LOCALITY AND ITS DISCONTENTS

One of the key principles behind the two fundamental theories, that scientists discovered in the 20th century — the standard model of particle physics and the theory of general relativity — is locality. Local interactions, local degrees of freedom, local field theories. This is what I would like to discuss now. 

What is locality?

Before we begin let us first ask what locality actually means? What is local?

Well, there are various ways to formulate this but it essentially means that degrees of freedom are assigned pointwise in space (and time) and that there is no limit to the degree that one can localise various quantities. 

Examples: take Einsteins theory of general relativity. Here there is no limit to the degree by which matter can be localised in space and hence no limit to how large the curvature of space and time can be. The extreme case is of course the black hole where the curvature of space and time becomes infinite as does the localisation of matter, i.e. we have a singularity where the entire mass of the black hole is concentrated in a single point. Or take quantum field theory, where the variables are operator valued distributions — operators assigned to each point in space and time — and where there is again no limit to how localised these quantum fields can be. This can for example be seen in the canonical commutation relations, where the commutator of two quantum fields is equated with a Dirac delta-function, which roughly speaking is a function that is zero everywhere except for a single point, where it is infinite.

In quantum field theory locality is regarded as one of the holiest of cows. It is simply one of the very cornerstones of this mathematical formalism, together with Lorentz invariance and a few other principles it forms the backbone of what we understand a quantum field theory to be. This is also seen in the axiomatic approaches to quantum field theory: the Osterwalder-Schrader axioms for the Euclidian theory and the Gårding-Wightman or the Haag-kastler axioms for the Lorentzian theory all have locality as a key ingredient.

A cow less holy

But there are two reasons why I do not believe that this cow is as holy as it appears to be:

First reason: we don’t actually know what a quantum field theory is. Not as a rigorous mathematical object. In fact, one of the Clay Institute’s Millenium problems is to formulate and prove the existence of a quantum field theory in four dimensions (a Yang-Mills theory, to be precise). This is considered one of the hardest and most inaccessible problems in contemporary mathematics and in mathematical physics.  

But since we don’t know what a quantum field theory is — and people have tried to solve this problem for many decades — it is seems natural to suspect, that some of our key assumptions about what a quantum field theory is, are wrong. And one of these key assumptions is, as already said, locality.

Second reason: there exist in fact a very convincing argument as to why locality cannot be realised in Nature. This argument, which combines quantum mechanics with Einsteins theory of general relativity, strongly suggest that measurements below the Planck length are operational meaningless. You cannot measure anything shorter than the Planck length — according to this argument. But if you cannot measure anything below a given scale then locality cannot be realised in Nature. 

What is that argument? Well, the argument is that when you measure shorter and shorter distances you will need test particles with shorter and shorter wavelengths or, equivalently, higher and higher energies. But that means that these test particle will begin to curve space and time and at some point this curvature will be large enough to form a black hole. And when that happens we will not be able to get a signal back from our measurement. The distance where this happens is precisely the Planck length.

Now, the two points, which I have listed here, are usually understood to have nothing to do with each other: The second point is generally believed to apply to a theory of quantum gravity; it is believed that a such a theory will impose a Planck length screening, i.e. it will make it impossible to measure anything below that scale. And regarding the first point, it is generally believed that the correct formation of quantum field theory has nothing to do with locality or the absence thereof. 

In fact, many experts would say that a quantum field theory is that, which satisfies the Wightman or Osterwalder-Schrader axioms. If one of these set of axioms are not satisfied, well, it simply is not a quantum field theory. When I have discussed this question with other theoretical physicist it is always the same story; almost everyone I know in theoretical physics regard locality and quantum field theory to be almost synonymous. 

Here is what I believe: 

  • First, I consider the problem of finding a precise formulation of quantum field theory (in four space-time dimensions) to be perhaps the most important unsolved question in theoretical physics. It is more important than quantum gravity and more important than the unification of the four fundamental forces. Why is that? Well, all the theories, that we today know describe reality, are field theories — the standard model; Einsteins theory of relativity — so if we don’t know how to formulate a quantum theory of fields, well, then it seems to me that this is where we ought to focus our attention before we move on to other questions such as a quantum theory of gravity. And the fact that so many extremely clever physicists have spent decades searching without finding a precise formulation of a quantum theory of fields is, to put it mildly, rather surprising. 
  • Second, I suspect that the solution to this problem involves letting go of locality.

What I suggest is to revisit the various axiomatic systems and analyse what a modification, that permits a Planck scale non-locality, would look like. 

I think that we already now can say one thing about such a modification and that is that it will involve a modification of the Lorentz symmetry. Why is that? Well, if the non-locality is spatial (i.e. involves the three spatial dimensions) then it cannot be compatible with the Lorentz symmetry, which involves, essentially, rotations in the four space-time dimensions. Instead we will probably have a Lorentz symmetry that has been modified with a scale transformation.

Is that really possible? I think so. The experimental bounds on the Lorentz symmetry permits a certain degree of modification at the Planck scale. Or, said differently, it is not known whether the Lorentz symmetry is an exact symmetry of Nature. There is experimental room for a modification of the Lorentz symmetry and that is a possibility that various experimental physicists take serious.

I might be wrong but I think that it is an idea worth considering.

Locality and quantum field theory 

But what is a quantum theory actually? Since we here discuss the formulation of quantum theory of fields then it might be a good idea to mention what a quantum theory actually looks like. 

So, a quantum theory consist of basically two ingredients: there is an algebra of observables — operators — which encode the canonical commutation relations, and there is a Hilbert space, where this algebra has a representation. 

That sounds easy, but when it comes to a field theory it is not. First of all there are usually a number of symmetries, which are present in the classical theory and which one requires to hold also in the corresponding quantum theory. One of these symmetries is the aforementioned Lorentz symmetry, another one could be a gauge symmetry if one is dealing with a gauge theory. Another challenge, and this is where it gets really hard, is to find the Hilbert space representation. In four space-time dimensions there exist no examples of physically relevant quantum field theories of the type that we are discussing here, that can be formulated non-perturbatively as an algebra of operators represented on a Hilbert space. We simply do not know how to do this.

— except that perhaps we do. 

Together with the mathematician Johannes Aastrup I have proposed a different approach to the problem of constructing a quantum theory of fields. What we suggest is to work with a slightly different kind of algebra of observables (which I have written about elsewhere), that involves instead of the Heisenberg relation a modification hereof. 

The Heisenberg relation is a relation that tells us something about how infinitesimal operators interact with each other. Think about quantum mechanics: here we have the position and the momentum operators and the statement that their commutator equals the Planck constant. Now, the momentum operator is usually represented as a derivation, which is the infinitesimal version of a translation. 

You can, however, also construct quantum mechanics in terms of the finite (not infinitesimal) operators. In that case we take instead of the position operator a suitable algebra of functions on the underlying space and instead of the momentum operator a translation operator on the underlying space. In this case the canonical commutation relation will be encoded in an integrated algebraic relation. If we call the first approach for the ‘infinitesimal’, then we can call this second approach for the ‘integrated’ ditto. 

Now, as long as the number of degrees of freedom is finite — as is the case with quantum mechanics — then these two formulations are equivalent, it makes no difference which one we work with. But when the number of degrees of freedom is infinite — as is the case with a field theory — then these two methods of formulating a quantum theory are no longer equivalent.

In which way do these two approaches differ when there are infinitely many degrees of freedom? Well, it turns out that their representation theories are different. 

What does that mean? It means that the integrated approach has Hilbert space representations, which do not exist for the infinitesimal approach. And the reason for this is that these representations are non-local.

Let me recapitulate. So we have two ways of formulating a quantum theory. First there is the ordinary way, where we have infinitesimal operators and a commutation relation and secondly there is the integrated fashion, that involves a translation operator. And it is the second approach that permits a non-trivial Hilbert space representation.

And note this: in the second approach the canonical commutation relations are only realised up to a Planck scale modification. The canonical commutation relations encode locality (due to the presence of the Dirac delta-function) but since the Hilbert space representation, that exist in this integrated approach, is non-local, the canonical commutation relations must have a modification.

Now, as far as i know nobody has tried to work with the non-local representations that Johannes and I have found (and if some of the readers of this blog has information on previous work that I am not aware of I would be very thankful for an email with the references).

So what does all this tell us?

I believe that this tells us one interesting thing: if we let go of locality, then non-trivial quantum field theories in four dimensions do exist.

Locality and UV regulators

Now, let me add a few more words about the Hilbert space representations that Johannes and I have found. First of all we are primarily working with an algebra, which we call the QHD algebra and which involves gauge fields. This means that we are working within the framework of a gauge theory, it could be Yang-Mills theory or something else. Secondly, the Hilbert space representations involve what amounts to a dampening of degrees of freedom associated to short distances. This is what some people will call a cut-off or an UV-regulator. Note, however, that this does not break any spatial symmetries (for instance rotations, if we have a flat space; our Hilbert space representations are isometric). 

The concept of a cut-off is of course something that everyone who has worked with perturbative quantum field theory will be familiar with. The whole point of perturbative quantum field theory is essentially to work out how the theory depends on the cutoff and how we can let it approach zero — this is what we call renormalisation theory.

This means that there are two ways to interpret the UV regulator in the Hilbert space representations, which we have found: either it is an unphysical cut-off that needs to be taken to zero or it is not. Either we are dealing with an alternative formulation of ordinary quantum field theory or we have found something that is substantially different.

What we suggest is to consider the second option, i.e. to consider whether the dampening effect, that comes with our Hilbert space representation, could be a physical consequence of the theory. That is, to interpret it as a physical Planck scale phenomena. Here it is important to say that this does not mean that there will be no renormalisation theory — that wouldn’t make sense — it simply means that we are not going to remove the cut-off completely. 

Does this make sense? Well, it might. Such an approach will, however, have physical consequences, one being the already mentioned modification of the Lorentz symmetry.

We are currently trying to get better understanding of the Hilbert space representations, which we have found. The idea to interpret the aforementioned non-locality as a physical reality immediately raises a number of critical questions. One issue is that there are in fact many ways to construct the UV-regulators in our Hilbert space representations and we do not know whether one of them could be singled out or whether there might be an equivalence between them. Another issue is the gauge symmetry, which a cut-off will normally break. But that is not acceptable if the cut-off is physical: a broken gauge symmetry is a no-go and thus this issue has to be circumvented. 

Setting these technical issues aside the main point, which I wish to make here, is that a quantum theory of fields — understood in the sense of an algebra of observables represented on a Hilbert space — does exist. We know that. Whether this construction will turn out to be of physical relevance still remains to be seen. But I can tell you this: the reviewers hate this idea when we try to publish it. I actually consider that response to be somewhat promising. If you have an idea that evokes strong emotional responses but where nobody is really able to explain to you precisely why its wrong, well, then you might actually be on to something.

With that I will end this post. 

Best wishes, Jesper