# A Is Poincare symmetry the real thing?

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1. Mar 29, 2016

### A. Neumaier

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

Last edited: Mar 29, 2016
2. Mar 29, 2016

### A. Neumaier

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.
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.

Last edited: Mar 29, 2016
3. Mar 29, 2016

### 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.

4. Mar 29, 2016

### A. Neumaier

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.

Last edited: Mar 29, 2016
5. Mar 29, 2016

### 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.

6. Mar 29, 2016

### A. Neumaier

Everything we do and know is only a long-distance approximation; hence it is appropriate to work in the corresponding limit.
How do you formulate relativistic causality and local commutation rules on a lattice (which is what we started with)?

7. Mar 29, 2016

### 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.

8. Mar 29, 2016

### A. Neumaier

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.

9. Mar 29, 2016

### A. Neumaier

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

10. Mar 29, 2016

### 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.)

11. Mar 29, 2016

### A. Neumaier

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.

Last edited: Mar 29, 2016
12. Mar 29, 2016

### atyy

Do Lorentz invariant and gauge invariant regularizations exist?

13. Mar 29, 2016

### 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.

14. Mar 29, 2016

### A. Neumaier

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.

15. Mar 31, 2016

### stevendaryl

Staff Emeritus
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?

16. Apr 1, 2016

### A. Neumaier

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

17. Apr 1, 2016

### 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.

18. Apr 1, 2016

### A. Neumaier

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!

19. Apr 1, 2016

### Ilja

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

20. Apr 1, 2016

### Demystifier

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.