in the center of a large spherical room.

Of course that for a sufficiently high rotation speed of the laser (but smaller than "c") and for a large enough room, the trace of light may move along the wall by a speed that is faster than light. But this trace can't carry any information from one place of the wall to another. It is just an illusion that "something is moving along the wall". In reality, the photons are moving from the center to the walls, and the speed is "c", not higher.

String theory

I just say a few words. String theory is not strictly a local quantum field theory - at least most experts think it's not, and the preservation of the information during a black hole evaporation is the most concise argument why it's not (because the causal structure of the spacetime has to be circumvented when the information tunnels out of the black hole: in this regime, the metric tensor must be emergent and one can't take the causal structure implied by it too seriously). But it preserves most consequences of locality that one can derive in quantum field theories.

You can reformulate string theory as a string field theory, i.e. a quantum field theory with infinitely many fields corresponding to energy eigenstates of a string. But the interactions of these fields won't be manifestly "local" if you assign the string fields to the "center of mass" of the string - simply because the string is an extended object and two strings may join even if their centers of mass don't coincide. The reason is similar to my explanation above why the experimenters (Lene Hau et al., even though she prefers to slow the light down) "seem to observe" superluminal signals.

However, in the case of the string, the superluminal propagation is a kind of an illusion, too. The evolution of one string is completely local - it is described by local equations ("wave equations") on the worldsheet. And the interactions of the strings are local, too. Well, they're local on the worldsheet - whether two strings join or not is always decided by the degrees of freedom around the interaction point only (and the interaction points on the two strings must correspond to the same point in spacetime if the interaction is allowed to occur: there's a delta-function in the vertex).

In this sense, it means that every violation of locality that you derive in perturbative string theory is an artifact of a wrong choice of degrees of freedom - you have effectively made an unnecessary nonlocal field redefinition that masked the locality in other variables.

And indeed, if you look at some analytic properties and inequalities (implied by unitarity combined with locality) that are satisfied by the scattering S-matrix, you may see that string theory perturbatively satisfies all of these conditions that can be derived from strictly local quantum field theories. String theory is just much more capable to saturate many of them: it is literally a "quantum field theory taken to the limit". While it shows that the conditions derived from general quantum field theories are damn real and important, it actually implies that the particular quantum field theories we have known before are just "provincial special cases". Quantum field theories are analogous to N = 2,3,5,7 while string theory also tells you that N can be 2^{42,643,801}-1.

Even if you do those non-local redefinitions, the nonlocality is typically just able to get you one string length away from the light cone, so to say. So string theory is "almost local" even in generic degrees of freedom, as long as the nonlocal field redefinition is linked to the string scale.

The previous sentence holds nonperturbatively, too. But it is somewhat less clear whether nonperturbative string theory may "strictly" preserve locality in some appropriate degrees of freedom. The answer is probably "No" and the preservation of the information during black hole evaporation is a reason. Still, it is tempting to expect that some "equally powerful" constraint to the locality has to survive even when the full-fledged quantum gravity regimes when the metric tensor becomes heavily affected by the quantum fluctuations.

At any rate, causality and locality are extremely strong conditions that must be checked in any "radically new" theory meant to describe the Universe. They must hold almost exactly. And all statements about the past and the future that we ever rationally make must boil down to direct or indirect analyses of our observations of the past, and the patterns in these observations. Building upon assumptions or "prophesies" about the future is no science.

And that's the memo.