# Is causality a formalised concept in physics?

I have never seen a “causality operator” in physics. When people invoke the informal concept of causality aren’t they really talking about consistency (perhaps in a temporal context)?

For example, if you allow material object velocities > c in SR you will be able to prove that at a definite space-time location the physical state of an object is undefined (for example, a light might be shown to be both on and off). This merely shows that SR is formally inconsistent if the v <= c boundary condition is violated, doesn’t it; despite there being a narrative saying FTL travel violates causality?

Note: this is a spinoff from the question: The transactional interpretation of quantum mechanics.

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I voted to close the question as "subjective and argumentative" because its author tries to force the visitors to give him a philosophically pleasing answer and totally ignores all the deep physics answers that are actually being written. – Luboš Motl Jan 25 '11 at 22:16
It seems to me that what you are asking boils down to a philosophical question about the nature of causality, which is a central one in philosophy of science starting from Hume. I'd suggest to pick a text on philosophy of science (e.g the one by Alex Rosenberg is good) and familiarize yourself with the debate, it's good stuff. – user566 Jan 25 '11 at 22:22

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@Lubos you can't retroactively decide a question is "subjective and argumentative" based on your negative view of the OP's response to the answers, including your own. @Moshe you seem to assume the OP doesn't already have a familiarity with the philosophy of science or has not already given the question of causality some consideration. Perhaps you could give @Nigel the benefit of doubt since he does state that he has a Phd in AI. – user346 Jan 26 '11 at 6:11

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Dear @Lubos it is hard to have a meaningful discussion with you when you dismiss viewpoints that you disagree with as "nonsensical philosophical gibberish". – user346 Jan 26 '11 at 7:30

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I'll try to summarise the overall discussion in my own answer tomorrow. This question originated from the concept of "retrocausality" in the previous cited question, where it was flagged as a rebuttal of the TIQM. So it's hard to justify taking an agnostic position and booting the question over to "the philosophers" as if physicists don't need to worry. (I see a lot of worrying in some of the answers!). Anyway, more later. – Nigel Seel Jan 26 '11 at 21:58
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## 9 Answers

Dear Nigel, causality is not an observable (quantity) with a value and a unit; so it is not identified with any operators.

Causality is a principle. In a broader scientific and colloquial context, causality is any property of the relationship between the cause and its effect.

However, in physics, we mean something more particular by causality. In classical physics, we mean the following proposition:

If a cause takes place at time $t_1$ and its effect at time $t_2$, then $t_1<t_2$ must hold.

In other words, the cause precedes its effect.

It's obvious that if the principle above would be violated, the world would become logically inconsistent. Events at time $t_1$ could cause some events at time $t_2$ which would generally cause different events at time $t_1$, producing contradictory answers to the question what happened at time $t_1$.

Looking at causality from a relativistic viewpoint

In the special theory of relativity, the statement above must still hold for the history of spacetime to be free of logical contradictions. However, special relativity is based on the principle of relativity that says

Laws of physics take the same form in all inertial frames - those that are in uniform motion relatively to one chosen inertial system.

This must be true for all laws, including the principle of causality itself.

If this principle of relativity is combined with the principle of locality above, we may actually derive a stronger statement. In relativity, the delay between two events depends on the inertial system: simultaneity of events is relative, we say. So two events may be chronologically ordered in the opposite way if you switch into a different inertial system. However, spacelike separated events remain spacelike separated events; and an event in the future (or past) light cone of another event stays in the same cone from the viewpoint of all inertial frames.

Applying the principle of relativity to the principle of causality, we may derive a stronger, relativistic principle of causality:

If a cause takes place at point $P_1$ in spacetime and if its effect takes place at point $P_2$ in spacetime, then $P_2$ must belong to the future light cone of $P_1$.

This is a stronger statement than the original one (about the ordering of $t_1$ and $t_2$): the relativistic causality implies the ordinary causality, but something more (it implies the non-relativistic condition from the viewpoint of all relativistic inertial frames). A cause is not only unable to affect its past, like in the non-relativistic causality; it is unable to affect the spacelike-separated points in the spacetime, too.

Any violation of the relativistic causality - which means that causes may only influence their future light cones - would lead to the same logical contradictions that I explained in the non-relativistic context. In particular, you wrote:

This merely shows that SR is formally inconsistent if the $v \leq c$ boundary condition is violated, doesn’t it; despite there being a narrative saying FTL travel violates causality?

Well, indeed. However, special relativity is demonstrably a valid theory of our spacetime (at least locally). So the "mere" inconsistency of special relativity that you mentioned, in a somewhat incomprehensibly dismissive tone, would automatically mean an inconsistency of the whole Universe which is a pretty serious problem. There's no doubt that there can't be any signals that move faster than light. Logical consistency is an omnipresent and unquestionable assumption in all of physics (and maths), so one is always allowed - and encouraged - to assume it. When we assume it, we may easily show that faster-than-light motion violates causality. In fact, relativistic causality is exactly what bans faster-than-light motion.

I am convinced that this text explains - and fully unmasks - all deeper and more foundational facts and arguments behind the notion of causality in physics.

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The problem that I have with this answer is that you are using natural-language terms (cause, effect) as a metalanguage to talk about the physics. Where in the mathematics is there some entity which you can point to and say 'that's intrinsically the concept of a 'cause' or 'that's the concept of an effect'? Formally speaking, how could this work when we can (for the purposes of this discussion) just time-reverse everything in our fundamental theories? I could be led to believe that our everyday concept of causality has more to do with the arrow of time and increasing entropy :-). – Nigel Seel Jan 25 '11 at 20:54
It is you who began to talk about causality, so I just replied. As I said at the very beginning, "causality" is not any quantity that can be expressed by a variable or an equation. Causality is a principle addressing natural-language terms. It's exactly what I wrote. Everyone who writes that it is something completely different is lying to you. ... And you can't reverse things in physics: it's the very point of causality that you can't. The fundamental laws are CPT-symmetric but the system of predictions in physics requires an arrow of time to define a globally consistent law of causality. – Luboš Motl Jan 25 '11 at 22:03
"I am convinced that this text explains - and fully unmasks - all deeper and more foundational facts and arguments behind the notion of causality in physics." .... @Lubos by "this text" do you mean your answer? – user346 Jan 26 '11 at 6:02
@Lubos The word locality appeared once in your answer, is it supposed to be causality instead of locality? – Revo Aug 23 '11 at 2:05
Lubos, In the line "If this principle of relativity is combined with the principle of locality above", I think you mean "causality above" ? – Revo Aug 11 '12 at 12:26

Causality becomes much more subtle whenever theories are statistical or probabilistic. When we see a correlation, it may be that one event caused the other, but it may be that there is a common cause or that there is just a chance correlation that would disappear if we do more of the same data gathering. For the notion of "common cause", the standard old-time reference is Reichenbach. Try the "Reichenbach's common cause principle" entry in the Stanford Encyclopedia of Philosophy, and other entries on causality therein. Often quite a good reference for Philosophy of Science.

The mathematics of causality in modern statistical physics is very simple, but it has subtle consequences. Quantum field theory distinguishes between the brute random fluctuations of the vacuum state (which are caused, if they are caused by anything, by the random fluctuations that were there in the past, which were caused, if they were caused ...) and causal relationships between measurements. If two measurements are associated with regions of space-time that are at space-like separation from each other, quantum field theories predict that correlations will be observed in the recorded data, but we say that ideal measurement devices do not cause such correlations. If we don't have measurement devices that are close enough to this ideal, we may have to make allowances for the non-ideal details of the real devices to make the theory match the data. Somewhat non-standardly, I would say that ideal quantum measurements that are at time-like separation from each other do cause some component of the correlations we observe in the recorded data from such measurements. That's different from ideal classical measurement devices, which record data while changing neither the physical state nor the data recorded by other ideal classical measurement devices. But this is a research project.

Apropos of the Transactional Interpretation starting point, the SEP entry "Action at a Distance in Quantum Mechanics" might be interesting.

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I think this is a very interesting comment and one which genuinely engages with what it even means to say that event A "causes" event B within the framework of a particular physical theory (QFT in your example). It's good that we're now focussing on operational concepts such as measurements and correlations: +1. – Nigel Seel Jan 26 '11 at 22:05

In the axiomatic approach to quantum field theory, sometimes also called local or algebraic quantum field theory, pioneered by Araki, Haag, Kastler, Bogoljiobov et. alt., causality is formalized as an axiom, most often called the "locality" axiom.

The idea is this: To every bounded open subset of Minkowski spacetime we associate an operator algebra, all selfadjoint elements of this algebra represent all observables of this region, that is everything that is measurable in this region. Then algebras associated to two spacelike separated regions are assumed to commute, this is the locality or causality axiom.

When two observables aka selfadjoint operators commute, this means of course that measuring aka observing the first will have no effect on measuring aka observing the second and vice versa, therefore there cannot be any causal relationship of the events of measuring them.

BTW: The Reeh-Schlieder theorem seems - intuitively - to violate causality/locality, so it is interesting to note that it is possible to prove this theorem without invoking the locality axiom. The reason for this is that the Reeh-Schlieder theorem is about entanglement effects which don't violate locality in the sense of SR.

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Interesting. However what you're describing seems to be the formalisation of the concept of a spacelike interval within a variant of QFT and then labelling this axiom with the word "causality" (or "non-causality" in fact). My question was intended to explore whether causality has a deeper, more foundational role in physics than as a synonym for non-spacelike separation. – Nigel Seel Jan 25 '11 at 20:31
@Nigel: No, as far as I know there is no deeper formalization of causality in physics, as Lubos already said. Maybe one could add that the time evolution of physical systems is described by hyperbolic evolution equations and that the causal relationship of events is preserved via the appropriate representations of the Poincare group, and that's it. – Tim van Beek Jan 26 '11 at 19:46
I don't think spacelike commutativity is the same thing as causality. Probably a much closer characterization is the primitive causality axiom stating that the algebra associated with the causal completion of a region is the same as the algebra associated with the region. – QGR Jan 27 '11 at 16:42

Perhaps the question is too broad. The fundamental principle underlying causality is locality. We can observe causality indirectly by observing locality. By this I mean that the very existence of a coordinate system is the very manifestation of locality.

Think of it this way; the motion of an observer 'causes' lengths to appear smaller. In other words, the 'effect' is the shortening of measured length. You can imagine that the relative motion is a cause and that length contraction is an effect.

In order to measure this effect we postulate a coordinate system. When we do this, it turns out, that we predict a finite maximal speed. Which happens to be the speed of light.

In Quantum Mechanics the issue of locality is completely defined by the Heisenberg Uncertainty Principle (HUP) which is equivalent to the treatment of locality in General Relativity.

Namely, the HUP allows you to measure precisely the position of a particle/field. In other words, the HUP allows you to define a coordinate system. This is sufficient to define locality and hence sufficent to embody causality.

Think of it this way; the existence of an observer causes a coordinate system to appear. In QM, the observer is the 'cause' and the coordinate system is an 'effect'.

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As the originator of this question I have reviewed and learned something from all the answers posted so far. I would like to summarise my own views here.

1. Where did this query come from? From a question about the Transactional Interpretation of Quantum mechanics (TIQM), where said theory’s reliance upon “retrocausality” (‘causality’ backwards in time) was held to be a fatal defect. This kind of causality-argument is common in physics: we say that faster-than-light travel in SR is ruled out because it would violate causality.

2. Such causality arguments are conducted in what you might call the “metalanguage of physics”: technical English which supports and explains formal results. However, the arbiter in the end is the maths, so how do we interpret the notion of causality within the formalism?

3. Physical theories are defined by mathematical relationships between entities (observed and unobserved) usually expressed by equations (think Schrodinger, Dirac, the Lorentz transformation). If we say that event E1 “causes” event E2, several answers here suggest that the interpretation of causality in the formal theory is that: (i) if E1 is postulated to occur then the theory logically implies that E2 must occur as well; (ii) E2 is within or on the future light-cone of E1 (we say “cause precedes effect”).

4. However, it’s possible that condition (ii) is too stringent. While logical entailment is obviously an essential part of any formalised theory, our smuggling in of the word “future” is already an extra assumption. Our fundamental theories do not impose a specific past-future direction on the time dimension. This means that if you reverse the film, the events you see are still consistent with our fundamental theories.

5. Sometimes people use causality-like words in the physics metalanguage without conventional time-ordering condition (ii). For example, a possible Feynman diagram for electron-positron scattering has a narrative that an electron travelling backwards in time from the future encounters a (normal) electron, they exchange a virtual photon and continue on their way scattered. The 'cause' of the scattering event was the arrival of the future electron. Many textbooks mention this way of thinking but we don’t mind because the underlying theory gives consistent results which accord with observation. Perhaps TIQM is like this despite its narrative of retrocausation.

6. So my conclusion is that we have to be careful about arguments concerning a theory’s validity relying upon causation arguments couched in physics’ metalanguage. It’s not a slam-dunk. Sometimes if a theory violates conventional “cause precedes effect” causation it indicates a breakdown in the underlying mathematics, normally inconsistency. At other times a 'causation' argument is just a way of talking about the entailment of the theory in an innovative or whimsical way, and the theory is actually OK. Go look at the maths.

NOTE: there is a whole separate discussion about why, in natural language, we think so naturally in terms of cause and effect. It links to discussions about the arrow of time and why we do seem to be unhappy about running the film backwards as a valid picture of reality. That is a whole separate issue but still, I suspect, part of physics judging by the number of recent books on the subject.

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Good clarifying points: I think though that a Stack lesson is that if the question was really about "retrocausality in TQM" that could have been made explicit in the question description. I have examined Cramer's 1986 paper in the light of your answer and found he covers these points too in defence of TQM. Specifically he has introduced "strong" and "weak" causality. Put simply he admits that TQM has retro strong causality but has no retro weak causality. He also mentions the arrow of time and the K meson problem associated with it. I'll amend my answer slightly too. – Roy Simpson Jan 27 '11 at 12:34

The Causality structure of a Spacetime might be best modelled by a partial order between events. An event is a primitive concept in spacetime theory, though one is welcome to question that primitiveness in a yet more fundamental theory. Penrose's Twistor Theory is another theory (this time geometric) in which spacetime points (events) are derived, not fundamental.

However given that we assume events, we can introduce a partial order between them. Introduce some conditions - or derive a null cone structure from the partial order. So one can ask for conditions on the PO to derive a null cone from it. Then from a Global perspective one can introduce further conditions to ensure that this PO does not have x < y and y < x. Of course this is the Closed Timelike Curve condition in General Relativity.

Now in GR/SR a "cause" of x is just anything in the past lightcone of x. In Quantum Theory this gets more interesting, particularly in the context of the TQM: here something needs to be "emitted" and "received" - much like a telephone message. TQM might not be correct, but this telecom analogy shows that one could go deeper into what a "cause" really is. I suspect that the place for such a theory is in the "Fundamental Framework for Quantum Gravity" - should such a revolutionary Framework come about.

Roy.

EDIT: It has become clear that this question was motivated by a criticism of retrocausality in a related post on Transactional Quantum Mechanics (TQM). Rather than continue with a general discussion of Causality in Physics I shall note that Cramer addresses this question in his papers (Transactional Interpretation of Quantum Mechanics, 1986 and cites therein). He introduces two notions of Causality:

Strong Causality Cause always precedes effect in any reference frame.

Weak Causality Ditto, but only for macroscopic observations and observer-to-observer communication.

Cramer's cancellation argument implies that there is no Weak Causality Violation in his theory/interpretation. If we accept this, then the question is whether it is acceptable to have strong causality violation in a theory of physics. I can envisage further debate on this.

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From a statistical perspective, whenever you can establish that two type of events (described as states (e1,e2) of a pair systems S1xS2 ) have high correlation (correlations basically stands for the probability of system S1 be in state e1 and S2 be in state e2 simultaneously), you can say that both events are 'causally related'

Being said that, there is no formal way to establish (at least from a purely statistical point of view) which event causes which one, other than your willingness to accept a particular convention of time order.

In QFT, There is a very formal notion of causality established by the conmutator of field operators as functions of space-time, for instance:

$$\left[ \phi(x_\mu),\phi(x'_\mu)\right]=\theta (x_\mu x'_\mu)$$

when this conmutator is zero it means there is no possible correlation between physical events. the quantity in the right side is different from zero when the two space-time points are time-like to each other (inside the light cone of influence)

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lurscher, try $[ \phi(x_\mu) , \phi(x'_\mu) ] = \Sigma ( x_\mu x'_\mu )$ – Roy Simpson Jan 25 '11 at 21:59
One interesting question here is whether the commutator is ever different from zero when the events are spacelike (also null) separated? Another complicating factor is whether one can use points (events) at all in an axiomatic formulation, but must use "test functions". As test functions are spread out the definition of timelike might be more subtle. – Roy Simpson Feb 4 '11 at 13:49
i agree, the current formalism is not entirely satisfactory. But it is a formal concept of causality nonetheless. Thanks for the edits! – lurscher Feb 4 '11 at 15:28

Great question. I'll try to answer.

In quantum-mechanical world the timeline is completely reversible, the system undergoes unitary evolution and there is no preferred time direction. There is no concept of causalty either. That's why it is possible to move faster than light in the quantum world.

For example, electron (and photon) can travel faster than light with some probability. But it can be interpreted as a virtual electron-positron pair appearing ahead of the propagating electron and then the positron from the pair annihilates with the propagating electron (the trick is that information cannot be transferred such way because you cannot be sure whether you spotted just a virtual particle from a vacuum fluctuation or an actual signal).

The causality problem arises from asymmetry of time in classical world. And the asymmetry of time arises from the irreversibility of collapse of the wavefunction. That is the FTL travel can lead to a controversy only when measurements are taken.

It is impossible to measure the quantum value and leave it as it is without collapse. Now imagine one can travel in time, measure the value in the future, then return to present. It would appear as if he made a measurement without collapsing the WF.

In fact the WF collapse is the only existing fundamental law which is not symmetric against the time direction. It is the cause of all other observed irreversible processes, including the second law of thermodynamics.

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"And the asymmetry of time arises from the irreversibility of collapse of the wavefunction." uh, no. Nothing arises from the collapse of the wavefunction because it is completely unphysical. Asymmetry arises from the simple observation that we know past and not future. It cannot be explained, it's just the only possibility to make sense of physics (or indeed, anything in life). – Marek Feb 21 '11 at 8:21
"Nothing arises from the collapse of the wavefunction because it is completely unphysical." - It is physical. At least in Copenhagen interpretation. "Asymmetry arises from the simple observation that we know past and not future." - we know? Asymmetry is already in the thermodynamics. Thermodynamical system cannot know anything. – Anixx Feb 21 '11 at 13:24
What is physical can't depend on the interpretation. If you are familiar with decoherence you should understand this basic point. Add asymmetry: yes, we know past, at least in principle (it's called history, by the way). But we don't know future. We can only predict it using the knowledge of the past. This fundamental assymetry (knowledge of past and prediction of future) underlies all the physics. Nothing to do with QM or TD in particular. And by the way, please use @-sign, so that I know you left a comment. – Marek Mar 28 '11 at 11:41
"What is physical can't depend on the interpretation." No. What is OBSERVED can't depend on interpretation. What is physical varies between interpretations heavily. "Add asymmetry: yes, we know past, at least in principle (it's called history, by the way). But we don't know future." Actually "knowledge of the past" is a Classical process, and cannot happen without measurement and wavefunction collapse. That is why this assymetry. Quantum system without collapse has no knowledge of the past and time-symmetric. – Anixx Mar 28 '11 at 13:11
Anixx is not wrong, just taking a nonstandard philosophical path through a certain well-trod thicket. – Ron Maimon Aug 15 '11 at 5:10

Causality as a consequence of finite energy:

0 - in the universe there is more than one object with finite energy content each.

"Reductio ad absurdum':

1 - If there is no limit on the speed of transmission of energy (light, electric force, gravitational ,...)

2 - then at the same time a moving object (with finite energy content) is here and there. You need twice the energy.

3 - then at the same time an object is here and in any other place.

4 - then that object must have an infinite amount of energy distributed throughout the universe.

5 - then there is no room or energy for more than that object in the universe.

5 contradicts 0, then 1 is false.

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