Is Poincare symmetry the real thing?

[A. Neumaier]

This is a continuation of a side issue from another thread.

Removing the cutoff (in the lattice case taking the continuum limit) is essential to get the correct Poincare symmetry. The approximate theories are only construction tools, not the real thing.

if Poincare symmetry would be the real thing, one would be able to construct models without infinities which have full Poincare symmetry, and one would not need such non-Poincare-symmetric "construction tools".

Your argument is very misinformed.

It is typical in mathematics that nice objects are first constructed in a messy way.
The real numbers have very nice properties but to construct them one needs artifacts that make the numbers appear to be complicated sets (Dedekind cuts, euqivalence classes of Cauchy sequences, etc.).
The exponential function has many nice properties, but to construct it one needs limits of simpler functions that do not have this property.

But I was talking about physics. Your example is also not very impressive. Essentially, it is an example of a simplification reached by going to some infinite limit. So that a mathematical simplification can be reached by going to limits. But what about reality?

Another reason why Poincare symmetry is the real thing is that it gives rise (via Noether's theorem) to the conservation laws on which all basic physics relies. Drop symmetries - and you have nothing left to guide your theory building.

Hm. Maybe you are an opponent of GR, given that you cannot get conservation laws via Noether, but have only some pseudotensors, which do not even allow a physical interpretation in agreement with the spacetime interpretation, which allows only tensors to have a physical meaning?

 

[A. Neumaier]

But I was talking about physics. Your example is also not very impressive. Essentially, it is an example of a simplification reached by going to some infinite limit. So that a mathematical simplification can be reached by going to limits. But what about reality?

You were talking not of physics in general but of construction, which is a mathematical procedure within physics. Physics needs the exponential function for translational symmetries, to be able to talk about continuous evolution - also a convenient infinite limit to make things natural and tractable. In physics, the exponential is either postulated with its properties, or constructed via its power series whose partial sums lack these properties. The same holds for quantum field theory and all other physics.

Everyone considers the simplifications reached by going to a natural limit - otherwise physics would be very awkward if not impossible.

Another reason why Poincare symmetry is the real thing is that it gives rise (via Noether's theorem) to the conservation laws on which all basic physics relies. Drop symmetries - and you have nothing left to guide your theory building.

Hm. Maybe you are an opponent of GR, given that you cannot get conservation laws via Noether, but have only some pseudotensors, which do not even allow a physical interpretation in agreement with the spacetime interpretation, which allows only tensors to have a physical meaning?

On the lattice, one also loses rotation and boost symmetry, and hence conservation of angular momentum. This is fully conserved in general relativity. The translation symmetry of the Poincare group becomes in GR enhanced to the much bigger diffeomorphism symmetry, and reduces to it in the small-distance limit relevant for almost all of physics.

Drop symmetries - and you have nothing left to guide your theory building.

 

[Ilja]

My point was not at all directed against the use of simplifications. Of course, we use them and have to use them. But one should avoid the idea that the simplification is the real thing. Reality is not obliged to care about human abilities. Don't forget that atomic theory is, at least in some aspects, in particular those connected with symmetries, more complex than the continuous condensed matter approximation.

And is there really nothing left if we drop Poincare symmetry? Not symmetries in general, but only Poincare symmetry. I think that, for example, the SM contains a lot of quite complex information to guide theory building.

 

[A. Neumaier]

the SM contains a lot of quite complex information

Among others the standard model contains the information encoded in Poincare symmetry. It is responsible for being able to talk about relativistic causality and local (spacelike) commutation rules. It is also responsible for selecting the form of the terms in the Lagrangian of the standard model. Without that the whole structure of the standard model breaks down. Thus Poincare symmetry is indispensable for the standard model.

 

[Ilja]

I disagree. Poincare symmetry is something quite natural for a long distance approximation of a wave equation. So, one can obtain it approximately without having to assume that it holds fundamentally. The SM is, anyway, only a long distance approximation, so to have Poincare symmetry only for long distances is sufficient for all one needs for the SM.

To talk about causality and local commutation rules would be also possible without this particular symmetry. So, there is nothing indispensable.

 

[A. Neumaier]

The SM is, anyway, only a long distance approximation

Everything we do and know is only a long-distance approximation; hence it is appropriate to work in the corresponding limit.

To talk about causality and local commutation rules would be also possible without this particular symmetry.

How do you formulate relativistic causality and local commutation rules on a lattice (which is what we started with)?

 

[Ilja]

Of course, we have observations only about the long distance limit. But the theory development is not forced to consider only long distance approximations. There was, in particular, a long time when microscopic atomic models have been considered, and long distance limits derived from them, at a time no empirical evidence about atoms was available yet. And without a lot of such research beyond the limits of observability at that time the success of atomic theory would have been, if not impossible, but at least seriously delayed.

Nobody aims to forbid to use the long distance limit. But is it inappropriate to go beyond this limit?

And you have talked about causality, not relativistic causality. Of course, one can consider lattice models in a preferred frame. Then, in this preferred frame, one can have causality as well as one-time local communication rules. Which is what we need for standard quantum field theory.

 

[A. Neumaier]

in this preferred frame, one can have causality as well as one-time local communication rules. Which is what we need for standard quantum field theory.

You are shifting ground. A preferred frame is ok only for nonrelativistic QFT.

But this excludes all subatomic physics and even QED, whose lattice version was the original context of this discussion. I was targeting the latter, and you were so too, at first, since you were talking about the standard model.

For studying photons and other elementary particles, velocities are so large that one needs a relativistic quantum field theory to match experiment. This is impossible on a lattice. Approximate lattice calculations in particle physics are usually done in a Euclidean 4D version with multiple lattice spacings, extrapolated to the continuum limit, and then related to the true, relativistic theory by means of the Osterwalder-Schrader theorem. The latter needs the full Poincare symmetry for its validity.

 

[A. Neumaier]

But the theory development is not forced to consider only long distance approximations.

This leads to physics beyond the standard model for which the present forum is not the right one. Please stay in context.

 

[Ilja]

No, a preferred frame is also ok for regularizations in relativistic QFT. And while there may be some particular cases where the regularized theory remains Lorentz-covariant (like Pauli-Villars regularization), the usual case is that the regularized theory has a preferred frame. In particular, this is what I would expect for lattice regularizations. And, as well, I would expect a preferred frame also for trans-Planckian physics.

(BTW, the thread is about "the real thing". And you have introduced the argument of "nothing left for theory building" if one gives up Poincare symmetry. So, the context clearly includes theory building. Which, once the SM is well established, goes beyond the SM. It was you who has decided to start it here. I would not object against a shift.)

 

[A. Neumaier]

you have introduced the argument of "nothing left for theory building" if one gives up Poincare symmetry.

I only stated that there is nothing left for theory building if one gives up symmetry. If you go to a more fundamental description including gravity you need to use even bigger symmetry groups than the Poincare group - which is a tiny subgroup of the symmetry group of general relativity, which is a semidirect product of the Lorentz group and the diffeomorphism group.

 

[atyy]

Do Lorentz invariant and gauge invariant regularizations exist?

 

[Ilja]

How can you be sure that for gravity one needs a bigger symmetry? Ok, GR is a theory with such bigger symmetry - if one really counts this as a symmetry, given that every classical physical theory allows a covariant formulation, so that covariance does not really show the existence of a symmetry.

But even if one thinks that the symmetry group of GR is greater, this is not a reason to believe that it should be greater. We have examples like atomic theory where the symmetries of the continuous approximation are greater.

 

[A. Neumaier]

Do Lorentz invariant and gauge invariant regularizations exist?

For regularization in the UV, yes. Causal perturbation theory works like that in the vacuum sector - but one cannot see the gauge origin of broken symmetries
Dimensional regularization, though less rigorous in mathematical terms, also has these properties, and is therefore used in most of the practical work.

 

[stevendaryl]

Just for clarification of the issues here: If spacetime were discrete (a lattice) then Poincare symmetry would only be approximate, and therefore presumably the conservation laws that follow from it (momentum and angular momentum) would also only be approximate. But presumably, it would be difficult to empirically distinguish between an exact conservation law and an approximate one, if the approximation was very good.

So is it really the case that we can empirically rule out lattice theories?

 

[A. Neumaier]

Just for clarification of the issues here: If spacetime were discrete (a lattice) then Poincare symmetry would only be approximate, and therefore presumably the conservation laws that follow from it (momentum and angular momentum) would also only be approximate.

How would you derive approximate angular momentum conservation from a lattice model?

 

[Ilja]

One can derive from the lattice model a continuum limit, and then derive angular momentum conservation for the continuum limit. This gives an approximate angular momentum conservation for the lattice model.

 

[A. Neumaier]

One can derive from the lattice model a continuum limit, and then derive angular momentum conservation for the continuum limit. This gives an approximate angular momentum conservation for the lattice model.

This shows that the continuum model is the real (more fundamental) thing, as it is needed to get an approximate conservation law for an approximate model!

 

[Ilja]

No, this does not prove any such thing. Approximations can be easily more symmetric than the real thing.

 

[Demystifier]

Ok, GR is a theory with such bigger symmetry - if one really counts this as a symmetry, given that every classical physical theory allows a covariant formulation, so that covariance does not really show the existence of a symmetry.

There is one big difference between non-trivial diffeomorphism invariance of GR and trivial general covariance of "every" theory. To write "every" theory in general covariant form, one typically needs a unit time-like vector field ##n^{\mu}(x)##, which is a fixed non-dynamical quantity. In GR, on the other hand, no such fixed non-dynamical structure exists.

 

[stevendaryl]

How would you derive approximate angular momentum conservation from a lattice model?

I don't know how you would derive it, but if the lattice dynamics is an approximation to the continuum dynamics, and the continuum dynamics conserves angular momentum, then that places limits on how badly angular momentum nonconservation could be in the lattice dynamics, doesn't it?

 

[A. Neumaier]

Yes, and it seems the only reasonable way to do so.

If the universe were almost exactly angular momentum conserving but not exactly, because the true description is on a lattice - the near exactness would need explanation. In the lattice action it would mean that there would be a fine-tuning problem much more severe than in grand unification, since a large number of possible coefficents would have to miraculously take the values that come from the discretization of a Lorentz invariant continuous action. While Ilja may find this highly plausible, I find it extremely weird.

 

[ShayanJ]

There is one big difference between non-trivial diffeomorphism invariance of GR and trivial general covariance of "every" theory. To write "every" theory in general covariant form, one typically needs a unit time-like vector field ##n^{\mu}(x)##, which is a fixed non-dynamical quantity. In GR, on the other hand, no such fixed non-dynamical structure exists.

Is there any paper or book that explains it in detail and more or less clarifies how exactly you can do this for any given theory?

 

[Ilja]

If the universe were almost exactly angular momentum conserving but not exactly, because the true description is on a lattice - the near exactness would need explanation. In the lattice action it would mean that there would be a fine-tuning problem much more severe than in grand unification, since a large number of possible coefficents would have to miraculously take the values that come from the discretization of a Lorentz invariant continuous action. While Ilja may find this highly plausible, I find it extremely weird.

What fine tuning you need if the natural and straightforward continuous approximation has an exact angular momentum conservation?

By the way, you can learn how this works in reality by considering the mechanical properties of silicon crystals. In the continuous limit, they are defined by a tensor field, and the symmetries of the lattice enforce that the typical way this tensor can be not rotationally invariant do not work. So it has to be rotationally invariant.

 

[Ilja]

There is one big difference between non-trivial diffeomorphism invariance of GR and trivial general covariance of "every" theory. To write "every" theory in general covariant form, one typically needs a unit time-like vector field ##n^{\mu}(x)##, which is a fixed non-dynamical quantity. In GR, on the other hand, no such fixed non-dynamical structure exists.

So what? It may be easily hidden. In my preferred ether interpretation of GR, the preferred coordinates - which are, indeed, fixed, non-dynamical objects, follow a natural evolution equation: The harmonic condition [tex]\square X^\mu = 0.[/tex]
So, in the covariant version they look quite dynamical.

 

[Demystifier]

Is there any paper or book that explains it in detail and more or less clarifies how exactly you can do this for any given theory?

I think the explanation in Rovelli's "Quantum Gravity" is quite good.

 

[Demystifier]

So what? It may be easily hidden. In my preferred ether interpretation of GR, the preferred coordinates - which are, indeed, fixed, non-dynamical objects, follow a natural evolution equation: The harmonic condition [tex]\square X^\mu = 0.[/tex]
So, in the covariant version they look quite dynamical.

From the description above, I would say it's dynamical, covariant and not fixed. But maybe we disagree on the meaning of the word "fixed".

 

[Ilja]

The point is that there is also the condition that the solutions [itex]X^\mu(x)[/itex] of this dynamical equation also have to define a valid global system of coordinates. So, if you start with arbitrary initial values, you will hardly obtain a valid solution.

Therefore I would not say that these fields are truly dynamical, they only look dynamical if one looks at the equation and ignores other conditions for defining a valid solution.

On the other hand, if you start with a particular valid solution [itex]\square X_0^\mu(x)=0[/itex] and consider only small modifications [itex]X^\mu(x) = X_0^\mu(x) + \delta X^\mu(x)[/itex], small in amplitude as well as frequency, these external non-dynamical restrictions play no role, except by defining the meaning of "small".

So, in a small enough environment of a valid solution the theory will be indistinguishable from a purely dynamical theory. Feel free to define the meaning of "fixed" vs. "dynamical", I would be interested to see the results.

 

[A. Neumaier]

What fine tuning you need if the natural and straightforward continuous approximation has an exact angular momentum conservation?

There are many actions invariant under the group of the lattice but not under the rotation group. All these must be excluded by fine-tuning. There is no renormalization criterion that would exclude the rotation symmetry-violating higher order terms of the action.

 

[Demystifier]

The point is that there is also the condition that the solutions ##X^{\mu}(x)## of this dynamical equation also have to define a valid global system of coordinates. So, if you start with arbitrary initial values, you will hardly obtain a valid solution.

Therefore I would not say that these fields are truly dynamical, they only look dynamical if one looks at the equation and ignores other conditions for defining a valid solution.

But isn't a similar feature present in standard field theories? Even though the equations of motion are dynamical, one puts restrictions on initial conditions, e.g. that the field must vanish at infinity. What's the difference between such standard restrictions and your restrictions?

 

[Ilja]

There are many actions invariant under the group of the lattice but not under the rotation group. All these must be excluded by fine-tuning. There is no renormalization criterion that would exclude the rotation symmetry-violating higher order terms of the action.

Why do you think so? The straightforward continuous limit is nothing else but what renormalization gives. Of course, with postulating some symmetries one can restrict the terms which are allowed in a renormalization procedure. But this does not give the other terms greater long distance effects. They may disappear in the long distance limit as well. What survives at long distances are only the lowest order terms.

 

[Ilja]

But isn't a similar feature present in standard field theories? Even though the equations of motion are dynamical, one puts restrictions on initial conditions, e.g. that the field must vanish at infinity. What's the difference between such standard restrictions and your restrictions?

Why do you ask me?

I think that your claim that there is a certain distinction between fixed and dynamical degrees of freedom has a problem with the preferred coordinates - the straightforward example of something fixed, as describing absolute space and time - fulfilling a quite dynamical-looking equation.

It was an aspect of my claim that covariance is nothing physical, given that every classical theory allows a covariant formulation.

If one, anyway, has to add some fixed boundary conditions even for really dynamical entities, your problem to distinguish them becomes even greater.